hw4 - norm on the input and output. (b) Intuition suggests...

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EECS 250 – Digital Signal Processing I Fall 2009 Homework 4 Due: Thurs. November 19 1. 4.2-4 2. 4.2-6 3. 4.2-14 4. 4.3-24 5. 4.3-25 6. Find the operator norm of a matrix A IR m × n where k·k 2 is used for IR n (the input space) and k·k is used for IR m (the output space). 7. The Sample and Hold Operator . Let h > 0 and define S h : L L via ( S h u )( t ) def = u ± h t h ”¶ , where b x c denotes the integer part of x . (a) Show that S h is a linear operator and that k S h u k ≤ k u k . Find k S h k using the L
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Unformatted text preview: norm on the input and output. (b) Intuition suggests that as h 0 ( i.e., the sampling gets faster), S h should approach the identity map I that satises ( Iu )( t ) = u ( t ) for all u ( t ). Calculate the operator norm k S h-I k as a function of the sampling period h . Does it converge to zero as h 0? Can you explain this? (Think in terms of the sampling theorem)....
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