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Chapter 2

Chapter 2. Introduction to spread-spectrum communications Part II. As discussed in Chapter 1 Part I, a spread spectrum modulation produces a transmitted spectrum much wider than the minimum bandwidth required. There are many ways to generate spread spectrum signals.

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Chapter 2

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  1. Chapter 2 Introduction to spread-spectrum communications Part II

  2. As discussed in Chapter 1 Part I, a spread spectrum modulation produces a transmitted spectrum much wider than the minimum bandwidth required. There are many ways to generate spread spectrum signals. • We are going to introduce some of the most common spread spectrum techniques such as direct sequence (DS) and frequency hop (FH). • Of course, one can also mix these spread spectrum techniques to form hybrids which have the advantages of different techniques. • Spread spectrum originates from military needs and finds most applications in hostile communication environments. • We will start by briefly looking at the advantage of spreading the spectrum in the presence of a Gaussian jammer as our motivation to study spread spectrum communications.

  3. 1.1 Motivation—a jamming analysis • Consider the transmission of a bit stream through an AWGN channel. • We employ BSPK modulation at the carrier frequency • The channel is also corrupted by an intentional jammer. • The received signal r(t), in complex envelope representation, is given by (1.1) (1.2) • s(t) is the transmitted signal. • n(t) is the AWGN with power spectrum • j(t) is the jamming signal. • T is the symbol duration. • Tcis the symbol pulse width. • P is the average transmitted power.

  4. From Section 1.7, we know that the spectrum of the transmitted signal s(t) is given by (2.3) • If we model the bits as iid random variables and Δas a uniform random variable on [0; T). • By examining the spectrum of the transmitted signal, a reasonable jamming strategy is to put all the jamming power PJinto the band coincides with the main lobe of the signal spectrum, i.e., from 2π/Tc to -2π/Tc rad/sec. • For simplicity, we assume that j(t) is a zero-mean WSS Gaussian random process with power spectrum otherwise. • Moreover, n(t) and j(t) are independent.

  5. Neglecting the jamming signal, the ML receiver is the matched filter receiver developed in Section 1.2. • We redraw the matched filter receiver in Figure 2.1 here for convenience. Figure 1.1: Matched filter receiver for BPSK data with jammer

  6. Let us consider the performance of the matched filter when the jamming signal is present. • Conditioning on Δ, the sampled output of the matched filter corresponding to the kth symbol is (1.4) (1.5) (1.6) (1.7)

  7. From the assumptions above, we know that jkand nk are independent zero-mean Gaussian random variables. • It remains to determine their variances. • The variance of nk is • For jk, we note that its variance is equal to the value of the autocorrelation function of the matched filter output component due to j(t) at 0. • Using the Fourier relationship between autocorrelation function and power spectral density, we have

  8. Now, we can calculate the symbol (bit) error probability of the communication system described above. • By symmetry, we know that the average symbol error probability is equal to the conditional symbol error probability given that, say, bk = 1. • Under the condition that bk = 1, the decision statistic Re[rk] is a Gaussian random variable with mean and variance • Therefore, the symbol error probability is where is the symbol energy.

  9. We suffer a loss in SNR by a factor of 1 + 0.9028PJTc/N0 with respect to the case where the jammer is not present. • There are two ways to reduce the loss in SNR. • For a bandwidth limited channel, we can increase the transmitted power P of the signal. • If power is the main constraint, we can reduce the pulse width Tc. This corresponds to spreading the spectrum of the transmitted signal. • In military applications, one consideration is that we do not want our enemies to intercept or detect our transmission. • The higher the transmission power the more susceptible is the transmission being intercepted.

  10. Therefore, we usually resort to spreading the spectrum of the transmitted signal instead of raising the transmission power. • This is the reason why spread spectrum is originally considered for military communications. • In terms of jamming immunity, the spreading method described above is far from desirable. • Simply reducing Tc is effective only for the continuous Gaussian jammer assumed above. • Since the continuous Gaussian jammer spreads its power across the whole symbol period, for a small Tc, we only need to integrate a small fraction of the symbol duration and, hence, pick up a small jamming energy.

  11. The discussion above brings out an important characteristic of spread spectrum communications. • In order for the receiver to perform properly, it has to know the transmission times of the pulses. • A sequence of pseudo-random transmission times is pre-assigned to both the transmitter and the receiver. • This sequence is generally referred to as a code. • We will see that all spread spectrum techniques contain some forms of pseudorandom codes.

