# Introduction to Random Signals and Noise (van Etten/Introduction to Random Signals and Noise) || Bandpass Processes

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5Bandpass Processes

Bandpass processes often occur in electrical engineering, mostly as a result of the bandpass

filtering of white noise. This is due to the fact that in electrical engineering in general, and

specifically in telecommunications, use is made of modulation of signals. These informa-

tion-carrying signals have to be filtered in systems such as receivers to separate them from

other, unwanted signals in order to enhance the quality of the wanted signals and to prepare

them for further processing such as detection.

Before dealing with bandpass processes we will present a summary of the description of

deterministic bandpass signals.

5.1 DESCRIPTION OF DETERMINISTIC BANDPASS SIGNALS

There are many reasons why signals are modulated. Doing so shifts the spectrum to a certain

frequency, so that a bandpass signal results (see Section 3.4). On processing, for example,

reception of a telecommunication signal, such signals are bandpass filtered in order to

separate them in frequency from other (unwanted) signals and to limit the amount of noise

power. We consider signals that consist of a high-frequency carrier modulated in amplitude

or phase by a time function that varies much more slowly than the carrier. For instance,

amplitude modulation (AM) signals are written as

st A 1 mt cos!0t 5:1

where A is the amplitude of the unmodulated carrier, mt is the low-frequency modulatingsignal and !0 is the carrier angular frequency. Note that lower case characters are used here,since in this section we discuss deterministic signals. Assuming that 1 mt is nevernegative, then st looks like a harmonic signal whose amplitude varies with the modulatingsignal.

A frequency-modulated signal is written as

st A cos !0t Z t

0

d

5:2

Introduction to Random Signals and Noise Wim C. van Etten# 2005 John Wiley & Sons, Ltd ISBN: 0-470-02411-9

The instantaneous angular frequency of this signal is !0 t and is found by differentiat-ing the argument of the cosine with respect to t. In this case the slowly varying function tcarries the information to be transmitted. The frequency-modulated signal has a constant

amplitude, but the zero crossings will change with the modulating signal.

The most general form of a modulated signal is given by

st at cos !0t t 5:3

In this equation at is the amplitude modulation and t the phase modulation, while thederivative dt=dt represents the frequency modulation of the signal. Expanding the cosineof Equation (5.3) yields

st at cost cos!0t sint sin!0t xt cos!0t yt sin!0t 5:4

with

xt4 at costyt4 at sint

5:5

The functions xt and yt are called the quadrature components of the signal. Signal xt iscalled the in-phase component or I-component and yt the quadrature or Q-component.They will vary little during one period of the carrier. Combining the quadrature components

to produce a complex function will give a representation of the modulated signal in terms of

the complex envelope

zt4 xt jyt at exp jt 5:6

When the carrier frequency !0 is known, the signal st can unambiguously be recoveredfrom this complex envelope. It is easily verified that

st Refzt exp j!0tg 12 zt exp j!0t zt expj!0t 5:7

where Refg denotes the real part of the quantity in the braces. Together with the carrierfrequency !0, the signal zt constitutes an alternative and complete description of themodulated signal. The expression zt expj!0t is called the analytic signal or pre-envelope.The complex function zt can be regarded as a phasor in the xy plane. The end of the phasormoves around in the complex plane, while the plane itself rotates with an angular frequency

of !0 and the signal st is the projection of the rotating phasor on a fixed line. If themovement of the phasor zt with respect to the rotating plane is much slower than the speedof rotation of the plane, the signal is quasi-harmonic. The phasor in the complex z plane has

been depicted in Figure 5.1.

It is stressed that zt is not a physical signal but a mathematically defined auxiliarysignal to facilitate the calculations. The name complex envelope suggests that there is

a relationship with the envelope of a modulated signal. This envelope is interpreted as

102 BANDPASS PROCESSES

the instantaneous amplitude of the signal, in this case at. Now the relationship is clear,namely

at jztj x2t y2t

pt argzt arctan yt

xt5:8

It is concluded that the complex envelope, in contrast to the envelope, not only comprises

information about the envelope at but also about the phase t.As far as detection is concerned, the quadrature component xt is restored by multiplying

st by cos!0t and removing the double frequency components with a lowpass filter:

st cos!0t 12 xt1 cos 2!0t 12 yt sin 2!0t 5:9

This multiplication operation can be performed by the modulator scheme presented in

Figure 3.5. After lowpass filtering, the signal produced will be xt=2. The second quadraturecomponent yt is restored in a similar way, by multiplying st by sin!0t and using alowpass filter to remove the double frequency components. A circuit that delivers an output

signal that is a function of the amplitude modulation is called a rectifier and such a circuit

will always involve a nonlinear operation. The quadratic rectifier is a typical rectifier; it has

an output signal in proportion to the square of the envelope. This output is achieved by

squaring the signal and reads

s2t 12x2t y2t 1

2x2t y2t cos 2!0t xtyt sin 2!0t 5:10

By means of a lowpass filter the frequency terms in the vicinity of 2!0 are removed, so thatthe output is proportional to jztj2 a2t x2t y2t. A linear rectifier, which mayconsist of a diode and a lowpass filter, yields at.

