Digital communication receivers P3

Since the information is encoded into the phase, the signal maintains a constant envelope which allows a band-limited signal’s amplifi- 211 Digital Communication Receivers: Synchronizat

PART D Passband Communication Over Time Invariant Channels

Heinrich Meyr, Marc Moeneclaey, Stefan A Fechtel Copyright  1998 John Wiley & Sons, Inc Print ISBN 0-471-50275-8 Online ISBN 0-471-20057-3

Chapter 3 Passband Transmission

In this chapter we briefly review the fundamentals of passband transmission over a time-invariant channel In Section 3.1 we describe the transmission methods

In Section 3.2 we introduce channel and transceiver models In the last section of this chapter we are concerned with the fundamental bounds of the outer receiver (see chapter on introduction and preview), the channel capacity This bound defines the ultimate transmission rate for error-free transmission We gain considerable insight by comparing this bound with the performance of any communication system Furthermore, studying the fundamental bounds on the outer receiver performance (channel capacity) and that of the inner receiver (variance of the parameter estimates) provides a deep understanding of the interdependence between the two parts

3.1 Transmission Methods

Passband transmission of digital information can be roughly separated into two main classes: noncoherent and coherent The first class uses so-called noncoherent modulation techniques, which do not require an estimate of the carrier frequency and phase

Noncoherent modulation techniques have significant disadvantages, in partic- ular a power penalty and spectral inefficiency when compared to the second class

of techniques employing coherent transmission

The most commonly used member of the class of coherent transmission tech- niques is pulse-amplitude modulation (PAM) Special cases of passband PAM are phase-shift keying (PSK), amplitude and phase modulation (AM-PM), and quadra- ture amplitude modulation (QAM) In a pulse-amplitude-modulated passband sig- nal the information is encoded into the complex amplitude values ura for a single pulse and then modulated onto sinusoidal carriers with the same frequency but a 90’ phase difference PAM is a linear technique which has various advantages

to be discussed later

Another class of coherent modulation techniques discussed is continuous- phase modulation (CPM) Since the information is encoded into the phase, the signal maintains a constant envelope which allows a band-limited signal’s amplifi-

211

Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing

Heinrich Meyr, Marc Moeneclaey, Stefan A Fechtel Copyright  1998 John Wiley & Sons, Inc Print ISBN 0-471-50275-8 Online ISBN 0-471-20057-3

cation without serious spectral spreading of the signal, because the main nonlinear effects of bandpass high-performance amplifiers, namely amplitude-to-amplitude (AM/AM) and amplitude-to-phase (AM/PM) conversion, are avoided

3.2 Channel and Transceiver Models

3.2.1 Linear Channel Model

The general linear model of a carrier-modulated digital communication system

is shown in Figure 3-l This model adequately describes a number of important channels such as telephone channels and microwave radio transmission

In Figure 3-1, u(t) represents any type of linearly modulated signal in baseband:

n The channel symbols (an) are chosen from an arbitrary signal set over the complex plane and gT(t) is the impulse response of the pulse-shaping filter T

is referred to as the channel symbol duration or period The pulse gT(t) contains

a (possibly slowly time varying) time shift EO with respect to the time reference

of a hypothetical observer

In the quaternary PSK modulator, a sign change in both in-phase and quadra- ture components causes a phase shift of 180’ If only one of the components, either the Z or Q component, changes its sign, a phase shift of &n/2 occurs This reduces the envelope fluctuations [l , p 239ffl which is useful when the signal undergoes nonlinear amplification,

Bits

we+‘) -B, +

!p-+$=+~~,

Amplitude Control

q=PH@p, O))l Figure 3-1 Linear Model of Passband Transmission: Linear Modulation

3.2 Channel and Transceiver Models 213

~-l * * n *-1 -~

I

I ,,,r,

I

L-^ I -L -I -~

wsp(f)

Al +

Figure 3-2 Linear Model of Passband Transmission: CPM

When the quadrature component is delayed with respect to the in-phase

component by T/2 only phase changes of &n/2 occur This modulation is called

offset QPSK (OQPSK) or staggered QPSK The signal u(t) in (3-l) for OQPSK

reads

u(t> = (p%J4 - z,(T) -4)

(0 (9) where the channel symbols a, , a, take values in the set { - 1, 1)

For nonlinear modulation schemes, the dashed block in Figure 3-l must be

replaced by that of Figure 3-2 For a CPM signal, the signal u(t) is

(3-3) The phase is given by

4(t) = 2nh zakq(t - IcT - E&!“) + 0

k

(3-3a)

where a = {a, } is the data sequence of A4-ary information symbols selected from

the alphabet fl, f3, f (M - l), h is the modulation index, and q(t) is some

normalized waveform The waveform q(t) may be represented as the integral of

some frequency pulse v(t)

t

q(t) = J V(T) dr (3-4)

