7 Interference suppression and CDMA overlay 7.1 NARROWBAND INTERFERENCE SUPPRESSION To get an initial insight into the problem, we assume that the received signal after frequency down conversion has the form x(i) = b · c(i) + J(i)+ n(i) (7.1) where at sampling instant iT c ,b is data, c(i) is the code, J(i) is the narrowband inter- ference, T c is the chip interval and n(i) is the Gaussian noise. The receiver structure is shown in Figure 7.1. For the two types of filters, from Figures 7.2 and 7.3 we define vectors of input samples and filter taps as follows: X i1  = [x i ,x i−1 ,x i−2 , .,x i−L ] T X i2  = [x i+M ,x i+M−1 , .,x i+1 ,x i−1 , .,x i−M ] T W 1  = [a 1 ,a 2 , .,a L ] T W 2  = [a −M ,a −M+1 , .,a −1 ,a 1 , .,a M ] T (7.2) where T stands for transpose. With this notation, the filter output signal can be represented as y if = x if − W T f · X if (7.3) where f = 1 for one-sided filter (1SF) and f = 2 for two-sided filter (2SF). In the sequel index, f can be dropped for simplicity whenever this does not cause any ambiguity. If the interfering signal is stronger than the sum of Gaussian noise and useful signal, then the whole process can be interpreted as the estimation of J(i) in the presence of an equivalent noise. In this case, equation (7.3) can be interpreted as the estimation error. Adaptive WCDMA: Theory And Practice. Savo G. Glisic Copyright ¶ 2003 John Wiley & Sons, Ltd. ISBN: 0-470-84825-1 192 INTERFERENCE SUPPRESSION AND CDMA OVERLAY Chip rate sampler Estimation filter Decision device Synchronized PN sequence c ( i ) Σ M = number of chips per symbol + x ( i ) n ( t ) + J ( t ) 2cos w 0 t ∫ T c 0 s ( t ) r ( t ) Figure 7.1 Receiver block diagram. T c T c T c x i x i + 1 x i + 2 x i + L a 1 y 1 a 2 a L + + −− − Figure 7.2 Single-sided transversal filter. Linear prediction filter. + − − − − T c T c T c T c x i + N x i + l x i − l x i − N x i a l a N a − l a − N y i + Figure 7.3 Two-sided transversal filter. NARROWBAND INTERFERENCE SUPPRESSION 193 The filter coefficients will be evaluated from the condition that the Mean-Square Error (mse) of the estimation is minimized. So, we first evaluate y 2 i = x 2 i − 2x i X T i W + W T X i X T i W (7.4) The mean value can be represented as ξ = E[y 2 i ] = E[x 2 i ] − 2E[x i X T i ]W + W T E[X i X T i ]W  = E[x 2 i ] − 2P T W + W T RW (7.5) where P T  = E[x i X T i ] R  = E[X i X T i ] = [ρ x (k − m)]; k, m = 1, .,M (7.6) where ρ x (k − m) is the signal covariance function. To minimize the estimation error, the filter tap weights are obtained from ∂E[y 2 i ] ∂a kf = 0  k 2 =−M, .,−1, 1, .,M k 1 = 1, .,L (7.7) It is straightforward to show that equation (7.7) results in −2P + 2RW 0 = 0 W 0 = R −1 P (7.8) where W 0 is the optimum tap weight vector. This equation is well known as the Wiener–Hopf equation. By taking z-transform of equation (7.3), the filter transfer function can be represented as A 1 (z) = 1 − L  k=1 a k z −k A 2 (z) = 1 − M  k=−M k=0 a k z −k (7.9) The signal-to-noise ratio (SNR) improvement factor G is defined as the ratio of the output SNR to the input SNR. G = (SNR) out (SNR) in (7.10) 194 INTERFERENCE SUPPRESSION AND CDMA OVERLAY 7.2 GENERALIZATION OF NARROWBAND INTERFERENCE SUPPRESSION In the previous section, it was shown that the optimum filter coefficients depend on the input signal correlation. So, if the interfering signal correlation function is specified, the closed-form solution for the SNR improvement factor can be obtained. This will be illustrated in this section by modeling the interference as a narrowband first-order autore- gressive process [1,2]. At the sampling instant iT c , after ideal frequency down conversion, the filter input signal, for these purposes, can be represented again by equation (7.1). We assume that instead of the chip-matched filter