Dorf, R.C., Wan, Z., Millstein, L.B., Simon, M..K. “Digital Communication” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 70 Digital Communication 70.1 Error Control Coding Block Codes?Convolutional Codes?Code Performance?Trellis- Coded Modulation 70.2 Equalization Linear Transversal Equalizers?Nonlinear Equalizers?Linear Receivers?Nonlinear Receivers 70.3 Spread Spectrum Communications A Brief History?Why Spread Spectrum??Basic Concepts and Terminology?Spread Spectrum Techniques?Applications of Spread Spectrum 70.1 Error Control Coding Richard C. Dorf and Zhen Wan Error correcting codes may be classified into two broad categories: block codes and tree codes. A block code is a mapping of k input binary symbols into n output binary symbols. Consequently, the block coder is a memoryless device. Since n > k, the code can be selected to provide redundancy, such as parity bits, which are used by the decoder to provide some error detection and error correction. The codes are denoted by (n, k), where the code rate R is defined by R = k/n. Practical values of R range from 1/4 to 7/8, and k ranges from 3 to several hundred [Clark and Cain, 1981]. Some properties of block codes are given in Table 70.1. A tree code is produced by a coder that has memory. Convolutional codes are a subset of tree codes. The convolutional coder accepts k binary symbols at its input and produces n binary symbols at its output, where the n output symbols are affected by v + k input symbols. Memory is incorporated since v > 0. The code rate is defined by R = k/n. Typical values for k and n range from 1 to 8, and the values for v range from 2 to 60. The range of R is between 1/4 and 7/8 [Clark and Cain, 1981]. Block Codes In block code, the n code digits generated in a particular time unit depend only on the k message digits within that time unit. Some of the errors can be detected and corrected if d 3 s + t + 1, where s is the number of errors that can be detected, t is the number of errors that can be corrected, and d is the hamming distance. Usually, s 3 t, thus, d 3 2t + 1. A general code word can be expressed as a 1 , a 2 ,...,a k , c 1 , c 2 ,...,c r . k is the number of information bits and r is the number of check bits. Total word length is n = k + r. In Fig. 70.1, the gain h ij (i = 1, 2,..., r, j = 1, 2,..., k) are elements of the parity check matrix H. The k data bits are shifted in each time, while k + r bits are simultaneously shifted out by the commutator. Cyclic Codes Cyclic codes are block codes 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. An encoding circuit with (n – k) shift registers is shown in Fig. 70.2. Richard C. Dorf University of California, Davis Zhen Wan University of California, Davis L. B. Milstein University of California M. K. Simon Jet Propulsion Laboratory ? 2000 by CRC Press LLC In Fig. 70.2, the gain g k s are the coefficients of the generator polynomial g(x) = x n–k + g 1 x n–k–1 + . . . + g n–k–1 x + 1. The gains g k are either 0 or 1. The k data digits are shifted in one at a time at the input with the switch s held at position p 1 . The symbol D represents a one-digit delay. As the data digits move through the encoder, they are also shifted out onto the output lines, because the first k digits of code word are the data digits themselves. As soon as the last (or kth) data digit clears the last (n – k) register, all the registers contain the parity-check digits. The switch s is now thrown to position p 2 , and the n – k parity-check digits are shifted out one at a time onto the line. TABLE 70.1 Properties of Block Codes Code a Property BCH Reed–Solomon Hamming Maximal Length Block length n = 2 m – 1, n = m(2 m – 1) bits n = 2 m – 1 n = 2 m – 1 m = 3, 4, 5, . . . Number of parity bits r = m2t bits r = m Minimum distance d 3 2t + 1 d = m(2t + 1) bits d = 3 d = 2 m – 1 Number of information bits k 3 n – mt k = m a m is any positive integer unless otherwise indicated; n is the block length; k is the number of information bits; t is the number of errors that can be corrected; r is the number of parity bits; d is the distance. FIGURE 70.1An encoding circuit of (n, k) block code. FIGURE 70.2An encoder for systematic cyclic code. (Source: B.P. Lathi, Modern Digital and Analog Communications, New York: CBS College Publishing, 1983. With permission.) ? 2000 by CRC Press LLC Examples of cyclic and related codes are 1.Bose–Chaudhuri–Hocquenhem (BCH) 2.Reed–Solomon 3.Hamming 4.Maximal length 5.Reed–Muller 6.Golay codes Convolutional Codes In convolutional code, the block of n code digits generated by the encoder in a particular time unit depends not only on the block of k message digits within that time unit but also on the block of data digits within a previous span of N – 1 time units (N >1). A convolutional encoder is illustrated in Fig. 70.3. Here k bits (one input frame) are shifted in each time, and concurrently n bits (the output frame) are shifted out, where n > k. Thus, every k-bit input frame produces an n-bit output frame. Redundancy is provided in the output, since n > k. Also, there is memory in the coder, since the output frame depends on the previous K input frames where K > 1. The code rate is R = k/n, which is 3/4 in this illustration. The constraint length, K, is the number of input frames that are held in the kK bit shift register. Depending on the particular convolutional code that is to be generated, data from the kK stages of the shift register are added (modulo 2) and used to set the bits in the n-stage output register. Code Performance The improvement in the performance of a digital communication system that can be achieved by the use of coding is illustrated in Fig. 70.4. It is assumed that a digital signal plus channel noise is present at the receiver input. The performance of a system that uses binary-phase-shift-keyed (BPSK) signaling is shown both for the case when coding is used and for the case when there is no coding. For the BPSK no code case, P e = Q (). For the coded case a (23,12) Golay code is used; P e is the probability of bit error—also called the bit error rate (BER)—that is measured at the receiver output. FIGURE 70.3Convolutional encoding (k = 3, n = 4, K = 5, and R = 3/4). 2(/EN bo ? 