1. 1 Chapter 1 Introduction to spread-spectrum communications Part I

2. 2 1.1 What is spread spectrum? Spread spectrum: –A modulation technique that produces a spectrum for the transmitted signal much wider than the usual bandwidth needed to convey a particular stream of information. Narrowband modulation: –A modulation technique that produces a transmitted signal with the usual bandwidth as opposed to a spread spectrum modulation. –BPSK, QPSK, QAM, and MSK are common examples of narrowband modulation techniques.

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4. 4 1.2 Why spread spectrum? Resistant to jamming and interference Difficult to intercept Better multipath resolution, i.e., resistant to fading Time and range measurement Code division multiple access

5. 5 1.3 What is code division multiple access (CDMA)? CDMA: –A multiple access scheme which allows multiple users to communicate simultaneously using the same frequency band by assigning different ‘codes’ to different users. Usually, CDMA is achieved by spread spectrum techniques. FDMA: –A multiple access scheme which allows multiple users to communicate simultaneously by assigning non-overlapping frequency bands to different users. TDMA: –A multiple access scheme which allows multiple users to communicate using the same frequency band by restricting different users to transmit in non-overlapping time slots.

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7. 7 Why CDMA? – Military needs – Larger capacity for wireless cellular systems Practical systems – GPS – IS-95 – W-CDMA, CDMA2000 – Numerous applications in the ISM band

8. 8 Review of Digital Communication Theory Maximum likelihood receiver –Assume that the communication channel is corrupted by an additive white Gaussian noise (AWGN) with two-sided power spectral density N 0 /2 W/Hz. –The transmitter sends a signal chosen from the set of M signals –Further assume that all the M signals are time-limited to [0; T], where T is called the symbol duration.

9. 9 –The received signal r(t) is given by

10. 10 Our goal is to develop a receiver which observes the received signal r(t) and determines which one of the M signals is being sent based on maximizing the likelihood function. Define what the likelihood function is. –By employing the Gram-Schmidt procedure, we can construct a set of N ( ) orthonormal functions (all are time limited to [0; T]) which spans the signal space formed by –We augment this set of functions by another set of orthonormal functions so that the augmented set forms an orthonormal basis for the space of square-integrable functions.

11. 11 Based on this representation, we can rewrite (1.1) as where

12. 12 is a sufficient statistic for determining which signal is being sent, i.e., determining the value of m. Rewriting (1.2) with these finite dimensional vectors, we have –n N is a zero mean Gaussian random vector whose covariance matrix is. The maximum likelihood (ML) receiver makes a decision (select ) which maximizes the likelihood function defined as the following conditional probability density function:

13. 13 ML receiver picks such that the squared Euclidean distance between the signal vector s mN and the receiver vector r N, is minimized.

14. 14 is called the correlation metric between the received signal r(t) and the transmitted signal s m (t). E m is the energy of the transmitted signal s m (t).

15. 15 1.2 Matched filter receiver The correlators in Figure 1.2 can be replaced by the linear filters and samplers as shown in Figure 1.3.

16. 16 The matched filter has the optimal property that it is the linear filter that maximizes the output signal-to-noise ratio (SNR).

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18. 18 Therefore, the matched filter s(T - t), among all linear filters, maximizes the output SNR.

19. 19 1.3 Signal space representation Suppose is an orthonormal basis for the signal space spanned by a set of square integrable signal waveforms We represent the signal waveforms by a set of M N-dimensional vectors with respect to the basis More precisely, for s m (t) is represented by the N-dimensional vector Given the basis, we can uniquely determine the signal s m (t) from the vector s m or vice versa.

20. 20 The notation denotes the Euclidean norm of a vector. The first identity states that the inner products in the function space and the vector space are equivalent. The second identity states that the squared distance in the function space is the same as the squared Euclidean distance in the vector space.

