radar-verfahren und -signalverarbeitung...radar fundamentals i digital pulse compression in the...

Radar-Verfahren und -Signalverarbeitung

- Lesson 2: RADAR FUNDAMENTALS I

Hon.-Prof. Dr.-Ing. Joachim Ender

Head of

Fraunhoferinstitut für Hochfrequenzphysik and Radartechnik FHR

Neuenahrer Str. 20, 53343 Wachtberg

[email protected]

Ender: Radarverfahren- 2 -

RADAR FUNDAMENTALS I

Coherent radar - quadrature modulator and demodulator

The QM transfers the

complex baseband signal to a

real valued RF signal.

Figure: Quadrature modulator Figure: Quadrature demodulator



Coherent radar - complex envelope

Real valued RF signal

Complex envelope,base band signal

Reference frequency (RF)

The QDM performs the inverse operation to that of the QM.

The real valued RF-signal may be regarded as a carrier of the base band-signal s(t), able to be transmitted as RF waves over long ranges.

Figure: Bypass of quadrature modulator and demodulator



Coherent radar - generic radar system

Traveling time

Traveling distance

Received signal

r Distance to a point scatterer

c0 Velocity of light

t Traveling time

N(t) White noise

s(t;r) Received waveform

a complex amplitude

Point target

Antenna

T/R Switch

r

*

QM

s(t)

f0

QDM

as(t;r)

N(t)

Z(t)

Figure: Radar system with baseband signals



Coherent radar - received waveform

Complex envelope

Received waveform

Wave length and

wave number:

Rk

R

c

Rff

0

0

0

00

2

22

t

0

0

0

00

2

k

f

c



Coherent radar - Fourier transform of the received waveform

Fourier transform:

Baseband frequency

Reference frequency

RF frequency

Wave number in range direction



Coherent radar - optimum receive filter

i.e.

f0

QDM

as(t;r)

N(t)

Z(t)h(t)Y(t)



Coherent radar - matched filter

Received

signal

Replica

Response of

matched filter

The pulse response of the optimum filter is equal to the time-inverted, complex conjugated signal

The maximum SNR is . This filter is called matched filter.



Coherent radar - correlation with the transmit signal



Coherent radar - point spread function



Coherent radar - matched filter, point spread function

Reflectivity of three point targets Output of the matched filter

Point spread function

Matched filtering means correlation with the transmit signal. The point spread function is the

reaction of the receive filter to the transmit signal.

The point spread function is equal to the autocorrelation of the transmit signal, if a matched filter

is used.

In this case it is the Fourier back transform of the magnitude-squared of the signal spectrum.



Definitions of resolution

b

cr

tc

r

Rayleigh2

2

Figure: Definitions of resolution



Pulse compression

The solution is to expand the bandwidth by modulation of the pulse.

The Rayleigh range resolution of a waveform with a rectangular spectrum S(f)

of bandwidth b is given by

without direct dependence on the pulse length.



Pulse compression

For the range resolution the bandwidth of the transmitted waveform is

decisive: .

The gain in range resolution with respect to a rectangular pulse of same

duration is called compression rate, which is equal to the time-bandwidth

product.

Two different waveforms s(t) and sRF (t) effect the same point spread function

for matched filtering, if sRF (t) is generated by s(t) by passing it through a

filter with a transfer function of magnitude 1.



Generation of high bandwidth signals

Analogue

Phase modulation with phase shifter

Frequency modulation by a VCO

SAW filter

Frequency multiples (non-linear devices,

extraction of higher harmonics)

Digital

Arbitrary wave form generator (AWG)

Direct digital synthesizer (DDS)

VCO coupled to DDSAWG principle

Re Im

Memory(writeable,

fast read out)

D/A D/AClock (e.g. 1 GHz)

de-

glitch

de-

glitchFilter


DDS principle

D/A D/AClock (e.g. 1 GHz)

de-

glitch

de-glitch

Filter

cos sin

Look-up table(fast read out)

read pointer

fast accu-mulator

(mod 2)

Fast logic(e.g. GaAs)

Memory

slow read out(e.g. 50 MHz)


Generation of high bandwidth signals


The reference signal can be

the designed (wanted) signal

the measured signal in a calibration mode

Receive signal

A/D

FFT

complex conj.

FFT

FFT

ze

Ze

za

S* S referencesignal s

Za

This part is performed only during calibration


Digital pulse compression in the frequency domain



Pulse compression in the time and in the frequency domain

h(t)=s*(-t)Compression

filter

Tt

Range com-

pressed data

For each range line

Range FFT

T

t Raw data

Tf

For each range line

T

t Raw data

H(f)=S*(f)

Tf

Range IFFT

Tt

Range com-

pressed data

Figure: Range compression with the matched filter.

