Tài liệu Digital communication receivers P2 - P1 pptx

The transmitted signal is a pulse-amplitude-modulated PAM signal: it consists of a sequence of time translates of a baseband pulse which is amplitude- modulated by a sequence of data sym

PART C Baseband Communications

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 2 Baseband Communications

2.1 Introduction to Digital Baseband Communication

In baseband communication, digital information is conveyed by means of a pulse train Digital baseband communication is used in many applications, such as Transmission at a few megabits per second (Mb/s) of multiplexed digitized voice channels over repeatered twisted-pair cables

Transmission of basic rate ISDN (16Okb/s) over twisted-pair digital subscriber lines

Local area networks (LANs) and metropolitan area networks (MANS) oper- ating at 10-100 Mb/s using coaxial cable or optical fiber

Long-haul high-speed data transmission over repeatered optical fiber

Digital magnetic recording systems for data storage

This chapter serves as a short introduction to digital baseband communication

We briefly consider important topics such as line coding and equalization, but without striving for completeness The reader who wants a more detailed treatment

of these subjects is referred to the abundant open literature, a selection of which

is presented in Section 2.1 S

2.1.1 The Baseband PAM Communication System

Baseband communication refers to the case where the spectrum of the trans- mitted signal extends from zero frequency direct current (DC) to some maximum frequency The transmitted signal is a pulse-amplitude-modulated (PAM) signal:

it consists of a sequence of time translates of a baseband pulse which is amplitude- modulated by a sequence of data symbols conveying the digital information to be transmitted

A basic communication system for baseband PAM is shown in Figure 2-l

At the transmitter, the sequence of information bits (bk) is applied to an encoder, which converts {bk} into a sequence (uk} of data symbols This conversion is called line coding, and will be considered in more detail in Section 2.1.3 The information bits assume the values binary zero (“0”) or binary one (“l”), whereas the data symbols take values from an alphabet of a size L which can be larger than

2 When L is even, the alphabet is the set {fl, f3, =t(L - l)}, for an odd L the alphabet is the set (0, f2, f4, =t(L - 1))

The data symbols enter the transmit filter with impulse response gT (t), whose Fourier transform is denoted by GT(w) The resulting transmit signal is given by

m

where VT is the symbol rate, i.e., the rate at which the data symbols are applied

to the transmit filter The impulse response g*(t) is called the baseband pulse of the transmit signal The quantity ET is a fractional unknown time delay between the transmitter and the receiver (1~1 5 3) The instants {H’} can be viewed as produced by a hypothetical reference clock at the receiver At the transmitter, the lath channel symbol ak: is applied to the transmit filter at the instant Kf + ET, which is unknown to the receiver, Figure 2-2 shows a baseband pulse g*(t) and

a corresponding PAM signal s(t), assuming that I, = 2

Figure 2-2 (a) Baseband PAM Pulse gT(t), (b) Binary PAM Signal s(t)

The channel is assumed to be linear It introduces linear distortion and adds noise The linear distortion (amplitude distortion and delay distortion) is char- acterized by the channel frequency response C(u) It causes a broadening of the transmitted pulses The “noise” is the sum of various disturbances, such as thermal noise, electronics noise, cross talk, and interference from other commu- nication systems

The received noisy PAM signal is applied to a receive filter (which is also called data filter) with frequency response GR(w) The role of this filter is to reject the noise components outside the signal bandwidth, and, as we will explain

in Section 2.1.2, to shape the signal The receive filter output signal y(t; E) is sampled at symbol rate l/T From the resulting samples, a decision (&) is made about the data symbol sequence (ah} The sequence (&k} is applied to a decoder, which produces a decision

11 8, on the information bit sequence { bk )

The signal at the output of the receive filter is given by

m where g(t) and n(t) are the baseband pulse and the noise at the receive filter output The Fourier transform G(w) of the baseband pulse g(t) is given by

