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Chapter 9 Signal Design For Band-Limited Channel 9-1 Characterization of Band-Limited Channels9-2 Signal Design For Band-Limited Channel 9-2-1 Design of Band-Limited Signals For No Intersymbol Interference-The Nyquist Criterion 9-2-2 Design of Band-Limited Signals with Controlled ISI-Partial-Response Signals 9-2-3 Data Detection for controlled ISI 9-2-4 Signal Design for Channel with Distortion 9-3 Probability of Error In Detection Of Pam 9-3-1 Probability of Error for Detection of PAM with Zero ISI 9-3-2 Probability of Error for Detection of Partial-Response Signals 9-3-3 Probability of Error for Optimum Signals in a Channel with Distortion 9-4 Modulation Codes For Spectrum ShapingChapter 9 Signal Design For Band-Limited ChannelIn this chapter,we consider the problem of signal design when the channel is band-limited to some specified bandwidth W Hz.Under this condition,the channel may be modeled as a linear filter having an equivalent lowpass frequency C(f)that is zero for|f|W.ReturnThe first topic that is treated is the design of the signal pulse g(t)in a linearly modulated signal,represented as that efficiently utilizes the total available channel bandwidth W.We shall see that when the channel is ideal for ,a signal pulse can be designed that allows us to transmit at symbol rates comparable to or exceeding the channel bandwidth W.On the other hand,when the channel is not ideal,signal transmission at a symbol rate equal to or exceeding W results in intersymbol interference(ISI)among a number of adjacent symbols.The second topic that is treated in this chapter is the use of coding to shape the spectrum of the transmitted signal and,thus,to avoid the problem of ISI.Return9-1 Characterization of Band-Limited ChannelsOf the various channels available for digital communications,telephone channels are by far the most widely used.Such channels are characterized as band-limited linear filters.This is certainly the proper characterization when frequency-division multiplexing(FDM)is used as a means for establishing channels in the telephone network.A band-limited channel such as a telephone channel will be characterized as a linear filter having an equivalent lowpass frequency response characteristic C(f).Its equivalent lowpass impulse response is denoted by c(t).ReturnThen,if a signal of the form is transmitted over a bandpass telephone channel,the equivalent lowpass received signal is Where the integral represents the convolution of c(t)with v(t),and z(t)denotes the additive noise.ReturnWithin the bandwidth of the channel,we may express the frequency response C(f)asWhere|C(f)|is the amplitude response characteristic and is the phase response characteristic.Furthermore,the envelope delay characteristic is defined asA channel is said to be nondistorting or ideal if the amplitude response|C(f)|is constant for all and is a linear function of frequency,i.e.,is a constant for all .On the other hand,if|C(f)|is not constant for all ,we say that the channel distorts the transmitted signal V(f)in amplitude,and,if is not constant for all ,we say that the channel distorts the signal V(f)in delay.ReturnThe extent of the intersymbol interference on a telephone channel can be appreciated by observing a frequency response characteristic of the channel.In addition to linear distortion,signals transmitted through telephone channels are subject to other impairments,specifically nonlinear distortion,frequency offset,phase jitter,impulse noise and thermal noise.(1)Nonlinear distortion in telephone channels arises from nonlinearities in amplifiers and compandors used in the telephone system.This type of distortion is usually small and it is very difficult to correct.(2)A small frequency offset,usually less than 5Hz,results from the use of carrier equipment in the telephone channel.ReturnSuch an offset cannot be tolerated in high-speed digital transmission systems that use synchronous phase-coherent demodulation.The offset is usually compensated for by the carrier recovery loop in the demodulator.(3)Phase jitter is basically a low-index frequency modulation of the transmitted signal with the low frequency harmonics of the power line frequency(50-60Hz).Phase jitter poses a serious problem in digital transmission of high rates.However,it can be tracked and compensated for,to some extent,at the demodulator.(4)Impulse noise is an additive disturbance.It arises primarily from the switching equipment in the telephone system.Thermal(gaussian)noise is also present at levels of 20-30 dB below the signal.ReturnThe degree to which one must be concerned with these channel impairments depends on the transmission rate over the channel and the modulation technique.