GEORGE ANSON HAMILTON (1843-1935 telegraphy captivated George Hamiltons interest while he was still a boy-to the extent that he built a small telegraph line himself, from sinking the poles to making the necessary apparatus. By the time he was 17, he was the manager of the telegraph office of the Atlantic great Western Railroad at Ravenna Ohio. Hamilton continued to hold managerial positions with telegraph companies until 187 when he became assistant to moses g. farmer in his work on general electrical apparatus and In 1875, Hamilton joined Western Union ssistant electrician and, for the next two years, worked with Gerritt Smith in establishing and maintaining the first quadruplex telegraph cir cuits in both America and England. He then focused on the development of the Wheatstone high-speed automatic system and was also the chief electrician on the Key West-Havana cable repair expedition. Hamilton left Western Union in 1889, however, to join Western Electric, George Anson Hamilton (1843-1935) where he was placed in charge of the production of fine electrical instruments until the time of his retirement. Courtesy of the IEEE Center for the History of Electrical Engineering. data, which are partitioned into k-bit and(m -k)-bit words where k 2 m. The k-bit words( frames)are convolutionally encoded into(n =k+ 1)-bit words so that the code rate is r= k /(k+ 1). The amplitude and phase are then set jointly on the basis of the coded n-bit word and the uncoded(m -k)-bit word. Almost dB of coding gain can be realized if coders of constraint length 9 are used. Defining Terms Block code: A mapping of k input binary symbols into n output binary symbols Convolutional code: A subset of tree codes, accepting k binary symbols at its input and producing n binary symbols at its output. Cyclic code: Block code such that another code word can be obtained by taking any one code word, shifting Tree code: Produced by a coder that has memory Related Topics 69.1 Modulat c2000 by CRC Press LLC
© 2000 by CRC Press LLC data, which are partitioned into k-bit and (m – k)-bit words where k ³ m. The k-bit words (frames) are convolutionally encoded into (n = k + 1)-bit words so that the code rate is R = k/(k + 1). The amplitude and phase are then set jointly on the basis of the coded n-bit word and the uncoded (m – k)-bit word. Almost 6 dB of coding gain can be realized if coders of constraint length 9 are used. Defining Terms Block code: A mapping of k input binary symbols into n output binary symbols. Convolutional code: A subset of tree codes, accepting k binary symbols at its input and producing n binary symbols at its output. Cyclic code: Block code such that another code word can be obtained by taking any one code word, shifting the bits to the right, and placing the dropped-off bits on the left. Tree code: Produced by a coder that has memory. Related Topics 69.1 Modulation • 70.2 Equalization GEORGE ANSON HAMILTON (1843–1935) elegraphy captivated George Hamilton’s interest while he was still a boy — to the extent that he built a small telegraph line himself, from sinking the poles to making the necessary apparatus. By the time he was 17, he was the manager of the telegraph office of the Atlantic & Great Western Railroad at Ravenna, Ohio. Hamilton continued to hold managerial positions with telegraph companies until 1873 when he became assistant to Moses G. Farmer in his work on general electrical apparatus and machinery. In 1875, Hamilton joined Western Union as assistant electrician and, for the next two years, worked with Gerritt Smith in establishing and maintaining the first quadruplex telegraph circuits in both America and England. He then focused on the development of the Wheatstone high-speed automatic system and was also the chief electrician on the Key West–Havana cable repair expedition. Hamilton left Western Union in 1889, however, to join Western Electric, where he was placed in charge of the production of fine electrical instruments until the time of his retirement. (Courtesy of the IEEE Center for the History of Electrical Engineering.) T
Transmitter (t) Encoder ource 燃叫器=}四 (a)Conventional Coding Technique Transmitter m bit word (parallel data) m-k bit word / ata k bit word nsk+1 bit word (b) FIGURE 70.5 Transmitters for conventional coding and for TCM References V.K. Bhargava, Forward error correction schemes for digital communications, IEEE Communication magazine 21,1983 G.C. Clark and J.B. Cain, Error-Correction Coding for Digital Communications, New York: Plenum, 1981 L W Couch, Digital and Analog Communication Systems, New York: Macmillan, 1990 B.P. Lathi, Modern Digital and Analog Communication, New York: CBS College Publishing, 1983 G Ungerboeck, " Channel coding with multilevel/phase signals, IEEE Transactions on Information Theory, vol IT-28(January), Pp 55-67, 1982. G. Ungerboeck,Trellis-coded modulation with redundant signal sets, Parts 1 and 2, IEEE Communications agazine, vol. 25, no. 2(February), Pp 5-21, 1987 L. Wei,"Rotationally invariant convolutional channel coding with expanded signal space--Part II: Nonlinear codes, IEEE Journal on Selected Areas in Communications, vol SAC-2, no 2, pp 672-686, 1984 Further Information For further information refer to IEEE Communications and IEEE Journal on Selected Areas in Communications. 