2000 by CRC Press LLC Trellis-Coded Modulation Trellis-coded modulation (TCM) combines multilevel modulation and coding to achieve coding gain without bandwidth expansion [Ungerboeck, 1982, 1987]. TCM has been adopted for use in the new CCITT V.32 modem that allows an information data rate of 9600 b/s (bits per second) to be transmitted over VF (voice frequency) lines. The TCM has a coding gain of 4 dB [Wei, 1984]. The combined modulation and coding operation of TCM is shown in Fig. 70.5(b). Here, the serial data from the source, m(t), are converted into parallel (m-bit) FIGURE 70.4Performance of digital systems—with and without coding. E b is the energy-per-bit to noise-density at the receiver input. The function Q(x) is Q(x) = (1/ x)e –x2/2 . TABLE 70.2Coding Gains with BPSK or QPSK Coding Gain (dB) Coding Gain (dB) Data Rate Coding Technique Used at 10 –5 BER at 10 –8 BER Capability Ideal coding 11.2 13.6 Concatenated Reed–Solomon and convolution (Viterbi decoding) 6.5–7.5 8.5–9.5 Moderate Convolutional with sequential decoding (soft decisions) 6.0–7.0 8.0–9.0 Moderate Block codes (soft decisions) 5.0–6.0 6.5–7.5 Moderate Concatenated Reed–Solomon and short block 4.5–5.5 6.5–7.5 Very high Convolutional with Viterbi decoding 4.0–5.5 5.0–6.5 High Convolutional with sequential decoding (hard decisions) 4.0–5.0 6.0–7.0 High Block codes (hard decisions) 3.0–4.0 4.5–5.5 High Block codes with threshold decoding 2.0–4.0 3.5–5.5 High Convolutional with threshold decoding 1.5–3.0 2.5–4.0 Very high BPSK: modulation technique—binary phase-shift keying; QPSK: modulation technique—quadrature phase- shift keying; BER: bit error rate. Source: V.K. Bhargava, “Forward error correction schemes for digital communications,” IEEE Communication Magazine, 21, 11–19, ? 1983 IEEE. With permission. 2p ? 2000 by CRC Press LLC 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 T ? 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 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 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, 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.) 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 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 much as possible to maximize the probability of correct decisions. FIGURE 70.5Transmitters for conventional coding and for TCM. ? 2000 by CRC Press LLC 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(t – nT) of the received signal are linearly weighted by equalizer coefficients (tap gains) c n 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(t 0 + kT) or z k are computed digitally, once per symbol, according to where N is the number of equalizer coefficients and t 0 denotes sample timing. The equalizer coefficients, c n , 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 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. 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 70.6Linear transversal equalizer. (Source: K. Feher, Advanced Digital Communications, Englewood Cliffs, N.J.: Prentice-Hall, 1987, p. 648. With permission.) z crt kT nt kn n N =+ = ? (–) – 0 0 1 d d e ck ert kTnT k n k 2 0 2 () (–)=+ ? 2000 by CRC Press LLC is used. Thus, the tap gains are updated according to c n (k + 1) = c n (k) – De k r(t 0 + kT – nT), n = 0, 1, . . ., N – 1 where c n (k) is the nth tap gain at time k, e k is the error signal, and D is a positive adaptation constant or step size, error signals e k = z k – q k can be computed at the equalizer output and used to adjust the equalizer coefficients to reduce the sum of the squared errors. Note q k = ?x k . 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 k = z k – q k is derived from the final (not necessarily correct) receiver estimate {q k } of the transmitted sequence {x k } where q k is the estimate of x k . 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 m (k + 1) = b m (k) + De k ? x k–m m = 1, . . ., M where ? x k is the kth symbol decision, b m (k) is the mth feedback coefficient at time k, and there are M feedback coefficients in all. The optimum LMS settings of b m , 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 70.7Automatic adaptive equalizer. (Source: K. Feher, Advanced Digital Communications, Englewood Cliffs, N.J.: Prentice-Hall, 1987, p. 651. With permission.) ? 2000 by CRC Press LLC Fractionally Spaced Equalizers The optimum receive filter in a linear modulation system is the cascade of a filter matched to the actual channel, with a transversal T-spaced equalizer [Forney, 1972]. The fractionally spaced equalizer (FSE), by virtue of its sampling rate, can synthesize the best combination of the characteristics of an adaptive matched filter and a T-spaced equalizer, within the constraints of its length and delay. A T-spaced equalizer, with symbol-rate sampling at its input, cannot perform matched filtering. A fractionally spaced equalizer can effectively compen- sate for more severe delay distortion and deal with amplitude distortion with less noise enhancement than a T-equalizer. A fractionally spaced transversal equalizer [Monsen, 1971] is shown in Fig. 70.9. The delay-line taps of such an equalizer are spaced at an interval t, which is less than, or a fraction of, the symbol interval T. The tap spacing t is typically selected such that the bandwidth occupied by the signal at the equalizer input is *f* < FIGURE 70.8Decision-feedback equalizer. (Source: K. Feher, Advanced Digital Communications, Englewood Cliffs, N.J.: Prentice-Hall, 1987, p. 655. With permission.) FIGURE 70.9Fractionally spaced equalizer. (Source: K. Feher, Advanced Digital Communications, Englewood Cliffs, N.J.: Prentice-Hall, p. 656. With permission.) ? 2000 by CRC Press LLC 1/2t: that is, t-spaced sampling satisfies the sampling theorem. In an analog implementation, there is no other restriction on t, and the output of the equalizer can be sampled at the symbol rate. In a digital implementation t must be KT/M, where K and M are integers and M > K. (In practice, it is convenient to choose t = T/M, where M is a small integer, e.g., 2.) The received signal is sampled and shifted into the equalizer delay line at a rate M/T, and one input is produced each symbol interval (for every M input sample). In general, the equalizer output is given by The coefficients of a KT/M equalizer may be updated once per symbol based on the error computed for that symbol according to Linear Receivers When the channel does not introduce any amplitude distortion, the linear receiver is optimum with respect to the ultimate criterion of minimum probability of symbol error. The conventional linear receiver consists of a matched filter, a symbol-rate sampler, an infinite-length T-spaced equalizer, and a memoryless detector. The linear receiver structure is shown in Fig. 70.10. In the conventional