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090206HomeWork4 - ECE 700 Digital Signal Processing Home...

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ECE 700 Digital Signal Processing Home Work 4 Due 2/16/09 1. (CWT ) Let CWT ( , ) X a τ be the Continuous Wavelet Transform of the signal x ( t ). Show that a. If y ( t ) = x ( t - t 0 ), then CWT CWT 0 ( , ) ( , ). Y a X a t τ τ = b. If 1 ( ) ( ), t g t f s s = then CWT CWT ( , ) ( , ). a G a X s s τ τ = 2. (Wavelets) Suppose that if φ ( t ) is an orthogonal scaling function with CTFT Show that ( ). Φ Ω 2 ( 2 ) 1 l l π =−∞ . Φ Ω + = 3. (Spline wavelets) The first order spline scaling function is defined as , 0 1 ( ) (2 ), 1 2 . 0, otherwise At t t A t t φ < = < a) Find the value of A so that ( ) 1. t φ = Sketch φ ( t ) for this A . b) Find the coefficients h[n] so that φ ( t ) satisfies the scaling equation ( ) 2 [ ] (2 ). n t h n t φ φ =−∞ n = c) Assume that [ ] ( 1) [ 1 ]. n g n h N = − − − n Find N and the wavelet ψ ( t ) such that ( ) 2 [ ] (2 ), n t g n t ψ φ =−∞ n = and ψ ( t ) is non-zero only in the interval [0, 2]. 1
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