Let us denote by {kT + ZT} the sequence of instants at which the sampler at the receive filter output is activated These sampling instants are shifted by an amount

tT with respect to the instants (IcT} produced by the hypothetical reference clock

of the receiver Then the lath sample is given by

m#O

g(mT - eT), and n( IcT + CT), while e = E - i denotes the difference, nor- malized by the symbol duration T, between the instant where the Kth symbol ak

is applied to the transmit filter and the Lth sampling instant at the receiver

In order to keep the decision device simple, receivers in many applications perform symbol-by-symbol decisions: the decision &k is based only on the sample

yk (e) Hence, only the first term of the right-hand side of (2-4) is a useful one, because it is the only one that depends on ck The second term is an intersymbol infelference (ISI) term depending on ck -m with m # 0, while the third term is

a noise term When the noise n(t) at the receive filter output is stationary, the statistics of the noise sample nk do not depend on the sampling instant On the other hand, the statistics of the useful term and the IS1 in (2-4) do depend on the sampling instant, because the PAM signal is cyclostationary rather than stationary Let us consider given sampling instants (IcT + gT} at the receive filter output

The symbol-by-symbol decision rule based on the samples YA! (e) from (2-4) is

m - 1 < yk(e)/go(e) 5 m + 1 m # k(L - 1) Cik = Y- 1 L - 2 < !/k(e)h’O(e)

(2-5) where the integer m takes on only even (odd) values when L is odd (even), This decision rule implies that the decision device is a slicer which determines the symbol value &k which is closest to & (e)/go( e) In the absence of noise, the baseband communication system should produce no decision errors A necessary and sufficient condition for this to be true is that the largest magnitude of the IS1 over all possible data sequences is smaller than 1 go(e) I, i.e., M(e) > 0 where M(e) is given by

W4 = I go(e) 1 - max thn) m;to am g-m e E ()I (2-6)

When M(e) < 0, the data symbol sequence yielding maximum IS1 will surely give rise to decision errors in the absence of noise, because the corresponding yk (e) is outside the correct decision region When M(e) > 0, a decision error can occur only when the noise sample nk from (2-4) has a magnitude exceeding M(e) ; M(e)

is called the noise margin of the baseband PAM system The noise margin can be visualized by means of an eye diagram, which is obtained in the following way Let us denote by ~e(t ; E) the receive filter output signal in the absence of noise, i.e.,

yo(t;&) = Cam g(t - mT- 0)

m

(2-7)

The PAM signal ~o(t ; E) is sliced in segments yo,i (t ; E), having a duration equal

to the symbol interval 2’:

The eye diagram is a display of the periodic extension of the segments yo,i(t; E)

An example corresponding to binary PAM (L = 2) is shown in Figure 2-3 As g(t) has a duration of three symbols, the eye diagram for binary PAM consists of

23 = 8 trajectories per symbol interval Because of the rather large value of g(r), much IS1 is present when sampling the eye at t = 0 The noise margin M(e) for a specific sampling instant is positive (negative) when the eye is open (closed)

at the considered instant; when the eye is open, the corresponding noise margin equals half the vertical eye opening,

The noise margin M(e) depends on the sampling instants IcT + 2T and the unknown time delay ET through the variable e = e - i The optimum sampling instants, in the sense of minimizing the decision error probability when the worst-

(a) (b>

Figure 2-3 (a) Baseband PAM Pulse g(t), (b) Eye Diagram for Binary PAM

case IS1 is present, are those for which M(e) is maximum Using the appropriate time origin for defining the baseband pulse g(t) at the receive filter output, we can assume without loss of generality that M(e) becomes maximum for e = 0 Hence, the optimum sampling instants are IcT+ ET, and e = E - E^ denotes the timing error normalized by the symbol interval The sensitivity of the noise margin M(e) to the normalized timing error e can be derived qualitatively from the eye diagram: when the horizontal eye opening is much smaller than the symbol interval T, the noise margin and the corresponding decision error probability are very sensitive

to timing errors

Because of the unknown delay 0 between the receiver and the transmitter, the optimum sampling instants { IcT + ET} are not known a priori to the receiver Therefore, the receiver must be equipped with a structure that estimates the value of

e from the received signal A structure like this is called timing recovery circuit or symbol synchronizer The resulting estimate 6 is then used to activate the sampler

at the instants (Kf + Z7’) The normalized timing error e = E - i should be kept small, in order to avoid the increase of the decision error probability, associated with a reduction of the noise margin M(e)

2.1.2 The Nyquist Criterion for Eliminating ISI

It is obvious that the shape of the baseband pulse g(t) and the statistics of the noise n(t) at the output of the receive filter depend on the frequency response

GR(w) of the receive filter Hence, the selection of GR(w) affects the error probability when making symbol-by-symbol decisions The task of the receive filter is to reduce the combined effect of noise and ISI

