2.0 signal processor analyses · to-pulse. for now we assume that the target is an ideal point...

© M. C. Budge, Jr., 2012 – [email protected] 1

2.0 SIGNAL PROCESSOR ANALYSES

2.1 INTRODUCTION

We now want to turn our attention to the analysis of signal processors. We will

be specifically concerned with analyzing the ability of signal processors to reject clutter

and improve signal-to-noise ratio (SNR). This is an extension of the waveform and

matched filter work of EE 619.

We do not want to discuss clutter and signal processor analyses in general terms.

Instead we want to discuss how one would perform specific analyses. To this end, we

will select a specific radar, target and clutter scenario, and specific signal processors. We

assume that the radar is ground based and has the job of detecting and tracking airborne

targets such as aircraft, helicopters and cruise missiles. We assume that the targets are

flying at low altitudes so that the radar is receiving returns from the target and ground. In

our case, the ground is the clutter source (termed ground clutter). We assume that the

radar is transmitting a pulsed (as opposed to CW or continuous wave) signal. We further

assume that the radar is transmitting an infinite or semi-infinite series of pulses with a

given pulse repetition interval (PRI). In some cases we will let the PRI vary from pulse-

to-pulse. For now we assume that the target is an ideal point target (SW0) although we

may allow a SW1 or SW3 target later on.

We will consider two types of signal processors: a moving target indicator or MTI

and a pulsed-Doppler signal processor. These are the two most common types of signal

processors in the type of radar indicated above. As an aside, the type of radar we are

considering would be an air defense or air traffic control radar that performs search

and/or track. These types of radars must contend with ground clutter (or sea clutter for

naval radars) while trying to perform these functions. The purpose of the signal

processor is to help the radar mitigate the ground clutter.

We will begin our studies by defining a ground clutter model. After this we will

develop equations that characterize the clutter, target and noise signals at the input to the

signal processor. Finally, we will discuss the characteristics MTI and pulsed-Doppler

signal processors and how they react to the target, clutter and noise signals.

2.2 CLUTTER MODEL

2.2.1 Radar Cross-Section (RCS) Model

A drawing that we will use to develop the ground clutter model is shown in

Figure 2-1. The top drawing represents a side view and the bottom drawing represents a

top view. For the initial development of the ground clutter model we assume that the

earth is flat. Later we will add a correction factor to account for the fact that the earth is

not flat.

The triangle and semicircle on the left represents the radar which is located a

height of h above the ground. When we discuss radar height in the context of clutter or,


more specifically, the signal the radar emits, we refer to the “height to the phase center of

the radar”. The phase center is usually taken to be the location of the feed for a reflector

antenna or the center of the phased array for a phased array antenna.

The dashed lines on the side and top views denote the 3-dB boundaries of the

antenna main beam. The angles E and

A denote the elevation and azimuth 3-dB

beamwidths, respectively.

Figure 2-1 – Geometry for Ground Clutter Model

The horizontal line through the antenna phase center is simply a horizontal

reference line, it is not the elevation angle to which the main beam is steered. The target

is located at a range of R from the radar and at an altitude of Th . The elevation angle

from the radar phase center to the target is

1sine Th h R . (2-1)

In the geometry of Figure 2-1 the clutter patch of interest is also located at a range

of R from the radar. In most applications, this is the region of clutter that is of interest

because we are interested in the clutter that competes with the target. However, for some

cases, most notably pulsed Doppler radars, the ground clutter that competes with the

target will not be at the target range, but at a much shorter range than the target range.


This will not pose a problem for the clutter model. It is developed so as to handle this

case.

The width of the clutter patch along the R direction is R . In most cases R is

taken as the range resolution of the radar. The reason for this is that almost all signal

processors quantize the incoming signal into range cells that have a width of one range

resolution cell. (Recall our discussions of detection and range cells from EE 619.) In

some cases a range resolution cell is large enough to cause problems in the accuracy of

the ground clutter model. In this case, R is taken to be smaller than a range resolution

cell. If this is necessary, the signal processor calculations must account for this by

integrating across multiple clutter range cells. A discussion of this is beyond the scope of

this course.

With a little thought, it is easy to see that the ground region that extends over R

at a range of R is an annulus centered on the radar. This is depicted in the top view where

a portion of the annulus is shown. For purposes of calculating the radar cross section

(RCS) of the ground in this annulus, the annulus is divided into two regions as indicated

in Figure 2-1. One of these is termed the main beam clutter region and represents the

ground area illuminated by the main beam of the radar. The other is termed the sidelobe

clutter region and represents the ground area illuminated by the sidelobes of the radar.

The standard assumption is that the sidelobe clutter region extends from 2 to 2 .

In other words, it is assumed that there are no clutter returns from the back of the radar.

As implied by the statements above, the ground clutter model incorporates the transmit

and receive antenna beam characteristics. In this development, we are assuming a

monostatic radar that uses the same antenna for transmit and receive.

The size of the clutter RCS will depend upon the size of the ground area

illuminated by the radar (the region discussed in the previous paragraph) and the

reflectivity of the ground. This reflectivity is denoted by the variable . Consistent with

the previous discussions of target RCS, one can think of clutter reflectivity as the ability

of the ground to absorb and re-radiate energy. In general, clutter reflectivity depends

upon the type of ground (soil, water, asphalt, gravel, sand, grass, trees, etc.) and its

roughness. It also depends upon moisture content and other related phenomena. Finally,

it also depends upon the angle to the clutter patch (R in Figure 2-1). Chapter 12 of

Skolnik’s Radar Handbook contains further discussions of . For our purposes, we will

use 0 . That is, we use a backscatter coefficient that doesn’t depend upon

R . Our

justification for this is that, in general R is relatively constant (and small). If we were

considering an airborne radar, we would need to revert to a model wherein varies with

R . Some of these models are discussed in Skolnik’s Radar Handbook. We would also

need to revert to the variable model for clutter that is very close to the radar.

However, this is not common in pulsed radars. (It is common in CW radars.)

We will use three values for : 20 dB , 30 dB and 40 dB .

These are fairly standard values currently used for radars that operate in the 5 to 10 GHz

range. The first case corresponds to moderate clutter and would be representative of

trees, fields and choppy water. The second value is light clutter and would be


representative of sand, asphalt and concrete. The third value is very light clutter and

would be representative of smooth ice and smooth water.

Table 2-1 – Ground Clutter Backscatter Coefficients

Backscatter Coefficient,

(dB)

Comment

-20 Moderate Clutter – Trees, fields,

choppy water

-30 Light Clutter – Sand, asphalt,

concrete

-40 Very Light Clutter – Smooth ice,

smooth water

With the above, we can write the RCS of the main beam ground area as

CMB CM AA d d (2-2)

where the various parameters are shown on Figure 2-1. An assumption in this equation is

that A is small so that

Ad can be taken as a straight line that is perpendicular to d .

Examination of the top part of Figure 2-1 shows that the clutter area is not located

at beam center. This means that the clutter patch is not being fully illuminated, in

elevation, by the main beam. To account for this we include a loss term that depends

upon the antenna pattern. Rather than have to use specific antenna patterns, we will

define a generic gain that is consistent with reasonable antennas. There are two of these.

One is

2

2.78.sinc 2.78

0 elsewhere

E EG

(2-3)

and the other is

2

2.77 EG e

. (2-4)

The first is a sinx/x pattern and the second is a Gaussian pattern. In both cases, is the

angle off of beam center and E is the elevation beamwidth. It will be left as an exercise

for you to show that both of these models are reasonably good within the mainbeam

region of the antenna pattern. Of the two, the second is easier to use because it is not a

piecewise function.

With this, we can modify the equation for the main beam clutter as

2 2 mCMB A B Rd dG (2-5)


where B is the elevation pointing angle of the main beam and 1sinR h R . The

reason for the plus sign on R in the above equation is that

R is negative (see Figure 2-

1) but we defined it as a positive angle via the equation 1sinR h R . In most

applications we assume that the main beam is pointed at the target so that B e .

We now need to examine the side lobe clutter. The basic approach is the same as

for the main beam clutter but in this case we need to account for the fact that the side lobe

clutter represents ground areas that are illuminated through the transmit antenna side

lobes and whose returns enter through the receive antenna sidelobes. The ground area of

concern is the semicircular annulus excluding the mainbeam region. Relatively speaking,

the ground area illuminated by the main beam is small compared to the ground area

illuminated by the side lobes. Because of this, it is reasonable to include the main beam

area in with the side lobe area. With this, the RCS of the clutter in the side lobe region is

2 2 mCSL SL d d (2-6)

where SL is the average antenna sidelobe level relative to the main beam peak. A typical

value for SL is -30 dB or 0.001 (see your antenna homework problems). This value could

be as high as -20 dB for “cheap” antennas and as low as -40 to -45 dB for “low side lobe

antennas”. The equation above includes a 2

SL term to account for the fact that the

clutter is in the sidelobes of the transmit and receive antenna.

To get the total clutter RCS from both the main lobe and the side lobes we make

the assumption that the clutter signals are random processes and that they are

uncorrelated from angle to angle. (We also assume that the clutter signals are

uncorrelated from range cell to range cell.) Since the clutter signals are uncorrelated

random processes, and since RCS is indicative of power, we can get the total clutter RCS

by adding the main beam and side lobe RCSs. Thus

22 2 mC CMB CSL B R AG SL d d . (2-7)

In this equation, the terms d and d are related to range, R, and range resolution, R , by

cos Rd R and cos Rd R .

For the final step we need a term to account for the fact that the earth is round and

not flat. We do this by including a pattern propagation factor. This pattern propagation

factor allows the clutter RCS to gracefully decrease as clutter cells move beyond the

radar horizon. David Barton has developed some sophisticated models to account for this

that, I believe, are in his latest book. He also provided a simple approximation that works

very well. Specifically, the defines a loss factor as

4

1 hL R R (2-8)

where hR is the radar horizon and is defined as

2 4 3h ER R h (2-9)


with 6,371,000 mER being the mean radius of the earth. The 4/3 factor in the above

equation invokes the so-called 4/3 earth model. This model states that, to properly

account for diffraction we need to increase the earth radius to effectively reduce its

curvature. This is discussed in Skolnik’s text book as well as other places. With the

pattern propagation factor, the equation for the clutter RCS becomes

22

2

4 m

1

B R A

C

h

G SL R R

R R

. (2-10)

Figure 2-2 contains a plot of clutter RCS for a typical scenario. In particular, the

radar uses a circular beam with a beamwidth of 1.5 degrees. Thus 1.5 180A E .

The sidelobe level is assumed to be -30 dB. The radar pulse width is 1 µs so that the

range resolution is 150 mR . The phase center of the antenna is at 5 mh . The

three curves of Figure 2-2 correspond to beam pointing angles (B ) of 0, ½ and 1

beamwidth above horizontal. The assumed value of backscatter coefficient is

20 dB .

Figure 2-2 – Sample Clutter RCS Plots – 20 dB

The first observation from Figure 2-2 is that the ground clutter RCS is quite large

for low beam elevation angles. This means that for low altitude targets at short ranges

(less than about 30 Km) the clutter will be larger than typical (6 to 10 dBsm) targets.

Thus, unless the radar includes signal processing to reduce the clutter returns, they will

dominate the target returns. At larger elevation angles the problem is less severe because

the ground is no longer being illuminated by the main beam.


The shape of the curves of Figure 2-2 requires some discussion. Examination of

the equation for clutter RCS indicates that the numerator term increases with increasing

range to the clutter because d depends directly upon R. However, for ranges past the

radar horizon, which is at 9.2 Km for this radar, the pattern propagation factor starts to

predominate and reduces the clutter RCS. This is what causes the curves of Figure 2-2 to

first increase and then decrease.

2.2.2 Clutter Spectral Model

To reduce the clutter return in the radar, it is necessary to have a clutter

characteristic that is different from the target so that that characteristic can be used as the

basis for designing a signal processor. The characteristic that will be used is Doppler

frequency. (Range and angle can’t be used because the target and clutter are at the same

range and angle.) Specifically, the signal processor will exploit the fact that the ground

clutter is at zero Doppler while the target is at some non-zero Doppler (usually).