A circuit giving an output signal that is proportional to the instantaneous frequency

deviation 0t is known as a discriminator. Its output is proportional to dImfln ztg=dt.

a(t )

(t )

Re {z (t )}

Im {z(t )}

0t

y (t )

x (t )

Figure 5.1 The phasor of st in the complex z plane

DESCRIPTION OF DETERMINISTIC BANDPASS SIGNALS 103

If the signal st comprises a finite amount of energy then its Fourier transform exists.Using Equation (5.7) the spectrum of this signal is found to be

S! 12

Z 11

zt expj!0t zt expj!0t expj!t dt 1

2Z! !0 Z! !0 5:11

where S! and Z! are the signal spectra (or Fourier transform) of st and zt,respectively, and * denotes the complex conjugate.

The quadrature components and zt vary much more slowly than the carrier and will bebaseband signals. The modulus of the spectrum of the signal, jS!j, has two narrow peaks,one at the frequency !0 and the other at !0. Consequently, st is called a narrowbandsignal. The spectrum of Equation (5.11) is Hermitian, i.e. S! S!, a condition that isimposed by the fact that st is real.

Quasi-harmonic signals are often filtered by bandpass filters, i.e. filters that pass

frequency components in the vicinity of the carrier frequency and attenuate other frequency

components. The transfer function of such a filter may be written as

H! Hl! !c Hl ! !c 5:12

where the function Hl! is a lowpass filter; it is called the equivalent baseband transferfunction. Equation (5.12) is Hermitian, because ht, being the impulse response of aphysical system, is a real function. However, the equivalent baseband function Hl! will notbe Hermitian in general. Note the similarity of Equations (5.12) and (5.11). The only

difference is the carrier frequency !0 and the characteristic frequency !c. For !c, an arbitraryfrequency in the passband of H! may be selected. In Equation (5.7) the characteristicfrequency !0 need not necessarily be taken as equal to the oscillator frequency. A shift in thecharacteristic frequency over ! !1 merely introduces a factor of expj!1t in thecomplex envelope:

zt expj!0t zt expj!1t expj!0 !1t 5:13

This shift does not change the signal st. From this it will be clear that the complexenvelope is connected to a specific characteristic frequency; when this frequency changes the

complex envelope will change as well. A properly selected characteristic frequency,

however, can simplify calculations to a large extent. Therefore, it is important to take the

characteristic frequency equal to the oscillator frequency. Moreover, we select the char-

acteristic frequency of the filter equal to that value, i.e. !c !0.Let us suppose that a modulated signal is applied to the input of a bandpass filter. It

appears that using the concepts of the complex envelope and equivalent baseband transfer

function the output signal is easily described, as will follow from the sequel. The signal

spectra of input and output signals are denoted by Si! and So!, respectively. It thenfollows that

So! Si! H! 5:14

104 BANDPASS PROCESSES

Invoking Equations (5.11) and (5.12) this can be rewritten as

So! 12Zo! !0 Zo! !0 1

2Zi! !0 Zi ! !0 Hl! !0 Hl ! !0 5:15

where Zi! and Zo! are the spectra of the complex envelopes of input and output signals,respectively. If it is assumed that the spectrum of the input signal has a bandpass character

according to Equations (4.40), (4.41) and (4.42), then the cross-terms Zi! !0 Hl !!0 and Zi ! !0Hl! !0 vanish (see Figure 5.2). Based on this conclusion Equation(5.15) reduces to

Zo! !0 Zi! !0 Hl! !0 5:16

or

Zo! Zi!Hl! 5:17

Transforming Equation (5.17) to the time domain yields

zot Z 11

hlzit d 5:18

with zot and zit the complex envelopes of the input and output signals, respectively. Thefunction hlt is the inverse Fourier transform of Hl! and represents the complex impulseresponse of the equivalent baseband system, which is defined by means of Hl! or the dualdescription hlt. The construction of the equivalent baseband system Hl! from the actualsystem is illustrated in Figure 5.3. This construction is quite simple, namely removing the

part of the function around !0 and shifting the remaining portion around !0 to thebaseband, where zero replaces the original position of !0. From this figure it is observed that

0 0

Hl*(0)

Zi*(0) Zi(0)

Hl(0)

0

Figure 5.2 A sketch of the cross-terms from the right-hand member of Equation (5.15)

DESCRIPTION OF DETERMINISTIC BANDPASS SIGNALS 105

in general Hl! is not Hermitian, and consequently hlt will not be a real function. Thismay not surprise us since it is not the impulse response of a real system but an artificially

constructed one that only serves as an intermediate to facilitate calculations.

