# hwa - 1. An example of convolution Consider a function f...

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Unformatted text preview: 1. An example of convolution Consider a function f (x) = cos x + cos 3x 1 1 (exp(ix) + exp(−ix)) + (exp(3ix + exp(−3ix)) = 2 2 Imagine passing this function through a low-pass ﬁlter, that multiplies the coeﬃcients of 1, exp(±ix) by 1, and multiplies all other Fourier components by zero. If we write ∞ f (x) = (cf )n exp(inx) n=−∞ then the action of the low-pass ﬁlter can be described as ∞ (cf )n (cg )n exp(inx) n=−∞ for 1 n = 0, ±1 0 n = 0, ±1 (c g )n = a) Show that ∞ (cg )n exp(inx) = 1 + 2 cos x n=−∞ b) For g (x) = 1 + 2 cos x, compute (f ∗ g )(x) ≡ 1 2π π −π f (s)g (x − s)ds c) Show that the convolution product above matches ∞ (cf )n (cg )n exp(inx) n=−∞ (See next page for another problem) 2. The values of a 2π -periodic function are sampled at four times: 0, π/2, π , 3π/2, where the function has values 0, 1, 0, 1, respectively. Compute the discrete-time Fourier series of this function. ...
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## This note was uploaded on 11/02/2011 for the course ECON 7125 taught by Professor Smith during the Spring '11 term at James Madison University.

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hwa - 1. An example of convolution Consider a function f...

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