21. 21 Consider the following signal set of the QPSK scheme:

22. 22 A simple basis for this signal set is Using this basis, the corresponding signal vectors are －

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24. 24 1.4 ML receiver error analysis A symbol error event occurs when the decision made by the receiver is different from the transmitted symbol. Let denotes the conditional symbol error probability given that s m (t) is being transmitted. P m denotes that probability that the transmitter sends s m (t). The average symbol error probability, P s, is given by For simplicity, we assume all the signals are equally likely to be transmitted, i.e., P m = 1/M. Then the problem reduces to evaluating

25. 25 1.4.2 BPSK For the case of BPSK (binary antipodal signaling), the matched filter receiver in Section 1.2 is the ML receiver. The receiver compares the sampled output Y of the matched filter to the threshold zero. If Y > 0, the receiver decides that s(t) = s 0 (t) is sent. Otherwise, it decides that s(t) = s 1 (t) is sent. From (1.14) and (1.15), we know that the noise sample Y n is a zero mean Gaussian random variable with variance

26. 26 Suppose s 0 (t) is being sent, then Y is a Gaussian random variable with mean E and variance

27. 27 1.4.2 General case (a geometric approach) Now assume that we employ M-ary signaling, i.e., the transmitter sends a signal out from the set Using the vector representation in Section 1.1, we know that the ML receiver decides that the m-th signal is sent when the Euclidean distance is the smallest among all the M signal vectors. If we draw the signal vectors as points in the constellation diagram as shown in Figure 1.6, the geometric meaning of the ML decision rule is that the signal s mN closest to the receiver vector r N is selected.

28. 28 A diagram showing the signal points and their corresponding decision regions is known as the Voronoi diagram of a modulation scheme.

29. 29 Equivalently, we can construct a decision region (based on the minimum distance principle) for each of the signal point in the constellation diagram. Decide a specific signal point is sent if the received vector r N falls into the corresponding decision region.

30. 30 Suppose s m (t), for some is being sent, and let R m denotes the decision region for s m (t). We make an error if the received vector r N falls outside R m. Therefore, the conditional symbol error probability given that s m (t) is sent,

31. 31 The first special case we consider is the binary signaling case (M = 2, ). It is intuitive that the decision regions for the signal points s 0 and s 1 are separated by the hyperplane half-way between the signal points and perpendicular to the line joining the two signal points. The next step is to evaluate the integral in (1.24). Suppose s 0 (t) is being sent, we know that where is the variance of an element of the noise vector n N.

32. 32 The next special case we consider is the QPSK example given in Section 1.3 (M = 4, N = 2). It is again obvious that the decision region for a signal point is the quadrant in which the signal point is located. Suppose s 0 (t) is being sent, then

33. 33 1.4.3 Union bound When the exact symbol error probability is too difficult to evaluate, we resort to bounds and approximations. One of such methods is the union bound. Suppose s 0 (t) is being transmitted, The event in (1.27) is exactly the same as the error event as if there were only two signals, s 0 (t) and s m (t) ( ), in the signal set.

34. 34 The union bound of the conditional symbol error probability as By averaging over all the signals, we obtain the union bound for the average symbol error probability as The union bound for the symbol error probability for the QPSK

35. 35 By symmetry, we have which is slightly larger than the exact symbol error probability given in (1.26).

36. 36 1.5 Complex envelope Very often in a communication system, we do not transmit the lowpass baseband signal directly. Instead, we mix the baseband signal with a carrier up to a certain frequency, which matches the electromagnetic propagation characteristic of the channel. As a result, the actual transmitted signal is a bandpass signal. In this section, we introduce the concept of complex envelope which provides a convenient way to represent bandpass signals.

37. 37 1.5.1 Narrowband signal Suppose s(t) is a (real-valued) bandpass signal with most of its frequency content concentrated in a narrow band in the vicinity of a center frequency f c. A sufficient condition is that the Fourier transform of s(t) satisfies We refer to this condition as the narrowband assumption. For a bandpass signal s(t) satisfying the narrowband assumption stated above, it can be shown that s(t) can be represented by an in-phase component x(t) and a quadrature component y(t).

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39. 39 Using (1.34) and (1.35), we can reconstruct the real-valued bandpass signal s(t) back from its complex envelope.

40. 40 How to obtain the complex envelope from the signal s(t) –The complex envelope as –The Fourier transform of s(t) is given

41. 41 The complex envelope is sometimes called the lowpass equivalent signal of s(t).

42. 42 1.5.2 Bandpass filter We can use the complex envelope in the previous section to represent the impulse response h(t) of a bandpass filter given that h(t) satisfies the narrowband assumption stated before. Hence, if is the complex envelope of h(t), then If a bandpass signal (satisfying the narrowband assumption) s i (t) is the input to the bandpass filter h(t), then the output from the filter s o (t) also satisfies the narrowband assumption and In the frequency domain,

43. 43 From (1.37), the Fourier transform of the complex envelope, of s o (t) is given by By taking inverse Fourier transform on both sides of (1.42), we obtain Hence, we can convolute the complex envelopes of h(t) and s i (t) and then convert the result back to obtain the output bandpass signal.