70.2 Equalization Richard C. Dorf and Zhen Wan In bandwidth-efficient digital communication systems the effect of each symbol transmitted over a time dispersive channel extends beyond the time interval used to represent that symbol. The distortion caused by the resulting overlap of received symbols is called intersymbol interference(ISI)[Lucky et al., 1968]. ISI arises in all pulse-modulation systems, including frequency-shift keying(FSK), phase-shift keying(PSK), and quadra ture amplitude modulation(QAM)[Lucky et al., 1968]. However, its effect can be most easily described for a baseband PAM system. The purpose of an equalizer, placed in the path of the received signal, is to reduce the ISI as mud to maximize the probability of correct decisions e 2000 by CRC Press LLC
© 2000 by CRC Press LLC References V.K. Bhargava,“Forward error correction schemes for digital communications,” IEEE Communication Magazine, 21, 1983. G.C. Clark and J.B. Cain, Error-Correction Coding for Digital Communications, New York: Plenum, 1981. L.W. Couch, Digital and Analog Communication Systems, New York: Macmillan, 1990. B.P. Lathi, Modern Digital and Analog Communication, New York: CBS College Publishing, 1983. G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Transactions on Information Theory, vol. IT-28 (January), pp. 55–67, 1982. G. Ungerboeck, “Trellis-coded modulation with redundant signal sets,” Parts 1 and 2, IEEE Communications Magazine, vol. 25, no. 2 (February), pp. 5–21, 1987. L. Wei, “Rotationally invariant convolutional channel coding with expanded signal space—Part II: Nonlinear codes,” IEEE Journal on Selected Areas in Communications, vol. SAC-2, no. 2, pp. 672–686, 1984. Further Information For further information refer to IEEE Communications and IEEE Journal on Selected Areas in Communications. 70.2 Equalization Richard C. Dorf and Zhen Wan In bandwidth-efficient digital communication systems the effect of each symbol transmitted over a time dispersive channel extends beyond the time interval used to represent that symbol. The distortion caused by the resulting overlap of received symbols is called intersymbol interference (ISI) [Lucky et al., 1968]. ISI arises in all pulse-modulation systems, including frequency-shift keying (FSK), phase-shift keying (PSK), and quadrature amplitude modulation (QAM) [Lucky et al., 1968]. However, its effect can be most easily described for a baseband PAM system. The purpose of an equalizer, placed in the path of the received signal, is to reduce the ISI as much as possible to maximize the probability of correct decisions. FIGURE 70.5 Transmitters for conventional coding and for TCM
T T T zk URE 70.6 Linear transversal equalizer. Source: K Feher, Advanced Digital Communications, Englewood Cliffs, N J tice- Hall, 1987, P 648. with permission. Linear Transversal equalizers Among the many structures used for equalization, the simplest is the transversal (tapped delay line or nonre- cursive)equalizer shown in Fig. 70.6. In such an equalizer the current and past values r() of the received signal are linearly weighted by equalizer coefficients(tap gains)cn and summed to produce the output. In the commonly used digital implementation, samples of the received signal at the symbol rate are stored in a digital shift register(or memory), and the equalizer output samples(sums of products)zt kT)or zg are computed digitally, once per symbol, according to z=∑cnr(t+kr-m) where Nis the number of equalizer coefficients and to denotes sample timing The equalizer coefficients, cm, n=0, 1,. N-1, may be chosen to force the samples of the combined channel and equalizer impulse response to zero at all but one of the NT-spaced instants in the span of the equalize Such an equalizer is called a zero-forcing(ZF)equalizer [Lucky, 1965 If we let the number of coefficients of a ZF equalizer increase without bound, we would obtain an infinite length equalizer with zero ISI at its output. An infinite-length zero-ISI equalizer is simply an inverse filter, which inverts the folded frequency response of the channel. Clearly, the ZF criterion neglects the effect of noise altogether. A finite-length ZF equalizer is approximately inverse to the folded frequency response of the channel a finite-length ZF equalizer is guaranteed to minimize the peak distortion or worst-case ISI only if the peak distortion before equalization is less than 100%[Lucky, 1965 The least-mean-squared(LMS)equalizer [Lucky et al., 1968] is more robust. Here the equalizer coefficients re chosen to minimize the mean squared error( MSe)the sum of squares of all the ISI terms plus the noise power at the output of the equalizer. Therefore, the LMS equalizer maximizes the signal-to-distortion ratio (S/D)at its output within the constraints of the equalizer time span and the delay through the equalizer. Before regular data transmission begins, automatic synthesis of the ZF or LMS equalizers for unknown channels may be carried out during a training period. During the training period, a known signal is transmitted and a ynchronized version of this signal is generated in the receiver to acquire information about the channel characteristics. The automatic adaptive equalizer is shown in Fig. 70.7. A noisy but unbiased estimate 2er(to +kr-nT) c2000 by CRC Press LLC
