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hw7 - Boston University Department of Electrical and...

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Boston University Department of Electrical and Computer Engineering ENG EC515 Digital Communication ( Fall 2009 ) Problem set 7 Assigned: Wed 3 Dec Due: Fri 11 Dec Suggested reading: Class-notes, handouts, and Ch. 7 textbook O ffi ce hours: M, W 4:00–6:00pm PHO438 Problem 7.1 Problem 7.1 Problem 7.2 Show that raised cosines (Page 167 of Handout) are Nyquist pulses. Problem 7.3 A real BPSK waveform x BPS K ( t ) = n - 1 l = 0 a l s ( t - lT ), a l ∈ {- A , + A } is transmitted over a channel that introduces ISI described by the impulse-response h ( t ) = 1 + 0 . 25 δ ( t - T ). The received signal is described by y ( t ) = ( x BPS K ? h )( t ) + Z ( t ) where Z ( t ) is a real zero-mean AWGN processes with psd σ 2 = N 0 / 2. The symbols a l are all independent and equally likely. For such a channel we know that the optimum receiver can base it’s decisions on samples of u ( t ) taken at time instants that are multiples of T where u ( t ) = y ( t ) ? s * ( - t ). By design, the pulse s ( t ) ? s * ( - t ) is Nyquist for the symbol rate 1 / T . Hence, the equivalent discrete-time channel model reduces to: u l = a l + 0 . 25 a l - 1 + Z l where the Z l ’s are real, i.i.d., Normal random variables with zero-mean and variance
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