in Figure 7.1 only a low-pass filter of bandwidth proportional to 1/T c is used to limit the noise. The interfering signal {J(i)} is assumed to be a wide sense stationary stochastic process with zero mean and covariance sequence {ρ i (k)}. At this point, we introduce notation (a, b) to be a set of integers between a and b including a and b and  0 (a, b), the same set excluding zero. The filter output signal can be represented as y(i) =  l∈ h(l)x(i − l) (7.11) where  is (0, M)or1SFand(−M,M) for 2SF h(l) =  −a l ,l = 0 a 0 = 1 (7.12) and a l is defined by equation (7.2). By substituting equation (7.1) into equation (7.11), we have y(i) = C 0 (i) + J 0 (i) + n 0 (i) (7.13) Decision variable U at the input of decision device in Figure 7.1 is formed by multiplying the filter output signal by code and can be resolved in three components U = N  i=1 y(i)c(i) = N  i=1 C 0 (i)c(i) + N  i=1 J 0 (i)c(i) + N  i=1 n 0 (i)c(i) = U 1 + U 2 + U 3 (7.14) Under the assumption that signal noise and narrowband interference are mutually inde- pendent, we have for the average values E[U 1 ] = b · N,E[U 2 ] = E[U 3 ] = 0 (7.15) GENERALIZATION OF NARROWBAND INTERFERENCE SUPPRESSION 195 and bearing in mind that b 2 = 1, we have for the variance var U 1 = N  m∈ 0 h 2 (m) var U 2 = N  m 1 ,m 2 ∈ h(m 1 )h(m 2 )ρ i (m 2 − m 1 ) var U 3 = N  m 1 ,m 2 ∈ h(m 1 )h(m 2 )ρ n (m 2 − m 1 ) (7.16) where ρ i ()andρ n ( ) are covariance functions of the interfering signal and the noise signal, respectively. For the covariance functions, we have cov{U i ,U j }=0 i = j(7.17) The signal-to-noise ratio at the filter output can be expressed as (SNR) 0  = E 2 [U ] var[U ] = N  m∈ 0 h 2 (m) +  m 1 ,  m 2 ∈ h(m 1 )h(m 2 )[ρ i (m 2 − m 1 ) + ρ n (m 2 − m 1 )] (7.18) When no suppression filter is used, h(0) = 1, and h(l) = 0forl = 0, and we have (SNR) n 0 = N ρ i (0) + ρ n (0) (7.19) The improvement factor in the performance due to the use of the filter is then the ratio of equations (7.18 and 7.19) G = ρ i (0) + ρ n (0)  m∈ 0 h 2 (m) + M  m 1 , M  m 2 ∈ h(m 1 )h(m 2 )[ρ i (m 2 − m 1 ) + ρ n (m 2 − m 1 )] (7.20) 7.2.1 Examples of the interfering signal For the signal x(i) given by equation (7.1), the covariance function ρ(i) can be expressed as ρ(i) = δ c (i) + ρ i (i) + ρ n (i) (7.21) 196 INTERFERENCE SUPPRESSION AND CDMA OVERLAY where δ c (i), the Kronecker delta, is the covariance sequence of the pseudonoise (PN) code. For ρ n (i) and ρ i (i), we will assume ρ n (i) = σ 2 n δ c (i) ρ i (i) = σ 2 i α | i | ;0<α<1 (7.22) where σ 2 n and σ 2 i are the noise variance and the interference variance, respectively. The power spectral density function φ i (ω) is obtained by the Fourier transform of ρ i (i) as φ i (ω) = (1 − α 2 )σ 2 i /2π | 1 − α exp(j ω) | 2 ; −π ≤ ω ≤ π = (1 − α 2 )σ 2 i /2π 1 + α 2 − 2α cos ω (7.23) and parameter α will characterize the shape of the spectra. The larger the α, the narrower the spectra, and vice versa. Bearing in mind equations (7.21 and 7.22), we have ρ(i) = (1 + σ 2 n )δ(i) + σ 2 i α | i | (7.24) It is straightforward to show that the Wiener–Hopf equation (7.8) for this case becomes a i (1 + σ 2 n ) + σ 2 i  m∈ 0 a m α | i−m | = σ 2 i α | i | (7.25) Solving the filter coefficients from this system of equations is conceptually straightfor- ward, but rather cumbersome and tedious work. Without going into any further details one can show that using equation (7.25) to evaluate coefficients a i and then substituting equation (7.22) in equation (7.23) we have for the filter improvement factor G 1SF = σ 2 n + σ 2 i σ 2 n + σ 2 i (1 − α 2 ) · (1 − αβ ) + (α − β)β 2M+1 (1 − αβ ) 2 − (α − β) 2 β 2M β = γ −  γ 2 − 1 γ = 1 2α  (1 + α 2 ) + σ 2 i (1 − α 2 ) 1 + σ 2 n  (7.26) GENERALIZATION OF NARROWBAND INTERFERENCE SUPPRESSION 197 A BCD 10 log 1/(s n 2 + s i 2 ) (A) Prediction