© 2000 by CRC Press LLC Linear Transversal Equalizers Among the many structures used for equalization, the simplest is the transversal (tapped delay line or nonrecursive) equalizer shown in Fig. 70.6. In such an equalizer the current and past values r(t – nT) of the received signal are linearly weighted by equalizer coefficients (tap gains) cn and summed to produce the output. In the commonly used digital implementation, samples of the received signal at the symbol rate are stored in a digital shift register (or memory), and the equalizer output samples (sums of products) z(t0 + kT) or zk are computed digitally, once per symbol, according to where N is the number of equalizer coefficients and t0 denotes sample timing. The equalizer coefficients, cn, n = 0, 1,. . .,N – 1, may be chosen to force the samples of the combined channel and equalizer impulse response to zero at all but one of the NT-spaced instants in the span of the equalizer. Such an equalizer is called a zero-forcing (ZF) equalizer [Lucky, 1965]. If we let the number of coefficients of a ZF equalizer increase without bound, we would obtain an infinitelength equalizer with zero ISI at its output. An infinite-length zero-ISI equalizer is simply an inverse filter, which inverts the folded frequency response of the channel. Clearly, the ZF criterion neglects the effect of noise altogether.A finite-length ZF equalizer is approximately inverse to the folded frequency response of the channel. Also, a finite-length ZF equalizer is guaranteed to minimize the peak distortion or worst-case ISI only if the peak distortion before equalization is less than 100% [Lucky, 1965]. The least-mean-squared (LMS) equalizer [Lucky et al.,1968] is more robust. Here the equalizer coefficients are chosen to minimize the mean squared error (MSE)—the sum of squares of all the ISI terms plus the noise power at the output of the equalizer. Therefore, the LMS equalizer maximizes the signal-to-distortion ratio (S/D) at its output within the constraints of the equalizer time span and the delay through the equalizer. Automatic Synthesis Before regular data transmission begins, automatic synthesis of the ZF or LMS equalizers for unknown channels may be carried out during a training period. During the training period, a known signal is transmitted and a synchronized version of this signal is generated in the receiver to acquire information about the channel characteristics. The automatic adaptive equalizer is shown in Fig. 70.7. A noisy but unbiased estimate: FIGURE 70.6 Linear transversal equalizer. (Source: K. Feher, Advanced Digital Communications, Englewood Cliffs, N.J.: Prentice-Hall, 1987, p. 648. With permission.) z c r t kT nt k n n N = + = Â ( – ) – 0 0 1 d d e c k e r t kT nT k n k 2 0 2 ( ) = + ( – )
Decision Device ek FIGURE 70.7 Automatic adaptive equalizer. Source: K Feher, Advanced Digital Communications, Englewood Cliffs, N J . is used. Thus, the tap gains are updated according to Cn(k+ 1)=c(k)-Aegr(t, + kT-nT), n=0,1, where c,(k)is the nth tap gain at time k, e, is the error signal, and A is a positive adaptation constant or step ize, error signals e=22-qu can be computed at the equalizer output and used to adjust the equalizer coefficients to reduce the sum of the squared errors. Note gk=xk The most popular equalizer adjustment method involves updates to each tap gain during each symbol interval. The adjustment to each tap gain is in a direction opposite to an estimate of the gradient of the mse with respect to that tap gain. The idea is to move the set of equalizer coefficients closer to the unique optimum set corresponding to the minimum MSE. This symbol-by-symbol procedure developed by widrow and Hoff [ Feher, 1987] is commonly referred to as the stochastic gradient method Adaptive Equalization After the initial training period (if there is one), the coefficients of an adaptive equalizer may be continually adjusted in a decision-directed manner. In this mode the error signal e,=42- qu is derived from the final(not necessarily correct) receiver estimate ak of the transmitted sequence x where qr is the estimate of xx. In normal operation the receiver decisions are correct with high probability, so that the error estimates are correct often enough to allow the adaptive equalizer to maintain