Left: direct convolution, right: processing in the frequency domain



Pulse compression - Anatomy of a chirp I

2exp)( tjt

trectts

s

ttf )(

Instantaneous frequency:

Frequency span

(= bandwidth for large ):stb

Time-bandwidth product:2

ss tbt

Rectangular chirp

t

R{s(t)}

ts/2-ts/2

t

f

f=t

ts/2-ts/2




Fourier transform of a rectangular chirp




Figure: Magnitude of the Fourier transforms of chirps with growing time basis




Fourier transform of a rectangular chirp

The magnitude of the Fourier transform of a rectangular chirp with rate has the

approximate shape of a rectangular function with bandwidth close to the frequency

span ts.

For infinite duration the Fourier transform is again a chirp with rate -1/ .


t

f

t

f

Chirp

F F-1

act

t|.|2

Spectrum

Power

Spectrum

Compression

result


Pulse compression - Compression of a chirp



De-ramping



Spatial interpretation of the radar signal and the receive filter

From the viewpoint of focusing to images (SAR), the spatial domain is the primary

one. We transform the temporal signals into spatial signals, dependent on the spatial

variable R = 2r via t -> R = c0 t. We will use the symbols s, h, p as functions of R.

Signal spectrum (wave number domain)

Transfer function (wave number domain)

Point spread function (wave number domain)

For the matched filter we get

The point spread function for matched filtering in the range domain is given by the

inverse Fourier transform of the power of the signal spectrum in the wave number

domain.



Matched filter / inverse filter / robustified filter

We regard a receive filter with transfer function



Matched filter / inverse filter / robustified filter

Figure: Robustified inverse filter



The k-set



The k-set

For the application of the inverse filter, the point spread function is equal to

the Fourier back transform of the indicator function of the carrier of the signal

spectrum (k-set)



Pre-processing to the normal form (inverse filter)

We regard a noise-free signal of a point scatterer at R=R0:



Coherent radar

Pulse repetition frequency:

PRF (~ 100 Hz ... 10 kHz)

Intrapulse sampling frequency:

fs (~ 10 MHz ... 1 GHz)

tf

TPRFF

s

s

1

1

T= 1ms: Covered range =150 km

t= 1ns: Range sampling = 15 cm Figure: Two time scales for pulse radar



Pre-processing to the normal form (inverse filter)

Figure: Pre-processing in the k-domain



Doppler effect

Basic component of the Doppler frequency:

Resolution of the waveform in spectral components:



Doppler effect

Object motion negligible during the wave's travelling time (stop and go

approximation):



Doppler effect

Exact expression

The Doppler frequency of a moving target is given by

For the stop-and-go approximation this is simplified to



Doppler effect

Effects caused by target motion

Phase rotation from pulse to pulse

Range migration

Phase modulation during one pulse

Intra-pulse time stretch / compression

Christian Andreas Doppler (29 November 1803 – 17 March 1853) was an Austrian

mathematician and physicist. He is most famous for describing what is now called the

Doppler effect, which is the apparent change in frequency and wavelength of a wave as

perceived by an observer moving relative to the wave's source.



Doppler effect - modulation and time expansion



Doppler effect - in the two-times domain

slow time

fast time

Figure: Doppler-effect for a pulse train



Range-Doppler processing

Range-Doppler

processing with

subsequent pulse

compression and

Doppler filtering

Range-Doppler

processing via the

double frequency

domain

Tk r

Pre-processed

data

Slow-time FFT

Fk r

Double fre-

quency data

Range IFFT

FR

Range-

Doppler data



Ambiguity function

Ambiguity function =

response of a matched

filter to a signal shifted

in time and Doppler

frequency

Figure: Ambiguity function

of a rectangular pulse



Ambiguity function

Figure: Ambiguity function of a chirp with Gaussian envelope



Doppler tolerance of a chirp

time of best fit

Doppler shifted signal

Matched filter

for f0

t0

t

f0t

f

Area of phase match

f0+

t0-ttime shift



Doppler tolerance of a chirp

Obviously, for the chirp waveform there is an ambiguity between Doppler

and range. If one of the two is known, the other variable can be measured

with high accuracy. The Doppler frequency may be measured over a sequence

of pulses and used for a correction of range.

The chirp wave form is Doppler tolerant, i.e. a Doppler shift of the echo with respect to

the reference chirp leads only to a moderate SNR loss corresponding to the non-

overlapping part of the signal spectra. A time shift is effected which is proportional to

the Doppler shift.



Ambiguity function

Figure: Ambiguity function of a train of 5 rectangular pulses



Ambiguity of range and Doppler

Doppler ambiguity:

PRFT

F

1 PRF modes:

Low PRF: unambiguous range,

ambiguous Doppler

High PRF: ambiguous range,

unambiguous Doppler

Medium PRF: in between

2

0crF

Range ambiguity:

Tc

r 2

0

Area of ambiguity rectangle:

For a pulse train repeated with T, the product of temporal and frequency ambiguity is

equal to 1.

The product of range ambiguity and Doppler ambiguity is equal to c0/2.

The product of range ambiguity and radial velocity ambiguity is c0/ = f.

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