linear receiver, a memoryless threshold detector is sufficient to minimize the probability of error; the equalizer response is designed to satisfy the zero-ISI constraint, and the matched filter is designed to minimize the effect of the noise while maximizing the signal. Matched Filter The matched filter is the linear filter that maximizes (S/N) out = s 2 0 (t)/n 2 0 (t) of Fig. 70.11 and has a transfer function given by where S(?) = F[s(t)] is the Fourier transform of the known input signal s(t) of duration T sec. P n (?) is the PSD of the input noise, t 0 is the sampling time when (S/N) out is evaluated, and K is an arbitrary real nonzero constant. A general representation for a matched filter is illustrated in Fig. 70.11. The input signal is denoted by s(t) and the output signal by s 0 (t). Similar notation is used for the noise. Nonlinear Receivers When amplitude distortion is present in the channel, a memoryless detector operating on the output of this receiver filter no longer minimizes symbol error probability. Recognizing this fact, several authors have investigated optimum or approximately optimum nonlinear receiver structures subject to a variety of criteria [Lucky, 1973]. FIGURE 70.10Conventional linear receiver. zcrtkT nKT M kn n N =+ ? è ? ? ? ÷ = ? 0 0 1 – – ck ck ertkT nKT M nN nnk ( ) ()– – , ,,..., –+= + ? è ? ? ? ÷ =101 0 D Hf K Sf Pf e n jt () *() () – = w 0 ? 2000 by CRC Press LLC Decision-Feedback Equalizers A DFE takes advantage of the symbols that have already been detected (correctly with high probability) to cancel the ISI due to these symbols without noise enhancement. A DFE makes memoryless decisions and cancels all trailing ISI terms. Even when the whitened matched filter (WMF) is used as the receive filter for the DFE, the DFE suffers from a reduced effective signal-to-noise ratio, and error propagation, due to its inability to defer decisions. An infinite-length DFE receiver takes the general form (shown in Fig. 70.12) of a forward linear receive filter, symbol-rate sampler, canceler, and memoryless detector. The symbol-rate output of the detector is then used by the feedback filter to generate future outputs for cancellation. Adaptive Filters for MLSE For unknown and/or slowly time-varying channels, the receive filter must be adaptive in order to obtain the ultimate performance gain from MLSE (maximum-likelihood sequence estimation). Secondly, the complexity of the MLSE becomes prohibitive for practical channels with a large number of ISI terms. Therefore, in a practical receiver, an adaptive receive filter may be used prior to Viterbi detection to limit the time spread of the channel as well as to track slow time variation in the channel characteristics [Falconer and Magee, 1973]. Several adaptive receive filters are available that minimize the MSE at the input to the Viterbi algorithm. These methods differ in the form of constraint [Falconer and Magee, 1973] on the desired impulse response (DIR) which is necessary in this optimization process to exclude the selection of the null DIR corresponding to no transmission through the channel. The general form of such a receiver is shown in Fig. 70.13. FIGURE 70.11Matched filter. FIGURE 70.12Conventional decision-feedback receiver. (Source: K. Feher, Advanced Digital Communications, Englewood Cliffs, N.J.: Prentice-Hall, 1987, p. 675. With permission.) ? 2000 by CRC Press LLC One such constraint is to restrict the DIR to be causal and to restrict the first coefficient of the DIR to be unity. In this case the delay (LT) in Fig. 70.13 is equal to the delay through the Viterbi algorithm and the first coefficient of {b k } is constrained to be unity. The least restrictive constraint on the DIR is the unit energy constraint proposed by Falconer and Magee [1973]. This leads to yet another form of the receiver structure as shown in Fig. 70.13. However, the adaptation algorithm for updating the DIR coefficients {b k } is considerably more complicated [Falconer and Magee, 1973]. Note that the fixed predetermined WMF and T-spaced prefilter combination of Falconer and Magee [1973] has been replaced in Fig. 70.13 by a general fractionally spaced adaptive filter. Defining Terms Equalizer: A filter used to reduce the effect of intersymbol interference. Intersymbol interference: The distortion caused by the overlap (in time) of adjacent symbols. Related Topic 70.1 Coding References L.W. Couch, Digital and Analog Communication Systems, New York: Macmillan, 1990. D.D. Falconer and F.R. Magee, Jr., “Adaptive channel memory truncation for maximum likelihood sequence estimation,” Bell Syst. Technical Journal, vol. 5, pp. 1541–1562, November 1973. K. Feher, Advanced Digital Communications, Englewood Cliffs, N.J.: Prentice-Hall, 1987. G.D. Forney, Jr., “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Information Theory, vol. IT-88, pp. 363–378, May 1972. R.W. Lucky, “Automatic equalization for digital communication,” Bell Syst. Tech. Journal, vol. 44, pp. 547–588, April 1965. R.W. Lucky, “A survey of the communication theory literature: 1968–1973,” IEEE Trans. Information Theory, vol. 52, pp. 1483–1519, November 1973. R.W. Lucky, J. Salz, and E.J. Weldon, Jr., Principles of Data Communication, New York: McGraw-Hill, 1968. P. Monsen, “Feedback equalization for fading dispersive channels,” IEEE Trans. Information Theory, vol. IT-17, pp. 56–64, January 1971. FIGURE 70.13 General form of adaptive MLSE receiver with finite-length DIR. (Source: K. Feher, Advanced Digital Communications, Englewood Cliffs, N.J.: Prentice-Hall, 1987, p. 684. With permission.) ? 