precise equalization. Moreover, a decision-directed adaptive equalizer can track slow variations in the channel characteristics or linear perturbations in the receiver front end, such as slow jitter in the sampler phase Nonlinear equalizers Decision-Feedback Equalizers A decision-feedback equalizer(DFE)is a simple nonlinear equalizer [Monsen, 1971], which is particularly useful for channels with severe amplitude distortion and uses decision feedback to cancel the interference from symbols which have already been detected. Fig 70.8 shows the diagram of the equalizer. The equalized signal is the sum of the outputs of the forward and feedback parts of the equalizer. The forward part is like the linear transversal equalizer discussed earlier. Decisions made on the equalized signal are fed back via a second transversal filter. The basic idea is that if the values of the symbols already detected are known (past decisions are assumed to be correct), then the ISI contributed by these symbols can be canceled exactly, by subtracting past symbol values with appropriate weighting from the equalizer output. The forward and feedback coefficients may be adjusted simultaneously to minimize the mse. The update equation for the forward coefficients is the same as for the linear equalizer. The feedback coefficients are adjusted according to b (k +1)=b(k)+ where A is the kth symbol decision, bm(k)is the mth feedback coefficient at time k, and there are M feedback coefficients in all. The optimum LMS settings of bm, m=1,., M, are those that reduce the ISI to zero, within the span of the feedback part, in a manner similar to a ZF equalizer e 2000 by CRC Press LLC
© 2000 by CRC Press LLC is used. Thus, the tap gains are updated according to cn(k + 1) = cn(k) – Dekr(t0 + kT – nT), n = 0, 1, . . ., N – 1 where cn(k) is the nth tap gain at time k, ek is the error signal, and D is a positive adaptation constant or step size, error signals ek = zk – qk can be computed at the equalizer output and used to adjust the equalizer coefficients to reduce the sum of the squared errors. Note qk = xˆk. The most popular equalizer adjustment method involves updates to each tap gain during each symbol interval. The adjustment to each tap gain is in a direction opposite to an estimate of the gradient of the MSE with respect to that tap gain. The idea is to move the set of equalizer coefficients closer to the unique optimum set corresponding to the minimum MSE. This symbol-by-symbol procedure developed by Widrow and Hoff [Feher, 1987] is commonly referred to as the stochastic gradient method. Adaptive Equalization After the initial training period (if there is one), the coefficients of an adaptive equalizer may be continually adjusted in a decision-directed manner. In this mode the error signal ek = zk – qk is derived from the final (not necessarily correct) receiver estimate {qk} of the transmitted sequence {xk} where qk is the estimate of xk. In normal operation the receiver decisions are correct with high probability, so that the error estimates are correct often enough to allow the adaptive equalizer to maintain precise equalization. Moreover, a decision-directed adaptive equalizer can track slow variations in the channel characteristics or linear perturbations in the receiver front end, such as slow jitter in the sampler phase. Nonlinear Equalizers Decision-Feedback Equalizers A decision-feedback equalizer (DFE) is a simple nonlinear equalizer [Monsen, 1971], which is particularly useful for channels with severe amplitude distortion and uses decision feedback to cancel the interference from symbols which have already been detected. Fig. 70.8 shows the diagram of the equalizer. The equalized signal is the sum of the outputs of the forward and feedback parts of the equalizer. The forward part is like the linear transversal equalizer discussed earlier. Decisions made on the equalized signal are fed back via a second transversal filter. The basic idea is that if the values of the symbols already detected are known (past decisions are assumed to be correct), then the ISI contributed by these symbols can be canceled exactly, by subtracting past symbol values with appropriate weighting from the equalizer output. The forward and feedback coefficients may be adjusted simultaneously to minimize the MSE. The update equation for the forward coefficients is the same as for the linear equalizer. The feedback coefficients are adjusted according to bm(k + 1) = bm(k) + Dek xˆ k–m m = 1, . . ., M where xˆk is the kth symbol decision, bm(k) is the mth feedback coefficient at time k, and there are M feedback coefficients in all. The optimum LMS settings of bm, m = 1, . . ., M, are those that reduce the ISI to zero, within the span of the feedback part, in a manner similar to a ZF equalizer. FIGURE 70.7 Automatic adaptive equalizer. (Source: K. Feher, Advanced Digital Communications, Englewood Cliffs, N.J.: Prentice-Hall, 1987, p. 651. With permission.)