Let us investigate the possibility of selecting the receive filter such that all IS1 is eliminated when sampling at the instants {kT + ET} It follows from (2-4) that IS1 vanishes when the baseband pulse g(t) at the receive filter output satisfies g(mT) = 0 for m # 0 As G( w is the Fourier transform of g(t), ) g (mT) is for all m given by

+w

G(w) exP (Mq g -W

Note that Gad(W) is periodic in w with period 27r/T It follows from (2-9) that

Tg (-mT) can be viewed as the mth coefficient in the Fourier-series expansion

of Gfid(w):

Gad(w) = T c g(-mT) exp (jmwT)

m=-co

(2-12) Using (2-12) and the fact that Gad(w) is periodic in w, we obtain

g(mT) =0 for m#O tj Gad(W) is constant for 10 1 < x/T

(2-13) This yields the well-known Nyquist criterion for zero ISI: a necessary and sufficient condition for zero IS1 at the receive filter output is that the folded Fourier transform Gfid(w) is a constant for Iw 1 < ?r/T

The Nyquist criterion for zero IS1 is sometimes referred to as the&st Nyquist

criterion A pulse satisfying this criterion is called an interpolation pulse or a Nyquist-I pulse Let us consider the case where G(w) is band-limited to some frequency B; i.e., G(w) = 0 for Iw( > 27rB

l When B < 1/(2T), G~,-J(w) = G(u) for 1~1 < n/T As G(w) = 0 for 27rB < IwI < r/T, G fld w cannot be constant for 1~1 < x/T Taking (2-3) ( ) into account, it follows that when the bandwidth of the transmit filter, of the channel or of the receive filter is smaller than 1/2T, it is impossible to find

a receive filter that eliminates ISI

When B = l/(273, Gm( w is constant only when, within an irrelevant ) constant of proportionality, G(w) is given by

e, yielding a horizontal eye opening of zero width

When B > 1/(2T), the baseband pulse g(t) which eliminates IS1 is no longer unique Evidently, all pulses that satisfy g(t) = 0 for It I 2 T eliminate ISI Because of their time-limited nature, these pulses have a large (theoretically infinite) bandwidth, so that they find application only on channels having a bandwidth B which is considerably larger than 1/(2T); an example is optical fiber communication with on-off keying of the light source When bandwidth

is scarce, one would like to operate at a symbol rate l/T which is only slightly less than 2 B This is referred to as narrowband communication

When 1/(2T) < B < l/T, the Nyquist criterion (2-13) is equivalent to imposing that G(w) has a symmetry point at w = ?r/T:

A widely used class of pulses with 1/(2T) < B < l/T that satisfy (2- 16) are the cosine rolloff pulses (also called raised cosine pulses), determined by

sin (?rt/T) cos (curt/T)

withO<cu< 1 Fora= 0, (2-17) reduces to (2-15) The Fourier transform G(w) of the pulse g(t) from (2-17) is given by

From the above discussion we conclude that a baseband PAM pulse g(t) that

(a)

09

Figure 2-4 Cosine Rolloff Pulses: (a) Baseband Pulse gT(t) , (b) Fourier Transform G(w) , (c) Eye Diagram for Binary PAM (25% Rolloff), (d) Eye Diagram for Binary PAM (50% Rolloff), (e) Eye Diagram for Binary PAM (100% Rolloff)

1

eliminates IS1 must satisfy the Nyquist criterion (2-13) In order that a receive filter exist that eliminates ISI, it is necessary that the available bandwidth exceed half the symbol rate

As the receive filter which eliminates (or at least substantially reduces) IS1 yields a baseband pulse g(t) whose folded Fourier transform Gad(w) is essentially flat, the receive filter is often referred to as a (linear) equalizer The equalizer must compensate for the linear distortion introduced by the channel and therefore depends on the channel frequency response C(w) When there is sufficient a priori knowledge about C(w), the equalizer can be implemented as a fixed filter However, when there is a rather large uncertainty about the channel characteristics, the equalizer must be made adaptive

Generally speaking, the equalizer compensates for the channel attenuation

by having a larger gain at those signal frequencies that are more attenuated by the channel However, the noise at those frequencies is also amplified, so that equalization gives rise to a noise enhancement The larger the variation of the channel attenuation over the frequency interval occupied by the transmit signal, the larger this noise enhancement