2000 by CRC Press LLC 70.3 Spread Spectrum Communications 1 L.B. Milstein and M.K. Simon A Brief History Spread spectrum (SS) has its origin in the military arena where the friendly communicator is (1) susceptible to detection/interception by the enemy and (2) vulnerable to intentionally introduced unfriendly interference (jamming). Communication systems that employ spread spectrum to reduce the communicator’s detectability and combat the enemy-introduced interference are respectively referred to as low probability of intercept (LPI) and antijam (AJ) communication systems. With the change in the current world political situation wherein the U.S. Department of Defense (DOD) has reduced its emphasis on the development and acquisition of new communication systems for the original purposes, a host of new commercial applications for SS has evolved, particularly in the area of cellular mobile communications. This shift from military to commercial applications of SS has demonstrated that the basic concepts that make SS techniques to useful in the military can also be put to practical peacetime use. In the next section, we give a simple description of these basic concepts using the original military application as the basis of explanation. The extension of these concepts to the mentioned commercial applications will be treated later on in the chapter. Why Spread Spectrum? Spread spectrum is a communication technique wherein the transmitted modulation is spread (increased) in bandwidth prior to transmission over the channel and then despread (decreased) in bandwidth by the same amount at the receiver. If it were not for the fact that the communication channel introduces some form of narrowband (relative to the spread bandwidth) interference, the receiver performance would be transparent to the spreading and despreading operations (assuming that they are identical inverses of each other). That is, after despreading the received signal would be identical to the transmitted signal prior to spreading. In the presence of narrowband interference, however, there is a significant advantage to employing the spread- ing/despreading procedure described. The reason for this is as follows. Since the interference is introduced after the transmitted signal is spread, then, whereas the despreading operation at the receiver shrinks the desired signal back to its original bandwidth, at the same time it spreads the undesired signal (interference) in bandwidth by the same amount, thus reducing its power spectral density. This, in turn, serves to diminish the effect of the interference on the receiver performance, which depends on the amount of interference power in the spread bandwidth. It is indeed this very simple explanation, which is at the heart of all spread spectrum techniques. Basic Concepts and Terminology To describe this process analytically and at the same time introduce some terminology that is common in spread spectrum parlance, we proceed as follows. Consider a communicator that desires to send a message using a transmitted power S Watts (W) at an information rate R b bits/s (bps). By introducing a SS modulation, the bandwidth of the transmitted signal is increased from R b Hz to W ss Hz where W ss @ R b denotes the spread spectrum bandwidth. Assume that the channel introduces, in addition to the usual thermal noise (assumed to have a single-sided power spectral density (PSD) equal to N 0 W/Hz), an additive interference (jamming) having power J distributed over some bandwidth W J . After despreading, the desired signal bandwidth is once again now equal to R b Hz and the interference PSD is now N J = J/W ss . Note that since the thermal noise is assumed to be white, i.e., it is uniformly distributed over all frequencies, its PSD is unchanged by the despreading operation and, thus, remains equal to N 0 . Regardless of the signal and interferer waveforms, the equivalent bit energy-to-total noise ratio is, in terms of the given parameters, 1 The material in this article was previously published by CRC Press in The Mobile Communications Handbook, Jerry P. Gibson, Editor-in-Chief, 1996. ? 2000 by CRC Press LLC (70.1) For most practical scenarios, the jammer limits performance and, thus, the effects of receiver noise in the channel can be ignored. Thus, assuming N J @ N 0 , we can rewrite Eq. (70.1) as (70.2) where the ratio J/S is the jammer-to-signal power ratio and the ratio W ss /R b is the spreading ratio and is defined as the processing gain of the system. Since the ultimate error probability performance of the communication receiver depends on the ratio E b /N J , we see that from the communicator’s viewpoint his goal should be to minimize J/S (by choice of S) and maximize the processing gain (by choice of W ss for a given desired information rate). The possible strategies for the jammer will be discussed in the section on military applications dealing with AJ communications. Spread Spectrum Techniques By far the two most popular spreading techniques are direct sequence (DS) modulation and frequency hopping (FH) modulation. In the following subsections, we present a brief description of each. Direct Sequence Modulation A direct sequence modulation c(t) is formed by linearly modulating the output sequence {c n } of a pseudorandom number generator onto a train of pulses, each having a duration T c called the chip time. In mathematical form, (70.3) where p(t) is the basic pulse shape and is assumed to be of rectangular form. This type of modulation is usually used with binary phase-shift-keyed (BPSK) information signals, which have the complex form d(t) exp{j(2pf c t + q c )}, where d(t) is a binary-valued data waveform of rate 1/T b bit/s and f c and q c are the frequency and phase of the data-modulated carrier, respectively. As such, a DS/BPSK signal is formed by multiplying the BPSK signal by c(t) (see Fig. 70.14), resulting in the real transmitted signal (70.4) Since T c is chosen so that T b @ T c , then relative to the bandwidth of the BPSK information signal, the bandwidth of the DS/BPSK signal 2 is effectively increased by the ratio T b /T c = W ss /2R b , which is one-half the spreading factor or processing gain of the system. At the receiver, the sum of the transmitted DS/BPSK signal and the channel interference I(t) (as discussed before, we ignore the presence of the additive thermal noise) are ideally multiplied by the identical DS modulation (this operation is known as despreading), which returns the DS/BPSK signal to its original BPSK form whereas the real interference signal is now the real wideband signal Re{(t)c(t)}. In the previous sentence, we used the word ideally, which implies that the PN waveform used for despreading at the receiver is identical to that used for spreading at the transmitter. This simple implication covers up a 2 For the usual case of a rectangular spreading pulse p(t), the PSD of the DS/BPSK modulation will have (sin x/x) 2 form with first zero crossing at 1/T c , which is nominally taken as one-half the spread spectrum bandwidth W ss . E N E NN SR NJW b t b J b = + = + 00s E N E N SR JW S J W R b t b J b b @= = ss ss ct cpt nT nc n () =- ( ) =-￥ ￥ ? xt ctdt j ft cc () = ()() + ( ) []{} Re exp 2pq ? 