A wireline channel has unit sample response h1[n] = e-an for n ≥ 0, and 0 otherwise,
where a > 0 is a real number. (As an aside, a = Ts/τ, where Ts is the sampling
rate and τ is the wire time constant. The wire resistance and capacitance prevent
fast changes at the end of the wire regardless of how fast the input is changing. We
capture this decay in time with exponential unit sample response e-an).
Ben Bitdiddle, an MIT student who recently got a job at WireSpeed Inc., is trying
to convince his manager that he can significantly improve the signaling speed (and
hence transfer the bits faster) over this wireline channel, by placing a filter with unit
sample response
h2[n] = Aδ[n] + Bδ[n – D],
at the receiver, so that
(h1 ∗ h2)[n] = δ[n].
(a) Derive the values of A, B and D that satisfy Ben’s goal.
(b) Sketch the frequency response of H2(Ω) and mark the values at 0 and ±π.
(c) Suppose a = 0.1. Then, does H2(Ω) behave like a (1) low-pass filter, (2) highpass filter, (3) all-pass filter? Explain your answer.
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