(1)For rates below 1800 bits/s(R/W1),bandwidth-efficient coded modulation techniques such as trellis-coded QAM,PAM,and PSK are employed.For such rates,special attention must be paid to linear distortion,frequency offset,and phase jitter.Linear distortion is usually compensated for by means of an adaptive equalizer.Phase jitter is handled by a combination of signal and some type of phase compensation at the demodulation.Return(4)At rates above 9600 bits/s,special attention must be paid not only to linear distortion,phase jitter,and frequency offset,but also to other channel impairments mentioned above.Besides the telephone channels,there are other physical channels that exhibit some form of time dispersion,and thus,introduce intersymbol interference.Radio channels such as shortwave ionospheric propagation(HF)and tropospheric scatter are two examples of time-dispersive channels.In these channels,time dispersion and,hence,intersymbol interference is result of multiple propagation paths with different path delays.ReturnThe number of paths and the relative time delays among the paths vary with time,and,for this reason,these radio channels are usually called time-variant multipath channels.The time-variant multipath conditions give rise to a wide variety of frequency response characteristics.Consequently the frequency response characterization that is used for telephone channels is inappropriate for time-variant multipath channels.Instead,these radio channels are characterized statistically,in terms of the average received signal power as a function of relative time delay and Doppler frequency.Return9-2 Signal Design For Band-Limited ChannelThe equivalent lowpass transmitted signal for several different types of digital modulation techniques had common formThe received signal can be represented asWhereAnd Z(t)represents the additive whites Gaussian noise.ReturnWe denote the output of the receiving filter asNow,if y(t)is sampled at times we haveOr,equivalently,(k=0，1，)The sample values can be expressed as Return(k=0，1，)Below,we consider the problem of signal design under the condition that there is no intersymbol interference at the sampling instants.We regard as an arbitrary scale factor,which we arbitrarily set equal to unity for convenience.Then The term represents the desired information symbol at the kth sampling instant,the termRepresents the ISI,and is the additive Gaussian noise variable at the kth sampling instant.Return9-2-1 Design of Band-Limited Signals For No Intersymbol Interference-The Nyquist Criterion We assume that the band-limited channel has ideal frequency response characteristics,i.e.,C(f)=1 for .Then the pulse x(t)has a spectral characteristic X(f)=,whereSince ReturnThe condition for no intersymbol interference isTHEOREM(NYQUIST).The necessary and sufficient condition for x(t)to satisfy is that its Fourier transform X(f)satisfyNow suppose that the channel has a bandwidth of W.Then C(f)0 for|f|W and,conseqently,X(f)=0 for|f|W.Return We distinguish three cases.There is no choice for X(f)to ensure B(f)T in this case and there is no way that we can design a system with no ISI.2.When T=1/2W,or,equivalently,1/T=2W(the Nyquist rate),the replications of X(f),separated by 1/T,are as shown in Figure 01.When T2W,since B(f)=consists of nonoverlapping replicas of X(f),separated by 1/T as shown in Figure 2.-W 0 W3.+W -W +W ReturnIn this case there exists only one X(f)that results in B(f)=T,namely,Which corresponds to the pulse3.When T1/2W,B(f)consists of overlapping replications of X(f)separated by 1/T,as shown in FigureIn this case,there exist numerous choices for X(f)such that B(f)T.Return9-2-2 Design of Band-Limited Signals with Controlled ISI-Partial-Response Signals As we have observed from our discussion of signal design for zero ISI,it is necessary to reduce the symbol rate 1/T below the Nyquist rate of 2W symbols/s to realize practical transmitting and receiving filters.On the other hand,suppose we choose to relax the condition of zero ISI and,thus,achieve a symbol transmission rate of 2W symbols/s.by allowing for a controlled amount of ISI,we can achieve this symbol rate.Return1.One special case that leads to(approximately),physically realizable transmitting and receiving filters is specified by the samplesThe corresponding pulse X(t)is given byIts spectrum is =This pulse is called a duobinary signal pulse.ReturnNote that the spectrum decays to zero smoothly,which means that physically realizable filters can be designed that approximate this spectrum very closely.Thus,a symbol rate of 2W is achieved.2.Another special case that leads to(approximately)physically realizable transmitting and receiving filter is specified by the samplesAnd its spectrum is ReturnIt is called a modified duobinary signal pulse.Note that the spectrum of this signal has a zero at f=0,making it suitable for transmission over a channel that does not pass d.c.3.One can obtain other interesting and physically realizable filter characteristics by selecting different values for the samples and more than two nonzero samples.In general,the class of bandlimited signals pulses that the form ReturnAnd their corresponding spectraare called partial-response signals when controlled ISI is purposely