In practice the clutter signal in the receiver is not concentrated at zero Doppler. In

fact, it has some spread about zero Doppler because of motion of objects (leaves, waves,

grass, etc.) that make up the clutter. For scanning radars (i.e. search radars) there will be

a Doppler spread of the return clutter signal caused by the fact that the radar beam is

moving across the clutter. A standard model for the spectrum of ground clutter is

2 22

12

fCf

fC

kC f e k f

(2-11)

where fC is the Doppler spread of the clutter, f is the Dirac delta function and k is

a constant that apportions the clutter power between the spectral line at zero and the

portion that is spread. The quantity fC is computed from 2fC v where v is the

velocity spread of the objects that make up the ground clutter. Sample values for v can

be obtained from Chapter 15 (page 15.9) of Skolnik’s Radar Handbook. To the author’s

knowledge, there is so set method of apportioning the spectrum between the impulse and

Gaussian parts. Most analysts completely ignore the impulse by setting 1k . This will

be the approach we will take. Some guesses at k for different conditions would be

0.1k for hard surfaces like sand, concrete, asphalt, ice, and smooth water; 0.5k for

fields and woods in light winds or for medium rough sea; 0.9k for rough seas or fields

and woods in high winds.

The first term in the clutter spectrum is somewhat justified in Papoulis1. In the

example cited he shows that if the density function for the velocity distribution of the

clutter is Gaussian (which is a reasonable assumption by the central limit theorem) then

the spectrum of the signal returned from the clutter will also be Gaussian.

1 Papoulis, Athanasios “Probability Random Variables and Stochastic Processes” Second Ed, McGraw Hill,

Example 10-4 on page 267


As the beam scans by the target its amplitude will vary. If we assume a Gaussian

beam shape, the amplitude variation of the returned clutter voltage will have a Gaussian

shape. Since the Fourier transform of a signal with a Gaussian shape is also a Gaussian

shape the spectrum will be described by a Gaussian function. Skolnik’s Radar Handbook

gives the form of this spectrum as

2 221

2

Sf

S

S

C f e

(2-12)

where

0.265S rf n , PRFrf , 2r A scann f T , Azimuth Beam WidthA and

scan periodscanT . To get the total spectrum due to the internal motion of the spectrum

and the scanning, one would convolve C f with SC f . The justification for this

convolution is discussed below and in Appendix B.

A tacit implication of the terminology used above is that the clutter is a stationary

random process. Also, it will be noted that

1SC f df C f df

. (2-13)

This means that the clutter spectrum is normalized to unity power. To get the actual

clutter spectrum one would multiply C f or SC f by the clutter power. The clutter

power would be computed from the clutter RCS and the radar range equation.

Now that we have a clutter model, we want to develop the equations that

characterize the clutter, noise and target spectra at the input to the signal processor. This

is the topic of the next section.

2.3 SIGNAL ANALYSIS

2.3.1 Background and Definitions

We want to develop equations that allow us to analyze what a radar signal

processor does to signals returned from clutter (or targets). We will eventually work in

the frequency domain, but we start in the time domain.

A simplified block diagram of a radar transmitter and receiver is shown in Figure

2-3. The block diagram contains only the elements essential to our development.

Specifically, it does not contain any of the intermediate frequency (IF) amplifiers and

filters, nor the mixers needed to up- and down-convert the various signals. We have not

lost any generality with this technique because we will use normalized, complex signal

notation. This allows us to ignore all IF processes. Recall the detection discussions from

EE 619.


Figure 2-3 – Transmitter, Receiver and Signal Processor

Complex signal notation has an advantage of being easy to manipulate since the

signals are represented by complex exponentials rather than sines and cosines.

Operations such as filtering, sampling, transforms, etc. are treated the same with complex

signals and real signals. The place where one must take care when using complex signals

is in non-linear operations such as mixing. For example, in the transmit mixer of the

previous figure we used LOv t whereas on the receive mixer we used LOv t . We knew

we needed to do this based on real signal analyses. Specifically, we performed real

signal analyses and used the results to determine what we needed to do with complex

signals.

As a caution, be very careful when using complex signal analysis with other

nonlinearities such as limiters, saturating amplifiers, squarers, diodes, etc. The rule-of-

thumb I use is to perform careful, real, analysis and look for ways to extend it to complex

signals. I find that I must revert to the real, IF signal, go through the non-linearity, and

then reconstruct the complex signal. A key point to remember is that the magnitude of

the complex signal is the magnitude of the IF signal and the phase of the complex signal

is the phase of the IF signal. In equation form, if

j t

cv t A t e

(2-14)

is a complex signal voltage then the corresponding real, IF voltage is

cosIF IFv t A t t t . (2-15)

Also, if

cos sincv t I t jQ t A t t jA t t (2-16)

then

cos sin

cos cos sin sin

IF IF IF

IF IF

v t I t t Q t t

A t t t A t t t

. (2-17)


Let us return to the problem at hand and define the signals of the previous figure.

pv t is the pulse train and is generally a complex, base-band signal. This means that its

energy, or power, is generally concentrated around 0 Hz, as opposed to some IF. It

should be noted that signals that have a Doppler frequency are usually considered base-

band signals, even though their energy is not truly concentrated around 0 Hz.

The cv t in the Equations 2-14 through 2-17 are base-band signals. On the other

hand, the IFv t are termed IF signals, or more generally, band-pass signals. They are

termed band-pass signals because their energy is centered around IF and thus looks like

the response of a band-pass filter.

The typical pv t of interest to us is an infinite sequence of rectangular pulses

with a width of p and a PRI (pulse repetition interval) of T. A graphical representation

of pv t (actually pv t ) is shown in the Figure 2-4.

Figure 2-4 – Depiction of pv t

In equation form, pv t is given by

rect rect rectp

k kp p p

t kT t tv t t kT i t

(2-18)

where

1

2

12

1rect

0

xx

x

(2-19)

t is the Dirac delta and denotes convolution. The notation k

denotes a

summation over all integers. This implies an infinite number of pulses. In practice,

radars use only a finite number of pulses. The techniques we develop for the case of an

infinite number of pulses will apply to a finite number of pulses provided the number of

pulses in a burst (i.e. a burst of N pulses) satisfies some constraints. We will discuss this

when we consider specific signal processor.

A more general form of pv t would be


p

k

v t p t i t p t kT (2-20)

where p t is a complex signal notation of a complicated waveform such as a phase-

coded pulse or an LFM pulse.

The STALO signal, LOv t , is of the form

c j tj t

LOv t e e

. (2-21)

In the above, 2c cf is the carrier frequency. t is termed the phase noise on the

STALO signal and represents the instability of the oscillator that generates the STALO

signal. As implied by its name, t is a random process. It is such that j t

e

is wide-

sense stationary (WSS), or at least this is the standard assumption. We will address the

phase noise later. Phase noise is included because it is often the limiting factor on signal

processor performance.

In most radars, LOv t also includes an amplitude noise component such that

1 c j tj t

LOv t A t e e

. However, A t is usually made very small by the radar

designer and is normally considered to have a much smaller influence on signal processor

performance than t . For this reason it is almost always ignored in signal processor

analyses. Having said this, it should be noted that modern STALOs are becoming so

stable that the amplitude noise will soon overtake phase noise as the limiting factor in

signal processor performance.

Tv t is the transmitted signal and is simply

T p LOv t v t v t . (2-22)

Sv t is a term we include to account for the fact that the antenna may be

scanning (which is generally taken to mean that the beam is rotating horizontally, as in a

search radar). If we are considering a tracking radar, 1Sv t . If the antenna is

scanning, the standard form of Sv t is

2 22 TSt

Sv t e

(2-23)

where

2

2 1

2.77

A SCANTS

T

. (2-24)

SCANT is the scan period (in sec) and A is the azimuth beamwidth (in radians).


In practice, sv t is a periodic function with a period of SCANT . However, since the time

period of interest, in terms of the signal processor, is small relative to SCANT , it is assumed

that the radar beam scans by the target only one time. Incidentally, the “time period of

interest, in terms of the signal processor” is termed a CPI, or coherent processing interval.

C t is the “clutter signal” and is our means of capturing the power spectrum

properties of the clutter, as we discussed earlier. C t is a random process that is usually

assumed to be WSS2. The power spectrum of the clutter is

2 22

12

fC

c

f

fC

C f R E C t C t

ke k f

. (2-25)

This is the form we discussed earlier (see Equation 2-11).

To complete our definitions, Rv t is the received signal after it goes through the

antenna. mv t is the output of the receiver’s mixer and MFv t is the matched filter

output. ov k is the sampled version of MFv t and is the signal that goes to the signal

processor. The matched filter is matched to a single pulse (i.e. it is matched to p t ) of

the original pulse train, pv t .

2.3.2 Signal Analysis

We start our analysis by noting that the mixer is a multiplication process. Thus,

the signal sent to the antenna is

T p LOv t v t v t . (2-26)

If the antenna is scanning, its pattern modulates the amplitude of Tv t . We model this

as a multiplication of Tv t by Sv t . Thus, the signal that leaves the antenna is

antT p LO Sv t v t v t v t . (2-27)

Recall that we set 1Sv t if we consider the tracking problem.

2 A note about stationarity: Realistically, none of the random processes we are dealing with are truly WSS.

However, over the CPI we can reasonably assume they are stationary. From random processes theory, we

know that if a process is stationary, in the wide sense, over a CPI then we can reasonably assume that it is

WSS. This stems from the fact that we are only interested in the random process over the CPI.


After the signal leaves the antenna, it propagates a distance of R to the clutter (or

target). We represent this propagation by incorporating a delay, which we denote as

2d , into antTv t . We should also include an attenuation of 1 R . However, we are

dealing with normalized signals for now so we can ignore it. We will consider the actual

power in the signal at a later time.

With the above, the signal that arrives at the clutter (or target) is

2

2 2 2

CT antT d

p d LO d S d

v t v t

v t v t v t

. (2-28)

The clutter “reflects” the signal back to the radar and imposes its temporal, or spectral,

characteristics on the reflected signal. We represent this operation by multiplying CTv t

by C t , the function that we use to represent the temporal (and spectral) properties of

the clutter. The fact that we represent the operation by multiplication is due to the fact

that the interaction of the signal with the target (clutter) is essentially a modulation

process. We derived this in EE619 when we found that the motion of the target caused a

shift in the frequency of the signal (Doppler shift) and that the amplitude of the return

signal was a function of the target RCS.

The signal reflected by the clutter is

2 2 2

CR CT

p d LO d S d

v t v t C t

v t v t v t C t

. (2-30)

and the signal back at the receive antenna is

2

2

antR CR d

p d LO d S d d

v t v t

v t v t v t C t

. (2-31)

This signal next picks up the scan modulation and is then heterodyned by the receiver

mixer to produce the matched filter input, mv t . In equation form,

2

m antR S LO

p d LO d S d d S LO

v t v t v t v t

v t v t v t C t v t v t

. (2-32)

We now want to study and manipulate this equation. We start by simplifying the

equation and making some approximations. Since the antenna will not move far over the

round trip delay, d , we can assume that Sv t doesn’t change much over

d . This

means that S d Sv t v t . With this we get

2 2m p d S d LO d LOv t v t v t C t v t v t . (2-33)

For our next step we want to look at the last two terms. We write the product as


c d d c c dj t j t j t j t j tj

LO d LOv t v t e e e e e e

(2-34)

where dt t t . t represents the total (transmit and receive) phase

noise on the radar. A standard assumption is that t is small relative to unity so that

1j t

e j t t

. With this mv t becomes

2 2m p d S dv t v t v t C t t . (2-35)

Note that we dropped the phase term, c dje

. We were able to do this because it is a

phase term that we can normalize away in future calculations.