If v(t) = 0 for t 2 T the CPM signal is called full response, otherwise partial response The equivalent lowpass signal u(t) has a constant envelope, which is

an advantage when a nonlinear amplifier is employed

Of great practical importance are so-called MSK (minimum shift keying) signals They are a special case of CPM signals obtained for

For MSK the frequency pulse is rectangular

v(t) = { y2T O<t<T

otherwise (full response) (3-6) For Gaussian MSK (GMSK) v(t) is the response of the Gaussian filter to a rectangular pulse of duration T (partial response)

An interesting property of MSK signals is that they can be interpreted as linearily modulated passband signals where the quadrature component is delayed with respect to the in-phase component by T [2 , Chapter 61 The following signal

is equivalent to (3-3):

with

c azn-l h(t - [2n- l]T - eoT)

10 0 O<t_<2T else

(3-7)

(3-W

and a2t-t) azn- 1 take on values f 1 Notice that the pulse h(t) is time-limited Hence, u(t) is not band-limited

The baseband signal is up-converted to passband by an oscillator

which models the physical channel frequency response as well as any filtering in the passband region The signal is subject to wideband additive Gaussian noise wgp(t) with flat spectral density No/2 (see Section 1.2.3 above and Chapter

3 of Volume 1)

After downconversion by a local oscillator exp[-j(wot + OR(t))] and lowpass filtering F(w) in order to remove double-frequency terms we obtain the signal

u(t) = &3[As(t) @ f(t) + w(t) @ f(t)1

= fh [As! (t> + n(t)] U-9)

where sf (t) is related to u(t) and n(t) is a normalized complex-valued additive Gaussian noise process (see Table 3-l)

3.2 Channel and Transceiver Models Table 3-1 Signals and Normalizations in the Linear Channel Model

215

Signal Transmittter u(t) = can gT(t - nT - EOT)

n

Linear modula- tion

w

=2rh c akq(t - IcT - eoT)

Nonlinear CPM modula- tion

signal Channel C(w) = + + wO)cBP(u + UO)

Receiver r(t) =

with

s(t) =

with 4) =

n(t) = rjw =

&[A+) + w(t)]

[u(t)ejeT(t) 8 c(t)] e-jeR(t)

fiRe( w(t)ejwot } r(t) @ f(t) = K~[Asj(t) + n(t)]

4) Q9 N

44 @ f(t)

input signal Noise WBP(t): Bandpass process with flat

spectral density NO /2

Using the bandpass system model just described, a baseband equivalent model

is now derived The baseband equivalent channel is defined as [see eq (l-97)]

Then Figures 3-l and 3-2 can be simplified to the model shown in Figure 3-3 which still has some traces of the mixer oscillator of the transmitter

n(t)

A

Figure 3-3 Normalized Baseband Model with Arbitrary 0~ (t) and 0~ (t)

If the phase fluctuations 6$(t) have a much smaller bandwidth than the channel frequency response C(w), we may interchange the operation of filtering and multiplication The useful signal s(t) in (3-10) now reads (see Figure 3-4

= u(t) QD c(t) ejeo(t) with @c(t) = eT(t) - OR(t) s ummarizing the previous operations, the received signal of (t ) = y(t)/( KRA) for linear modulators equals

n(t) 7(t) = Sf (t) + A

= c a,g(t - nT - EOT) ejeo(t) + F

n

(3-12)

where g(t) is the pulse form seen by the receiver

I I exP[i(e,(t) - +#))I

n(t)

A

Figure 3-4 Normalized Baseband Model with Slowly Varying Phase t&,(t)

3.2 Channel and Transceiver Models 217

and n(t) is a noise process with power spectral density

St&) = I~(~)12Stu(~) (3- 14)

The index 0 in (~0, 00) denotes the actual (unknown) values of the synchronization parameters which must be estimated by the receiver

For easy reference, the signals in the channel model are summarized in Table 3-l

3.2.2 Nonlinear Channel Model

The linear channel model described in the previous section is not applicable

to satellite transmission, as shown in Figure 3-5 It consists of two earth stations (TX and Rx) usually far from each other, connected by a repeater traveling in the sky (satellite) through two radio links A functional block diagram of the system

of Figure 3-5 is shown in Figure 3-6

The block labeled HPA represents the earth station power amplifier with a nonlinear input-output characteristics of the saturation type The TX filter limits the bandwidth of the transmitted signal whereas the input filter in the transponder limits the amount of uplink noise The block TWT represents the satellite’s on-

