book Mobk087 August 3, 2007 13:15 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION 35 Conjecture: The infinite matrix sum Y = R +QR +Q 2 R +Q 3 R +··· enumerates all paths of all lengths between inner states and end states. Specifically, y jk =  l∈L ( j,k ) B ( l, z ) , (5.16) where L ( j, k ) is the set of all paths beginning at state j and ending at state k. For the system defined by Figs. 13 and 14, the average time required to reject an incorrect cell is given by (5.11), where L denotes all paths between state 1 and states 0 or 6. Thus, L = L ( 1, 0 ) + L ( 1, 6 ) (5.17) and (5.11) becomes T da =  l∈L ( 1, 0 )  dB ( l, z ) dz  z−1 +  l∈L ( 1, 6 )  dB ( l, z ) dz  z=1 =    d dz  l∈L ( 1, 0 ) B ( l, z ) + d dz  l∈L ( 1, 6 ) B ( l, z )    z=1 =  dy 10 dz + dy 16 dz  z=1 . (5.18) Since the derivative of a matrix is the matrix of derivatives, the last line of (5.18) can be obtained from dY dz .Now Y = R +QR + Q 2 R +···=  I + Q + Q 2 +···  R = ( I − Q ) −1 R. (5.19) We may use the matrix derivative relationship d dz A −1 =−A −1  d dz A  A −1 (5.20) to obtain d dz Y = d dz ( I − Q ) −1 R = ( I − Q ) −1  d dz R  +  d dz ( I − Q ) −1  R = ( I − Q ) −1  d dz R  − ( I − Q ) −1  d dz ( I − Q )  ( I − Q ) −1 R. (5.21) book Mobk087 August 3, 2007 13:15 36 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION It is convenient to define the matrix T =       T 1 0 T 2 . . . 0 T n       (5.22) where n is the number of internal states in the state transition diagram. Thus, it is seen that  d dz R  z=1 =  TR  z=1 (5.23) and  d dz ( I − Q )  z=1 =  −TQ  z=1 . (5.24) Using these equations in the last equation of (5.21) with z = 1, we obtain the following:  d dz Y  z=1 =  ( I − Q ) −1  TR  + ( I − Q ) −1  TQ  ( I − Q ) −1 R  z=1 =  ( I − Q ) −1 T  R +Q ( I − Q ) −1 R  z=1 =  ( I − Q ) −1 T  I + Q ( I − Q ) −1  R  z=1 =  ( I − Q ) −1 T  I + Q  I + Q + Q 2 +···  R   z=1 =  ( I − Q ) −1 T  I + Q + Q 2 +···  R   z=1 =  ( I − Q ) −1 T ( I − Q ) −1 R  z=1 . (5.25) Using the result of (5.18), the average time to reject an incorrect phase cell is the sum of the elements of the first row of the final matrix of (5.25). Since incorrect phase cells are being considered, the transition probabilities are computed assuming noise only conditions. All thresholds and integration times are assumed to be known. The probability of detection is the probability of passing from state 1 to “enter tracking mode”. Arrival at the code tracking mode also implies that the system will eventually arrive at state 6. The probability of this is unity since there is no other path to an end state and the system is guaranteed to eventually reach an end state. Therefore, in the example system of Fig. 13, this book Mobk087 August 3, 2007 13:15 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION 37 0 00 p z 1 10 T pz 1 12 T pz 2 23 T pz 3 34 T p z 4 45 T p z 3 33 T pz 2 21 T pz FIGURE 15: State-transition diagram for Examples 7 and 8. probability is P d =  l∈L ( 1, 6) ) Pr ( l ) =  l∈L ( 1, 6) ) B ( l, z ) z=1 = ( y 16 ) z=1 . (5.26) Thus, P d is the element of the first row and second column of ( Y ) z=1 =  ( I − Q ) −1 R  z=1 , (5.27) where the transition probabilities are now evaluated under signal plus noise conditions. Example 7. Evaluate the matrix expressions for T da and P d for the state transition diagram of Fig. 15. Algebraic expressions are desired. Solution: The following matrices are obtained from the state transition diagram: R   z=1 =    p 10 0 00 0 p 34    ; Q   z=1 =    0 p 12 0 p 21 0 p 23 00p 33    . (5.28) From the Q-matrix, we obtain [ I − Q ] z=1 =    1 −p 12 0 −p 21 1 −p 23 001− p 33    (5.29) book Mobk087 August 3, 2007 13:15 38 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION from which  ( I − Q ) −1  z=1 = 1 1 − p 12 p 21        1 p 12 p 12 p 23 1 − p 33 p 21 1 p 23 1 − p 33 00 1 − p 12 p 21 1 − p 33        . (5.30) Define the T-matrix as a