We further simplify the mv t equation by shifting the time origin by d . This

yields

2' 2m m d p S d d dv t v t v t v t C t t . (2-36)

We argued earlier that Sv t changes slowly relative to d so that 2 2

S d Sv t v t .

Also, C t and t are WSS random processes. This means that their means and

autocorrelations do not depend on time origin. Thus, we can replace 2dC t with

C t and dt with t and not change their means and autocorrelations (the

autocorrelation is what we eventually use to find the power spectrum of mv t ). With

this we get

2

m p Sv t v t v t C t t . (2-37)

We dropped the prime and reverted to the notation mv t for convenience.

The next steps in our derivation is to process mv t through the matched filter and

to then sample the matched filter output via the analog-to-digital converter (ADC) (see

Figure 2-3). Before we do this we need to examine mv t more closely. If we substitute

for pv t from Equation 2-20 into Equation 2-37 we get

2

2

m S

k

S

k

v t v t C t t p t kT

p t kT v t C t t

. (2-38)

We recall that C t and t are WSS random processes. Because of this, the

product 2

Sr t v t C t t is also a random process. However, because of the fact

that Sv t is periodic r t is not WSS. As is shown in Appendix B, r t is

cyclostationary. As a result, we can use the averaged statistics of r t and treat it as a

WSS process in the following development. With this we write


m

k

v t p t kT r t (2-39)

where we treat r t as if it was a WSS random process.

We note that, because of the p t kT term, mv t is not stationary. This is

something we will need to deal with in the following discussions.

If we represent the impulse response of the matched filter as m t , we can write

the output of the matched filter as

MF m

k

v t m t v t m t p t kT r t . (2-40)

We normally derive m t by saying that the matched filter is matched to some

signal q t . Recalling matched filter theory, this means that we can write

m t q t . (2-41)

In this application, the matched filter is termed a single-pulse matched filter. The

matched filter is often matched to the transmit pulse, p t so that

q t p t . (2-42)

However, in some instances, notably when p t is an LFM pulse, m t includes an

amplitude taper to reduce range sidelobes. In this case q t will not exactly equal p t .

In the remainder of this derivation we will use the more general form of Equation 2-41.

Substituting Equation 2-41 into Equation 2-40 yields

MF

k

v t q t p t kT r t (2-43)

or

MF

k

v t q t p kT r d

(2-44)

where we have replaced the convolution notation ( ) by the integral it represents.

Figure 2-5 contains depictions of mv , q t and MFv t for the case

where p t is an unmodulated pulse and m t is matched to p t (i.e. q t p t ).

As expected from matched filter theory, MFv t is a series of triangle shaped pulses

whose amplitudes depend upon r t .

Since mv t is a non-stationary random process so is MFv t . This makes MFv t

difficult to deal with since we do not have very sophisticated mathematical tools and


procedures that would allow us to efficiently analyze non-stationary random processes.

Fortunately, because of the ADC we don’t need to deal directly with MFv t . We will

only work with samples of MFv t . As stated below, and proved in Appendix A, the

presence of the ADC greatly simplifies our ability to work with its output.

In Figure 2-3, and ensuing discussions, we assume that the ADC produces digital

outputs and that the signal processor is a digital signal processor (DSP). This is an

accurate model for modern radars since almost all of them employ DSPs. However, the

analyses to be discussed also apply, with minor modifications, to older radars that use

analog signal processors.

Figure 2-5 – Depictions of mv (top plot), q t (center plot) and MFv t

(bottom plot)

We note that, consistent with most DSP analyses, we are ignoring the amplitude

quantization performed by the ADC. We will only be concerned with its time

quantization, or time sampling. In this sense, we are assuming that the ADC is an

infinite-bit ADC, which is simply a sampler.

We will assume that we are sampling the matched filter once per PRI (every T

seconds) on the peak of the matched filter response. The fact that we assume we are

sampling the matched filter output on its peaks allows us to use the radar-range equation

to compute the SNR and the CNR at the signal processor input. Recall that the

assumption associated with the radar-range equation is that SNR is computed at the peak

of the matched filter output. That is, SNR is the peak signal power at the matched filter

output divided by the average noise power at the matched filter output.

If we find that the radar timing is such that it does not sample the matched filter

output on its peaks, we incorporate a range-gate, or range straddling, loss in the radar-

range equation (see EE619 notes). However, we still assume the radar samples the

matched filter output at its peaks.


With the above, the signal that is sent to the signal processor is

o MFv k v kT (2-45)

where we assume (see Figure 2-5) that the matched filter response peaks occur at t kT .

Since MFv t is a continuous-time random process, ov k is a discrete-time

random process. As shown in Appendix A, while MFv t is not stationary, ov k is a

WSS random process. This means that we can write the autocorrelation of ov k as

1 2 1 2 1 2,o o o o oR k k E v k v k R k k R k (2-46)

where 1 2k k k . This carries the further implication that we can find the power

spectrum of ov k and use this to perform our signal processor analyses in the frequency

domain.

In Appendix A we show that the power spectrum of the ADC output is given by

2

o r

l

S f K MF f l T S f l T (2-47)

where

r sS f C f C f f (2-48)

and sC f , C f and f are discussed below.

MF f is the matched-range, Doppler cut of the cross ambiguity function of

p t and q t , the signal to which the matched filter, M t , is matched. Specifically,

2j ftMF f p t q t e dt

. (2-49)

For uncoded pulses, phase coded pulses and LFM pulses that don’t incorporate range

sidelobe reduction, MF f is of the form (see EE619 notes and homework)

sinc pMF f f (2-50)

where p is the uncompressed pulse width.

The scanning function, SC f , is of the form shown earlier (see Equation 2-12).

That is


2 221

2

Sf

S

S

C f e

(2-51)

where 2

0.265s

A scanT

. If the radar antenna is not scanning, SC f reduces to

SC f f . (2-52)

The clutter spectrum, C f was given earlier (see Equations 2-11 and 2-25) as

2 22

12

fCf

fC

kC f e k f

. (2-53)

As indicated earlier, f represents the phase noise spectrum of the radar. A

reasonable expression for f is

0f f (2-54)

where 0 is termed the phase noise sideband level.

0 is caused by noise in the stable

local oscillator (STALO) circuitry. It has the units of dBc/Hz which means dB relative to

the power in the carrier of the radar, measured in a 1 Hz bandwidth. Typical values for

0 are -125 to -140 dBc/Hz for radars that use STALOs that employ very narrow band

filters or phase locked loops (such as Klystron-based STALOs); around -110 dBc/Hz for

radars that use frequency multiplied or digitally synthesized STALOs; around -90

dBc/Hz for radars that use Magnetron transmitters. I should caution you that these

numbers are based on my experiences with specific hardware. Other people have

reported better (meaning lower) and worse (meaning higher) values of phase noise for

radars similar to those I have worked with. There is still a fairly large controversy over

phase noise levels. However, the general consensus seems to be that modern, well

designed radars that use good STALOs have phase noise values in the vicinity of -125 to

-135 dBc/Hz. Some advanced radar designs appear to be pushing phase noise to -150 to -

160 dBc/Hz.

If we ignore phase noise, f reduces to

f f . (2-55)

f is the center spectral line or carrier and represents a pure sinusoid

The final term in the equation for oS f is the constant K. This constant is

chosen such that

2

c rP K MF f S f df

. (2-56)

Where cP is the clutter power calculated from the radar-range equation and the clutter

model that we developed earlier. The integral simply states that the total power in the


matched filter output, for a single received pulse, is the integral of its power spectrum

over all frequencies. If we have correctly specified the terms 2

MF f and rS f , we

should get that

cK P . (2-57)

A few notes on the forms of the terms of rS f :

1. All of the terms contain either impulses, Gaussian functions or constants. We

chose them that way because of the next three notes and the fact that they

represent the “real world” fairly well.

2. The convolution of two Gaussian functions is a Gaussian function with a variance

equal to the sum of the variances of the two Gaussian functions. That is, if

2 2

12

1

1

1

2

fG f e

and 2 2

22

2

2

1

2

fG f e

then

2 2 2

1 22

1 2 2 2

1 2

1

2

fG f G f e

.

3. The convolution of an impulse with any other function is the other function

shifted so that it is centered on the impulse. That is:

0 0G f f f G f f .

4. The convolution of a constant with any other function is a constant with a value

equal to the product of the original constant and the area under the function. That

is: G f K K G f df

.

As a final note, the spectrum equations we have derived here are general and can

be used to represent any clutter, or target, by changing C f . As an example, to

represent a target with non-zero Doppler frequency, we would choose dC f f f

, where df is the Doppler frequency of the target. When we use the model to represent

targets, we usually ignore phase noise and the spectrum spreading caused by scanning.

2.4 SIGNAL PROCESSOR ANALYSES

2.4.1 Introduction and Background

Now that we have an equation for the clutter (and target) spectrum at the ADC

output, we want to turn our attention to considering how to use it to perform signal

processor analyses. As indicated earlier, we will consider digital signal processors. For

now, we assume we have a signal processor with a z-transfer function of H z and

equivalent frequency response of


2

2

j fTz eH f H z

. (2-57)

We note that we are using the form of frequency response usually used in analyzing

random processes because, by assumption, our clutter (and target) signal at the input to

the signal processor is a random process.

The standard way of performing digital signal processor analyses, in the

frequency domain, is to find oS f from the sum given earlier, multiply it by H f and

integrate the result over (-1/2T, 1/2T] to find the power at the output of the signal

processor. Recall that we use this approach because, for digital signals, the only valid

frequency region is (-1/2T, 1/2T].

We propose a different approach here. Rather than use oS f over (-1/2T, 1/2T]

we use 2

MF rS f MF f S f over , . We also use H f over , . As

before, we multiply these and integrate to find the power; except that this time we

integrate over , . With this approach we are “unfolding” oS f and H f and

then “refolding” them when we find the power. This approach had the advantage of

avoiding the oS f sum and allowing us to work with only MFS f . It has the added

advantage allowing us to analyze staggered waveforms without having to derive a new

set of equations. Staggered waveforms are waveforms whose pulse repetition interval

(PRI) changes from pulse to pulse.

We want to digress to show that the approach we propose is valid, in terms of

computing the power out of the signal processor. We start by noting that the power at the

output of the digital signal processor is

1 2

1 2

T

out o

T

P T H f S f df

. (2-58)

We substitute for oS f and bring H f inside of the sum to yield

1 2 1 2

1 2 1 2

1T T

out MF MF

l lT T

P T H f S f l T df H f S f l T dfT

. (2-59)

Note that we canceled the T’s. We next note that H f is periodic with a period of 1 T .

This allows us to replace H f with H f l T since H f H f l T . Doing

this and reversing the order of summation and integration results in:

1 2

1 2

T

out MF

l T

P H f l T S f l T df

. (2-60)

In each of the integrals of the sum, we make the change of variables f l T to get

1 2

1 2

lT T

out MF

l lT T

P H S d

. (2-61)


Finally, we recognize the above as an infinite sum of non-overlapping integrals, which

we can write as a single integral over , . Specifically,

out MFP H f S f df

(2-62)

which is the desired result. Note that we changed the variable of integration from

back to f.

2.4.2 Moving Target Indicator (MTI)

We are now ready to consider our first signal processor: a moving target indicator,

or MTI. An MTI is a high-pass digital filter that is designed to reject clutter, but not

targets that are moving. A block diagram of a two-pulse MTI is shown in Figure 2-4. It

is termed a two-pulse MTI because it operates on two pulses at a time. It successively

subtracts the returns from two adjacent pulses. For signal processor buffs, it is a first-

order, non-recursive, high-pass, digital filter.

