7.15 Channel equalization. We suppose that u1, . . . , um is asignal (time series) that is trans- mitted (for example by radio).A receiver receives the signal y = c ∗ u, where the n-vector c iscalled the channel impulse response. In most applications n issmall, e.g., under 10, and m is much larger. An equalizer is ak-vector h that satisfies h∗c ≈ e1, the first unit vector of lengthn + k − 1. The receiver equalizes the received signal y byconvolving it with the equalizer to obtain z = h ∗ y.
(a) How are z (the equalized received signal) and u (theoriginal transmitted signal) related? Hint. Recall that h∗(c∗u) =(h∗c)∗u.
(b) Numerical example. Generate a signal u of length m = 50,with each entry a random value that is either −1 or +1. Plot u andy = c ∗ u, with c = (1,0.7,−0.3). Also plot the equalized signal z= h ∗ y, with
h = (0.9, −0.5, 0.5, −0.4, 0.3, −0.3, 0.2, −0.1).