  12. 2.2 Direct sequence spread spectrum • One non-trivial way of spreading the spectrum of the transmitted signal is to modulate the data signal by a high rate pseudo-random sequence of phase-modulated pulses before mixing the signal up to the carrier frequency for transmission. • This spreading method is called direct sequence spread spectrum (DS-SS). • More precisely, suppose the data signal is (1.11) • is the symbol sequence. • T is the symbol duration. • Note that all the signals hereare complex envelopes unless otherwise indicated.

  13. We modulate the data signal b(t) by a spreading signal a(t) which is given by (1.12) • is called the signature sequence • is called the chip waveform, which is time limited to [0; Tc). • We impose that condition that T = NTc, where N, which is usually referred to as the processing gain or the spreading gain, is the number of chips in a symbol and Tc is the separation between consecutive chips. • We normalize the energy of the chip waveforms to Tc. • The spread spectrum signal is given by (1.13) where is the largest integer which is smaller than or equal to x

  14. This general model for DS-SS contains many different modulation and spreading schemes. • Some of the common examples are listed in Table 1.1. • For example, a pictorial description of the BPSK modulation with BPSK spreading scheme is given in Figure 1.2. Table 1.1

  15. Figure 1.2: BPSK modulation and BPSK spreading scheme

  16. To obtain the power spectrum of the spread spectrum signal, we model the spreading elements al as iid zero-mean random variables with and the propagation delay as a uniform random variable. • Moreover, we also normalize the average symbol energy to PT, i.e., • Then the power spectrum of the spread spectrum signal s(t) is (1.14) where is the Fourier transform of the chip waveform • The power spectra of the spread signals for the four schemes shown in Table 1.1 are all given by (1.15)

  17. Comparing this to the power spectrum of the original data signal (1.16) • we see that the spectrum is spread N times wider by the direct sequence technique in (1.15). • In practice, the spreading sequences are pseudo-random. • We will discuss, in Chapter 2, different ways to generate sequences which have properties close to those of random sequences.

  18. In an AWGN channel, the ML receiver for the spread spectrum signal is the matched filter receiver shown in Figure 1.3. • We note that the matched filter is time-varying unless the spreading sequence is periodic with period N. • For the kth symbol, the impulse response of the matched filter hk(t) is given by (1-17) Figure 1.3: Matched filter receiver for the kth symbol of the DS-SS signal

  19. For BPSK modulation, the decision device gives decision • For QPSK modulation, the decision device gives decision • Alternatively, we can implement the matched filter receiver as shown in Figure 1.4. Figure 1.4: Equivalent implementation of the matched filter receiver in Figure 1.3

  20. 1.3 Frequency hop spread spectrum • Another common method to spread the transmission spectrum of a data signal is to (pseudo) randomly hop the data signal over different carrier frequencies. • This spreading method is called frequency hop spread spectrum (FH-SS). • Usually, the available band is divided into non-overlapping frequency bins. • The data signal occupies one and only one bin for a duration Tc and hops to another bin afterward. • When the hopping rate is faster than the symbol rate (i.e., T > Tc), the FH scheme is referred to as fast hopping. • Otherwise, it is referred to as slow hopping. • A typical FH-SS transmitter and the corresponding receiver are shown in Figures 1.6 and 1.7, respectively.

  21. Figure 1.6: Transmitter for FH-SS

  22. Figure 1.7: Receiver for FH-SS

  23. Because it is practically difficult to build coherent frequency synthesizers, modulation schemes, such as M-ary FSK, which allow noncoherent detection are usually employed for the data signal. • For M-ary FSK, the data signal can be expressed as (1.25) • The frequency synthesizer outputs a hopping signal (1.26)

  24. This means that there are L frequency bins in the FH-SS system. • T = NTc for fast hopping, Tc = NT for slow hopping. • For fast hopping , the FH-SS signal is given by (1.27) • For slow hopping , the FH-SS signal is given by (1.28)

  25. The orthogonality requirement for the FSK signals forces the separation between adjacent FSK symbol frequencies be at least 2π/Tc for fast hopping, or 2π/T for slow hopping. • Hence, the minimum separation between adjacent hopping frequencies is 2Mπ/Tc for fast hopping, or 2Mπ/T for slow hopping. • For example, Figure 1.8 depicts the operation of a fast FH-SS system with 2-FSK modulation (M = 2), 8 hopping bins (L = 8), and 2 hops per symbol (T = 2Tc, N = 2).

  26. Figure 1.8Fast FH-SS system with 2-FSK modulation, 8 hopping bins, and 2 hops per symbol

  27. To obtain the power spectrum of the M-ary FH-SS signal, we model the phases as iid random variables uniformly distributed on [0; 2π). • The hopping frequencies are modelled as iid random variables taking values from the set with equal probabilities. • The FSK symbol frequencies are iid random variables taking values from the set with equal probabilities. • The delay is assumed to be uniformly distributed on [0, Tc) for fast hopping, or [0; T) for slow hopping. • We also assume that all the random variables mentioned above are independent.