Figure 2-6 – Two-Pulse MTI

A time domain model of the filter is

1SP o ov k v k v k . (2-63)

Note that if ov k K then 1 0SP o ov k v k v k K K . Thus, the MTI

perfectly cancels DC, or zero-frequency signals. If we take z-transform of both sizes we

get

1

SP o oV z V z z V z (2-64)

which can be solved to yield the filter transfer function as

11

SP

U

o

V zH z z

V z

(2-65)

where we use the subscript U to denote the fact that the filter transfer function is un-

normalized. We will discuss normalization of the MTI shortly. From Equation 2-65 we

can find the filter frequency response as


2 2

2 22 1 2

2 2

2

1 1

2 sin

4sin

j fT j fT

j fT

U U z e z e

j fT j fT j fT j fT

H f H z z e

e e e e j fT

fT

. (2-66)

A plot of UH f is shown in Figure 2-7 for the case where T = 400 s.

Figure 2-7 – Frequency Response of an un-normalized 2-pulse MTI

2.4.2.1 MTI Response Normalization

Before we turn our attention to computing the clutter rejection capabilities of an

MTI we need to normalize the MTI response to something. Without normalization, it is

difficult to quantify the clutter rejection capabilities of the MTI because we have no

reference. The instinct is to say that the clutter rejection is a measure of the clutter power

out of the MTI relative to the clutter power into the MTI. However, we can make this

anything we want by changing the gain of the MTI. To avoid this problem, we normalize

the gain of the MTI so that it has a noise gain of unity. In this way we can easily

compare the CNR at the output of the MTI to the CNR at the input since we have noise

power as a common reference. In a similar fashion, we will be able to characterize the

SNR improvement, or degradation, through the MTI.

To carry out the computations we consider that the MTI is digital and work in the

digital domain. We assume that the noise into the MTI is white and has a power, and

power spectrum, (the power and power spectrum of white noise in digital systems is

equal) of

0N pP kT F , (2-67)

which is the effective noise power computed from the radar range equation. The

assumption of white noise is good because the bandwidth of the noise at the matched

filter output is large compared to the sampling frequency.


Earlier we wrote the equation for the MTI time response equation as

1SP o ov k v k v k . (2-68)

We want to add a gain to this equation and adjust the gain so that the noise power out of

the MTI is the same as the noise power into the MTI. Thus, we rewrite the MTI equation

as

1SP MTI o ov k K v k v k (2-69)

and find the value of MTIK such that the noise power out of the MTI,

2

Nout SPnP E v k , (2-70)

is equal to the noise power, NP , into the MTI.

If we let ov k n k in the above equation we get

1SPn MTIv k K n k n k . (2-71)

We note that the noise power into the MTI is

2

NP E n k . (2-72)

We can then write the noise power at the output of the MTI as

2 22

2 22 2

2 2

22 2

1

1

1 1

2 2

Nout SPn MTI

MTI MTI

MTI MTI

MTI MTI N

P E v k E K n k n k

E K n k E K n k

E K n k n k E K n k n k

E K n k K P

. (2-73)

In the above, the cross expectations on the third line are zero because of the assumption

that n k is white. The fact that 2 22 2 1MTI MTIE K n k E K n k comes from the

assumption that n k is WSS. From the above, it is apparent that for Nout NP P we must

have that 1 2MTIK . If we apply this to our previous derivation, it is easy to see that

2 22sinMTI UH f K H f fT (2-74)

rather than the 24sin fT we derived earlier. A plot of the normalized H f is

shown in Figure 2-8.


Figure 2-8 – Normalized Frequency Response of a 2-pulse MTI

Note: we can also find NoutP from

1 2 1 2

1 2 1 2

T T

Nout N N N

T T

P T P H f df P T H f df P

. (2-75)

The proof that

1 2

1 2

1

T

T

T H f df

(2-76)

for the normalized version of H f , is left as an exercise.

2.4.2.2 MTI Clutter Performance

Now that we have normalized our MTI we want to compute its clutter attenuation

and SCR improvement. (Skolnik and other authors also call SCR improvement,

Improvement Factor, a term that we will also use.) We start with clutter attenuation.

Clutter attenuation is defined as the ratio of the CNR at the input to the MTI to the CNR

at the output of the MTI. The CNR at the input to the MTI is the CNR given by the radar

range equation . The CNR at the output of the MTI is the clutter power out of the

(normalized) MTI divided by the noise power at the output of the MTI. However, the

noise power at the output of the MTI is equal to the noise power at the input. Thus, the

clutter attenuation is the ratio of the clutter power at the input to the MTI divided by the

clutter power at the output of the MTI. In equation form

in c N c

out cout N cout

CNR P P PCA

CNR P P P . (2-77)

In this equation cP is the input clutter power and is the same

cP we used earlier.

Similarly, NP is the noise power we used earlier. The definitions of the rest of the

variables should be obvious.


For the next step, we want to eliminate the explicit dependence on cP . The clutter

power out of the (normalized) MTI is

cout MFP H f S f df

. (2-78)

But,

2

MF c rS f P MF f S f . (2-79)

Thus

2

cout c r cP P H f MF f S f df PG

(2-80)

and

2

1c c

c

c r

P PCA

PG GP H f MF f S f df

. (2-81)

This means that, to compute clutter attenuation, we only need to compute

2

rG H f MF f S f df

. (2-82)

We can assume that 1MF f for practical radars. With this we get

rG H f S f df

. (2-83)

For our first computation of clutter attenuation, we will ignore the phase noise and

let f f . Using the forms for SC f and C f given earlier we can write

2 22 2

2 22 2

22

22

11

2 2

11

2 2

fCS

S

ff

r

S C

ff

s

kS f e e k f f

ke k e

T

T

(2-84)

where 2 2 2

fC S T.

Lets further simplify our problem by assuming that 1k . With this we get

2 221

2

f

rS f e

T

T

. (2-85)

It should be obvious how one would extend this to the case where 1k .


If we substitute this, and the equation for H f , into the equation for G we get

2 222 1

2sin2

fG fT e df

T

T

. (2-86)

We can simplify this further by observing that, over the region of f where rS f is non

zero, we can approximate sin fT by fT . Over the rest of f, rS f is very small so

that the approximation to sin fT is not very important (i.e., fT is good enough).

With this, G becomes

2 2 2 2

2

22 222 1

22 2

f ffTG e df T f e df

T T

T T

. (2-87)

From random variable theory we recognize the term in parentheses as 2

T . This gives

2 2 22G T T

, (2-88)

and

2

2 2 2

12

2 2

PRFCA

T

T T

. (2-89)

We next want to look at improvement factor, or SCR improvement. Improvement

in defined as the SCR out of the MTI divided by the SCR into the MTI, averaged over all

Doppler frequencies of interest. The need for averaging comes from the fact that the

signal power out of the MTI will depend upon the target Doppler frequency. Indeed, if

we look at the frequency response plot in Figure 2-8 we note that the gain of the MTI

varies from 0 to 2 w/w. In order to remove this frequency dependency from the final

answer, we average across frequency. It should be noted that some people quote SCR

improvement as that measured at the peak response of the MTI. This is called peak SCR

improvement.

From the frequency response of Equation 2-74, the signal gain through the MTI,

averaged over one cycle, is unity. Also, recall that we normalized the MTI so that its

noise gain was also unity. With this, the SNR gain through the MTI is unity. That is,

out inSNR SNR . With this and the clutter attenuation results from above we get

out out out inSCR

in in in out

SCR SNR CNR CNRI CA

SCR SNR CNR CNR . (2-90)

Thus, because of the normalization we have performed, the SCR improvement is equal to

the clutter attenuation. It should be noted that the peak SCR improvement is 2CA in this

case since the peak gain through the MTI is 2 w/w.


2.4.2.3 An Example

Let’s compute the clutter attenuation for an example radar. This radar has a

carrier frequency of 8 GHz and uses a PRI of 400 µs. We assume that the clutter is

wooded hills in a 20 knot wind. From Table 15.1 on page 15.9 of Skolnik’s Radar

Handbook, the appropriate standard deviation on clutter velocity is 0.22 m/sv . From

this we can derive the frequency spread of the clutter as

2 2 0.22

11.7 Hz0.0375

vfC

. (2-91)

If we assume we are in a tracking environment and ignore scanning for now, we get

11.7 HzfC T and

2

2 2313 w/w or 33.6 dB2

scr

PRFCA I

T

. (2-92)

As an extension, lets assume the same radar parameters but use a scanning radar.

We assume that the radar has a 2 second scan period. We use the beam width associated

with the example of Figure 2-2, i.e. 1.5 degA . With this, the standard deviation on

the spectrum due to scanning is

2

0.265 31.8 HzS

A scanT

. (2-93)

The combination of the clutter spread and scanning gives a total spectrum spread of

2 2 33.9 Hzfc s T . (2-94)

The resulting clutter attenuation is

2

2 276 w/w or 24.4 dB2

scr

PRFCA I

T

. (2-95)

Let us carry this example further and examine SNR, CNR, and SIR. In addition to

the aforementioned parameters we assume the following for the radar

Peak Power is 100 Kw

Noise Figure is 6 dB

Total Losses for the target and clutter are 13 dB

The height of the antenna phase center is 5 m

The rms antenna side lobes are 30 dB below the peak gain

The clutter backscatter coefficient is -20 dB

The target RCS is 6 dBsm


The ranges of interest are 2 Km to 50 Km

From the beam width and the equations from EE619 we compute the antenna gain as

25,000 11,111 w/w 40.5 dBA EG .

Using these parameters, the SNR, CNR, and SIR vs. R at the matched filter output

is as shown in Figure 2-9. It will be noted that the SNR is reasonable but the SIR is much

too low to support detection and track.

Figure 2-9 – SNR, CNR, SIR at Matched Filter Output

Figure 2-10 contains plots similar to those of Figure 2-9 for the two cases (non-

scanning and scanning) where and MTI is used. As can be seen, the MTI significantly

reduced the CNR and allowed the SIR to approach the SNR. Thus, in these cases the

radar should be able to do a reasonable job of detecting and tracking the target.


Figure 2-10 – SNR, CNR, SIR at MTI Output

2.4.2.4 Phase Noise

We next want to examine how to handle phase noise in the MTI. Referring to

Equations 2-48 and 2-79, we can see that if we use 0f f we can write

MFS f as

2

2 2

0

MF c S

c S c

co c

S f P MF f C f C f f

P MF f C f C f P MF f

S f S f

. (2-96)


We considered the first term above and will now turn our attention to the second

term,

2

0c cS f P MF f . (2-97)

We note that the total power in cS f is

2

0 0c c c pP S f df P MF f df P

. (2-98)

We also note that, if p T , the spectrum of cS f is wide relative the sample

frequency (PRF) and thus that we can assume that cS f is the spectrum of white noise

with a power of P . Since we have normalized the MTI so that it has unity gain for white

noise, the phase noise component of the clutter at the output of the MTI will be P . If we

combine this with the clutter power contributed by the coS f term we now find that the

total clutter power out of the MTI is

0cout c c pP PG P (2-99)

and the resulting CA is

0 0

1c

c c p p

PCA

PG P G

. (2-100)

To get a feeling for the impact of phase noise on MTI signal processors, let’s

revisit the previous example and plot clutter attenuation vs. the phase noise level, 0 .

This plot is shown in Figure 2-11. It will be noted that, for the clutter only case, the

phase noise doesn’t start degrading the clutter attenuation until the phase noise level is

above about -105 dBc/Hz. For the case where the scanning effects are included, the

phase noise doesn’t degrade clutter attenuation until the phase noise level exceeds about -

95 dBc/Hz. As we increase the order of the MTI processor, we will see that phase noise

starts to contribute more to the overall degradation in clutter attenuation.


Figure 2-11 – Phase Noise Effects on Clutter Attenuation

2.4.2.5 Higher Order MTI Processors

If we want to make the radar of the previous example operate in a noise limited

environment we would need more than the 33.6 dB of clutter attenuation offered by the

2-pulse MTI. This leads us to ask the question of how much the clutter attenuation could

we obtain if we used a 3 or 4 pulse MTI. We address this issue now.

To obtain an n-pulse MTI we cascade n-1, 2-pulse MTIs. Specifically, if the

transfer function of a 2-pulse MTI is H z , the transfer function of an n-pulse MTI is

1n

n MTIH z K H z

(2-101)

where the constant MTIK is included to normalize nH z so that it provides unity noise

gain.