2.1.3 Line Coding

In many baseband communication systems, there is some frequency value B beyond which the linear distortion rapidly increases with frequency For example, the attenuation (in decibels) of a twisted-pair cable is proportional to the square- root of the frequency (skin effect) This sets a limit on the bandwidth that can

be used for baseband transmission: if the transmit signal contained components at too high frequencies, equalization of the severe linear distortion would give rise

to a large noise enhancement, yielding a considerable performance degradation

In addition, in many applications the transmitted signal is severely distorted near zero frequency, because of transformer coupling or capacitive coupling, which reject DC In these cases, the transmit signal should have low spectral content near zero frequency, in order to avoid excessive linear distortion and a resulting performance degradation

It is clear from the above that we must control the transmit spectrum in order to avoid both high frequencies and frequencies near DC We recall from Section 1.1.4 that the power spectrum S,(w) of the transmit signal s(t) from (2-l) is given by

where Sa ( ejwT) is the power spectrum of the data symbol sequence Hence, the control of the spectrum is achieved by acting on the transmit pulse g(t) and/or on the spectrum of the data symbol sequence The latter is achieved by means of line coding: line coding involves the specification of the encoding rule that converts the information bit sequence {bk} into the sequence {ak} of data symbols, and therefore affects the spectrum of the data symbol sequence

In addition to spectrum control, line coding should also provide timing information Most symbol synchronizers can extract a reliable timing estimate from the received noisy PAM signal, only when there are sufficient data symbol transitions When there are no data symbol transitions (suppose uk = 1 for all E), the receive filter output signal in the absence of noise is given by

YO(CE) = E g t-kT-ET) (

= ;-j-j q!) exp [j?z& -q] (2-20) -00

Note that ya(t; 6) is periodic in t with period T, so that one could be tempted to conclude that timing information can easily be extracted from yo(t; E) However, when G(w) = 0 for Iwl > 27rB with B < l/T (as is the case for narrowband communication), the terms with k # 0 in (2-20) are zero, so that yo~(t; E) contains only a DC component, from which obviously no timing information can be derived The same is true for wideband pulses with G(27rm/T) = 0 for m # 0 (as is the case for rectangular pulses with duration T) Hence, in order to guarantee sufficient timing information, the encoding rule should be such that irrespective of the bit sequence (bk}, the number of successive identical data symbols in the sequence {CQ} is limited to some small value

As the spectrum SQ (eiWT) is periodic in w with period 27r/T, it follows from (2-19) that the condition S,(w) = 0 for IwI > 27rB implies GT(w) = 0 for 1~1 > 27rB Hence, frequency components above w = 2nB can be avoided only

by using a band-limited transmit pulse According to the Nyquist criterion for zero ISI, this band-limitation on GT(w) restricts the symbol rate l/T to l/T < 2 B

A low spectral content near DC is achieved when Ss(0) = 0 This condition

is fulfilled when either GT( 0) = 0 or So ( ej”) = 0

When GT(O) = 0 and l/T is required to be only slightly less than 2B (i.e., narrowband communication), the folded Fourier transform GH&) of the baseband pulse at the receive filter output is zero at w = 0 This indicates that equalization can be performed only at the expense of a rather large noise enhancement The zero in the folded Fourier transform Gfid(U) and the resulting noise enhancement can be avoided only by reducing the symbol rate l/T below the bandwidth B Hence, using a transmit pulse gT(t) with

GT(O) = 0 is advisable only when the available bandwidth is sufficiently large, but not for narrowband applications

In the case of narrowband communication, the recommended solution to obtain S8 (0) = 0 is to take SQ (ej ‘) = 0 Noting that S, (ej”) can be expressed as

it follows that Sa (ej”) = 0 when the encoding rule is such that the magnitude

of the running digital sum, RDS (n), given by

or alternating zeroes and ones) which cause long strings of identical data symbols

at the encoder output In order that the transmitted signal contains sufficient timing information, the probability of occurrence of such binary strings at the encoder input should be made very small This can be accomplished by means

of a scrambler Basically, a scrambler “randomizes” the binary input sequence

by modulo-2 addition of a pseudo-random binary sequence At the receiver, the original binary sequence is recovered by adding (modulo-2) the same pseudo- random sequence to the detected bits