filter 3 taps Symmetric interpolation filter (B) 3 taps (C) 5 taps (D) Upper bound −20 −15 −10 −5 1050 SNR/chip W/O filtering (dB) 0 1 2 3 4 5 6 7 8 9 SNR improvement (dB) Figure 7.4 Improvement factor for a first-order autoregressive interference with α = 0.9; σ 2 n = 0. for the single-sided filter (1SF) and G 2SF = σ 2 n + σ 2 i σ 2 n + σ 2 i (1 − α 2 ) (1 − αβ ) + (α − β)β 2M+1 (1 − αβ )(1 + α 2 − 2αβ ) − (α − β)(2α − β − α 2 β)β 2M (7.27) for the two-sided filter (2SF), where β and γ are the same as in equation (7.26). As an illustration, Figure 7.4 presents several curves for the filter improvement factor G with the given set of the signal and filter parameters (Wiener optimum W/O). Curve D, designated as upper bound, is obtained for M →∞. For the analysis of the mutual influence of Code Division Multiple Access (CDMA) and narrowband communications network, we will assume the interfering signal to occupy a multiple frequency band that can be represented as φ i (ω) =      σ 2 i /2πp, ω ∈ A j ,j = 1, .,J 0,ω/∈ J  j=1 Aj (7.28) where the intervals A j s are disjoint and their total length  J j=1 |A j |=2πp for some 0 <p<1. The jammer occupies a pth fraction of the signal band. By using the same procedure as in the previous case, numerical results are shown in Figure 7.5 for p = 20%. 198 INTERFERENCE SUPPRESSION AND CDMA OVERLAY A B −20 −15 −10 −5 1050 SNR/chip W/O filtering (dB) (A) Predictive filter (B) Interpolative 0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 SNR improvement (dB) Figure 7.5 Upper bounds on improvement factor for a multiband interference with 20% bandwidth occupancy. 7.3 RECURSIVE SOLUTIONS FOR THE FILTER COEFFICIENTS For the evaluation of the optimum filter coefficients, defined by equation (7.8) a matrix inversion is required. This is a computationally intensive operation, and for practical appli- cations a form of recursive algorithm is preferred. An option is to solve equation (7.8) by using the recursive procedure. An example is Levinson’s algorithm that can be found in textbooks on signal processing. Another option is to build up a recursive algorithm that will evaluate an improved set of filter coefficients in each step. Within this section, we will discuss the method of steepest descent and the least mean square (LMS) algorithm. The method of steepest descent uses gradients of the performance surface in seeking its min- imum. For this reason, we will first extend a little bit of theory presented in this section. 7.3.1 The gradient and the Wiener solution The gradient of the mse function defined by equation (7.7) will be denoted as ∇=−2P + 2RW (7.29) When we set the gradient to zero, we get the optimal Wiener–Hopf solution defined by equation (7.8). Putting back equations (7.8) to (7.5) gives the minimum mse ξ min = E[x 2 i ] − P T W 0 (7.30) RECURSIVE SOLUTIONS FOR THE FILTER COEFFICIENTS 199 Now, if equation (7.30) is used back in equation (7.5) we have ξ = ξ min + (W − W 0 ) T R(W − W 0 )(7.31) This can be further expressed as ξ = ξ min + V T RV (7.32) where V  = W − W 0 (7.33) is the difference between W and the optimal values W 0 . Differentiation of equation (7.32) gives another form of the gradient ∇ = 2RV (7.34) If Q is the orthonormal modal matrix of symmetric and positive-definite matrix R and  is its diagonal matrix of eigenvalues  = diag[λ 1 ,λ 2 , .,λ n ] (7.35) then we can write R = QQ −1 = QQ T (7.36) Now equation (7.32) becomes ξ = ξ min + V T QQ −1 V(7.37) If we use notation V   = Q −1 V → V = QV  (7.38) equation (7.37) can be expressed as ξ = ξ min + V  T V  (7.39) and the primed coordinates are therefore the principal axes of the quadratic surface. In the same way, we may apply transformation (7.38) to vector W itself to get W  = Q −1 W → W = QW  (7.40) 7.3.2 The steepest descent