The specific transfer functions for 2-, 3-, 4-, and 5-pulse MTIs are

1

2

21 1 2

3

31 1 2 3

4

41 1 2 3 4

5

1

1 1 2

1 1 3 3

1 1 4 6 4

MTI

MTI MTI

MTI MTI

MTI MTI

H z K z

H z K z K z z

H z K z K z z z

H z K z K z z z z

(2-102)

It will be noted that the coefficients of the powers of z are binomial coefficients (see the

CRC Handbook) with alternating signs.


Following the method we used for the 2-pulse MTI, we can compute the

normalizing coefficient as

2

2

1

1MTI n

m

m

K

a

(2-103)

where thema are the MTI coefficients (binomial coefficients) given above. Specific

values of 2

MTIK for the 2-, 3-,4- and 5-pulse MTI are 1/2, 1/6, 1/20 and 1/70, respectively. 2

MTIK for an n-pulse MTI with binomial coefficients is given by

2 1

2

1

2 2 1 1 !!

2 1 !!

nn

m

m

na

n

(2-104)

or

2

2 1

2 1 !!

2 2 1 1 !!MTI n

nK

n

. (2-105)

where 2 1 !! 1 3 5 2 1m m and 2 !! 2 4 2 , 0 !! 1m m .

The above values are summarized in Table 3.

Table 2-3 - 2

MTIK for Various Size MTIs

Number of pulses in MTI – n 2

MTIK

2 1/2

3 1/6

4 1/20

5 1/70

n 2 1

2 1 !!

2 2 1 1 !!n

n

n


If we extend the results of our 2-pulse analysis, we can write the normalized

frequency response of an n-pulse MTI as

2 12 2sinn

n MTIH f K fT

. (2-106)

Figure 2-12 contains plots of the normalized frequency responses of 3- and 4-pulse

MTIs. It will be noted that peaks of the response become narrower and the valleys

become wider as the order of the MTI increases. This means that we should expect

higher CA and SCR improvement as the order of the MTI increases.

Figure 2-12 – Normalized Frequency Response of a 3- and 4- pulse MTI

We can compute the clutter attenuation for the general n-pulse MTI by extending

the work we did for the two pulse MTI.3 We again use the approximation that

sin fT fT . With this we get that

3 In this derivation we are using the clutter spectrum with the assumption that k=1 and no phase noise. See

the derivation for the 2-pulse MTI.


1

CAG

(2-107)

where G now becomes

2 2

2 1

222

2

n

f

MTI

fTG K e df

T

T

. (2-108)

Evaluation of this integral yields

2 12 2 1 1 !! 2n

MTIG K n T

T (2-109)

where 2 1 !! 1 3 5 2 1m m . We can write the clutter attenuation as

2 2 1

2 1

2

1 1 1

2 1 1 !! 2

1

22 1 1 !!

n

MTI

n

MTI

CAG K n T

PRF

K n

T

T

. (2-110)

As with the 2-pulse MTI case, we can show that the signal power averaged across

all expected target velocities is equal to one so that the average SNR gain through the

MTI is unity. With this, the SCR improvement, as before, is

SCRI CA . (2-111)

Specific values of CA and SCRI for a 3- and 4-pulse MTI are

4

3 3 22

SCR

PRFI CA

T

(2-112)

and

6

4 4

4

3 2SCR

PRFI CA

T

. (2-113)

If we revisit the previous example, we find that the CA for the non scanning case

is 64.3 dB for the 3-pulse MTI and 93.1 dB for the 4-pulse MTI. The CA for the

scanning case are 45.8 dB for the 3-pulse MTI and 65.4 dB for the 4-pulse MTI. Since

the clutter attenuations are so high for the non-scanning case, it is likely that phase noise


will become the limiting factor on CA for the 3- and 4-pulse MTIs. This is left as a

homework assignment.

As a closing note, the most common order MTIs in use are 2- and 3- pulse MTIs.

On rare occasions one will encounter a radar that uses a 4-pulse MTI and one almost

never encounters a radar that uses a 5- or higher-pulse MTI. The reason for this is that

higher order MTIs can’t achieve their theoretical potential because of phase noise, timing

jitter, instrumentation errors, round-off errors and the like. Therefore, there is usually no

reason to use higher than a 3-pulse MTI.

2.4.2.6 Staggered PRIs

Examination of the MTI frequency response plots of presented earlier indicate

that the SNR gain through the MTI can vary considerably with target Doppler frequency.

This is quantified in Figure 2-13 below which is a plot of the percent of time that the

MTI gain will be above some level. For example, the MTI gain will be above -5 dB 73%

of the time for the 2-pulse MTI, and 60% and 52% of the time for the 3- and 4- pulse

MTIs. If we say, arbitrarily, that the MTI is blind when the gain drops below -5 dB, we

can say that the 2-pulse MTI is blind 27% of the time and the 3- and 4-pulse MTIs are

blind 40% and 48% of the time. We would like to improve this situation. A method of

doing this is to use staggered PRIs. That is, we use waveforms in which the spacing

between pulses changes on a pulse-to-pulse basis. Through this approach we “break up”

the orderly structure of the MTI frequency response and “fill in” the nulls. We also

reduce the peaks in the frequency response. The net effect is to provide an MTI

frequency response that doesn’t have deep nulls and large peaks but, rather, a somewhat

constant level. The response still has the null at zero frequency and thus still provides

clutter rejection.


Figure 2-13 – Percent of Time MTI Gain Above Specified Levels

To illustrate how to work with staggered waveforms we consider a 3-pulse MTI

and a waveform that alternates between two PRIs of 1T and

2T . A sketch of the

waveform is shown in Figure 2-14. This type of waveform is termed a two-position

stagger because it uses two PRIs. An n-position stagger would use n PRIs, some of

which could be the same.

Figure 2-14 – Two-Position Stagger Waveform

To determine the frequency response of an MTI with a staggered waveform we

work in the continuous time domain and imply sampling by using impulse functions.

Thus, the impulse response of a 3-pulse MTI with binomial coefficients and the

waveform above is:

1 1 22MTIh t K t t T t T T . (2-114)

If we adjust the time origin so that it is centered on the middle pulse we get

1 22MTIh t K t T t t T . (2-115)


This will make some of the math to come a little easier.

To find H f for this filter we find the Fourier transform of h t and take its

magnitude squared. That is,

2

H f h t h t h t

(2-116)

or

1 2 1 22 2 2 22

2

1 2 1 2

2 2

6 4cos 2 4cos 2 2cos 2

j fT j fT j fT j fT

MTI

MTI

H f K e e e e

K fT fT f T T

. (2-117)

2 1 6MTIK as for the regular, 3-pulse MTI. Figure 2-15 contains a plot of H f for a 3-

pulse and PRIs of 385 and 415 s. The operating frequency of the radar is 8 GHz, which

was used to convert frequency (f in the above equation) to range-rate as shown on the

plot. The plot also contains the response of the MTI for the unstaggered waveform. It

will be noted that the use of the stagger fills-in the nulls that are present in the

unstaggered case.

Figure 2-15 – 3-pulse MTI Response With and Without Stagger

The response with the staggered waveform still has a considerable variation in

MTI gain as a function of range-rate. This is due to the fact that we only used a 2-

position stagger. According to your Skolnik (Page 15.36 in the Radar Handbook) the use

of a 4-position stagger would provide a better response. However, the responses he

shows are not much better than the one in Figure 2-15.


The use of a 2-position stagger with a 3-pulse MTI is a sort of “matched”

condition. In other words, the complete characteristics and effects of the stagger are

captured by looking at three pulses. Had we used a 4-position stagger we would use only

three pulses at a time and would be able to capture only two PRIs at a time. This means

that the frequency response of the MTI actually varies with time as different sets of three

pulses are processed through the MTI. You will look at this phenomenon in a future

homework. To get around this time variation of MTI response we usually determine the

response to the different sets of PRIs and then average the results. The average is done

on H f . Thus, if we had a 4-position stagger with PRIs of 1T ,

2T , 3T and

4T we

would find: 1H f using three pulses with PRIs of 1T and

2T , 2H f using three

pulses with PRIs of 2T and

3T , 3H f using three pulses with PRIs of 3T and

4T and

4H f using three pulses with PRIs of 4T and

1T . We would then form the averaged

response as

1 2 3 4 4H f H f H f H f H f . (2-118)

To determine the clutter attenuation of an MTI with a staggered waveform we use

the same formulas as for the unstaggered case. To find the SNR gain through the MTI

we find the average signal gain from the MTI response and use this as the SNR gain. We

can do this because we have still normalized the MTI so that it provides unity noise gain.

We often find the average MTI gain via the “eyeball” method; we estimate it from the

plot. A better method would be to numerically average the gain (in w/w) across the

range-rates of interest. The MTI gain indicated via the “eyeball” method for the response

below is about 0 dB. The calculated gain is -0.08 dB.

2.4.3 Pulsed-Doppler Processors

2.4.3.1 Introduction

We now want to turn our attention to pulsed-Doppler signal processors. The

exact origin of the phrase “pulsed-Doppler” is not clear. It probably derives from the fact

that early pulsed-Doppler radars did CW processing using pulsed waveforms.

Specifically, classical CW radars work primarily in the frequency (and angle) domain

whereas pulsed radars work primarily in the time (and angle) domain. It is assumed that

the phrase pulsed-Doppler was coined when designers started using pulsed radars that

worked primarily in the frequency, or Doppler, domain. Early pulsed-Doppler radars

used a 50% duty cycle pulsed waveform and had virtually no range resolution capability,

only Doppler resolution. The use of a pulsed waveform was motivated by the desire to

use only one antenna and to avoid isolation problems caused by CW operation. Modern

pulsed-Doppler radars are actually high PRF pulsed radars with duty cycles in the 10%

range. They are used for both range and Doppler measurement. Most pulsed-Doppler

radars are ambiguous in range and unambiguous in Doppler. However, because of

waveform constraints, some are ambiguous in both range and Doppler. For purposes of


our signal processor analyses we will consider pulsed-Doppler waveforms that are

unambiguous in Doppler.

The reason for using pulsed-Doppler radars is not clear. One of the claims is that

they provide better clutter rejection capabilities than pulsed radars. However, they need

to do so since they must contend with more clutter than do pulsed radars. It is believed

that designers may be forced to pulsed-Doppler radars when the radar must operate at

long ranges. However, this is not clear. An argument is that long-range radars must use

low PRFs for range unambiguous operation. Since clutter attenuation in MTI processors

decreases as the PRF decreases such radars might not be able to achieve suitable

performance against short range targets in clutter. However, with current computer

capabilities and transmitter flexibility, one could use higher PRF waveforms for short

ranges and lower PRFs at longer ranges. This way it would be possible to obtain good

MTI performance against short range targets. The MTI performance against long range

targets would not be good. However, the clutter will be past the radar horizon at long

ranges and good MTI performance may not be necessary.

Another possible reason for using pulsed-Doppler radars is that they provide the

capability of measuring target Doppler frequency, and thus have the ability to

discriminate on the basis of Doppler frequency. The former could be helpful in an ECM

environment because it provides a cross-check on the range-rate measured by the target

tracker. This, in turn, could help in attempting to counter range-gate deception jamming.

Pulsed-Doppler radars could be helpful in rejecting weather clutter and chaff

because of their ability to provide good Doppler measurement capability. Also, because

of the potential of using narrow bandwidth Doppler filters, the radar could be less

susceptible to noise jamming.

2.4.3.2 Pulsed-Doppler Clutter

The ground clutter environment in pulsed-Doppler radars is generally more severe

than in pulsed radars that are unambiguous in range. This has to do with the fact that, in

pulsed-Doppler radars, the signal returned from long-range targets must compete with

clutter at short ranges. Figure 2-16 is an attempt to illustrate this. In this figure, the solid

triangle is a target return from the first (leftmost) pulse in the string of pulses. The

dashed triangles are returns from the same target but different pulses. The solid, curved

line through the solid triangle represents the clutter from the pulse immediately preceding

the triangle. (The other, dashed, curved lines are clutter returns from other pulses.) The

significance of what signal comes from which pulse has to do with range attenuation.