2000 by CRC Press LLC multitude of tasks that a practical DS receiver must perform. In particular, the receiver must first acquire the PN waveform. That is, the local PN random generator that generates the PN waveform at the receiver used for despreading must be aligned (synchronized) to within one chip of the PN waveform of the received DS/BPSK signal. This is accomplished by employing some sort of search algorithm which typically steps the local PN waveform sequentially in time by a fraction of a chip (e.g., half a chip) and at each position searches for a high degree of correlation between the received and local PN reference waveforms. The search terminates when the correlation exceeds a given threshold, which is an indication that the alignment has been achieved. After bringing the two PN waveforms into coarse alignment, a tracking algorithm is employed to maintain fine alignment. The most popular forms of tracking loops are the continuous time delay-locked loop and its time-multiplexed version of the tau-dither loop. It is the difficulty in synchronizing the receiver PN generator to subnanosecond accuracy that limits PN chip rates to values on the order of hundreds of Mchips/s, which implies the same limitation on the DS spread spectrum bandwidth W ss . Frequency Hopping Modulation A frequency hopping (FH) modulation c(t) is formed by nonlinearly modulating a train of pulses with a sequence of pseudorandomly generated frequency shifts {f n }. In mathematical terms, c(t) has the complex form (70.5) where p(t) is again the basic pulse shape having a duration T h , called the hop time and {f n } is a sequence of random phases associated with the generation of the hops. FH modulation is traditionally used with multiple- frequency-shift-keyed (MFSK) information signals, which have the complex form exp{j[2p(f c + d(t))t]}, where d(t) is an M-level digital waveform (M denotes the symbol alphabet size) representing the information frequency modulation at a rate 1/T s symbols/s (sps). As such, an FH/MFSK signal is formed by complex multiplying the MFSK signal by c(t) resulting in the real transmitted signal (70.6) In reality, c(t) is never generated in the transmitter. Rather, x(t) is obtained by applying the sequence of pseudorandom frequency shifts {f n } directly to the frequency synthesizer that generates the carrier frequency f c (see Fig. 70.15). In terms of the actual implementation, successive (not necessarily disjoint) k-chip segments of a PN sequence drive a frequency synthesizer, which hops the carrier over 2 k frequencies. In view of the large bandwidths over which the frequency synthesizer must operate, it is difficult to maintain phase coherence from hop to hop, which explains the inclusion of the sequence {f n } in the Eq. (70.5) model for c(t). On a short term basis, e.g., within a given hop, the signal bandwidth is identical to that of the MFSK information modulation, which is typically much smaller than W ss . On the other hand, when averaged over many hops, the signal bandwidth is equal to W ss , which can be on the order of several GHz, i.e., an order of magnitude larger than that of implementable DS bandwidths. The exact relation between W ss , T h , T s and the number of frequency shifts in the set {f n } will be discussed shortly. FIGURE 70.14A DS-BPSK system (complex form). ct j f pt nT n nn h () =+ ( ){} - ( ) =-￥ ￥ ? exp 2pf xt ct j f dtt c () = () + ()( ) []{}{} Re exp 2p ? 2000 by CRC Press LLC At the receiver, the sum of the transmitted FH/MFSK signal and the channel interference I(t) is ideally complex multiplied by the identical FH modulation (this operation is known as dehopping), which returns the FH/MFSK signal to its original MFSK form, whereas the real interference signal is now the wideband (in the average sense) signal Re{I(t)c(t)}. Analogous to the DS case, the receiver must acquire and track the FH signal so that the dehopping waveform is as close to the hopping waveform c(t) as possible. FH systems are traditionally classified in accordance with the relationship between T h and T s . Fast frequency- hopped (FFH) systems are ones in which there exists one or more hops per data symbol, that is, T s = NT h (N an integer) whereas slow frequency-hopped (SFH) systems are ones in which there exists more than one symbol per hop, that is, T h = NT s . It is customary in SS parlance to refer to the FH/MFSK tone of shortest duration as a “chip”, despite the same usage for the PN chips associated with the code generator that drives the frequency synthesizer. Keeping this distinction in mind, in an FFH system where, as already stated, there are multiple hops per data symbol, a chip is equal to a hop. For SFH, where there are multiple data symbols per hop, a chip is equal to an MFSK symbol. Combining these two statements, the chip rate R c in an FH system is given by the larger of R h = 1/T h and R s = 1/T s and, as such, is the highest system clock rate. The frequency spacing between the FH/MFSK tones is governed by the chip rate R c and is, thus, dependent on whether the FH modulation is FFH or SFH. In particular, for SFH where R c = R s , the spacing between FH/MFSK tones is equal to the spacing between the MFSK tones themselves. For noncoherent detection (the most commonly encountered in FH/MFSK systems), the separation of the MFSK symbols necessary to provide orthogonality 3 is an integer multiple of R s . Assuming the minimum spacing, i.e., R s , the entire spread spectrum band is then partitioned into a total of N t = W ss /R s =W ss /R c equally spaced FH tones. One arrangement, which is by far the most common, is to group these N t tones into N b = N t /M contiguous, nonoverlapping bands, each with bandwidth M R s = M R c ; see Fig. 70.16(a). Assuming symmetric MFSK modulation around the carrier frequency, then the center frequencies of the N b = 2 k bands represent the set of hop carriers, each of which is assigned to a given k-tuple of the PN code generator. In this fixed arrangement, each of the N t FH/MFSK tones corresponds to the combination of a unique hop carrier (PN code k-tuple) and a unique MFSK symbol. Another arrangement, which provides more protection against the sophisticated interferer (jammer), is to overlap adjacent M -ary bands by an amount equal to R c ; see Fig. 70.16(b). Assuming again that the center frequency of each band corresponds to a possible hop carrier, then since all but M – 1 of the N t tones are available as center frequencies, the number of hop carriers has been increased from N t /M to N t – (M – 1), which for N t @ M is approximately an increase in randomness by a factor of M. 3 An optimum noncoherent MFSK detector consists of a bank of energy detectors each matched to one of the M frequencies in the MFSK set. In terms of this structure, the notion of orthogonality implies that for a given transmitted frequency there will be no crosstalk (energy spillover) in any of the other M-1 energy detectors. FIGURE 70.15An FH-MFSK system. ? 