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Chapter 1 Introduction to spread-spectrum communications Part I 1 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. 2 3 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 4 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. 5 6 • Why CDMA? – Military needs – Larger capacity for wireless cellular systems • Practical systems – GPS – IS-95 – W-CDMA, CDMA2000 – Numerous applications in the ISM band 7 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 N0/2 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. 8 – The received signal r(t) is given by 9 • 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 (N M) 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. 10 • Based on this representation, we can rewrite (1.1) as where 11 • 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 – nN is a zero mean Gaussian random vector whose covariance N matrix is 20 I . • The maximum likelihood (ML) receiver makes a decision (select ) which maximizes the likelihood function defined as the following conditional probability density function: 12 • ML receiver picks such that the squared Euclidean distance between the signal vector smN and the receiver vector rN, is minimized. 13 • is called the correlation metric between the received signal r(t) and the transmitted signal sm(t). Em is the energy of the transmitted signal sm(t). 14 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. 15 • The matched filter has the optimal property that it is the linear filter that maximizes the output signal-to-noise ratio (SNR). 16 17 • Therefore, the matched filter s(T - t), among all linear filters, maximizes the output SNR. 18 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 sm(t) is represented by the N-dimensional vector • Given the basis, we can uniquely determine the signal sm(t) from the vector sm or vice versa. 19 • 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. 20 • Consider the following signal set of the QPSK scheme: 21 • A simple basis for this signal set is - • Using this basis, the corresponding signal vectors are 22 23 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 sm(t) is being transmitted. • Pm denotes that probability that the transmitter sends sm(t). • The average symbol error probability, Ps, is given by • For simplicity, we assume all the signals are equally likely to be transmitted, i.e., Pm = 1/M. • Then the problem reduces to evaluating 24 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) = s0(t) is sent. Otherwise, it decides that s(t) = s1(t) is sent. • From (1.14) and (1.15), we know that the noise sample Yn is a zero mean Gaussian random variable with variance 25 • Suppose s0(t) is being sent, then Y is a Gaussian random variable with mean E and variance 26 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 smN closest to the receiver vector rN is selected. 27 • A diagram showing the signal points and their corresponding decision regions is known as the Voronoi diagram of a modulation scheme. 28 • 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 rN falls into the corresponding decision region. 29 • Suppose sm(t), for some is being sent, and let Rm denotes the decision region for sm(t). • We make an error if the received vector rN falls outside Rm. • Therefore, the conditional symbol error probability given that sm(t) is sent, 30 • The first special case we consider is the binary signaling case (M = 2, N 2 ). • It is intuitive that the decision regions for the signal points s0 and s1 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 s0(t) is being sent, we know that where nN. is the variance of an element of the noise vector 31 • 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 s0 (t) is being sent, then 32 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 s0 (t) is being transmitted, • The event in (1.27) is exactly the same as the error event as if there were only two signals, s0 (t) and sm (t) ( m 1), in the signal set. 33 • 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 34 • By symmetry, we have which is slightly larger than the exact symbol error probability given in (1.26). 35 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. 36 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 fc. • 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). 37 38 • • Using (1.34) and (1.35), we can reconstruct the real-valued bandpass signal s(t) back from its complex envelope s (t ) . 39 • How to obtain the complex envelope s (t ) from the signal s(t) – The complex envelope as – The Fourier transform S ( ) of s(t) is given 40 • The complex envelope s (t ) is sometimes called the lowpass equivalent signal of s(t). 41 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 h (t ) is the complex envelope of h(t), then • If a bandpass signal (satisfying the narrowband assumption) si(t) is the input to the bandpass filter h(t), then the output from the filter so(t) also satisfies the narrowband assumption and • In the frequency domain, 42 • From (1.37), the Fourier transform of the complex envelope, s0 (t ) of so(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 si(t) and then convert the result back to obtain the output bandpass signal. 43 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 nx(t) and ny(t) are zero-mean jointly WSS processes. • If n(t) is Gaussian, nx(t) and ny(t) are jointly Gaussian. 44 • Let us define the complex envelope n(t ) of the random process n(t) 45 • 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 46 • The complex envelope of n(t) is given by • Using the result above and (1.37), the power spectral density n(t ) of the complex envelope is given by • Taking inverse Fourier transform, we get 47 • 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. 48 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 49 • 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. 50 • 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. 51 52 • 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. 53 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 54 – ak’s are independent identically distributed (iid) random variables with mean zero and variance A2. – bk’s are also iid random variables with mean zero and variance B2. – 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. 55 • Our objective is to evaluate the autocorrelation function of s(t). • First, let us modelΔas a random variable which is uniformly distributed on [0; Ts), 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. 56 • Similarly, we have 57 • Therefore, the process s(t) is WSS and • The power spectral density (power spectrum) of s(t) is given by 58 • We consider the BPSK scheme where • Let A2 = 2. • We consider two cases: – Ts = T, the power spectrum is – Ts = T/10, the power spectrum is 59 60 1.8 References [1] J. G. Proakis, Digital Communications, 3rd Ed., McGraw-Hill, Inc., 1995. 61