Continued part 1, part 2 of ebook Fundamentals of digital communication provides readers with contents including: Chapter 6 Informationtheoretic limits and their computation; Chapter 7 Channel coding; Chapter 8 Wireless communication; Appendix A Probability, random variables, and random processes; Appendix B The Chernoff bound;... 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and getwhy That’s some I can’t sleep.’ really Hecall shook myself his head an Oxford ‘I want man.’ to wait Tomhere glanced till Daisy around goes to to see bed if we Good mirrored night,his oldunbelief sport.’ He But put wehis were hands all looking in his coat at Gatsby pockets‘Itand was turned an opportunity back eagerly theytogave his scrutiny to someofofthe thehouse, officersasafter though the Armistice,’ my presence he marred continued the ‘We sacredness could gooftothe any vigil of the So universities I walked away in England and left him or France.’ standing I wanted there intothe getmoonlight—watching up and slap him on the overback noth-I ing had Free one of eBooks those renewals at Planet of eBook.com complete 157 faithChapter in him that I couldn’t I’d experisleep enced all night; before.a Daisy fog-horn rose, was smiling groaning faintly, in- cessantly and went to onthe thetable Sound, ‘Open and the I tossed whiskey, half-sick Tom,’between she ordered grotesque ‘And reality I’ll make and you savage a mintfrightening julep Then dreams you won’t Toward seemdawn so stupid I heard to yourself a taxi go upLook Gatsby’s at thedrive mint!’and ‘Wait immediately a minute,’ Isnapped jumped out Tom, of ‘Ibed want andtobegan ask Mr to Gatsby dress—I one feltmore that Iquestion.’ had something ‘Go on,’ to Gatsby tell him,said something politely.to‘What warn kind him about of a row and are morning you trying would to cause be tooinlate my Crossing house anyhow?’ his lawnThey I sawwere that out his front in thedoor openwas at last still and openGatsby and hewas wasconleaning tent.against ‘He isn’ta causing table in the a row.’ hall,Daisy heavylooked with dejection desperately or sleep from one ‘Nothing to thehappened,’ other ‘You’re he said causing wanly a row ‘I waited, Please and have about a little fourself o’clock control.’ she ‘Self camecontrol!’ to the window repeated and Tom stood incredulously there for a minute ‘I suppose and the thenlatest turned thing outisthe to light.’ sit back Hisand house let Mr hadNobody never seemed from Nowhere so enormous make love to me to as your it did wife that Well, night if that’s when the we hunted idea youthrough can count the great me out rooms Nowadays for cig- arettes people We begin pushed by sneering aside curtains at familythat life were and family like pavilions institutions andand felt next over they’ll innumerable throw everyfeet ofthing dark overboard wall for electric and have light switches—once intermarriage between I tumbled black with and a sort of splash upon the keys of a ghostly piano The amount of dust everywhere and the rooms were musty as though they hadn’t been aired for many days I found the humidor on an unfamiliar table with two stale dry cigarettes inside Throwing open the French windows of the drawing-room we sat smoking out into the darkness August 13, 2007 5:55 p.m CUP/FOD Page-252 9780521874144c06 CHAPTER Information-theoretic limits and their computation Information theory (often termed Shannon theory in honor of its founder, Claude Shannon) provides fundamental benchmarks against which a communication system design can be compared Given a channel model and transmission constraints (e.g., on power), information theory enables us to compute, at least in principle, the highest rate at which reliable communication over the channel is possible This rate is called the channel capacity Once channel capacity is computed for a particular set of system parameters, it is the task of the communication link designer to devise coding and modulation strategies that approach this capacity After 50 years of effort since Shannon’s seminal work, it is now safe to say that this goal has been accomplished for some of the most common channel models The proofs of the fundamental theorems of information theory indicate that Shannon limits can be achieved by random code constructions using very large block lengths While this appeared to be computationally infeasible in terms of both encoding and decoding, the invention of turbo codes by Berrou et al in 1993 provide implementable mechanisms for achieving just this Turbo codes are random-looking codes obtained from easy-to-encode convolutional codes, which can be decoded efficiently using iterative decoding techniques instead of ML decoding (which is computationally infeasible for such constructions) Since then, a host of “turbo-like” coded modulation strategies have been proposed, including rediscovery of the low-density parity check (LDPC) codes invented by Gallager in the 1960s These developments encourage us to postulate that it should be possible (with the application of sufficient ingenuity) to devise a turbo-like coded modulation strategy that approaches the capacity of a very large class of channels Thus, it is more important than ever to characterize information-theoretic limits when setting out to design a communication system, both in terms of setting design goals and in terms of gaining intuition on design parameters (e.g., size of constellation to use) The goal of this chapter, therefore, is to provide enough exposure to Shannon theory to enable computation of capacity benchmarks, with the focus on the AWGN channel and some variants There is no attempt to give a complete, 252 and getwhy That’s some I can’t sleep.’ really Hecall shook myself his head an Oxford ‘I want man.’ to wait Tomhere glanced till Daisy around goes to to see bed if we Good mirrored night,his oldunbelief sport.’ He But put wehis were hands all looking in his coat at Gatsby pockets‘Itand was turned an opportunity back eagerly theytogave his scrutiny to someofofthe thehouse, officersasafter though the Armistice,’ my presence he marred continued the ‘We sacredness could gooftothe any vigil of the So universities I walked away in England and left him or France.’ standing I wanted there intothe getmoonlight—watching up and slap him on the overback noth-I ing had Free one of eBooks those renewals at Planet of eBook.com complete 157 faithChapter in him that I couldn’t I’d experisleep enced all night; before.a Daisy fog-horn rose, was smiling groaning faintly, in- cessantly and went to onthe thetable Sound, ‘Open and the I tossed whiskey, half-sick Tom,’between she ordered grotesque ‘And reality I’ll make and you savage a mintfrightening julep Then dreams you won’t Toward seemdawn so stupid I heard to yourself a taxi go upLook Gatsby’s at thedrive mint!’and ‘Wait immediately a minute,’ Isnapped jumped out Tom, of ‘Ibed want andtobegan ask Mr to Gatsby dress—I one feltmore that Iquestion.’ had something ‘Go on,’ to Gatsby tell him,said something politely.to‘What warn kind him about of a row and are morning you trying would to cause be tooinlate my Crossing house anyhow?’ his lawnThey I sawwere that out his front in thedoor openwas at last still and openGatsby and hewas wasconleaning tent.against ‘He isn’ta causing table in the a row.’ hall,Daisy heavylooked with dejection desperately or sleep from one ‘Nothing to thehappened,’ other ‘You’re he said causing wanly a row ‘I waited, Please and have about a little fourself o’clock control.’ she ‘Self camecontrol!’ to the window repeated and Tom stood incredulously there for a minute ‘I suppose and the thenlatest turned thing outisthe to light.’ sit back Hisand house let Mr hadNobody never seemed from Nowhere so enormous make love to me to as your it did wife that Well, night if that’s when the we hunted idea youthrough can count the great me out rooms Nowadays for cig- arettes people We begin pushed by sneering aside curtains at familythat life were and family like pavilions institutions andand felt next over they’ll innumerable throw everyfeet ofthing dark overboard wall for electric and have light switches—once intermarriage between I tumbled black with and a sort of splash upon the keys of a ghostly piano The amount of dust everywhere and the rooms were musty as though they hadn’t been aired for many days I found the humidor on an unfamiliar table with two stale dry cigarettes inside Throwing open the French windows of the drawing-room we sat smoking out into the darkness August 13, 2007 5:55 p.m 253 CUP/FOD Page-253 9780521874144c06 6.1 Capacity of AWGN: modeling and geometry or completely rigorous, exposition For this purpose, the reader is referred to information theory textbooks mentioned in Section 6.5 The techniques discussed in this chapter are employed in Chapter in order to obtain information-theoretic insights into wireless systems Constructive coding strategies, including turbo-like codes, are discussed in Chapter We note that the law of large numbers (LLN) is a key ingredient of information theory: if X1 Xn are i.i.d random variables, then their empirical average X1 + · · · + Xn /n tends to the statistical mean X1 (with probability one) as n → under rather general conditions Moreover, associated with the LLN are large deviations results that say that the probability of O1 deviation of the empirical average from the mean decays exponentially with n These can be proved using the Chernoff bound (see Appendix B) In this chapter, when I invoke the LLN to replace an empirical average or sum by its statistical counterpart, I implicitly rely on such large deviations results as an underlying mathematical justification, although I not provide the technical details behind such justification Map of this chapter In Section 6.1, I compute the capacity of the continuous and discrete-time AWGN channels using geometric arguments, and discuss the associated power-bandwidth tradeoffs In Section 6.2, I take a more systematic view, discussing some basic quantities and results of Shannon theory, including the discrete memoryless channel model and the channel coding theorem This provides a framework for capacity computations that I use in Section 6.3, where I discuss how to compute capacity under input constraints (specifically focusing on computing AWGN capacity with standard constellations such as PAM, QAM, and PSK) I also characterize the capacity for parallel Gaussian channels, and apply it for modeling dispersive channels Finally, Section 6.4 provides a glimpse of optimization techniques for computing capacity in more general settings 6.1 Capacity of AWGN channel: modeling and geometry In this section, I discuss fundamental benchmarks for communication over a bandlimited AWGN channel Theorem 6.1.1 For an AWGN channel of bandwidth W and received power P, the channel capacity is given by the formula P C = W log2 + bit/s N0 W (6.1) Let me first discuss some implications of this formula, and then provide some insight into why the formula holds, and how one would go about achieving the rate promised by (6.1) and getwhy That’s some I can’t sleep.’ really Hecall shook myself his head an Oxford ‘I want man.’ to wait Tomhere glanced till Daisy around goes to to see bed if we Good mirrored night,his oldunbelief sport.’ He But put wehis were hands all looking in his coat at Gatsby pockets‘Itand was turned an opportunity back eagerly theytogave his scrutiny to someofofthe thehouse, officersasafter though the Armistice,’ my presence he marred continued the ‘We sacredness could gooftothe any vigil of the So universities I walked away in England and left him or France.’ standing I wanted there intothe getmoonlight—watching up and slap him on the overback noth-I ing had Free one of eBooks those renewals at Planet of eBook.com complete 157 faithChapter in him that I couldn’t I’d experisleep enced all night; before.a Daisy fog-horn rose, was smiling groaning faintly, in- cessantly and went to onthe thetable Sound, ‘Open and the I tossed whiskey, half-sick Tom,’between she ordered grotesque ‘And reality I’ll make and you savage a mintfrightening julep Then dreams you won’t Toward seemdawn so stupid I heard to yourself a taxi go upLook Gatsby’s at thedrive mint!’and ‘Wait immediately a minute,’ Isnapped jumped out Tom, of ‘Ibed want andtobegan ask Mr to Gatsby dress—I one feltmore that Iquestion.’ had something ‘Go on,’ to Gatsby tell him,said something politely.to‘What warn kind him about of a row and are morning you trying would to cause be tooinlate my Crossing house anyhow?’ his lawnThey I sawwere that out his front in thedoor openwas at last still and openGatsby and hewas wasconleaning tent.against ‘He isn’ta causing table in the a row.’ hall,Daisy heavylooked with dejection desperately or sleep from one ‘Nothing to thehappened,’ other ‘You’re he said causing wanly a row ‘I waited, Please and have about a little fourself o’clock control.’ she ‘Self camecontrol!’ to the window repeated and Tom stood incredulously there for a minute ‘I suppose and the thenlatest turned thing outisthe to light.’ sit back Hisand house let Mr hadNobody never seemed from Nowhere so enormous make love to me to as your it did wife that Well, night if that’s when the we hunted idea youthrough can count the great me out rooms Nowadays for cig- arettes people We begin pushed by sneering aside curtains at familythat life were and family like pavilions institutions andand felt next over they’ll innumerable throw everyfeet ofthing dark overboard wall for electric and have light switches—once intermarriage between I tumbled black with and a sort of splash upon the keys of a ghostly piano The amount of dust everywhere and the rooms were musty as though they hadn’t been aired for many days I found the humidor on an unfamiliar table with two stale dry cigarettes inside Throwing open the French windows of the drawing-room we sat smoking out into the darkness August 13, 2007 5:55 p.m 254 CUP/FOD Page-254 9780521874144c06 Information-theoretic limits and their computation Consider a communication system that provides an information rate of R bit/s Denoting by Eb the energy per information bit, the transmitted power is P = Eb R For reliable transmission, we must have R < C, so that we have from (6.1): ER R < W log2 + b N0 W Defining r = R/W as the spectral efficiency, or information rate per unit of bandwidth, of the system, we obtain the condition Eb r r < log2 + N0 This implies that, for reliable communication, the signal-to-noise ratio must exceed a threshold that depends on the operating spectral efficiency: Eb 2r − > (6.2) N0 r “Reliable communication” in an information-theoretic context means that the error probability tends to zero as codeword lengths get large, while a practical system is deemed reliable if it operates at some desired, nonzero but small, error probability level Thus, we might say that a communication system is operating dB away from Shannon capacity at a bit error probability of 10−6 , meaning that the operating Eb /N0 for a BER of 10−6 is dB higher than the minimum required based on (6.2) Equation (6.2) brings out a fundamental tradeoff between power and bandwidth The required Eb /N0 , and hence the required power (assuming that the information rate R and noise PSD N0 are fixed) increases as we increase the spectral efficiency r, while the bandwidth required to support a given information rate decreases if we increase r Taking the log of both sides of (6.2), we see that the spectral efficiency and the required Eb /N0 in dB have an approximately linear relationship This can be seen from Figure 6.1, which plots achievable spectral efficiency versus Eb /N0 (dB) Reliable communication is not possible above the curve In comparing a specific coded modulation scheme with the Shannon limit, we compare the Eb /N0 required to attain a certain reference BER (e.g., 10−5 ) with the minimum possible Eb /N0 , given by (6.2) at that spectral efficiency (excess bandwidth used in the modulating pulse is not considered, since that is a heavily implementationdependent parameter) With this terminology, uncoded QPSK achieves a BER of 10−5 at an Eb /N0 of about 9.5 dB For the corresponding spectral efficiency r = 2, the Shannon limit given by (6.2) is 1.76 dB, so that uncoded QPSK is about 7.8 dB away from the Shannon limit at a BER of 10−5 A similar gap also exists for uncoded 16QAM As we shall see in the next chapter, the gap to Shannon capacity can be narrowed considerably by the use of channel coding For example, suppose that we use a rate 1/2 binary code (1 information bit/2 coded bits), with the coded bits mapped to a QPSK constellation (2 coded bits/channel use) Then the spectral efficiency and getwhy That’s some I can’t sleep.’ really Hecall shook myself his head an Oxford ‘I want man.’ to wait Tomhere glanced till Daisy around goes to to see bed if we Good mirrored night,his oldunbelief sport.’ He But put wehis were hands all looking in his coat at Gatsby pockets‘Itand was turned an opportunity back eagerly theytogave his scrutiny to someofofthe thehouse, officersasafter though the Armistice,’ my presence he marred continued the ‘We sacredness could gooftothe any vigil of the So universities I walked away in England and left him or France.’ standing I wanted there intothe getmoonlight—watching up and slap him on the overback noth-I ing had Free one of eBooks those renewals at Planet of eBook.com complete 157 faithChapter in him that I couldn’t I’d experisleep enced all night; before.a Daisy fog-horn rose, was smiling groaning faintly, in- cessantly and went to onthe thetable Sound, ‘Open and the I tossed whiskey, half-sick Tom,’between she ordered grotesque ‘And reality I’ll make and you savage a mintfrightening julep Then dreams you won’t Toward seemdawn so stupid I heard to yourself a taxi go upLook Gatsby’s at thedrive mint!’and ‘Wait immediately a minute,’ Isnapped jumped out Tom, of ‘Ibed want andtobegan ask Mr to Gatsby dress—I one feltmore that Iquestion.’ had something ‘Go on,’ to Gatsby tell him,said something politely.to‘What warn kind him about of a row and are morning you trying would to cause be tooinlate my Crossing house anyhow?’ his lawnThey I sawwere that out his front in thedoor openwas at last still and openGatsby and hewas wasconleaning tent.against ‘He isn’ta causing table in the a row.’ hall,Daisy heavylooked with dejection desperately or sleep from one ‘Nothing to thehappened,’ other ‘You’re he said causing wanly a row ‘I waited, Please and have about a little fourself o’clock control.’ she ‘Self camecontrol!’ to the window repeated and Tom stood incredulously there for a minute ‘I suppose and the thenlatest turned thing outisthe to light.’ sit back Hisand house let Mr hadNobody never seemed from Nowhere so enormous make love to me to as your it did wife that Well, night if that’s when the we hunted idea youthrough can count the great me out rooms Nowadays for cig- arettes people We begin pushed by sneering aside curtains at familythat life were and family like pavilions institutions andand felt next over they’ll innumerable throw everyfeet ofthing dark overboard wall for electric and have light switches—once intermarriage between I tumbled black with and a sort of splash upon the keys of a ghostly piano The amount of dust everywhere and the rooms were musty as though they hadn’t been aired for many days I found the humidor on an unfamiliar table with two stale dry cigarettes inside Throwing open the French windows of the drawing-room we sat smoking out into the darkness August 13, 2007 5:55 p.m Figure 6.1 Spectral efficiency as a function of Eb /N0 (dB) The large gap to capacity for uncoded constellations (at a reference BER of 10−5 ) shows the significant potential benefits of channel coding, which I discuss in Chapter Page-255 9780521874144c06 6.1 Capacity of AWGN: modeling and geometry Spectral efficiency r (in bit /channel use) 255 CUP/FOD 7.8 dB gap 16 QAM at BER = 10−5 7.8 dB gap QPSK at BER = 10−5 −2 10 12 14 16 Eb/N0 (in dB) is r = 1/2 × = 1, and the corresponding Shannon limit is dB We now know how to design turbo-like codes that get within a fraction of a dB of this limit The preceding discussion focuses on spectral efficiency, which is important when there are bandwidth constraints What if we have access to unlimited bandwidth (for a fixed information rate)? As discussed below, even in this scenario, we cannot transmit at arbitrarily low powers: there is a fundamental limit on the smallest possible value of Eb /N0 required for reliable communication Power-limited communication As we let the spectral efficiency r → 0, we enter a power-limited regime Evaluating the limit (6.2) tells us that, for reliable communication, we must have Eb > ln −16 dB minimum required for reliable communication N0 (6.3) That is, even if we let bandwidth tend to infinity for a fixed information rate, we cannot reduce Eb /N0 below its minimum value of −16 dB As we have seen in Chapters and 4, M-ary orthogonal signaling is asympototically optimum in this power-limited regime, both for coherent and noncoherent communication Let me now sketch an intuitive proof of the capacity formula (6.1) While the formula refers to a continuous-time channel, both the proof of the capacity formula, and the kinds of constructions we typically employ to try to achieve capacity, are based on discrete-time constructions and getwhy That’s some I can’t sleep.’ really Hecall shook myself his head an Oxford ‘I want man.’ to wait Tomhere glanced till Daisy around goes to to see bed if we Good mirrored night,his oldunbelief sport.’ He But put wehis were hands all looking in his coat at Gatsby pockets‘Itand was turned an opportunity back eagerly theytogave his scrutiny to someofofthe thehouse, officersasafter though the Armistice,’ my presence he marred continued the ‘We sacredness could gooftothe any vigil of the So universities I walked away in England and left him or France.’ standing I wanted there intothe getmoonlight—watching up and slap him on the overback noth-I ing had Free one of eBooks those renewals at Planet of eBook.com complete 157 faithChapter in him that I couldn’t I’d experisleep enced all night; before.a Daisy fog-horn rose, was smiling groaning faintly, in- cessantly and went to onthe thetable Sound, ‘Open and the I tossed whiskey, half-sick Tom,’between she ordered grotesque ‘And reality I’ll make and you savage a mintfrightening julep Then dreams you won’t Toward seemdawn so stupid I heard to yourself a taxi go upLook Gatsby’s at thedrive mint!’and ‘Wait immediately a minute,’ Isnapped jumped out Tom, of ‘Ibed want andtobegan ask Mr to Gatsby dress—I one feltmore that Iquestion.’ had something ‘Go on,’ to Gatsby tell him,said something politely.to‘What warn kind him about of a row and are morning you trying would to cause be tooinlate my Crossing house anyhow?’ his lawnThey I sawwere that out his front in thedoor openwas at last still and openGatsby and hewas wasconleaning tent.against ‘He isn’ta causing table in the a row.’ hall,Daisy heavylooked with dejection desperately or sleep from one ‘Nothing to thehappened,’ other ‘You’re he said causing wanly a row ‘I waited, Please and have about a little fourself o’clock control.’ she ‘Self camecontrol!’ to the window repeated and Tom stood incredulously there for a minute ‘I suppose and the thenlatest turned thing outisthe to light.’ sit back Hisand house let Mr hadNobody never seemed from Nowhere so enormous make love to me to as your it did wife that Well, night if that’s when the we hunted idea youthrough can count the great me out rooms Nowadays for cig- arettes people We begin pushed by sneering aside curtains at familythat life were and family like pavilions institutions andand felt next over they’ll innumerable throw everyfeet ofthing dark overboard wall for electric and have light switches—once intermarriage between I tumbled black with and a sort of splash upon the keys of a ghostly piano The amount of dust everywhere and the rooms were musty as though they hadn’t been aired for many days I found the humidor on an unfamiliar table with two stale dry cigarettes inside Throwing open the French windows of the drawing-room we sat smoking out into the darkness August 13, 2007 5:55 p.m 256 CUP/FOD Page-256 9780521874144c06 Information-theoretic limits and their computation 6.1.1 From continuous to discrete time I now consider an ideal complex WGN channel bandlimited to −W/2 W/2 If the transmitted signal is st, then the received signal yt = s ∗ ht + nt where h is the impulse response of an ideal bandlimited channel, and nt is complex WGN We wish to design the set of possible signals that we would send over the channel so as to maximize the rate of reliable communication, subject to a constraint that the signal st has average power at most P To start with, note that it does not make sense for st to have any component outside of the band −W/2 W/2, since any such component would be annihilated once we pass it through the ideal bandlimited filter h Hence, without loss of generality, st must be bandlimited to −W/2 W/2 for an optimal signal set design We now recall the discussion on modulation degrees of freedom from Chapter in order to obtain a discrete-time model By the sampling theorem, a signal bandlimited to −W/2 W/2 is completely specified by its samples at rate W , si/W Thus, signal design consists of specifying these samples, and modulation for transmission over the ideal bandlimited channel consists of invoking the interpolation formula Thus, once we have designed the samples, the complex baseband waveform that we send is given by i si/Wp t − st = (6.4) W i= where pt = sincWt is the impulse response of an ideal bandlimited pulse with transfer function Pf = W1 I− W2 W2 As noted in Chapter 2, this is linear modulation at symbol rate W with symbol sequence si/W and transmit pulse pt = sincWt, which is the minimum bandwidth Nyquist pulse at rate W The translates pt − i/W form an orthogonal basis for the space of ideally bandlimited functions, so that (6.4) specifies a basis expansion fo st For signaling under a power constraint P over a (large) interval To , the transmitted signal energy should satisfy To st2 dt ≈ PTo Let Ps = s1/W2 denote the average power per sample Since energy is preserved under the basis expansion (6.4), and we have about To W samples in this interval, we also have To WPs p2 ≈ PTo For pt = sincWt, we have p2 = 1/W , so that Ps = P That is, for the scaling adopted in (6.4), the samples obey the same power constraint as the continuous-time signal and getwhy That’s some I can’t sleep.’ really Hecall shook myself his head an Oxford ‘I want man.’ to wait Tomhere glanced till Daisy around goes to to see bed if we Good mirrored night,his oldunbelief sport.’ He But put wehis were hands all looking in his coat at Gatsby pockets‘Itand was turned an opportunity back eagerly theytogave his scrutiny to someofofthe thehouse, officersasafter though the Armistice,’ my presence he marred continued the ‘We sacredness could gooftothe any vigil of the So universities I walked away in England and left him or France.’ standing I wanted there intothe getmoonlight—watching up and slap him on the overback noth-I ing had Free one of eBooks those renewals at Planet of eBook.com complete 157 faithChapter in him that I couldn’t I’d experisleep enced all night; before.a Daisy fog-horn rose, was smiling groaning faintly, in- cessantly and went to onthe thetable Sound, ‘Open and the I tossed whiskey, half-sick Tom,’between she ordered grotesque ‘And reality I’ll make and you savage a mintfrightening julep Then dreams you won’t Toward seemdawn so stupid I heard to yourself a taxi go upLook Gatsby’s at thedrive mint!’and ‘Wait immediately a minute,’ Isnapped jumped out Tom, of ‘Ibed want andtobegan ask Mr to Gatsby dress—I one feltmore that Iquestion.’ had something ‘Go on,’ to Gatsby tell him,said something politely.to‘What warn kind him about of a row and are morning you trying would to cause be tooinlate my Crossing house anyhow?’ his lawnThey I sawwere that out his front in thedoor openwas at last still and openGatsby and hewas wasconleaning tent.against ‘He isn’ta causing table in the a row.’ hall,Daisy heavylooked with dejection desperately or sleep from one ‘Nothing to thehappened,’ other ‘You’re he said causing wanly a row ‘I waited, Please and have about a little fourself o’clock control.’ she ‘Self camecontrol!’ to the window repeated and Tom stood incredulously there for a minute ‘I suppose and the thenlatest turned thing outisthe to light.’ sit back Hisand house let Mr hadNobody never seemed from Nowhere so enormous make love to me to as your it did wife that Well, night if that’s when the we hunted idea youthrough can count the great me out rooms Nowadays for cig- arettes people We begin pushed by sneering aside curtains at familythat life were and family like pavilions institutions andand felt next over they’ll innumerable throw everyfeet ofthing dark overboard wall for electric and have light switches—once intermarriage between I tumbled black with and a sort of splash upon the keys of a ghostly piano The amount of dust everywhere and the rooms were musty as though they hadn’t been aired for many days I found the humidor on an unfamiliar table with two stale dry cigarettes inside Throwing open the French windows of the drawing-room we sat smoking out into the darkness August 13, 2007 5:55 p.m 257 CUP/FOD Page-257 9780521874144c06 6.1 Capacity of AWGN: modeling and geometry When the bandlimited signal s passes through the ideally bandlimited complex AWGN channel, we get yt = st + nt (6.5) where n is complex WGN Since s is linearly modulated at symbol rate W using modulating pulse p, we know that the optimal receiver front end is to pass the received signal through a filter matched to pt, and to sample at the symbol rate W For notational convenience, we use a receive filter transfer function GR f = I− W2 W2 which is a scalar multiple of the matched filter P ∗ f = Pf = W1 I− W2 W2 This ideal bandlimited filter lets the signal st through unchanged, so that the signal contributions to the output of the receive filter, sampled at rate W , are si/W The noise at the output of the receive filter is bandlimited complex WGN with PSD N0 I− W2 W2 , from which it follows that the noise samples at rate W are independent complex Gaussian random variables with covariance N0 W To summarize, the noisy samples at the receive filter output can be written as yi = si/W + Ni (6.6) where the signal samples are subject to an average power constraint si/W2 ≤ P, and Ni are i.i.d., zero mean, proper complex Gaussian noise samples with Ni2 = N0 W Thus, we have reduced the continuous-time bandlimited passband AWGN channel model to the discrete-time complex WGN channel model (6.6) that we get to use W times per second if we employ bandwidth W We can now characterize the capacity of the discrete-time channel, and then infer that of the continuous-time bandlimited channel 6.1.2 Capacity of the discrete time AWGN channel Since the real and imaginary part of the discrete-time complex AWGN model (6.6) can be interpreted as two uses of a real-valued AWGN channel, we consider the latter first Consider a discrete-time real AWGN channel in which the output at any given time Y = X + Z (6.7) where X is a real-valued input satisfying X ≤ S, and Z ∼ N0 N is realvalued AWGN The noise samples over different channel uses are i.i.d This is an example of a discrete memoryless channel, where pY X is specified for a single channel use, and the channel outputs for multiple channel uses are conditionally independent given the inputs A signal, or codeword, over such a channel is a vector X = X1 Xn T , where Xi is the input for the ith channel use A code of rate R bits per channel use can be constructed by designing a set of 2nR such signals Xk k = 1 2nR , with each signal and getwhy That’s some I can’t sleep.’ really Hecall shook myself his head an Oxford ‘I want man.’ to wait Tomhere glanced till Daisy around goes to to see bed if we Good mirrored night,his oldunbelief sport.’ He But put wehis were hands all looking in his coat at Gatsby pockets‘Itand was turned an opportunity back eagerly theytogave his scrutiny to someofofthe thehouse, officersasafter though the Armistice,’ my presence he marred continued the ‘We sacredness could gooftothe any vigil of the So universities I walked away in England and left him or France.’ standing I wanted there intothe getmoonlight—watching up and slap him on the overback noth-I ing had Free one of eBooks those renewals at Planet of eBook.com complete 157 faithChapter in him that I couldn’t I’d experisleep enced all night; before.a Daisy fog-horn rose, was smiling groaning faintly, in- cessantly and went to onthe thetable Sound, ‘Open and the I tossed whiskey, half-sick Tom,’between she ordered grotesque ‘And reality I’ll make and you savage a mintfrightening julep Then dreams you won’t Toward seemdawn so stupid I heard to yourself a taxi go upLook Gatsby’s at thedrive mint!’and ‘Wait immediately a minute,’ Isnapped jumped out Tom, of ‘Ibed want andtobegan ask Mr to Gatsby dress—I one feltmore that Iquestion.’ had something ‘Go on,’ to Gatsby tell him,said something politely.to‘What warn kind him about of a row and are morning you trying would to cause be tooinlate my Crossing house anyhow?’ his lawnThey I sawwere that out his front in thedoor openwas at last still and openGatsby and hewas wasconleaning tent.against ‘He isn’ta causing table in the a row.’ hall,Daisy heavylooked with dejection desperately or sleep from one ‘Nothing to thehappened,’ other ‘You’re he said causing wanly a row ‘I waited, Please and have about a little fourself o’clock control.’ she ‘Self camecontrol!’ to the window repeated and Tom stood incredulously there for a minute ‘I suppose and the thenlatest turned thing outisthe to light.’ sit back Hisand house let Mr hadNobody never seemed from Nowhere so enormous make love to me to as your it did wife that Well, night if that’s when the we hunted idea youthrough can count the great me out rooms Nowadays for cig- arettes people We begin pushed by sneering aside curtains at familythat life were and family like pavilions institutions andand felt next over they’ll innumerable throw everyfeet ofthing dark overboard wall for electric and have light switches—once intermarriage between I tumbled black with and a sort of splash upon the keys of a ghostly piano The amount of dust everywhere and the rooms were musty as though they hadn’t been aired for many days I found the humidor on an unfamiliar table with two stale dry cigarettes inside Throwing open the French windows of the drawing-room we sat smoking out into the darkness August 13, 2007 5:55 p.m 258 CUP/FOD Page-258 9780521874144c06 Information-theoretic limits and their computation having an equal probability of being chosen for transmission over the channel Thus, nR bits are conveyed over n channel uses Capacity is defined as the largest rate R for which the error probability tends to zero as n → Shannon has provided a general framework for computing capacity for a discrete memoryless channel, which I discuss in Section 6.3 However, I provide here a heuristic derivation of capacity for the AWGN channel (6.7), that specifically utilizes the geometry induced by AWGN Sphere packing based derivation of capacity formula For a transmitted signal Xj , the n-dimensional output vector Y = Y1 Yn T is given by Y = Xj + Z Xj sent where Z is a vector of i.i.d N0 N noise samples For equal priors, the MPE and ML rules are equivalent The ML rule for the AWGN channel is the minimum distance rule ML Y = arg Y − Xk 2 1≤k≤2nR Now, the noise vector Z that perturbs the transmitted signal has energy Z2 = N i=1 Zi2 ≈ nZ12 = nN where we This implies that, if we draw a sphere of √ have invoked the LLN j radius nN around a signal X , then, with high probability, the received vector Y lies within the sphere when Xj is sent Calling such a sphere the “decoding sphere” for Xj , the minimum distance rule would lead to very small error probability if the decoding spheres for different signals were disjoint We now wish to estimate how many such decoding spheres we can come up with; this gives the value of 2nR for which reliable communication is possible Since X is independent of Z (the transmitter does not know the noise realization) in the model (6.7), the input power constraint implies an output power constraint Y = X +Z2 = X +Z2 +2XZ = X +Z2 ≤ S +N (6.8) Invoking the law of large numbers again, the received signal energy satisfies Y2 ≈ nS + N so that, with high probability, the received signal vector lies within an n-dimensional sphere with radius Rn = nS + N The problem of signal design for reliable communication now boils down to packing disjoint decod√ ing spheres of radius rn = nN within a sphere of radius Rn , as shown in Figure 6.2 The volume of an n-dimensional sphere of radius r equals Kn r n , and getwhy That’s some I can’t sleep.’ really Hecall shook myself his head an Oxford ‘I want man.’ to wait Tomhere glanced till Daisy around goes to to see bed if we Good mirrored night,his oldunbelief sport.’ He But put wehis were hands all looking in his coat at Gatsby pockets‘Itand was turned an opportunity back eagerly theytogave his scrutiny to someofofthe thehouse, officersasafter though the Armistice,’ my presence he marred continued the ‘We sacredness could gooftothe any vigil of the So universities I walked away in England and left him or France.’ standing I wanted there intothe getmoonlight—watching up and slap him on the overback noth-I ing had Free one of eBooks those renewals at Planet of eBook.com complete 157 faithChapter in him that I couldn’t I’d experisleep enced all night; before.a Daisy fog-horn rose, was smiling groaning faintly, in- cessantly and went to onthe thetable Sound, ‘Open and the I tossed whiskey, half-sick Tom,’between she ordered grotesque ‘And reality I’ll make and you savage a mintfrightening julep Then dreams you won’t Toward seemdawn so stupid I heard to yourself a taxi go upLook Gatsby’s at thedrive mint!’and ‘Wait immediately a minute,’ Isnapped jumped out Tom, of ‘Ibed want andtobegan ask Mr to Gatsby dress—I one feltmore that Iquestion.’ had something ‘Go on,’ to Gatsby tell him,said something politely.to‘What warn kind him about of a row and are morning you trying would to cause be tooinlate my Crossing house anyhow?’ his lawnThey I sawwere that out his front in thedoor openwas at last still and openGatsby and hewas wasconleaning tent.against ‘He isn’ta causing table in the a row.’ hall,Daisy heavylooked with dejection desperately or sleep from one ‘Nothing to thehappened,’ other ‘You’re he said causing wanly a row ‘I waited, Please and have about a little fourself o’clock control.’ she ‘Self camecontrol!’ to the window repeated and Tom stood incredulously there for a minute ‘I suppose and the thenlatest turned thing outisthe to light.’ sit back Hisand house let Mr hadNobody never seemed from Nowhere so enormous make love to me to as your it did wife that Well, night if that’s when the we hunted idea youthrough can count the great me out rooms Nowadays for cig- arettes people We begin pushed by sneering aside curtains at familythat life were and family like pavilions institutions andand felt next over they’ll innumerable throw everyfeet ofthing dark overboard wall for electric and have light switches—once intermarriage between I tumbled black with and a sort of splash upon the keys of a ghostly piano The amount of dust everywhere and the rooms were musty as though they hadn’t been aired for many days I found the humidor on an unfamiliar table with two stale dry cigarettes inside Throwing open the French windows of the drawing-room we sat smoking out into the darkness August 13, 2007 5:55 p.m 259 CUP/FOD Page-259 9780521874144c06 6.1 Capacity of AWGN: modeling and geometry Figure 6.2 Decoding spheres √ of radius rn = nN are packed insidea sphere of radius Rn = nS + N rn Rn and the number of decoding spheres we can pack is roughly the following ratio of volumes: Kn nS + Nn Kn Rnn = ≈ 2nR √ Kn rnn Kn nN n Solving, we obtain that the rate R = 1/2 log2 1 + S/N I shall show in Section 6.3 that this rate exactly equals the capacity of the discrete-time real AWGN channel (It is also possible to make the sphere packing argument rigorous, but we not attempt that here.) I now state the capacity formula formally Theorem 6.1.2 Capacity of discrete-time real AWGN channel The capacity of the discrete-time, real AWGN channel (6.7) is CAWGN = log2 1 + SNR bit/channel use (6.9) where SNR = S/N is the signal-to-noise ratio Thus, capacity grows approximately logarithmically with SNR, or approximately linearly with SNR in dB 6.1.3 From discrete to continuous time For the continuous-time bandlimited complex baseband channel that we considered earlier, we have 2W uses per second of the discrete-time real AWGN channel (6.7) With the normalization we employed in (6.4), we have that, per real-valued sample, the average signal energy S = P/2 and the noise energy and getwhy That’s some I can’t sleep.’ really Hecall shook myself his head an Oxford ‘I want man.’ to wait Tomhere glanced till Daisy around goes to to see bed if we Good mirrored night,his oldunbelief sport.’ He But put wehis were hands all looking in his coat at Gatsby pockets‘Itand was turned an opportunity back eagerly theytogave his scrutiny to someofofthe thehouse, officersasafter though the Armistice,’ my presence he marred continued the ‘We sacredness could gooftothe any vigil of the So universities I walked away in England and left him or France.’ standing I wanted there intothe getmoonlight—watching up and slap him on the overback noth-I ing had Free one of eBooks those renewals at Planet of eBook.com complete 157 faithChapter in him that I couldn’t I’d experisleep enced all night; before.a Daisy fog-horn rose, was smiling groaning faintly, in- cessantly and went to onthe thetable Sound, ‘Open and the I tossed whiskey, half-sick Tom,’between she ordered grotesque ‘And reality I’ll make and you savage a mintfrightening julep Then dreams you won’t Toward seemdawn so stupid I heard to yourself a taxi go upLook Gatsby’s at thedrive mint!’and ‘Wait immediately a minute,’ Isnapped jumped out Tom, of ‘Ibed want andtobegan ask Mr to Gatsby dress—I one feltmore that Iquestion.’ had something ‘Go on,’ to Gatsby tell him,said something politely.to‘What warn kind him about of a row and are morning you trying would to cause be tooinlate my Crossing house anyhow?’ his lawnThey I sawwere that out his front in thedoor openwas at last still and openGatsby and hewas wasconleaning tent.against ‘He isn’ta causing table in the a row.’ hall,Daisy heavylooked with dejection desperately or sleep from one ‘Nothing to thehappened,’ other ‘You’re he said causing wanly a row ‘I waited, Please and have about a little fourself o’clock control.’ she ‘Self camecontrol!’ to the window repeated and Tom stood incredulously there for a minute ‘I suppose and the thenlatest turned thing outisthe to light.’ sit back Hisand house let Mr hadNobody never seemed from Nowhere so enormous make love to me to as your it did wife that Well, night if that’s when the we hunted idea youthrough can count the great me out rooms Nowadays for cig- arettes people We begin pushed by sneering aside curtains at familythat life were and family like pavilions institutions andand felt next over they’ll innumerable throw everyfeet ofthing dark overboard wall for electric and have light switches—once intermarriage between I tumbled black with and a sort of splash upon the keys of a ghostly piano The amount of dust everywhere and the rooms were musty as though they hadn’t been aired for many days I found the humidor on an unfamiliar table with two stale dry cigarettes inside Throwing open the French windows of the drawing-room we sat smoking out into the darkness August 13, 2007 5:55 p.m 260 CUP/FOD Page-260 9780521874144c06 Information-theoretic limits and their computation N = N0 W/2, where P is the power constraint on the continuous-time signal Plugging in, we get Ccont−time = 2W P log2 1 + bit/s N0 W which gives (6.1) As the invocation of the LLN in the sphere packing based derivation shows, capacity for the discrete-time channel is achieved by using codewords that span a large number of symbols Suppose, now, that we have designed a capacity achieving strategy for the discrete-time channel; that is, we have specified a good code, or signal set A codeword from this set is a discretetime sequence si We can now translate this design to continuous time by using the modulation formula (6.4) to send the symbols si = si/W Of course, as we discussed in Section 2, the sinc pulse used in this formula cannot be used in practice, and should be replaced by a modulating pulse whose bandwidth is larger than the symbol rate employed A good choice would be a square root Nyquist modulating pulse at the transmitter, and its matched filter at the receiver, which again yields the ISI-free discrete-time model (6.6) with uncorrelated noise samples In summary, good codes for the discrete-time AWGN channel (6.6) can be translated into good signal designs for the continuous-time