From Equations (5.17) and (5.18) it is observed that the relation between the output of a

bandpass filter and the input (mostly a modulated signal) is quite simple. The signals are

completely determined by their complex envelopes and the characteristic frequency !0.Using the latter two equations the transmission is reduced to the well-known transmission of

a baseband signal. After transforming the signal and the filter transfer function to equivalent

baseband quantities the relationship between the output and input is greatly simplified,

namely a multiplication in the frequency domain or a convolution in the time domain. Once

the complex envelope of the output signal is known, the output signal itself is recovered

using Equation (5.7). Of course, using the direct method via Equation (5.14) is always

allowed and correct, but many times the method based on the equivalent baseband quantities

is simpler, for instance in the case after the bandpass filtering envelope detection is applied.

This envelope follows immediately from the complex envelope.

5.2 QUADRATURE COMPONENTS OF BANDPASS PROCESSES

Analogously to modulated deterministic signals (as described in Section 5.1), stochastic

bandpass processes may be described by means of their quadrature components. Consider

the process

Nt At cos!0t t 5:19

with At and t stochastic processes. The quadrature description of this process is readilyfound by rewriting this latter equation by applying a basic trigonometric relation

Nt At cost cos!0t At sint sin!0t Xt cos!0t Yt sin!0t 5:20

0 0

0

Hl*(0)

Hl()

Hl(0)

0

Figure 5.3 Construction of the transfer function of the equivalent baseband system

106 BANDPASS PROCESSES

In this expression the quadrature components are defined as

Xt4 At costYt4 At sint 5:21

From these equations the processes describing the amplitude and phase of Nt are easilyrecovered:

At4X2t Y2t

pt4 arctan Yt

Xt 5:22

The processes Xt and Yt are stochastic lowpass processes (or baseband processes), whileEquation (5.20) presents a general description of bandpass processes. In the sequel we will

derive relations between these lowpass processes Xt and Yt, and the lowpass processeson the one hand and the bandpass process Nt on the other. Those relations refer to meanvalues, correlation functions, spectra, etc., i.e. characteristics that have been introduced in

previous chapters.

Once again we assume that the process Nt is wide-sense stationary with a mean value ofzero. A few interesting properties can be stated for such a bandpass process. The properties

are given below and proved subsequently.

Properties of bandpass processes

If Nt is a wide-sense stationary bandpass process with mean value zero and quadraturecomponents Xt and Yt, then Xt and Yt have the following properties:

1. Xt and Ytare jointly wide-sense stationary: 5:232. EXt EYt 0 5:243. EX2t EY2t EN2t 5:254. RXX RYY 5:265. RYX RXY 5:276. RXY0 RYX0 0 5:287. SYY! SXX! 5:298. SXX! LpfSNN! !0 SNN! !0g 5:309. SXY! j LpfSNN! !0 SNN! !0g 5:31

10. SYX! SXY! 5:32

In Equations (5.30) and (5.31), Lpfg denotes the lowpass part of the expression in thebraces.

QUADRATURE COMPONENTS OF BANDPASS PROCESSES 107

Proofs of the properties:

Here we shall briefly prove the properties listed above. A few of them will immediately be

clear. Expression (5.29) follows directly from Equation (5.26), and Equation (5.32) from

Equation (5.27). Moreover, using Equation (2.48), Equation (5.28) is a consequence of

Equation (5.27).

Invoking the definition of the autocorrelation function, it follows after some manipulation

from Equation (5.20) that

RNNt; t 12fRXXt; t RYYt; t cos!0 RXYt; t RYXt; t sin!0 RXXt; t RYYt; t cos!02t RXYt; t RYXt; t sin!02t g 5:33

Since we assumed that Nt is a wide-sense stationary process, Equation (5.33) has to beindependent of t. Then, from the last term of Equation (5.33) it is concluded that

RXYt; t RYXt; t 5:34

and from the last but one term of Equation (5.33)

RXXt; t RYYt; t 5:35

Using these results it follows from the first two terms of Equation (5.33) that

RXXt; t RXX RYY 5:36

and

RXYt; t RXY RYX 5:37

thereby establishing properties 4 and 5. Equation (5.33) can now be rewritten as

RNN RXX cos!0 RXY sin!0 5:38

If we substitute 0 in this expression and use Equation (5.26), property 3 follows.The expected value of Nt reads

ENt EXt cos!0t EYt sin!0t 0 5:39

This equation is satisfied only if

EXt EYt 0 5:40

so that now property 2 has been established. However, this means that now property 1

has been proved as well, since the mean values of Xt and Yt are independent of t

108 BANDPASS PROCESSES

(property 2) and also the autocorrel...

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