2000 by CRC Press LLC For FFH, where R c = R h , the spacing between FH/MFSK tones is equal to the hop rate. Thus, the entire spread spectrum band is partitioned into a total of N t = W ss /R h = W ss /R c equally spaced FH tones, each of which is assigned to a unique k-tuple of the PN code generator that drives the frequency synthesizer. Since for FFH there are R h /R s hops per symbol, then the metric used to make a noncoherent decision on a particular symbol is obtained by summing up R h /R s detected chip (hop) energies, resulting in a so-called noncoherent combining loss. Time Hopping Modulation Time hopping (TH) is to spread spectrum modulation what pulse position modulation (PPM) is to information modulation. In particular, consider segmenting time into intervals of T f seconds and further segment each T f interval into M T increments of width T f /M T . Assuming a pulse of maximum duration equal to T f /M T , then a time hopping spread spectrum modulation would take the form FIGURE 70.16(a)Frequency distribution for FH-4FSK —nonoverlapping bands. Dashed lines indicate lo cation of hop frequencies. FIGURE 70.16(b)Frequency distribution for FH-4FSK — overlapping bands. ? 2000 by CRC Press LLC (70.7) where a n denotes the pseudorandom position (one of M T uniformly spaced locations) of the pulse within the T r -second interval. For DS and FH, we saw that multiplicative modulation, that is the transmitted signal is the product of the SS and information signals, was the natural choice. For TH, delay modulation is the natural choice. In particular, a TH-SS modulation takes the form (70.8) where d(t) is a digital information modulation at a rate 1/T s . Finally, the dehopping procedure at the receiver consists of removing the sequence of delays introduced by c(t), which restores the information signal back to its original form and spreads the interferer. Hybrid Modulations By blending together several of the previous types of SS modulation, one can form hybrid modulations that, depending on the system design objectives, can achieve a better performance against the interferer than can any of the SS modulations acting alone. One possibility is to multiply several of the c(t) wideband waveforms [now denoted by c (i) (t) to distinguish them from one another] resulting in a SS modulation of the form (70.9) Such a modulation may embrace the advantages of the various c (i) (t), while at the same time mitigating their individual disadvantages. Applications of Spread Spectrum Military Antijam (AJ) Communications.As already noted, one of the key applications of spread spectrum is for antijam communications in a hostile environment. The basic mechanism by which a direct sequence spread spectrum receiver attenuates a noise jammer was illustrated in Sec. 70.3. Therefore, in this section, we will concentrate on tone jamming. Assume the received signal, denoted r(t), is given by (70.10) where x(t) is given in Eq. (70.4), A is a constant amplitude, (70.11) and n w (t) is additive while Gaussian noise (AWGN) having two sided spectral density N 0 /2. In Eq. (70.11), a is the amplitude of the tone jammer and q is a random phase uniformly distributed in [0, 2p]. If we employ the standard correlation receiver of Fig. 70.17, it is straightforward to show that the final test statistic out of the receiver is given by (70.12) ct pt n a M T n n T f () =-+ ? è ? ? ? ÷ é ? ê ê ù ? ú ú =-￥ ￥ ? xt ct dt j f cT () =- ()( ) + ( ) []{} Re exp 2pf ct ct i i () = () () ? rt Axt It nt w () = () + () + () It ft c () =+ ( ) apqcos2 gT AT ctt NT bb b T b () =+ () + () ò aqcos d 0 ? 2000 by CRC Press LLC where N(T b ) is the contribution to the test statistic due to the AWGN. Noting that, for rectangular chips, we can express (70.13) where (70.14) is one-half of the processing gain. it is straightforward to show that, for a given value of q, the signal-to-noise- plus-interference ratio, denoted by S/N total , is given by (70.15) In Eq. (70.15), the jammer power is (70.16) and the signal power is (70.17) If we look at the second term in the denominator of Eq. (70.15), we see that the ratio J/S is divided by M. Realizing that J/S is the ratio of the jammer power to the signal power before despreading, and J/MS is the ratio of the same quantity after despreading, we see that, as was the case for noise jamming, the benefit of employing direct sequence spread spectrum signalling in the presence of tone jamming is to reduce the effect of the jammer by an amount on the order of the processing gain. Finally, one can show that an estimate of the average probability of error of a system of this type is given by (70.18) where (70.19) FIGURE 70.17 ct t T c c T i i M b ( ) = ò ? = d 0 1 M T T b c = D S N N E J MS b total = + ? è ? ? ? ÷ 1 2 0 2 cos q J = D a 2 2 S A = D 2 2 P S N d e =- ? è ? ? ? ÷ ò 1 2 0 2 p fq p total f p xe yy x ( ) = - -￥ ò D 1 2 22 d ? 2000 by CRC Press LLC If Eq. (70.18) is evaluated numerically and plotted, the results are as shown in Fig. 70.18. It is clear from this figure that a large initial power advantage of the jammer can be overcome by a sufficiently large value of the processing gain. Low-Probability of Intercept (LPI).The opposite side of the AJ problem is that of LPI, that is, the desire to hide your signal from detection by an intelligent adversary so that your transmissions will remain unnoticed and, thus, neither jammed nor exploited in any manner. This idea of designing an LPI system is achieved in a variety of ways, including transmitting at the smallest possible power level, and limiting the transmission time to as short an interval in time as is possible. The choice of signal design is also important, however, and it is here that spread spectrum techniques become relevant. The basic mechanism is reasonably straightforward; if we start with a conventional narrowband signal, say a BPSK waveform having a spectrum as shown in Fig. 70.19(a), and then spread it so that its new spectrum is as shown in Fig. 70.19(b), the peak amplitude of the spectrum after spreading has been reduced by an amount on the order of the processing gain relative to what it was before spreading. Indeed, a sufficiently large processing gain will result in the spectrum of the signal after spreading falling below the ambient thermal noise level. Thus, there is no easy way for an unintended listener to determine that a transmission is taking place. That is not to say the spread signal cannot be detected, however, merely that it is more difficult for an adversary to learn of the transmission. Indeed, there are many forms of so-called intercept receivers that are specifically designed to accomplish this very task. By way of example, probably the best known and simplest to implement is a radiometer, which is just a device that measures the total power present in the received signal. FIGURE 70.18 ? 