diagonal matrix with T 1 , T 2 , T 3 as main diagonal elements. Then dY dz     z=1 =  ( I − Q ) −1 T ( I − Q ) −1 R  z=1 = 1 ( 1 − p 12 p 21 ) 2         1 p 12 p 12 p 23 1 − p 33 p 21 1 p 23 1 − p 33 00 1 − p 12 p 21 1 − p 33             T 1 00 0 T 2 0 00T 3     ×         1 p 12 p 12 p 23 1 − p 33 p 21 1 p 23 1 − p 33 00 1 − p 12 p 21 1 − p 33             p 10 0 00 0 p 34     = 1 ( 1 − p 12 p 21 ) 2 ×          p 10 (p 12 p 21 T 2 + T 1 ) − [ p 12 p 21 T 3 + p 33 ( T 1 + T 2 ) −T 1 −T 2 −T 3 ] p 12 p 23 p 34 ( 1 − p 33 ) 2 p 10 p 21 ( T 1 + T 2 ) − [ p 12 p 21 ( p 33 T 1 −T 1 + T 3 ) + p 33 T 2 − T 2 − T 3 ] p 23 p 34 ( 1− p 33 ) 2 0 ( 1− p 12 p 21 ) 2 p 34 T 3 ( 1 − p 33 ) 2          . (5.31) Taking the sum of the elements in the first row, we obtain T da = p 10 ( p 12 p 21 T 2 + T 1 ) ( 1 − p 12 p 21 ) 2 − [ p 12 p 21 T 3 + p 33 ( T 1 + T 2 ) − T 1 − T 2 − T 3 ] p 12 p 23 p 34 ( 1 − p 12 p 21 ) 2 ( 1 − p 33 ) 2 . (5.32) book Mobk087 August 3, 2007 13:15 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION 39 This can be simplified to T da = p 10 ( T 1 + p 12 p 21 T 2 ) ( 1 − p 12 p 21 ) 2 + p 12 p 23 p 34 ( T 1 + T 2 ) ( 1 − p 12 p 21 ) 2 ( 1 − p 33 ) + p 12 p 23 p 34 T 3 ( 1 − p 12 p 21 )( 1 − p 33 ) 2 . (5.33) Recall that noise-alone conditions are used to compute the various transition probabilities. To get P d ,wecompute Y | z=1 =  ( I − Q ) −1 R  z=1 = 1 1 − p 12 p 21         p 10 p 12 p 23 p 34 1 − p 33 p 10 p 1 p 23 p 34 1 − p 33 0 p 34 ( 1 − p 12 p 21 ) 1 − p 33         (5.34) and take the element in the upper-right-hand corner as P d .Thisgives P d = p 12 p 23 p 34 ( 1 − p 12 p 21 )( 1 − p 33 ) = p 12 p 23 1 − p 12 p 21 ; p 34 = 1 − p 33 . (5.35) Recall the signal plus noise conditions are used to compute the various transition proba- bilities. Example 8. Assume the parameters of Example 6, namely, a code clock frequency of f c = 3MHzand10log 10 ( P/N 0 ) = 46 dBHz. The propagation delay uncertainty is ±1.2ms and the code is stepped in 1 / 2 -chip steps. However, the double-dwell system of Fig. 15 is used with time–bandwidth products for integrations 1, 2, and 3 chosen as 4, 10, and 50, respectively. The bandwidth is still 24 kHz, giving integration times of T 1 = 1.67 × 10 −4 s, T 2 = 4.17 ×10 −4 s, and T 3 = 2.083 ×10 −3 s. The thresholds are chosen to give probabili- ties of detection of p 12 = p 23 = 0.9andp 33 = 0.99. Use these to compute the correspond- ing thresholds, then the probabilities of false alarm, and finally the mean synchronization time. Solution: Use of the marcumq function in MATLAB results in the following: BT = 4; V T = 11.5000; P fa = 0.1749 = p 12 ; P d = 0.8959 = p  12 BT = 10; V T = 37.0000; P fa = 0.0117 = p 23 ; P d = 0.9003 = p  23 BT = 50; V T = 201.5000; P fa = 8 ×10 −11 = p 33 ; P d = 0.9902 = p  33 p 10 = 1 − p 12 ; p 21 = 1 − p 23 ; p 34 = 1 − p 33 ; p 45 = 1 p  10 = 1 − p  12 ; p  21 = 1 − p  23 ; p  34 = 1 − p  33 ; p  45 = 1(primed probabilities are s + n). book Mobk087 August 3, 2007 13:15 40 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION The R and Q matrices for noise-alone conditions are R   z=1 =    1 −0.1749 0 00 01−8 ×10 −11    =    0.8251 0 00 01−8 ×10 −11    ; Q   z=1 =    00.1749 0 1 −0.0117 0 0.0117 008×10 −11    ≈    00.1749 0 0.9883 0 0.0117 008×10 −11    . Compute dY dz     z=1 =  ( I − Q ) −1 T ( I − Q ) −1 R  z=1 =    0.0002879 0.0000069 0.0006953 0.0000371 00.0020833    . The mean time to synchronization is the sum of the elements in the first row, or T da ≈ 0.000295 s = 295 µs. The question is whether readjustment of the thresholds or integrations times can decrease this. Computer evaluation shows that there is an optimum set of thresholds and integration times. For signal plus noise conditions, the R and Q matrices are R    z=1 =    1 −0.8959 0 00 01−0.9902    =    0.1041 0 00 00.0098    ; Q    z=1 =    00.8959 0 1 −0.9003 0 0.9003 000.9902    ≈    00.8959 0 