introduced by selecting two or more nonzero samples from the set .The resulting signal pulses allow us to transmit information symbols at the Nyquist rate of 2W symbols/s.The corresponding power spectral density is Return9-2-3 Data Detection for controlled ISIIn this section,we describe two methods for the information symbols at the receiver when the received signal contains controlled ISI.One is a symbol-by-symbol detection method that is relatively easy to implement.The second method is based on the maximum-likelihood criterion for detecting a sequence of symbols.Return1.Symbol-by-Symbol Suboptimum Detection Error propagation can be avoided by precoding the data at the transmitter instead of eliminating the controlled ISI by subtraction at the receiver.The precoding is performed on the binary data sequence prior to modulation.From the data sequence of 1s and 0s that is to be transmitted,a new sequence ,called the precoded sequence,is generated.For the duobinary signal,the precoded sequence is defined as ,m=1,2 the noise-free samples at the output of the receiving filter are given by ReturnConsequently,Since ,it follows that the data sequence is obtained from using the relation Consequently,if ,then ,and if ,then An example that illustrates the precoding and decoding operations is given in Table1.Data sequence1 11010010001101Precoded sequence0101100011110110Transmitted sequence-11-111-1-1-11111-111-1received sequence00020-2-202220020Decoded sequence111010010001101Table1 Binary signaling with duobinary pulses ReturnIn this case is compared with the two thresholds set at+1 and-1.The data sequence is obtained according to the detection rule The extension from binary PAM to multilevel PAM signaling using the duobinary pulses is straightforward.In this case the M-level amplitude sequence results in a(noise-free)sequence which has 2M-1 possible equally spaced levels.The amplitude levels are determined from the relationWhere is the precoded sequence that is obtained from an M-level data sequence according to the relation In the absence of noise,the samples at the output of the receiving filter may be expressed as Hence,Since ,it follows that An example illustrating multilevel precoding and decoding is given in Table2.Data sequence0013120332010Precoded sequence00012331211322Transmitted sequence-3-3-3-1133-11-1-1311received sequence-6-6-4046200-2242Decoded sequence0013120332010Table2 Four-level signal transmission with duobinary pulses In the presence of noise,the received signal-plus-noise is quantized to the nearest of the possible signal levels and the rule given above is used on the quantized values to recover the data sequence.In the case of the modified duobinary pulse,the controlled ISI is specified by the values x(n/2W)=-1,for n=1,x(n/2W)=1 for n=-1,and zero otherwise.Consequently,the noise-free sampled output from the receiving filter is given asFrom these relations,it is easy to show that the detection rule for recovering the data sequence from in the absence of noise is The precoding of the data at the transmitter makes it possible to detect the received data on a symbol-by-symbol basis without having to look back at previously detected symbols.Thus,error propagation is avoided.Return2.Maximum-Likelihood Sequence Detection(1)partial-response waveforms are signal waveforms with memory,this memory is conveniently represented by a trellis.In general,for M-ary modulation,the number of trellis states is .(2)the optimum maximum-likelihood(ML)sequence detector selects the most probable path through the trellis upon observing the received data sequence at the sampling instants t=mT,m=1,2,Return(3)in general,each node in the trellis will have M incoming paths and M corresponding metrics.One out of the M incoming paths is selected as the most probable,based on the values of the metrics and the other M-1 paths and their metrics are described.The surviving path at each node is then extended to M new paths,one for each of the M possible input symbols,and the search process continues.This is basically the Viterbi algorithm for performing the trellis search.ReturnIn this section,we perform the signal design under the condition that the channel distorts the transmitted signal.1.We assume that the channel frequency response C(f)is known for and that C(f)=0 for.The criterion for the optimization of the filter response and is the maximization of the SNR at the output of the demodulation filter or equivalently,at the input to the detector.The additive channel noise is assumed to be gaussian with power spectral density.Figure 9-1-12 illustrates the overall system under consideration.9-2-4 Signal Design for Channel with Distortion Return2.For the signal component at the output of the demodulator,we must satisfy the conditionWhere is the desired frequency response of the cascade of the modulator,channel,and demodulator,and is a time delay that is necessary to ensure the physical realizability of the modulation and demodulation filters.FIGURE 