44. 44 1.5.3 Narrowband process Suppose n(t) is a wide-sense stationary (WSS) process with zero mean and power spectral density If satisfies the narrowband assumption, then n(t) is called a narrowband process. n(t) can also be written as where n x (t) and n y (t) are zero-mean jointly WSS processes. If n(t) is Gaussian, n x (t) and n y (t) are jointly Gaussian.

45. 45 Let us define the complex envelope of the random process n(t)

46. 46 If we treat the autocorrelation function as a bandpass signal, then is its complex envelope. Hence, we can use the results in Section 1.5.1 to convert between and. A common example of narrowband process is the bandpass additive Gaussian noise n(t) with zero mean and power spectral density n(t) can be written as

47. 47 The complex envelope of n(t) is given by Using the result above and (1.37), the power spectral density of the complex envelope is given by Taking inverse Fourier transform, we get

48. 48 For the case where bandpass transmitted signals are sent through a channel corrupted by n(t) and the bandwidths of the transmitted signals are much smaller than the carrier frequency, we approximate in (1.53) by This means that the lowpass equivalent of the additive bandpass Gaussian noise looks white to the lowpass equivalents of the transmitted signals.

49. 49 1.6 Noncoherent receiver As we mentioned before, since most communication systems transmit bandpass signals instead of baseband ones, we focus on this kind of signals and use the complex envelopes to represent them here. Again, we consider the simple case of a non-dispersive channel, for which we can model the received signal as –Where A > 0 represents the channel gain (attenuation) –θrepresents the carrier phase shift due to propagation delay, local oscillator mismatch, and etc. –n(t) is the complex AWGN with autocorrelation function

50. 50 Suppose the receiver knows the value of θ, the problem reduces to the one in Section 1.1. Hence we can use the correlation receiver in Figure 1.2 to detect the received signal r(t). Generally, receivers that make use of the phase information are referred to as coherent receivers. Therefore, the correlation receiver in Figure 1.2 and the matched filter receiver in Figure 1.4 are coherent receivers. For coherent reception, we need to estimate the carrier phase. –This estimation can sometimes be hard to perform, and inaccurate estimation of the carrier phase will significantly degrade the performance of the coherent ML receiver.

51. 51 One alternative to coherent reception is to avoid using the phase information. To do so, we model the carrier phase as a random variable uniformly distributed on [0; 2π). Following steps similar to those in Section 1.2, we can develop the ML receiver for this case. –The resulting receiver is known as the noncoherent ML receiver. –For the case where the transmitted signals have equal energies. –The ML receiver assumes the simple form shown in Figure 1.8. –This receiver is usually referred to as the envelope receiver or the square-law receiver.

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53. 53 It is difficult to evaluate the symbol error probability for a general M-ary signal set received by the noncoherent ML receiver. For the special case of equal-energy binary orthogonal signals, we state that the average symbol error probability (assuming equal a priori probabilities) is given by where E is the signal energy.

54. 54 1.7 Power spectrum In this section, we consider a more realistic model in which a train of pulses are transmitted. For simplicity, we ignore the white noise and assume that the (complex envelope of the) received signal is given by

55. 55 –a k ’s are independent identically distributed (iid) random variables with mean zero and variance A 2. –b k ’s are also iid random variables with mean zero and variance B 2. –The two data streams are independent. –Δ can be interpreted as the propagation delay – are the pulses for the in-phase and quadrature channels, respectively. –s(t) is a zero-mean random process. This model almost covers all practical quadrature modulation schemes.

56. 56 Our objective is to evaluate the autocorrelation function of s(t). First, let us modelΔas a random variable which is uniformly distributed on [0; T s ), and is independent to both Then the autocorrelation function of s(t) is given by The last equality in (1.59) follows from the fact that the two data streams consist of zero-mean independent random variables.

57. 57 Similarly, we have

58. 58 Therefore, the process s(t) is WSS and The power spectral density (power spectrum) of s(t) is given by

59. 59 We consider the BPSK scheme where Let A 2 = 2. We consider two cases: –T s = T, the power spectrum is –T s = T/10, the power spectrum is

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61. 61 1.8 References [1] J. G. Proakis, Digital Communications, 3rd Ed., McGraw-Hill, Inc., 1995.