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hw2 - k x-y k H ∞ ≤ 01(d Show that k x k ∞ ≤ k x k...

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EECS 250 – Digital Signal Processing I Fall 2009 Homework 2 Due: Thurs. October 15 1. 2.3-39 2. 2.4-42 3. 2.5-43 4. 2.7-47 5. For parts (a) and (b) you may use Matlab if you wish. (a) In IR 2 , sketch the unit balls for the p -norms when p = 1 , 1 . 5 , 2 , 3 , . (b) Sketch the unit ball for p = 0 . 5; i. e. , sketch x = x 1 x 2 : | x 1 | 1 / 2 + | x 2 | 1 / 2 1 . Does the formula for the p -norm yield a norm when p = 0 . 5? (c) Show that for every x IR 2 , k x k = lim p →∞ k x k p . (d) Show that for every x IR 2 , sup k λ k p 1 x T λ = k x k q , where p and q are conjugate: 1 p + 1 q = 1 and p 1. (Don’t forget the p = 1 and p = cases. When p 6 = 1 and p 6 = , you may wish to use H¨older’s inequality). 6. The so-called H norm is important in the field of robust control theory. For IR n , it is defined as k x k H def = sup 0 θ< 2 π fl fl fl fl fl n X k =1 x k e - jkθ fl fl fl fl fl . (a) Show that k x k H is a norm on IR n . (Parseval’s theorem may be useful here). (b) Show that k [ - 1 1 1] T k H = 5. (c) If the elements of the vectors x and y represent the impulse responses of two FIR filters, what is the engineering significance of
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Unformatted text preview: k x-y k H ∞ ≤ . 01? (d) Show that k x k ∞ ≤ k x k H ∞ ≤ n k x k ∞ . Show that both inequalities are taken on for some (different) choice of x . Hint: for the first inequality, use x m = 1 2 π Z 2 π ˆ n X k =1 x k e-jkθ ! e jmθ dθ . 7. Show that the norms k u k 1 = Z 1 | u ( t ) | dt k u k ∞ = sup {| u ( t ) | : 0 ≤ t ≤ 1 } are not equivalent on the vector space of continuous functions from [0 , 1] into IR . 8. Let V be the following vector space of sequences: V = { x [ k ] : x [ k ] ∈ IR, k ≥ , | x [ k ] | < ∞} . Define k u k ∞ = sup k | u [ k ] | and k u k w = sup k w [ k ] | u [ k ] | for some given “weighting” sequence w [ k ]. (a) What conditions on w make k · k w a norm on V ? (b) What conditions on w make k · k w a norm on V equivalent to k · k ∞ ?...
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