2000 by CRC Press LLC In the case of our intercept problem, even though we have lowered the power spectral density of the transmitted signal so that it falls below the noise floor, we have not lowered its power (i.e., we have merely spread its power over a wider frequency range). Thus, if the radiometer integrates over a sufficiently long period of time, it will eventually determine the presence of the transmitted signal buried in the noise. The key point, of course, is that the use of the spreading makes the interceptor’s task much more difficult, since he has no knowledge of the spreading code and, thus, cannot despread the signal. Commercial Multiple Access Communications. From the perspective of commercial applications, probably the most important use of spread spectrum communications is as a multiple accessing technique. When used in this manner, it becomes an alternative to either frequency division multiple access (FDMA) or time division multiple access (TDMA) and is typically referred to as either code division multiple access (CDMA) or spread spectrum multiple access (SSMA). When using CDMA, each signal in the set is given its own spreading sequence. As opposed to either FDMA, wherein all users occupy disjoint frequency bands but are transmitted simultaneously in time, or TDMA, whereby all users occupy the same bandwidth but transmit in disjoint intervals of time, in CDMA, all signals occupy the same bandwidth and are transmitted simultaneously in time; the different waveforms in CDMA are distinguished from one another at the receiver by the specific spreading codes they employ. Since most CDMA detectors are correlation receivers, it is important when deploying such a system to have a set of spreading sequences that have relatively low-pairwise cross-correlation between any two sequences in the set. Further, there are two fundamental types of operation in CDMA, synchronous and asynchronous. In the former case, the symbol transition times of all of the users are aligned; this allows for orthogonal sequences to be used as the spreading sequences and, thus, eliminates interference from one user to another. Alternatively, if no effort is made to align the sequences, the system operates asynchronously; in this latter mode, multiple access interference limits the ultimate channel capacity, but the system design exhibits much more flexibility. CDMA has been of particular interest recently for applications in wireless communications. These applica- tions include cellular communications, personal communications services (PCS), and wireless local area net- works. The reason for this popularity is primarily due to the performance that spread spectrum waveforms display when transmitted over a multipath fading channel. To illustrate this idea, consider DS signalling. As long as the duration of a single chip of the spreading sequence is less than the multipath delay spread, the use of DS waveforms provides the system designer with FIGURE 70.19 ? 2000 by CRC Press LLC one or two options. First, the multipath can be treated as a form of interference, which means the receiver should attempt to attenuate it as much as possible. Indeed, under this condition, all of the multipath returns that arrive at the receiver with a time delay greater than a chip duration from the multipath return to which the receiver is synchronized (usually the first return) will be attenuated because of the processing gain of the system. Alternately, the multipath returns that are separated by more than a chip duration from the main path represent independent “looks” at the received signal and can be used constructively to enhance the overall performance of the receiver. That is, because all of the multipath returns contain information regarding the data that is being sent, that information can be extracted by an appropriately designed receiver. Such a receiver, typically referred to as a RAKE receiver, attempts to resolve as many individual multipath returns as possible and then to sum them coherently. This results in an implicit diversity gain, comparable to the use of explicit diversity, such as receiving the signal with multiple antennas. The condition under which the two options are available can be stated in an alternate manner. If one envisions what is taking place in the frequency domain, it is straightforward to show that the condition of the chip duration being smaller than the multipath delay spread is equivalent to requiring that the spread bandwidth of the transmitted waveform exceed what is called the coherence bandwidth of the channel. This latter quantity is simply the inverse of the multipath delay spread and is a measure of the range of frequencies that fade in a highly correlated manner. Indeed, anytime the coherence bandwidth of the channel is less than the spread bandwidth of the signal, the channel is said to be frequency selective with respect to the signal. Thus, we see that to take advantage of DS signalling when used over a multipath fading channel, that signal should be designed such that it makes the channel appear frequency selective. In addition to the desirable properties that spread spectrum signals display over multipath channels, there are two other reasons why such signals are of interest in cellular-type applications. The first has to do with a concept known as the reuse factor. In conventional cellular systems, either analog or digital, in order to avoid excessive interference from one cell to its neighbor cells, the frequencies used by a given cell are not used by its immediate neighbors (i.e., the system is designed so that there is a certain spatial separation between cells that use the same carrier frequencies). For CDMA, however, such spatial isolation is typically not needed, so that so-called universal reuse is possible. Further, because CDMA systems tend to be interference