Chapter 5 Problems
5-1 A matched filter has the frequency response
1?e?j2πfT
H(f)= j2πf
a) Determine the impulse response h(t) corresponding to H(f).
b) Determine the signal waveform to which the filter characteristic is matched.
5-4 A binary digital communication system employs the signals
s0(t)=0,0≤t≤T s1(t)=A,0≤t≤T
for transmitting the information. This is called on-off signaling. The demodulator cross-correlates the received signal r(t) with s(t) and samples the output of the correlator at t + T.
a) Determine the optimum detector for an AWGN channel and the optimum threshold, assuming that the signals are equally probable.
b) Determine the probability of error as a function of the SNR. How does on-off signaling compare with antipodal signaling?
5-8 The two equivalent low-pass signals shown in Figure P5.8 are used to transmit a binary sequence over an additive white Gaussian noise channel. The received signal can be expressed as
rl(t)=si(t)+z(t),0≤t≤Ti=1,2
where z(t) is a zero-mean Gaussian noise process with autocorrelation function
1φzz(τ)=E[z*(t)z(t+τ)]=N0δ(τ) 2
a) Determine the transmitted energy in s1(t)and s2(t)and the cross-correlation
coefficientρ12·
b) Suppose the receiver is implemented by means of coherent detection using two
matched filters, one matched tos1(t)and the other tos2(t). Sketch the equivalent
low-pass impulse responses of the matched filters.
FIGURE P5.8
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