Since the target is at a range of tgtR it will have a range attenuation of 4

tgtR . The clutter in

the target range cell is at a range of clutR and will undergo a range attenuation of 3

clutR

(recall that clutter attenuation varies as 3R ). Now, since tgt clutR R the target will

undergo much more attenuation than the clutter. The result of this is that the SCR at the

input to the signal processor in pulsed-Doppler radars is much lower than for the same

scenario in pulsed radars.


Figure 2-16 – Target and Clutter Returns in a Pulsed Doppler Radar

As an illustration of the difference in SCR ratios in pulsed and pulsed-Doppler

radars, Figure 2-17 contains plots of CNR and SCR for a radar similar to the one we

considered in the previous MTI example. These CNRs, and SCRs are termed single-

pulse CNRs and SCRs.

The bottom two curves correspond to the case where the radar uses a 50 KHz PRF

pulsed-Doppler waveform (with 1- s pulses) and the top two curves correspond to the

case where the radar uses the 2.5 KHz PRF of the previous examples. The pulsed-

Doppler curves were obtained by folding the clutter power calculated for a single pulse.

That is, if 1cP R is the clutter power due to a single pulse, the clutter power due to a

string of high PRF pulses is

1 2cpd c

k

P R P R kcT

. (2-119)

In the above equation, the sum can usually be limited to a small number of terms since

1cP R drops off rapidly past the radar horizon. Also, in practice 1 0cP R for

2 2pR c . When generating cpdP R , one only considers ranges of

, 1 , 02 2

p p

c cR kT k T k

. (2-120)

This accounts for the fact that the receiver is shut off during the transmit pulse and for

one pulse width before the transmit pulse. This is what gives rise to the blank regions in

the CNR and SCR plots of Figure 2-17.


Figure 2-17 – CNR and SCR for a Pulsed and Pulsed-Doppler Radar

In Figure 2-17 it will be noted that the CNR steadily decreases with target range

for the low PRF case. However, the CNR stays high for the pulsed-Doppler case.

Similarly, the SCR for the pulsed case rises and eventually goes above 0 dB at ranges

greater than about 23 Km. On the other hand, the SCR for the pulsed Doppler case

steadily decreases with increasing target range. The result of all of this is that the pulsed-

Doppler signal processor must provide considerably more clutter rejection capability than

the MTI processors we previously considered.

2.4.3.3 Signal Processor Configuration

The actual implementation of a particular, pulsed-Doppler signal processor will

depend upon the specific radar design and whether the signal processor is in the search

receiver or the track receiver. We will discuss some sample configurations later. For

purposes of analyzing what the signal processor does to the signal, noise and clutter we

can use the somewhat generic signal processor chain shown in Figure 2-18.


Figure 2-18 – Pulsed Doppler Signal Processor

For our purposes, the signal processor starts with the single-pulse matched filter

followed by the ADC. In practical systems, the ADC includes an anti-aliasing filter to

limit the bandwidth of the signals that are sampled by the ADC. Since we did not include

the anti-aliasing filter in our original development of Section 2.3 we will not include it

here. It turns out that this will not affect our analysis results. As a reminder, the ADC

samples the matched filter output once per PRI, every T , on the peak of the matched

filter response.

The high-pass filter following the ADC is used to reduce the clutter power that is

located near zero Doppler. In addition to reducing clutter power it also serves to reduce

the dynamic range requirements on the band-pass filter following the high-pass filter.

The high-pass filter is almost always included in ground based radars because of these

dynamic range considerations. It is almost never included in airborne radars because one

cannot guarantee that the clutter will be a zero Doppler.

Sometimes the high-pass filter is implemented before the ADC to limit the

dynamic range of the signal into the ADC. In the past it was thought that the dynamic

range of the ADC needed to be greater than the SCR at the ADC input. However, recent

analyses indicated that this is not the case. In any event, it turns out that the analyses

presented herein do not depend upon whether the high-pass filter is before or after the

ADC since we will not consider ADC quantization or dynamic range in these analyses.

(ADC quantization and dynamic range will be discussed later.)

The final device in the signal processing chain is the band-pass filter. This filter

usually has a small bandwidth (200 to 1000 Hz) and is centered on the target Doppler

frequency.

As a note, the implementation of pulsed-Doppler signal processors has evolved

over the years from all analog to all, or almost all, digital. The evolution has generally

been driven by the speed, availability and cost of ADCs and digital signal processing

components. Older radars (pre 1980’s or so) used all analog signal processors. Radars

designed between about 1980 and a few years ago used a mix of digital and analog

components. Pulsed-Doppler radars being designed today are almost exclusively digital.

Some go to the digital domain at the matched filter output, as in Figure 2-18. Others go

to the digital domain at the IF amplifier output and implement the matched filter in the

digital domain. Future plans (hopes) are to use phased array radars with solid-state

transmit-receive (TR) modules, digitize the signals at the output of each TR module and

do the beam forming and signal processing in the digital signal processor. This concept

goes by the name of space-time signal processing and has been touted for many years as

having the potential of offering huge (but somewhat vaguely stated) potential in all


aspects of interference (clutter, jamming, etc.) rejection. Hardware technology is not

quite at the state where it can support space-time signal processing.

2.4.3.4 Analysis Techniques

The analysis of digital, pulsed-Doppler signal processors is performed using

techniques very similar to those used in analyzing MTI processors. Specifically, one

multiplies the spectrum of the ADC output, oS f , by the G f ’s of the signal

processor and integrates the resulting spectrum, using Equation 2-62, to find the signal,

noise and clutter power at the output of the signal processor. These operations are

usually performed on the computer because of the difficulty of analytically evaluating the

integral of the (rather complicated) output spectrum.

2.4.3.4.1 Signal

For the case of the target, we assume that the signal is a sinusoid at some Doppler

frequency of df so that

rs s dS f P f f (2-121)

where sP is the signal power computed from the radar-range equation. With this, the

power spectrum at the output of the matched filter is

2

MFs s d dS f P MF f f f . (2-122)

From Equation 2-62, with

H BH f G f G f (2-123)

the signal power at the output of the signal processor is

sout S sP G P . (2-124)

where

2

S d H d B dG MF f G f G f . (2-125)

In most applications, the main lobe of the matched-range Doppler cut of the ambiguity is

wide relative to target Doppler so that 2

1dMF f . Also, the target Doppler is

frequency is normally in the pass band of the HPF and the BPF is centered very close to

the target Doppler frequency. This means that 1H dG f and 1B dG f . Combining

these results in the observation that 1SG . In practice we account for the fact that the

various terms of Equation 2-125 are not exactly unity by including a loss term in the

radar range equation. The most likely term to be less than unity is B dG f since it is not

easy to perfectly match the center frequency of the BPF to the target Doppler. The

resulting loss term for this is the Doppler straddling loss we discussed in EE619.


2.4.3.4.2 Noise

For noise we have

0rnS f kT F (2-126)

and

2

0MFnS f kT F MF f . (2-127)

In these equations 0kT F is the receiver noise power spectral density from the radar-range

equation. The definition of rnS f implies that the receiver noise, at the input to the

matched filter, is white. The noise power out of the signal processor is given by

2

0Nout H BP kT F MF f G f G f df

. (2-128)

From earlier, we know that the noise power into the signal processor (out of the ADC) is

0N pP kT F (2-129)

so that we can write NoutP as

2

Nout N p H B N NP P MF f G f G f df G P

(2-130)

which results in

2

N p H BG MF f G f G f df

. (2-131)

2.4.3.4.3 Clutter

For the clutter we use the clutter model of Equation 2-53 with 1k . We will

assume that the radar is not scanning so that we do not need to include a scanning

spectrum. We use the phase noise spectrum of Equation 2-54. With this, we get

0 0rc c c cS f PC f f PC f P , (2-132)

and

2 2

0MFc c cS f PC f MF f P MF f . (2-133)

The clutter power at the output of the signal processor is thus


2

2

0

cout c H B

c H B

P P MF f C f G f G f df

P MF f G f G f df

. (2-134)

We can write Equation 2-134 as

0cout c C N pP P G G (2-135)

with

2

C H BG MF f C f G f G f df

(2-136)

and NG as defined in Equation 2-131.

In these equations, cP is the folded clutter power, cpdP R given by Equation 2-

119.

All of the integrals above are over the limits of -∞ to ∞. Clearly, it is not possible

to numerically integrate over these limits. However, given that most of the area of the

2

MF f function is in the central lobe, it is usually sufficient to integrate over limits of

about 3 ,3p p or 5 ,5p p .

2.4.3.4.4 SNR and SCR Improvement.

If we combine Equations 2-124 and 2-130 we see that the SNR at the signal

processor output is

sout S S Sout SNR

Nout N N N

P G P GSNR SNR G SNR

P G P G (2-137)

where SNRG is the SNR gain of the signal processor and SNR is the single-pulse SNR

computed from the radar-range equation.

If we combine Equations 2-124 and 2-135 we see that the SCR at the signal

processor output is

0

sout S sout SCR

cout c C N p

P G PSCR I SCR

P P G G

(2-138)

where

SNR

SCRCNR

(2-139)


with SNR the single-pulse SNR from the radar-range equation and CNR is the “single-

pulse” CNR at the output of the matched filter, also from the radar-range equation (see

Section 3.4.3.2). Alternately, one could directly use the “single-pulse” SCR (again, see

Section 3.4.3.2).

In Equation 2-138, SCRI is the SCR improvement offered by the signal processor.

From Equation 2-138 it is given by

0 0

1

1 1

SSCR

C N p SCR SNR p

GI

G G G G

(2-140)

where SNRG is as define above and

SCRG represents the ability of the signal processor to

reject the central line clutter (not the clutter that enters through the phase noise). SCRG is

given as

SSCR

C

GG

G (2-141)

where CG is computed from Equation 2-136. As we will see in the following example,

SCRG is usually very large.

2.4.3.5 Example

As an example of how to perform a pulsed-Doppler signal processor analysis we

will consider the example discussed in connection with Figure 2-17. The parameters of

the radar are similar to those we used in the MTI example with the exception that the

peak power is 10 KW instead of 100 KW and the PRF is 50 KHz rather than 2.5 KHz.

We also reduced the antenna height to 3 m and parked the beam at 0.75º in elevation (1/2

beamwidth). The pertinent parameters are:

Peak Power= 10 Kw

Operating Frequency = 8 GHz

Noise Figure = 4 dB

PRF = 50 KHz (PRI = 20 µs)

Pulse Width = 1 µs

Total Losses for the target and clutter = 6 dB

Height of the antenna phase center = 3 m

rms antenna side lobes are 30 dB below the peak gain

Clutter backscatter coefficient = -20 dB

Target RCS = 6 dBsm


Ranges of interest are 2 Km to 50 Km

Figure 2-19 contains plots of SNR, CNR, SCR and SIR at the matched filter output for this

example. It will be noted that, at long ranges, the SNR is lower than desired. Also the,

the SCR is very low. As result of this, the overall SIR is also very low. This means that

the signal processor needs to offer a fairly reasonable SNR increase and a considerable

increase in SCR.

Figure 2-19 – SNR, CNR, SCR and SIR at Matched Filter Output

We want our radar to be able to track targets with range-rates down to about 20

m/s. This means that we must choose the cutoff frequency of the high-pass filter (HPF)

to be


min2 2 201067Hz

0.0375ch

Rf

. (2-142)

We will let 1000 Hzchf . We will use a 5th

-order, Butterworth HPF. Thus we have

10

10 10

11

1 1

ch

H

ch ch

G f

(2-143)

with

tan fT (2-144)

and

tanch chf T . (2-145)

We note that this is an approximation to the frequency response of a 5th

-order, digital

Butterworth filter. It has the flat pass and stop band shapes associated with Butterworth

filters. Its response is a periodic function of 1 T , which is expected since it is a digital

filter.