limited, for those applications involving voice transmission, an additional gain in the capacity of the system can be achieved by the use of voice activity detection. That is, in any given two-way telephone conversation, each user is typically talking only about 50% of the time. During the time when a user is quiet, he is not contributing to the instantaneous interference. Thus, if a sufficiently large number of users can be supported by the system, statistically only about one-half of them will be active simultaneously, and the effective capacity can be doubled. Interference Rejection. In addition to providing multiple accessing capability, spread spectrum techniques are of interest in the commercial sector for basically the same reasons they are in the military community, namely their AJ and LPI characteristics. However, the motivations for such interest differ. For example, whereas the military is interested in ensuring that systems they deploy are robust to interference generated by an intelligent adversary (i.e., exhibit jamming resistance), the interference of concern in commercial applications is unintentional. It is sometimes referred to as co-channel interference (CCI) and arises naturally as the result of many services using the same frequency band at the same time. And while such scenarios almost always allow for some type of spatial isolation between the interfering waveforms, such as the use of narrow-beam antenna patterns, at times the use of the inherent interference suppression property of a spread spectrum signal is also desired. Similarly, whereas the military is very much interested in the LPI property of a spread spectrum waveform, as indicated in Sec. 70.3, there are applications in the commercial segment where the same charac- teristic can be used to advantage. To illustrate these two ideas, consider a scenario whereby a given band of frequencies is somewhat sparsely occupied by a set of conventional (i.e., nonspread) signals. To increase the overall spectral efficiency of the band, a set of spread spectrum waveforms can be overlaid on the same frequency band, thus forcing the two sets of users to share common spectrum. Clearly, this scheme is feasible only if the mutual interference that one set of users imposes on the other is within tolerable limits. Because of the interference suppression properties ? 2000 by CRC Press LLC of spread spectrum waveforms, the despreading process at each spread spectrum receiver will attenuate the components of the final test statistic due to the overlaid narrowband signals. Similarly, because of the LPI characteristics of spread spectrum waveforms, the increase in the overall noise level as seen by any of the conventional signals, due to the overlay, can be kept relatively small. Defining Terms Antijam communication system: A communication system designed to resist intentional jamming by the enemy. Chip time (interval): The duration of a single pulse in a direct sequence modulation; typically much smaller than the formation symbol interval. Coarse alignment: The process whereby the received signal and the despreading signal are aligned to within a single chip interval. Dehopping: Despreading using a frequency-hopping modulation. Delay-locked loop: A particular implementation of a closed-loop technique for maintaining fine alignment. Despreading: The notion of decreasing the bandwidth of the received (spread) signal back to its information bandwidth. Direct sequence modulation: A signal formed by linearly modulating the output sequence of a pseudorandom number generator onto a train of pulses. Direct sequence spread spectrum: A spreading technique achieved by multiplying the information signal by a direct sequence modulation. Fast frequency-hopping: A spread spectrum technique wherein the hop time is less than or equal to the information symbol interval, i.e., there exist one or more hops per data symbol. Fine alignment: The state of the system wherein the received signal and the despreading signal are aligned to within a small fraction of a single chip interval. Frequency-hopping modulation: A signal formed by nonlinearly modulating a train of pulses with a sequence of pseudorandomly generated frequency shifts. Hop time (interval): The duration of a single pulse in a frequency-hopping modulation. Hybrid spread spectrum: A spreading technique formed by blending together several spread spectrum tech- niques, e.g., direct sequence, frequency-hopping, etc. Low-probability-of-intercept communication system: A communication system designed to operate in a hostile environment wherein the enemy tries to detect the presence and perhaps characteristics of the friendly communicator’s transmission. Processing gain (spreading ratio): The ratio of the spread spectrum bandwidth to the information data rate. Radiometer: A device used to measure the total energy in the received signal. Slow frequency-hopping: A spread spectrum technique wherein the hop time is greater than the information symbol interval, i.e., there exists more than one data symbol per hop. Spread spectrum bandwidth: The bandwidth of the transmitted signal after spreading. Spreading: The notion of increasing the bandwidth of the transmitted signal by a factor far in excess of its information bandwidth. Search algorithm: A means for coarse aligning (synchronizing) the despreading signal with the received spread spectrum signal. Tau-dither loop: A particular implementation of a closed-loop technique for maintaining fine alignment. Time-hopping spread spectrum: A spreading technique that is analogous to pulse position modulation. Tracking algorithm: An algorithm (typically closed loop) for maintaining fine alignment. Related Topics 69.1 Modulation and Demodulation ? 73.2 Noise Reference J.D. Gibson, The Mobile Communications Handbook, Boca Raton, FL: CRC Press, 1996. ? 2000 by CRC Press LLC Further Information M.K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum Communications Handbook, New York: McGraw Hill, 1994 (previously published as Spread Spectrum Communications, Computer Science Press, 1985). R.E. Ziemer and R. L. Peterson, Digital Communications and Spread Spectrum Techniques, New York: Macmillan, 1985. J.K. Holmes, Coherent Spread Spectrum Systems, New York: John Wiley & Sons, 1982. R.C. Dixon, Spread Spectrum Systems, 3rd ed., New York: John Wiley & Sons, 1994. C.F. Cook, F. W. Ellersick, L. B. Milstein, and D. L. Schilling, Spread Spectrum Communications, IEEE Press, 1983. ? 2000 by CRC Press LLC