Problem 5:

2.7

(a) The signal x(t) is a complex signal with a real part and an imaginary part:

x(t) = x1(t) + jxo(t).

The (matched) filter has a real and imaginary part and is given by

h(t) = hi(t) + jho(t).

Only the real part of the output of the matched filter is of interest.

(a) Find the real part of the output of the matched filter in terms of the real and imaginary part of the input and the real and imaginary

parts of the impulse response:

131

y(t) = R

x {Sh(1 = T)x(T)dT}.

That is, express y(t) in terms of h/(t), ho(t), xi(t), xo(t)./n(b) Suppose that

xi(t) = pr.(t) + pr.(t - Tc)+ pr.(t−2Tc) - pr.(t - 3Tc)

xo(t) = PT.(t) + Pr.(t-Te) - Pr.(t-2T) + pr.(t-3Tc)

which are sequences of pulses each of duration T = T/4 shown in Figure 2.73. The matched filter is given by

h(t) = x*(T-t)

= x₁(Tt)- jxo(T-1)

hi(t) = x1(Tt)

ho(t) = -xo(T-1).

Find the output of the matched filter and plot. Hint: Plot each term individually, then plot the total. The output should be a function of

time beginning at time 0 and ending at time T = 8Tc.