We typically want to choose the bandwidth of the band-pass filter (BPF) to be as

small as possible since this sets the ultimate SNR and SCR improvement of the radar.

The lower limit is generally set by track requirements, target decorrelation time, number

of BPFs required to cover a PRF, desired Doppler resolution and dwell time (in phased

array radars). The first two set ultimate lower limits of about 10 Hz. The second two set

more practical limits of 200 to 1000 Hz. We will choose a bandwidth of 1000 Hz in this

example. This choice gives a Doppler resolution of about 20 m/s and means we will need

50 filters to cover our PRF of 50 KHz. If we are using a phased array radar, it would

impose a dwell time minimum of about 4 ms (4/(Doppler filter bandwidth)). We assume

that the BPF is a 3rd

order Butterworth filter. We further assume that it is centered on the

target Doppler frequency of 8000 Hz. (which corresponds to a target range-rate of 150

m/s). With this, the frequency response of the BPF is

6

1

12

B

fT

cb

G f

(2-146)

where

tanfT Tf f T (2-147)

and

tancb cbf T . (2-148)

In these equations,

8000 HzTf (2-149)

is the frequency to which the BPF is tuned and


1000 Hzcbf (2-150)

is the bandwidth of the BPF. As with HG f , BG f is an approximation of the

frequency response of a 6th

-order Butterworth, band pass filter in that it has flat pass and

stop bands.4

We will use the same clutter spectrum model we used in the MTI analyses.

Specifically, we let

2 221

2

fcf

fc

C f e

. (2-151)

with

11.7 Hzfc . (2-152)

Note that this assumes that 1k . We will treat the phase noise level, 0 , as a parameter

for now.

Since our pulse is a 1- s, unmodulated pulse we get

2 2sinc pMF f f . (2-153)

With the above, we get that the signal power out of the signal processor is

2

sout s d H d B d sP P MF f G f G f P , (2-154)

which implies that the signal gain through the signal processor, SG , (see Equation 2-125)

is unity. Some thought will confirm the veracity of this statement in that the signal

processor is tuned to the signal.

The noise gain of the signal processor is can be found from Equation 2-131 as,

2

0.021N p H BG MF f G f G f df

. (2-155)

Combining this with SG yields a SNR gain of

1

47.6 or 16.8 dB0.021

SSNR

N

GG

G . (2-156)

4 The BPF of Equation 2-146 is a 3

rd order BPF with complex coefficients. It is centered at +8000 Hz.

Obviously, this is not realizable with standard hardware. It’s equivalent filter, with real coefficients, is a 6th

order BPF. This filter has responses centered at +8000 Hz and -8000 Hz. Thus the doubling of the filter

order.


The clutter gain through the signal processor can be found from Equation 2-136

as

2 241.63 10C H BG MF f C f G f G f df

. (2-157)

This results in a central-line SCR improvement of

24

24

16.14 10 or 238 dB

1.63 10

SSCR

C

GG

G

(2-158)

which, as indicated earlier, is very large. From Equation 2-140 we can compute the SCR

improvement as

24

0 0

1 1

1 1 1.63 10 0.021SCR

SCR SNR p p

IG G

(2-159)

If we assume a phase-noise level of -120 dBc/Hz we get

12 24 824

6

7

1 1

10 1.63 10 2.1 101.63 10 0.021

10

4.76 10 or 76.8 dB

SCRI

. (2-160)

Clearly, the predominant term in SCRI is the phase noise term. This is typical of pulsed-

Doppler radars and signal processors and is why designers strive to make the STALOs in

these radars very stable and quiet.

Figure 2-20 contains plots similar to those of Figure 2-19 at the output of the

signal processor. It will be noted that the SNR improvement provides for good values of

SNR at ranges out to about 50 Km. SCR and SIR at these ranges is not as good as hoped.

There are ranges where SIR drops below 13 to 15 dB (the “standard” rule-of-thumb that

we use for “reasonable” detection and track performance) for certain ranges. Because of

this, we would like to either obtain more SCR improvement from the radar or reduce the

clutter input to the radar.

From Equations 2-159 and 2-160 we see that we could increase SCRI by

decreasing the phase noise. If we were able to decrease the phase noise to -135 dBc/Hz,

SCRI would increase to about 92 dB and the performance of our radar, at long ranges,

would become good.

Another means of improving the SCR at the signal processor output would be to

decrease the clutter power at the input to the radar. There are several ways that the clutter

power into the radar could be reduced. One might be to use a smaller beamwidth.

Another might be to place the radar on a tower and reduce the antenna sidelobes.


3.4.3.6 Variations

The analysis presented above assumed that the signal processor was digital. In

some applications the signal processor is analog or incorporates analog components. A

block diagram of an all-analog signal processor is shown in Figure 2-21. It will be noted

that this block diagram is similar to the block diagram of Figure 2-18 except that the

ADC is replaced by a sampler and a band limit filter. The combination of the sampler

and the band limit filter is often referred to as a sample-and-hold device or as a range

gate.

We can approach the analysis of an analog signal processor via several paths.

One would be to fold the spectrum at the matched filter output using Equation 2-47. One

would then use this folded spectrum with the frequency responses of the analog filters

(band limit, high pass, band pass) filters of Figure 2-21 to find the appropriate spectra at

the output of the signal processor. To find the powers out of the signal processor one

would simply integrate these spectra.

Figure 2-20 – SNR, CNR, SCR and SIR at Signal Processor Output


An alternate approach, would be to use the theory associated with discrete-time

systems and samplers to “fold” the frequency responses of the analog filters. One would

then use the techniques of Section 3.4.3.4 to perform the analyses.

Figure 2-21 – Analog Signal Processor

In still another variation, some pulsed-Doppler signal processors include a

combination of analog and digital signal processing. An example might be where the

band limit and high pass filters are analog filters and the band pass filter is digital.

Again, the analysis could be approached by folding the spectrum at the output of the

matched filter or using discrete time system theory to fold the frequency responses of the

analog filters (the digital filter response would already be folded). If the spectrum is

folded, it would need to be refolded after passing through the analog filter. The refolded

spectrum would then be multiplied by the (folded) spectrum of the band pass filter.

However, in this case, the power out of the signal processor would need to be computed

using the Equation 2-58 since both the signal and filter would be represented in the

discrete-time domain.

In some cases the band pass filter is implemented with a FFT. In fact, if one uses

an N-point FFT one has a bank of N band pass filters. To perform the analyses using the

techniques of this section, one would compute the frequency response of the appropriate

FFT tap and use this in the analysis. This frequency response can be found by

recognizing that the FFT provides samples of the Fourier transform of the weights

applied to the input taps of the FFT. Thus, one could compute the Fourier transform of

the input weights, shift it to the frequency of the FFT tap of interest and square the result

to get its frequency response, as we have defined frequency response in this section.

Keep in mind that the frequency response of the FFT tap is periodic with a period of 1 T .

When computing the Fourier transform one would need to be sure that the resulting

response was periodic.

An assumption of the analyses presented in this section is that the waveform

consists of an infinite number of pulses. In phased array radars this will not be the case;

the waveforms will be finite duration. This introduces complications in the design of the

signal processor, and possibly its analysis. The reason for this has to do with transients in

the filters of the signal processor. Generally, the designer is careful to set the filter

bandwidths, and use time gating, such that the transients have settled fairly well and one


can treat the analysis as if the waveform consisted of an infinite number of pulses. That

is, one can use continuous time analysis techniques.

2.4.5 ADC Effects

In both the MTI and pulsed-Doppler signal processors we assumed that the ADC

had an infinite number of bits and an infinite dynamic range. In essence, we analyzed the

processors as if the ADC was simply a sampler. We now want to address the impact that

a realistic ADC will have on performance. In particular, we want to account for the

number of bits in the ADC, the quantization and internal ADC noise and ADC dynamic

range.

Until very recently, the rule of thumb used to characterize the impact of the ADC

on SCR improvement was to say that the ADC imposed an absolute limit on performance

of

6 1 dBSCR bitI N (2-161)

where bitN is the number of bits in the ADC. In the past, this has resulted in design

constraints that were unnecessary.

Engineers at Dynetics, Inc. have shown that, while SCRI is influenced by the

number of bits in the ADC, the hard limit on SCR improvement given by Equation 2-161

is not valid. A more representative equation for SCRI that includes the effects

SCRG ,

phase noise and the ADC is

0

SCR SNR ADCSCR

SNR ADC SCR ADC p SCR NADC p s

G G PI

G P G P G P F

. (2-162)

where several of the terms are familiar from Section 2.4.3.4. ADCP is the level of the

clutter at the ADC input relative to the ADC saturation level. It is normally taken to be -

6 dB to assure that the Swerling nature of clutter doesn’t occasionally cause ADC

saturation. The presence of ADCP implies that there is some type of gain control that

monitors the clutter level into the ADC and adjusts the gain to keep the clutter power 6

dB below ADC saturation.

The term NADCP accounts for the quantization noise of the ADC, the ADC internal

noise and any additional dither noise that is added to the ADC input to assure linear

operation of the ADC. This sentence raises an important issue concerning the ADC. In

order for the ADC to preserve the relative sized of signal, clutter and noise after

quantization, there must always be sufficient noise at the ADC input. Generally, only

quantization noise is not sufficient since it is too small. In modern ADCs with a large

number of bits (>10) the internal noise is usually sufficient. If it is not, dither noise must

be added to the input to the ADC. A reasonable value of NADCP is

26 1 log 10

10 bitN q

NADCP

(2-163)

where bitN is the number of bits in the ADC q is the number of quantization levels of the

ADC noise. Typical values of q are 1 to 3 (note: if we say noise toggles the lsb of the


ADC, q=1; if it toggles the lower two bits, q=3.) If the ADC specifications give an

ADC SNR then NADCP , in dB, is the negative of that SNR. If the negative of the SNR is

not equal to or greater that the NADCP of Equation 2-163 with 0q , then dither noise of

½ to 1 quanta ( 12 to 1q ) should be added to the ADC input. However, adding too

much dither noise to will degrade SCRI .

sF is the ADC sample rate. It is normally taken to be the modulation bandwidth

of the waveform if range gating is performed after the ADC, as in a full digital processor.

For an unmodulated pulse, 1s pF . If the radar uses IF sampling with digital down

conversion, sF can be much larger than the modulation bandwidth.

The SCRI equation above is written in term used for the pulsed-Doppler signal

processor. It is also applicable to the MTI processor with 1SNRG and SCRG CA .

A derivation of Equation 2-163 is given in Appendix C.


APPENDIX A – DERIVATION OF EQUATION 2-47

In this appendix we want to go through the derivation of Equation 2-47 of the

notes. We start with the output of the matched filter as

MF

l

v t q t p lT r d

(A-1)

where p t represents one of the transmit pulses and q t is the pulse to which the

single-pulse matched filter is matched. We assume that r t is a WSS random process

with an autocorrelation of rR and a power spectral density of rS f . It will be

recalled that r t is not WSS for the case of clutter. However, it is shown in Appendix

B that it is wide-sense cyclostationary. Because of this, it can be represented by its

averaged autocorrelation, which is then used in this development. In this instance, r t

represents some hypothetical WSS random process whose autocorrelation is equal to the

averaged autocorrelation of the actual random process.

As indicated in the notes, we will assume that the ADC samples the output of the

matched filter, MFv t , once per PRI, T , at the peak of the matched filter response. We

also, without loss of generality, assume that the matched filter peaks occur at t kT .