0.0997 0 0.9003 000.9902    . We compute the matrix Y | z=1 =  ( I − Q ) −1 R  z=1 under signal plus noise conditions to find P d as the upper-right-hand element. Carrying out the numerical calculations, we get P d = 0.8857. Using signal plus noise transition probabilities in (5.35) as a check, we get P d = p  12 p  23 1 − p  12 p  21 = 0.8959 ×0.9003 1 −0.8959 ( 1 −0.9003 ) = 0.8857, which is exactly the same as obtained with direct matrix calculations. book Mobk087 August 3, 2007 13:15 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION 41 5.2 Matched Filter Code Acquisition [1, 12] A matched filter has impulse response which is the delayed time reverse of the signal to which it is matched. Thus, for a time-limited signal s ( t ) that is zero for t < 0andt > T,theimpulse response of the filter matched to this signal is h ( t ) = s ( t 0 −t ) , where t 0 is the time of peak output and is usually chosen to make the filter causal. Thus, for causality, t 0 ≥ T in this case. For the choice of t 0 = T, the output of the matched filter for an arbitrary input x ( t ) is y ( t ) = ∞  −∞ h ( τ ) x ( t −τ ) dτ = T  0 s ( T −τ ) x ( t −τ ) dτ, (5.36) which, at time t = T,is y ( T ) = T  0 s ( T −τ ) x ( T −τ ) dτ = T  0 s ( λ ) x ( λ ) dλ. (5.37) This is a well-known property of a matched filter—its output at the optimum sampling time is the cross-correlation between the input and the signal towhich it is matched. If the input is x ( t ) = s ( t ) +n ( t ) , where n ( t ) is the white noise of the two-sided power spectral density of N 0 /2, the output is y ( T ) = T  0 s (λ) [ s (λ) + n ( λ )] dλ = T  0 s 2 ( λ ) dλ + N = E s + N, (5.38) where E s = T  0 s 2 ( λ ) dλ is the signal energy and N is a zero-mean random variable withvariance σ 2 N = N 0 E s 2 . Thus, the peak signal squared to mean-square noise at the output is SNR out = E 2 s ( N 0 E s /2 ) = 2E s N 0 (5.39) which is a well-known property of matched filters [11]. Since a matched filter performs the function of a correlator, the correlation operation in the acquisition circuitry of a DSSS receiver may be replaced by a matched filter. A conceptual block diagram for such a receiver is shown in Fig. 16. The upper three boxes constitute the matched filter code acquisition circuitry. The remaining boxes represent the fine code tracking and data detection. The bandpass matched filter (it could be realized at quadrature baseband) is matched to K chips of the spreading code modulating the IF carrier, so its impulse response book Mobk087 August 3, 2007 13:15 42 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION Envelope detector Threshold comparator Bandpass matched filter Spreading code generator Received signal plus noise Start pulse V T Spreading code phase detector and tracking loop filter Spreading code clock generator On-time despreader and data detector Early code Late code On-time code Demodulated data FIGURE 16: DSSS receiver using matched filter code acquisition [1]. is h ( t ) = 2c ( KT c −t ) cos ω IF t, 0 ≤ t ≤ KT c . (5.40) For signal alone at the input, its output is y ( t ) = ∞  −∞ h ( τ ) s in ( t −τ ) dτ = KT c  0 [ 2c ( KT c −τ ) cos ω IF τ ]  √ 2Pc ( t −τ − t d ) cos [ ω IF ( t −τ ) +φ ]  dτ = √ 2P KT c  0 c ( KT c −τ ) c ( t −τ − t d ) { cos ( ω IF t +φ ) +cos [ ω IF ( t −2τ ) +φ ] } dτ ≈ √ 2P  KT c  0 c ( KT c −τ ) c ( t −τ − t d ) dτ  cos ( ω IF t +φ ) = √ 2PR c , KT c ( KT c −t + t d ) cos ( ω IF t +φ ) , (5.41) where R c , KT c ( λ ) is the code correlation function over a duration of KT c and data modulation has been ignored for the time being for simplicity. The output of the envelope detector is the book Mobk087 August 3, 2007 13:15 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION 43 envelope or √ 2P   R c , KT c ( KT c −t + t d )   . The time at which this function is a maximum is used as an indication of the epoch at which the local code is to