algorithm The method of steepest descent updates the filter coefficients in accordance with W i+1 = W i + µ(−∇ i )(7.41) 200 INTERFERENCE SUPPRESSION AND CDMA OVERLAY where µ is a convergence factor that controls the stability and the rate of adaptation and ∇ i is the gradient at the ith iteration. Using the equations (7.34–7.40) in equation (7.41) we have V  i+1 = (I − 2µ)V  i (7.42) which after successive iterations for V  i becomes V  i = (I − 2µ) i V  in (7.43) where V  in is the initial difference between W and W 0 V  in = W  in − W  0 (7.44) From equation (7.43) one can see that for each component k of the vector V’, the transients will be geometric with the geometric ratio r k = (1 − 2µλ k )(7.45) For convergence, it is necessary that | r max | = | 1 − 2µλ max | < 1 (7.46) leading to the conditions 1 − 2µλ max > 1 1 − 2µλ max < 1 (7.47) which results into 1/λ max >µ>0 (7.48) In order to determine the time constant of the transients, an exponential envelope is fitted to a geometric sequence. If the time is normalized to the iteration cycle time, constant τ k can be determined from r k = (1 − 2µλ k ) ∼ = exp  − 1 τ k  = 1 − 1 τ k + 1 2!τ 2 k − 1 3!τ 3 k +··· ∼ = 1 − 1 τ k (7.49) leading to τ k ∼ = 1 2µλ k (7.50) On the basis of this, the time constant for the process can be defined as the maximum value of parameter τ k τ = max k τ k = 1 2µλ min (7.51) [...]... two components The first component is caused by the propagation of gradient noise and the second one by the response of the adaptive process to the random variations of W0i caused by a nonstationary input signal In what follows, we will show that increasing the time constant of the adaptive process diminishes the propagation of gradient noise but at the same time increases the lag error that results from... interference-suppression filter under a worst-case jamming condition IEEE Trans Commun., COM-34, 13–21 3 Amoroso, F (1983) Adaptive A/D converter to suppress CW interference in DSPN spreadspectrum communications IEEE Trans Commun., COM-31, 1117–1123 4 Amoroso, F and Bricker, J L (1986) Performance of the adaptive A/D converter in combined CW and Gaussian interference IEEE Trans Commun., COM-34, 209–213 5 Schilling,... of a transform domain processing radiometer for DS spread spectrum signals with adaptive narrowband interference exciser Presented at the IEEE International Conference on Communications, June, 1985 14 Gervargiz, J., Das, P., Milstein, L B., Moran, J and Mckee, O (1986) Implementation of DS-SS intercept receiver with an adaptive narrowband interference exerciser using transform domain processing and... 638–640 44 Saulnier, G I., Das, P and Milstein, L B (1984) Suppression of narrowband interference in a PN spread-spectrum receiver using a CTD-based adaptive filter IEEE Trans Commun., COM-32, 1227–1232 45 Saulnier, G I., Das, P and Milstein, L B (1985) An adaptive digital suppression filter for direct sequence spread-spectrum communications IEEE J Select Areas Commun., SAC-3(5), 676–686 46 Saulnier, G... 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(1985) An approximate statistical analysis of the Widrow LMS algorithm with application to narrowband interference rejection IEEE Trans Commun., COM33, 121–130 22 Ketchum, J W and Proakis, J G (1982) Adaptive algorithms for estimating and suppressing narrowband interference in PN spread-spectrum systems IEEE Trans Commun., COM-30, 913–924 23 Ketchum, J W (1984) Decision feedback techniques for interference . In this case, equation (7.3) can be interpreted as the estimation error. Adaptive WCDMA: Theory And Practice. Savo G. Glisic Copyright ¶ 2003 John Wiley. i V in (7.60) 204 INTERFERENCE SUPPRESSION AND CDMA OVERLAY As long as the adaptive process is convergent, which is defined by equations (7.46–7.48), the