  28. With these assumptions, we can show that the power spectrum of the FH-SS signal s(t) is given by • for fast hopping (1.29) • for slow hopping (1.30) • Therefore, the spectrum of the original data signal b(t) is approximately spread by a factor of LN for fast hopping, or by a factor of L for slow hopping.

  29. 1.4 Time hop spread spectrum • In time hop spread spectrum (TH-SS), we spread the spectrum by modulating the data signal by a pseudo-random pulse-position-modulated spreading signal. • Suppose the data signal b(t) is (1-31) • We modulate the data signal by the spreading signal (1-32) where • The resulting spread spectrum signal s(t) is given by (1-33)

  30. To obtain the power spectrum of the TH-SS signal s(t), we model the data symbols bk as iid zero mean random variables with • The pulse location indices ak are assumed to be iid random variables taking values from {0, 1, …, N} with equal probabilities. • The propagation delay is modelled as a uniform random variable on [0; T) as usual. • It can be shown that the power spectral density of the spread spectrum signal s(t) is (1.34)

  31. The matched filter receiver for this spreading method is shown in Figure 1.9. • The sampler is controlled by a timing circuit which is in turn driven by the pseudo-random pulse-location code. • We note that there are other types of TH-SS techniques. • For example, one can use pulse-position modulation for the data signal. • As a result, the spread spectrum signal will be purely pulse-position modulated. • The hopping scheme is similar to the M-ary FSK FH-SS system with frequency bins replaced by time bins.

  32. Figure 1.9: Matched filter receiver for TH-SS

  33. 1.5 Multicarrier spread spectrum • In FH-SS, only one of many possible frequencies is transmitted at a time. • The other extreme is that we transmit all the possible frequencies simultaneously. • The resulting spreading method is called multicarrier spread spectrum (MC-SS). • More precisely, suppose the data signal b(t) is given by (1-35) • We modulate the data signal by the spreading signal (1-36)

  34. The resulting spread spectrum signal is (1.37) • The data and spreading sequences are phase-modulated as in the case of DS-SS. • The carrier frequencies should be chosen so that signals at different frequencies do not interfere each other. • The minimum frequency separation is 2π/T. • To obtain the power spectrum of the spread spectrum signal, we model the spreading elements an,k as iid zero-mean random variables with and the propagation delay as a uniform random variable on [0; T).

  35. We also normalize the average symbol energy to PT by setting • Then the power spectrum of the spread spectrum signal s(t) is (1.38) • The matched filter receiver for MC-SS is shown in Figure 3.10. • When the operation of the correlator branches in Figure 1.10 can be approximately performed by a single FFT. • Hence the matched filter receiver can be implemented very efficiently.

  36. Figure 1.10: Matched filter receiver for MC-SS

  37. To see this, consider the output of the nth correlator for the kth symbol in Figure 1.10 and denote it by zn,k. • Then, (1.39) • In above, we divide the interval into N sub-intervals of length Tc. • In the lth sub-interval, for l = 0, 1,…, N-1, we approximate (1.40) (1.41)

  38. We note that (1.41) says that for each k, the sequence is approximately the DFT (scaled) of the sequence • Therefore, we can implement the correlator branches approximately by sampling r(t) at the chip rate and passing the samples through an FFT circuit. • Similar approximations can also be applied on the transmitter side.

  39. 3.5 References [1] R. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to Spread Spectrum Communications, Prentice Hall, Inc., 1995. [2] M. B. Pursley, “Performance evaluation for phase-coded spread-spectrum multiple-access communication — Part I: System analysis,” IEEE Trans. Commun., vol. 25, no. 8, pp. 795–799, Aug. 1977. [3] R. A. Scholtz, “Multiple access with time-hopping impulse modulation,” Proc. MILCOM ’93, pp. 11-14, Boston, MA, Oct. 1993. [4] N. Yee, J. M. G. Linnartz, and G. Fettweis, “Multi-carrier CDMA in indoor wireless radio networks,” IEICE Trans. Commun., vol. E77-B, no. 7, pp. 900–904, Jul. 1994. [5] S. Kondo and L. B. Milstein, “Performance of multicarrier DS CDMA systems,” IEEE Trans. Commun., vol. 44, no. 2, pp. 238–246, Feb. 1996. [6] R. L. Pickholtz, L. B. Milstein, and D. L. Schilling, “Spread spectrum for mobile communications,” IEEE Trans. Veh. Technol., vol. 40, no. 2, pp. 313–321, May 1991. [7] D. Torrieri, “Principles of spread spectrum communications theory,” Springer 2005.

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