h7 - ECE 459 Handout 7 Fall 2000 HOMEWORK ASSIGNMENT 6...

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ECE 459 Fall 2000 Handout # 7 November 17, 2000 HOMEWORK ASSIGNMENT 6 Reading: Lecture notes (lectures 19-23), papers/books referenced in lecture notes. Due Date: Thursday, December 7, 2000 (in class) 1. Performance analysis of MF detection for asynchronous band-limited CDMA . The received signal on the reverse link in the BPSK/CDMA system can be written as y ( t ) = K X k =1 X i = -∞ p E b ,k z ( b i/N c ) k c k,i g T c ( t - iT c - τ k ) e k + w ( t ) where w ( t ) is CCG( N 0 ), N is the processing gain, and g T c ( t ) is a unit energy chip waveform with the auto-correlation R g ( d ) = R -∞ g T c ( t ) g T c ( t - dT c ) dt . The pulse shape is assumed to have zero inter-chip interference (ICI) under perfect sampling, so that R g ( m - n ) = δ [ m - n ]. Note that the chip sequences { c k,i } i = -∞ are not divided into blocks of length N – they are assumed to be long (random) binary sequences taking values in {- 1 N , + 1 N } . The matched filter statistic for detecting the bit z (0) 1 of user 1 is given by X MF = Re (* y ( t ) , e - 1 " N - 1 X m =0 c 1 ,m g T c ( t - mT c - τ 1 ) # +) and ˆ z (0) 1 = sgn( X MF ). (i) Assuming w.l.o.g. that τ 1 = 0, show that X MF = A 1 z (0) 1 + K X k =2 N - 1 X m =0 c 1 ,m y k [ m ] + w 1 where w 1 ∼ N (0 , N 0 / 2), A k = p E b ,k and y k [ m ] = X i = -∞ A k z ( b i/N c ) k c k,i R g ( i + d k - m ) cos( φ k - φ 1 ) , with d k = τ k /T c . (ii) Moments. Define X k = N - 1 m =0 c 1 ,m y k [ m ] . Assume that the bits and chip sequences are all i.i.d.
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