Problem 2. The signal z(t) received by a binary communication system can be expressed as z(t) = Zapat-k)+w(t) where a = +1, an equally likely and independent binary sequence, and w(t) is white Gaussian noise with spectral density S f) = N,/2. The pulse shape pe(t) is as shown below. a) Write down and sketch the causal matched filter impulse response which maximizes the sample SNR at its output at time t=T. b) Obtain and sketch the response of the matched filter in part a to an input pulse p.lt). c) Obtain the maximum instantaneous signal to noise ratio at the output of the matched filter. Relate this result to the energy of the pulse peo). d) Sketch the signal part of the matched filter output for the sequence (1,1,-1} (assume the input is zero before and after the sequence). pe(t) 21/3