9-2-12 System model for the modulation and demodulation filters.Return3.the noise at the output of the demodulation filter may be expressed as Where n(t)is the input to the filter.Since n(t)is zero-mean Gaussian,v(t)is zero-mean Gaussian,with a power spectral density For simplicity,we consider binary PAM transmission.Then,The sampled output of the matched filter is represents the noise term,which is zero-mean Gaussian with variance ReturnConsequently,the probability of error is The probability of error is minimized by maximizing the ratio .Let us consider two possible solutions for the case in which the additive Gaussian noise is white,so that .1.First,suppose that we precompensate for the total channel distortion at the transmitter,so that the filter at the receiver is matched to the received signal.In this case,the transmitter and receiver filters have the magnitude characteristics The phase characteristic of the channel frequency response C(f)may also be compensated at the transmitter filter.For these filter characteristics,the average transmitted power is ReturnAnd,hence,The noise at the output of the receiver filter is and,hence,the SNR at the detector is2.Suppose we split the channel compensation equally between the transmitter and receiver filters,i.e.,The phase characteristic of C(f)may also be split equally between the transmitter and receiver filters.In this case,the average transmitter power is ReturnAnd the noise variance at the output of the receiver filter isHence,the SNR at the detector isWhen we express the SNR in term of the average transmitter power,there is a loss incurred due to channel distortion.In the case of the filters given by Equation ,the loss is In the case of the filters given by Equation ,the loss is ReturnEXAMPLE 1.Let us determine the transmitting and receiving filters given by Equation for a binary communication system that transmits data at a rate of 4800bits/s over a channel with frequency(magnitude)response where W=4800Hz.The additive noise is zero-mean white Gaussian with spectral density .since W=1/T=4800,we use a signal pulse with a raised cosine spectrum and .Thus,Then,and ,otherwise.FIGURE2.Frequency response of an optimum transmitter filter.9-3 Probability of Error In Detection Of PamIn this section,we evaluate the performance of the receiver for demodulating and detecting an M-ary PAM signal in the presence of additive,while,gaussian noise at its input.First,we consider the case in which the transmitter and receiver filters and are designed for zero ISI.Then,we consider the case in which and are designed such that is either a duobinary signal or a modified duobinary signal.Return9-3-1 Probability of Error for Detection of PAM with Zero ISIIn the absence of ISI,the received signal sample at the output of the receiving matched filter has the formWhere And is the additive gaussian noise that has zero mean and variance In general,takes one of M possible equally spaced amplitude values with equal probability.Given a particular amplitude level,the problem is to determine the probability of error.ReturnThe problem of evaluating the probability of error for digital PAM in a band-limited,additive white gaussian noise channel,in the absence of ISI,is identical to the evaluation of error probability for M-ary PAM.The final result that is obtained from the derivation is ReturnBut is the average energy per symbol and is the average energe per bit.Hence,In the treatment of PAM given in this chapter,we imposed the additional constraint that the transmitted signal is band-limited to the bandwidth allocated for the channel.Consequently,the transmitted signal pulses were designed to be band-limited and to have zero ISI.Return9-3-2 Probability of Error for Detection of Partial-Response Signals In this section,we determine the probability of error for detection of digital M-ary PAM signaling using duobinary and modified duobinary pulses.The channel is assumed to be an ideal bandlimited channel with additive white gaussian noise.The model for the communication system is shown in Fig.9-3-1.We consider two types of detectors.The first is the symbol-by-symbol detector and the second is the optimum ML sequence detector described in the previous section.ReturnFIGURE 9-3-1 Block diagram of modulator and demodulator for partial-response signals.Return1.Symbol-by-symbol DetectorAt the transmitter,the M-level data sequence is precoded as described previously.The precoder output is mapped into one of M possible amplitude levels.Then the transmitting filter with frequency response has an outputThe partial-response function X(f)is divided equally between the transmitting and receiving filters.Hence,the receiving filter is matched to the transmitted pulse,and the cascade of the two filters results in the frequency characteristic The matched filter output is sampled at t=nT=n/2W and the sampled are fed to the decoder.ReturnFor the duobinary signal,the output of the matched filter at the sampling instant m