be delayed for tracking of the incoming code. If there is a frequency error between the incoming signal and the matched filter center frequency (due to Doppler shift, say), then the output is degraded by a factor dependent on the frequency error and integration time [1]. If the frequency error is zero, there is still degradation due to the correlation being over only a portion of the code duration and also because of possible data transitions during the correlation duration. In spiteof these possibledegradations, matched filtercode acquisition is attractive because it speeds up the average acquisition time by a factor of roughly K. Matched filter acquisition is often implemented digitally. If N is the number of samples taken per chip, then for an initial code uncertainty of M chips it can be shown that, under the ideal conditions of P d = 1andP fa = 0, the average acquisition time is [1] T s ≈ T c N + MT c 2 . (5.42) This is a considerable savings over serial search. The schematic of a digitally implemented matched filter realization for DSSS code acquisition is shown in Fig. 17. Typical correlation function envelopes are shown in Fig. 18 for the 11-chip code {1–111–1111–1–1–1} for two samples per chip. A repetition of four transmitted bits is shown, where each bit contains a code repetition. An exact analysis of matched filter acquisition is complex and the reader is referred to the literature [12]. Example9. Compare serialsearch and digital matched filter acquisition under ideal conditions of P d = 1andP fa = 0, an initial code uncertainty of 10,000 half-chips, and a chip rate of 1Mchip/s. For the matched filter case, assume N = 2 samples/chip. Assume an integration time equivalent to 100 chips for the serial search case. Solution: The given chip rate means that T c = 1 µs and the integration time is T i = 10 −4 s. For P d = 1andP fa = 0, (5.2) simplifies to T s =  C + 1 2  T i = 10,001 2  10 −4  = 0.50005 s. For the matched filter case M = 10,000/2 = 5000 (the matched filter steps in chip intervals) and T s ≈ T c N + MT c 2 = 10 −6 2 + ( 5000 )  10 −6  2 = 1 2 ( 5001 )  10 −6  = 0.0025 s. The advantage of matched filter acquisition in terms of acquisition time is clear. book Mobk087 August 3, 2007 13:15 44 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION FIGURE 17: Digital implementation of a noncoherent matched filter for acquisition [1]. 5.3 Tracking in Spread Spectrum Once the locally generated code has been stepped to within a fraction of a chip (typically 1 / 2 chip) of the received code, the receiver is switched to the tracking mode and the local code is tracked to within a very small fraction of the incoming code (ideally local and re- ceived codes are coincident, but noise will cause jitter around this ideal value). To see how this might be accomplished, consider the idealized case of a received signal consisting of a spreading code plus noise (i.e., data and modulation on a carrier are being ignored for now): x r ( t ) = √ Pc ( t − T d ) +n ( t ) , (5.43) where P is the average power of the input signal component. The mechanism shown by the block diagram of Fig. 19 will track this signal if the received and local codes are within a chip of each other. To show that this is the case, consider the time average of the output of the . Q ) −1 R  z=1 . (5. 25) Using the result of (5. 18), the average time to reject an incorrect phase cell is the sum of the elements of the first row of the final matrix of (5. 25) . Since incorrect. Q ) −1 R. (5. 21) book Mobk087 August 3, 2007 13: 15 36 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION It is convenient to define the matrix T =       T 1 0 T 2 . . . 0 T n       (5. 22) where. book Mobk087 August 3, 2007 13: 15 FUNDAMENTALS OF SPREAD SPECTRUM MODULATION 35 Conjecture: The infinite matrix sum Y = R +QR +Q 2 R +Q 3 R +··· enumerates all paths of all lengths between inner