Data and signal analysis Assignment
Problem 3. Consider a filter hn.
a) Find the Fourier transform of its autocorrelation sequence
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b) Show that if (hk一n,hlkl〉=5ln, then
Heiw)) =1, tw.
c) Show that if |H(eiw) | = 1, then
d) Show that if |H(eiw)| = 1, Ww, then {hlk – ni, n E 7} is an orthonormal basis for e2(2).
Problem 4. Consider two waveforms 40|n| and 41 [n] and two waveform abo|n] and a/1|n] in 22 (Z). Let ho[n] and
hi In] be two filters such that holn|=4g|-n| and hiln|=4jl-nl, and so|n| and gilnj two filters such that
so[n= won and =
a) Show that if (o[nl, yo[n
2k]) = 8[k] then Ho (z) Go (z) + Ho(-z)Go(-z) = 2 and that if
= 8|k| then H1 (z) GI(z) + H1(-z)GI(-z)
b) Show that if (wolnl, 41 n
= 0 for all k € Z then H1 (z) Go (z) + H1 (-z)G(-z) = 0 and that if
(abi’ni, coin
= 0 for all k € Z then Ho (z) GI(z) + Ho(-
– z) G1 (-z) = 0.
Using the results of a) and b) show that if (yo[n], yo[n
(o[n], S1[n – 2k]) = 0 for all k € Z, and (21[n, coIn – 2k])
= 0 for all k € , then
Ho(z) Go (Z) + HI (2) GI (2) = 2 and Ho(-2) GO (2) + HI (=2)G(2) = 0.
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