Binary Antipodal Signaling

In the case of binary antipodal signaling, the channel symbol c&k equals +l

or -1 when the corresponding information bit bk is a binary one or binary zero, respectively Unless the Fourier transform G(w) of the baseband pulse at the receive filter output satisfies G(27rm/T) # 0 for at least one nonzero integer

m, this type of line coding does not provide sufficient timing information when the binary information sequence contains long strings of zeroes or ones The occurrence of such strings can be made very improbable by using scrambling Also, in order to obtain a zero transmit spectrum at DC, one needs GT(O) = 0 Consequently, the magnitude of the running digital sum is limited, so that the codes yield no DC,

Quaternary Line Codes

In the case of quaternary line codes, the data symbol alphabet is the set {f l,f3} An example is 2BIQ, where the binary information sequence is subdivided in blocks of 2 bits, and each block is translated into one of the four levels fl or f3 The 2BlQ line code is used for the basic rate ISDN (data rate

of 160 kb/s) on digital subscriber lines

2.1.4 Main Points

In a baseband communication system the digital information is conveyed by means of a pulse train, which is amplitude-modulated by the data The channel

is assumed to introduce linear distortion and to add noise This linear distortion broadens the transmitted pulses; the resulting unwanted pulse overlap gives rise

to ISI

The receiver consists of a receive filter, which rejects out-of-band noise Data detection is based upon receive filter output samples, which are taken once per symbol These samples are fed to a slicer, which makes symbol-by-symbol decisions The decisions are impaired by IS1 and noise that occurs at the sampling instants

The receive filter output should be sampled at the instants of maximum noise margin These optimum sampling instants are not a priori known to the receiver A timing recovery circuit or symbol synchronizer is needed to estimate the optimum sampling instants from the received noisy PAM signal

The receive filter must combat both noise and ISI According to the Nyquist criterion, the receive filter should produce a pulse whose folded Fourier transform

is essentially flat, in order to substantially reduce the ISI Such a filter is called

an equalizer When the system bandwidth is smaller than half the symbol rate, equalization cannot be accomplished

In many applications, the channel attenuation is large near DC and above some frequency B In order to avoid large distortion, the transmit signal should have negligible power in these regions This is accomplished by selecting a transmit pulse with a bandwidth not exceeding B, and by means of proper line coding to create a spectral zero at DC Besides spectrum control, the line coding must also provide a sufficient number of data transitions in order that the receiver is able

to recover the timing

2.1.5 Bibliographical Notes

Baseband communication is well covered in many textbooks, such as [ l]-[6] These books treat equalization and line coding in much more detail than we have done Some interesting topics we did not consider are mentioned below

Ternary Line Codes

In the case of ternary line codes, the data symbols take values from the set I-2,0, $2) In the following, we will adopt the short-hand notation (- , 0, +} for the ternary alphabet

A simple ternary line code is the alternate mark inversion (AMI) code, which

is also called bipolar The AM1 encoder translates a binary zero into a channel symbol 0, and a binary one into a channel symbol + or - in such a way that polarities alternate Because of these alternating polarities, it is easily verified that the running digital sum is limited in magnitude, so that the transmit spectrum

is zero at DC Long strings of identical data symbols at the encoder output can occur only when a long string of binary zeroes is applied to the encoder This yields a long string of identical channel symbols 0 The occurrence of long strings

of binary zeroes can be avoided by using a scrambler The AM1 decoder at the

Table 2-l 4B3T Line Code [l]

Binary input block

receiver converts the detected ternary symbols into binary symbols 0 is interpreted

as binary zero, whereas + and - are interpreted as binary one The AM1 code uses one ternary symbol to transmit one bit of information Hence the efficiency of the AM1 code, as compared to transmitting statistically independent ternary symbols, equals l/ log, 3 z 0.63 AM1 is widely used for transmission of multiplexed digitized voice channels over repeatered twisted pair cables at rates of a few megabits per second

A higher efficiency and more timing information than (unscrambled) AM1 are obtained when using ternary block codes, which map blocks of k bits to blocks of

n ternary symbols; such codes are denoted as kBnT As an example, we consider the 4B3Tcode, which maps 4 bits to 3 ternary symbols according to Table 2-l Note that there are two “modes”: when the running digital sum is negative (nonnegative), the entries from the first (second) column are used In this way, the magnitude of the running digital sum is limited, which guarantees a spectral zero at DC Also, it is not possible to have more than five successive identical