With this we can write the output of the ADC as

o MF t kTl

v k v t q kT p lT r d

. (A-2)

If we assume that p t and q t are of the form

' rectp

tp t p t

(A-3)

where

1 1 2

rect0 1 2

xx

x

(A-4)

and that 2p T , then all of the terms of the summation of Equation A-2 are zero except

for the case where l k . With this Equation A-2 reduces to

ov k q kT p kT r d

. (A-5)

To find the power spectrum of ov k we must first show that ov k is WSS. To

that end we form


1 2 1 2

1 1

2 2

1 1 2 2

,o o oR k k E v k v k

E q k T p k T r d

q t k T p t k T r t dt

q k T p k T q t k T p t k T

E r r t d dt

. (A-6)

From previous discussions we note that

rE r r t R t (A-7)

and

1 2 1 1 2 2,o rR k k q k T p k T q t k T p t k T R t d dt

. (A-8)

By making use of

2j f

r rR S f e df

(A-9)

we can write

2

1 2 1 1 2 2,j f t

o rR k k q k T p k T q t k T p t k T S f e df d dt

. (A-10)

We now make the change of variables, ,t d d to yield

1 2 2 2

2

1 1

,o r

j f

R k k q t k T p t k T S f

q t k T p t k T e d dfdt

. (A-11)

We next make the change of variables 1t k T d d and get

12

1 2 2 2,j f t k T

o rR k k q t k T p t k T S f q p e d dfdt

. (A-12)

Rearranging yields


12

1 2

2 2

2 2

,j fk T

o r

j ft j f

R k k S f e

q t k T p t k T e dt q p e d df

. (A-13)

For the next change of variable we let 2t k T d dt to yield

1 22 2 2

1 2,j f k k T j f j f

o rR k k S f e q p e d q p e d df

.(A-14)

The first thing to note about Equation A-14 is that the right side is a function of

1 2k k , and constitutes the proof that ov k is WSS. The next thing to note is that the

two integrals in the brackets are conjugates of each other. Finally, from ambiguity

function theory, we recognize that

2 0,j f

pqp q e d f

(A-15)

where 0,pq f is the matched-range, Doppler cut of the cross ambiguity function of

p t and q t . In the remainder we will use the notation 0,pq f MF f . With all

of the statements in this paragraph we can write

2 2j fkT

o rR k MF f S f e df

. (A-16)

We next want to find the power spectrum of ov k . We could do this by taking

the discrete-time Fourier transform of oR k . However, the math associated with this

will probably be quite involved. We will take a more indirect approach.

Let v t be a WSS random process with an autocorrelation of R and a power

spectrum of S f . Further assume that we can sample v t to get ov k . That is

o t kTv k v t

. (A-17)

ov k is the same as the random process defined by Equation A-2.

From random processes theory we can write

o kTR k R

. (A-18)

Further, from the theory of discrete-time signals and their associated Fourier transforms,

if S f is the power spectrum of v t we can write the power spectrum of ov k as


1

o

m

S f S f m TT

. (A-19)

From this same theory we can write

1 2

2

1 2

T

j kfT

o o

T

R k T S f e df

(A-20)

If we substitute Equation A-19 into A-20 we get

1 2

2

1 2

1T

j kfT

o

mT

R k T S f m T e dfT

(A-21)

or

1 2

2

1 2

T

j kfT

o

m T

R k S f m T e df

. (A-22)

We now make the change of variables x f m T to get

1 2

2

1 2

1 2

2 2

1 2

1 2

2

1 2

T m T

j k x m T T

o

m T m T

T m T

j km j kxT

m T m T

T m T

j kxT

m T m T

R k S x e dx

e S x e dx

S x e dx

. (A-23)

Where we made use of the fact that 2 1j kme .

We recognize that the last term is an infinite summation of integrals over non-

overlapping intervals, and that the total of the non-overlapping intervals cover the range

of ,x . With this we can write

2j kfT

oR k S f e df

(A-24)

where we have let x f .

If we compare Equation A-24 to Equation A-16 we have

2

rS f MF f S f . (A-25)

With this and Equation A-19 we arrive at the desired result that the power spectrum of

the signal at the ADC output is

21

o r

m

S f MF f m T S f m TT

. (A-26)


APPENDIX B – PROOF THAT r t IS WS CYCLOSTATIONARY

In this appendix we show that the process

2

sr t c t c t t (B-1)

is wide-sense cyclostationary (WSCS). We first note that since c t is zero-mean so is

r t . For ease of notation, we replace the symbol 2

sc t by the symbol 2sc t . To show

that r t is WSCS we must show that

s sE r t kT r t kT E r t r t (B-2)

for some sT . That is, we must show that the autocorrelation of r t is a periodic

function of t .

We recall that c t and t are WSS random processes. Thus the product

c t t is also WSS. The function 2sc t is a deterministic function and is periodic

with a period of sT where

sT is the scan period of the antenna. If we form

,rR t E r t r t (B-3)

we get

2 2

2 2

,r s s

s s c

R t c t c t E c t c t E t t

c t c t R R

(B-4)

where we have made use of the fact that c t and t are independent and WSS. In a

similar fashion we can write

2 2,r s s s s s cR t kT c t kT c t kT R R . (B-5)

But, since 2sc t is periodic with a period of sT we have

2 2s s sc t kT c t (B-6)

and thus that

, ,r s rR t kT R t (B-7)

which says that r t is WSCS.

From the theory of WSCS random processes we can use the averaged

autocorrelation of r t to characterize the average behavior of r t . Specifically, in

place of ,rR t we use


1

,

s

r r

s T

R R t dtT

(B-8)

where the integral notation means to perform the integration over one period of ,rR t .

As a note, what Equation B-8 is saying is that, on average, a system will respond to r t

in the same manner that it will respond to a WSS process that has the autocorrelation

rR . In the notes we have dispensed with the overbar and used the notation rR .

Also, in Appendix A and the notes we assumed that r t was a WSS random process.

What we really mean is that the r t used in those places is a WSS random process

whose autocorrelation, rR is equal to the averaged autocorrelation, rR of the

actual, WSCS random process.


APPENDIX C – DERIVATION OF EQUATION 2-163

Figure C-1 contains a block diagram of the receiver/ADC/signal processor

configuration we will use in this analysis. In this configuration it is assumed that the

ADC is sampling the IF signal and that the signal processor contains the matched filter,

digital down conversion to baseband and any other signal processor components such as

the clutter rejection filters and coherent integrators. This model will also apply to the

case where the ADC is sampling a baseband signal and the matched filter and/or down

conversion is performed in the block labeled IF section.

Figure C-1 – Block Diagram Used in the Derivation

In Figure C-1, SP and

CP are the (normalized) signal and clutter powers at the

output of the IF section and are the powers computed from the radar range equation (see

EE619 notes and the clutter model discussions of Section 2.2 and the examples). SP and

CP are single-pulse powers.

kTP is the (normalized) receiver noise power at the output of the IF section and is

the noise power term that is also encountered in the radar range equation (see EE619

notes). It is given by

0kT nP kT F B (C-1)

where k is Boltzman’s constant, 0 290 KT is the reference temperature,

nF is the

system noise figure and 1 pB is the effective noise bandwidth of the radar. p is the

uncompressed pulsewidth.

P is the component of clutter that is manifest through the phase noise sidebands

of the transmitter and local oscillator (LO) (see Section 2.4). If we assume a flat phase

noise spectrum with a power spectral density of 0 w/Hz relative to the transmit power

then P is given by

0CP P B (C-1)

where CP and B are as defined earlier.

The ADC normalizer is a gain, normG , that is used to adjust the total signal-plus-

clutter-plus-noise power so as to avoid ADC saturation. Generally, normG is selected so


that the power level, ADCP , at the ADC input is 3 to 6 dB below ADC saturation. This

should provide enough of a margin to avoid saturation in the presence of fluctuations in

the signal, clutter and noise. In this analysis, it is assumed, without loss of generality,

that the ADC saturates at a normalized power of 1 watt. With this, the normalizer gain is

given by

ADCnorm

S C kT

PG

P P P P

(C-3)

where ADCP is between

6 1010 and

3 1010.

The standard performance metrics used in radar and signal processor analyses are

signal-to-noise ratio (SNR), signal-to-clutter ratio (SCR) and signal-to-interference ratio

(SIR). These will be the performance metrics used here.

The SCR at the ADC input is defined as

norm S SADC

norm C C

G P PSCR SCR

G P P (C-4)

where SCR it the single-pulse SCR derived from the radar-range-equation (see Section

2.4). It has been shown that if sufficient noise is present in the ADC (and the ADC is not

allowed to saturate), SCR is preserved through the ADC. This means that the SCR at the

signal processor input is also the SCR of Equation C-4. That is,

SSPin

C

PSCR SCR

P . (C-5)

It will be assumed that the signal processor provides an SCR gain of SCRG as

discussed in Section 2.4.3.4. To repeat the statements of Section 2.4.3.4, SCRG is not to

be confused with signal-to-clutter improvement (SCRI ). The latter includes factors such

as phase noise and, as will be shown, ADC noise. SCRG is a measure of how well the

signal processor can reject clutter when the transmit and LO signals are perfect sinusoids

(no phase noise) and the ADC has an infinite number of bits and infinite dynamic range.

The noise into the signal processor consists of the receiver noise (kTP ), the phase

noise ( P ) and the ADC noise, which is represented by NADCP in Figure C-1. The phase

noise term is included as part of the noise, and not the clutter, because the signal

processor acts on it the same as it does on receiver noise.

The ADC noise consists of noise that is generated by the ADC circuitry,

quantization noise and, in some instances, intentionally injected dither noise. The

presence of ADC noise is critical to the preservation of SCR through the ADC. Without

it, the SCR at the output of the ADC would drop to zero (w/w, not dB) once the SCR at

the input to the ADC dropped below the dynamic range of the ADC.


A standard way of representing ADC noise is in terms of ADC quanta, q. One

can think of q as the count of the ADC. Thus, if the least significant bit (lsb) of the ADC

is 1 and the rest are 0, 1q . If the two lsb’s are 1 and the rest are zero, 3q (binary 11

= decimal 3). For an N-bit ADC with q quanta of ADC noise and an ADC saturation

level of 1 watt,

26 1 log 10

10N q

NADCP

(C-6)

where 1N is used to reflect the fact that the most significant bit (msb) of an N-bit ADC

is normally taken to be a sign bit.

With the above, the total noise power spectral density at the input to the signal

processor can be written as

0 0T norm n norm C NADC sN G kT F G P P F . (C-7)

The division by sF in the last term of Equation C-7 reflects the assumption that the ADC

noise power is uniformly distributed over a bandwidth equal to the ADC sample rate of

sF . For the reasonable assumption that sF is greater than the effective noise bandwidth,

B, the effective total noise power into the signal processor is

0 0T norm n norm C NADC sP G kT F B G P B P B F . (C-8)

With Equation C-8, we can write the SNR at the input to the signal processor as

0 0

norm SSPin

norm n norm C NADC s

G PSNR

G kT F B G P B P B F

. (C-9)

Given that the SNR gain of the signal processor is SNRG , the SNR at the signal

processor output is

0 0

SNR norm Sout SNR SPin

norm n norm C NADC s

G G PSNR G SNR

G kT F B G P B P B F

. (C-10)

The SIR at the signal processor output can be determined from

1 1 1

out out outSIR SNR SCR (C-11)

with Equation C-10 and out SCR S CSCR G P P .

In this form, Equation C-11 is not very easy to use. However, if we make the

reasonable assumption that the signal into the normalizer is clutter limited, we can write

ADCnorm

C

PG

P (C-12)

and

0 0

SNR ADC S Cout

ADC n C ADC NADC s

G P P PSNR

P kT F B P P B P B F

(C-13)


which can be substituted into Equation C-11 to yield

0 01 ADC n C ADC NADC s C

out SNR ADC S C SNR S

P kT BF P P B P B F P

SIR G P P P G P

. (C-14)

After some manipulation, Equation C-14 can be written as

1 1 1

out SNR SCRSIR G SNR I SCR (C-15)

where the SCR improvement, SCRI is given by

0

SCR SNR ADCSCR

SNR ADC SCR ADC SCR NADC s

G G PI

G P G P B G P B F

(C-16)

which is Equation 2-163.

2.0 signal processor analyses · to-pulse. for now we assume that the target is an ideal point...

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