dDiff - ECE 301, A Digital Differentiator Demonstration...

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Unformatted text preview: ECE 301, A Digital Differentiator Demonstration Input: Given an band-limited input x(t) with bandwidth WM = 2.5π . Sample it with sampling period T = 2/5 (ωs = 5π ). Let x[n] denote the sampled discrete-time (digital) array. Goal: Design a discrete-time processing h[n] satisfying the following. Let y [n] denote the output of the discrete-time system: y [n] = x[n] ∗ h[n]. Use perfect band-limited reconstruction to generate a continuous signal y (t). We desire that y (t) being the firstorder derivative of x(t). (Note that all we can handle/manipuate is the samples x[n], not the original signal x(t).) Example: x(t) = sin(2πt) Original signal 1.5 1 0.5 0 −0.5 −1 −1.5 −4 −3 −2 −1 0 1 2 3 4 What you really have is the sampled values: 1.5 1 0.5 0 −0.5 −1 −1.5 −4 −3 −2 −1 0 1 2 3 4 Perfect reconstruction without any processing: 1.5 1 0.5 0 −0.5 −1 −1.5 −4 −3 −2 −1 0 1 2 3 4 Naive Differentiator (in discrete time) + Perfect Reconstruction: y [n] = 1 2 x[n] − x[n − 1] x[n + 1] − x[n] + T T (1) 1.5 1 0.5 0 −0.5 −1 −1.5 −4 −3 −2 −1 0 1 2 3 4 True Digital Differentiator + Perfect Reconstruction: (see lecture notes) h[n] = (−1)n nT 0 if n = 0 . if n = 0 (2) 6 4 2 0 −2 −4 −6 −4 −3 −2 −1 0 1 2 3 4 Comparison: x(t) = sin(2πt) and x (t) = 2π cos(2πt). Reconstructed original x(t), a naive differentiator y1 (t), and the true differentiator ˆ y2 (t). 1.5 1 0.5 0 −0.5 −1 −1.5 −4 −3 −2 −1 0 1 2 3 4 6 4 2 0 −2 −4 −6 −4 −3 −2 −1 0 1 2 3 4 ...
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dDiff - ECE 301, A Digital Differentiator Demonstration...

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