bandlimited AWGN channel using practical linear modulation techniques; this corresponds to using translates of a square root Nyquist pulse as an orthonormal basis for the signal space It is also possible to use an entirely different basis: for example, orthogonal frequency division multiplexing, which I discuss in Chapter 8, employs complex sinusoids as basis functions In general, the use of appropriate signal space arguments allows us to restrict attention to discrete-time models, both for code design and for deriving information-theoretic benchmarks Real baseband channel The preceding observations also hold for a physical (i.e., real-valued) baseband channel That is, both the AWGN capacity formula (6.1) and its corollary (6.2) hold, where W for a physical baseband channel refers to the bandwidth occupancy for positive frequencies Thus, a real baseband signal st occupying a bandwidth W actually spans the interval −W W, with the constraint that Sf = S ∗ −f Using the sampling theorem, such a signal can be represented by 2W real-valued samples per second This is the same result as for a passband signal of bandwidth W , so that the arguments I have made so far, relating the continuous-time model to the discrete-time real AWGN channel, apply as before For example, suppose that we wish to find out how far uncoded binary antipodal signaling at BER of 10−5 is from Shannon capacity Since we transmit at bit per sample, the information rate is 2W bits per second, corresponding to a spectral efficiency of r = R/W = This corresponds limit of 1.8 dB Eb /N0 , using to a Shannon (6.2) Setting the BER of Q 2Eb /N0 for binary antipodal signaling to and getwhy That’s some I can’t sleep.’ really Hecall shook myself his head an Oxford ‘I want man.’ to wait Tomhere glanced till Daisy around goes to to see bed if we Good mirrored night,his oldunbelief sport.’ He But put wehis were hands all looking in his coat at Gatsby pockets‘Itand was turned an opportunity back eagerly theytogave his scrutiny to someofofthe thehouse, officersasafter though the Armistice,’ my presence he marred continued the ‘We sacredness could gooftothe any vigil of the So universities I walked away in England and left him or France.’ standing I wanted there intothe getmoonlight—watching up and slap him on the overback noth-I ing had Free one of eBooks those renewals at Planet of eBook.com complete 157 faithChapter in him that I couldn’t I’d experisleep enced all night; before.a Daisy fog-horn rose, was smiling groaning faintly, in- cessantly and went to onthe thetable Sound, ‘Open and the I tossed whiskey, half-sick Tom,’between she ordered grotesque ‘And reality I’ll make and you savage a mintfrightening julep Then dreams you won’t Toward seemdawn so stupid I heard to yourself a taxi go upLook Gatsby’s at thedrive mint!’and ‘Wait immediately a minute,’ Isnapped jumped out Tom, of ‘Ibed want andtobegan ask Mr to Gatsby dress—I one feltmore that Iquestion.’ had something ‘Go on,’ to Gatsby tell him,said something politely.to‘What warn kind him about of a row and are morning you trying would to cause be tooinlate my Crossing house anyhow?’ his lawnThey I sawwere that out his front in thedoor openwas at last still and openGatsby and hewas wasconleaning tent.against ‘He isn’ta causing table in the a row.’ hall,Daisy heavylooked with dejection desperately or sleep from one ‘Nothing to thehappened,’ other ‘You’re he said causing wanly a row ‘I waited, Please and have about a little fourself o’clock control.’ she ‘Self camecontrol!’ to the window repeated and Tom stood incredulously there for a minute ‘I suppose and the thenlatest turned thing outisthe to light.’ sit back Hisand house let Mr hadNobody never seemed from Nowhere so enormous make love to me to as your it did wife that Well, night if that’s when the we hunted idea youthrough can count the great me out rooms Nowadays for cig- arettes people We begin pushed by sneering aside curtains at familythat life were and family like pavilions institutions andand felt next over they’ll innumerable throw everyfeet ofthing dark overboard wall for electric and have light switches—once intermarriage between I tumbled black with and a sort of splash upon the keys of a ghostly piano The amount of dust everywhere and the rooms were musty as though they hadn’t been aired for many days I found the humidor on an unfamiliar table with two stale dry cigarettes inside Throwing open the French windows of the drawing-room we sat smoking out into the darkness August 13, 2007 5:55 p.m 261 CUP/FOD Page-261 9780521874144c06 6.1 Capacity of AWGN: modeling and geometry 10−5 , we find that the required Eb /N0 is 9.5 dB, which is 7.7 dB away from the Shannon limit There is good reason for this computation looking familiar: we obtained exactly the same result earlier for uncoded QPSK on a passband channel This is because QPSK can be interpreted as binary antipodal modulation along the I and Q channels, and is therefore exactly equivalent to binary antipodal modulation for a real baseband channel At this point, it is worth mentioning the potential for confusion when dealing with Shannon limits in the literature Even though PSK is a passband technique, the term BPSK is often used when referring to binary antipodal signaling on a real baseband channel Thus, when we compare the performance of BPSK with rate 1/2 coding to the Shannon limit, we should actually be keeping in mind a real baseband channel, so that r = 1, corresponding to a Shannon limit of dB Eb /N0 (On the other hand, if we had literally interpreted BPSK as using only the I channel in a passband system, we would have gotten r = 1/2.) That is, whenever we consider real-valued alphabets, we restrict ourselves to the real baseband channel for the purpose of computing spectral efficiency and comparing Shannon limits For a passband channel, we can use the same real-valued alphabet over the I and Q channels (corresponding to a rectangular complex-valued alphabet) to get exactly the same dependence of spectral efficiency on Eb /N0 6.1.4 Summarizing the discrete-time AWGN model In previous chapters, I have used constellations over the AWGN channel with a finite number of signal points One of the goals of this chapter is to be able to compute Shannon theoretic limits for performance when we constrain ourselves to using such constellations In Chapters to 5, when sampling signals corrupted by AWGN, we model the discrete-time AWGN samples as having variance