Figure 3-5 Model of a Satellite Link a)

Nonlinear Power

I-

Complex

Sym~s b,)

I Receiver

Figure 3-6 Model of a Satellite Link b)

board amplifier Owing to the satellite’s limited power resources, this amplifier is usually operated at or near saturation to obtain maximum power efficiency The output filter limits the bandwidth of the transmitted signal again whereas the Rx filter limits the downlink noise

The assessment of the error performance for this (and any other) nonlinear channel is difficult While refined mathematical tools and general comprehensive theorems are available in the linear case, only very special categories of problems can be analyzed in nonlinear situations There are mainly two avenues to ana- lyze satellite systems The first approach makes simplifications of the problems such that an analytical approach is feasible The other approach uses computer simulation This later approach is the one taken in this book

3.3 Channel Capacity of Multilevel/Phase Signals

Information theory answers two fundamental questions:

1 What is the ultimate data compression? (answer: the entropy H)

2 What is the ultimate transmission rate of communication? (answer: the channel capacity C)

We gain a considerable insight comparing these boundaries with the performance

of any communication system design

The communication model studied in information theory is shown in Figure 3-7 This discrete-time channel is defined to be a system consisting of an input alphabet X and an output alphabet Y and a probability matrix p(y 1 x) that expresses the probability of observing the output y given that we send x If the outcome yi depends only on the symbol xi the channel is said to be memoryless

We consider now the memoryless Gaussian channel when yi is the sum of input xi and noise 12;:

Yi = Xi + 126 (3-15)

Source _3 Source ) Channel

Encoder Encoder X

Discrete Channel

Sink - Source - Channel

Decoder Decoder Y

L

Figure 3-7 Discrete Model of a Communication System

3.3 Channel Capacity of Multilevel/Phase Signals 219

t C (BITS/SYMBOL)

64-QAM 32-AMPM 16-QAM 8-AMPM 8-PSK 4-PSK 2-PSK

SNR (dB)

30 Figure 3-8 Channel Capacity for a Discrete-Valued

Input ai and Continuous-Valued Output yi

The noise ni is drawn from an i.i.d Gaussian distribution with variance b: The channel capacity C for this channel can be computed for a discrete input channel signal as well as for a continuous-valued input

For a discrete input we denote the input by oi instead of Xi where oi is

a complex-valued PAM symbol The channel capacity has been computed by Ungerboeck [3] for the case where all symbols are equally probable The results are displayed in Figure 3-8

In Figure 3-8 (After Ungerboeck [3], the ultimate bound for a continuous- valued input alphabet xi is shown:

c = log, 1+ p

( 4 >

Formula 3-16 is one of the most famous formulas in communication

important to understand the conditions under which it is valid:

(3- 16) theory It is

(i) xi is a complex Gaussian random variable with an average power constraint

El [ xi I] 2 <P

(ii) ai equals the variance of a complex Gaussian random variable

The capacity C is measured in bits per channel use, which is the same as bits per symbol transmitted The capacity is a function of the average signal-to-noise ratio P/a: where P is the average power

Real and imaginary parts of (3-15) each describe a real-valued Gaussian channel The capacity of a real-valued channel is given by

(3- 17)

where PI = E{ x3}, xi is a real Gaussian random variable and a;/2 is the variance of the real noise sample

The total capacity of both channels is the sum of the capacity of the individual channels Since the sum is maximized if the available power P is equally split,

we obtain

.=2CD=2(;)log,(l+-$)

(3-18)

which is eq (3-16)‘

From Figure 3-8 we also observe that for any SNR we loose very little

in capacity by choosing a discrete input alphabet ai as long as the alphabet is sufficiently large The higher the SNR the larger the alphabet This result gives a solid theoretical foundation for the practical use of discrete-valued symbols

In a real physical channel a time-continuous waveform s(t, a) corresponding

to the sequence a is transmitted If the signal s(t , a) is band-limited of bandwidth

B the sampling theorem tells us that the signal is completely characterized by samples spaced T = 1/(2B) apart Thus, we can send at most 2B symbols per second The capacity per symbol then becomes the capacity per second

(baseband)

(3- 19)

A complex alphabet is always transmitted as a passband signal Looking at the sampling theorem we notice that W = l/T symbols per second can be transmitted Hence,

(passband) (3-20)

Thus the capacity formula (in bits per second) is applicable to both cases However, the proper bandwidth definition must be used (see Figure 3-9)

‘If the symbol ak is complex, one speaks of two-dimensional modulation If ak is real, of one speaks

of one-dimensional modulation The term originates from the viewpoint of interpreting the signals as elements in a vector space