# lab2 - ECE425 Lab 2 FALL 2006 SINC lNTERPOLATlON Week of...

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Unformatted text preview: ECE425 Lab 2 FALL 2006 SINC lNTERPOLATlON Week of Sept 11 — Sept 15, 2006 INTRODUCTION: In this lab, you are expected to produce four Mattab figures. The first three of these will reproduce figures that are in the sampling notes on the electronic blackboard. You will be using sinc interpolation, equation (12a) of the notes. Note that this equation is a sine series. Like a Fourier series, the sine series is a weighted sum of basis functions The weights are the given sample values, and basis functions are sincs, with a width that depends on a chosen cutoff frequency, displaced in time relative to each other. Note that while we consider this to be a convolution-like process, you should just sum the sincs and don't be tempted to use Matlab's- convolve function. For the purpose of plotting sincs that represent continuous waveforms, a time spacing of 0.01 is suggested. 1. TEXTBOOK FIGURE Reproduce Figure 3 of the sampling notes. Feel free to change some details. 2. ALIASING AND ANTI-IMAGING FAILURES Reproduce Figures 40 and 4d of the sampling notes. 3. TRUNCATION ERROR Consider a length 50 discrete sequence: x=[zeros(1,10) cos(2*pi*[0:29]/3.1) zeros(’l ,10)]; Plot this as a discrete sequence. Use plot, not stem, to keep things uncluttered, but plot a symbol instead of a line. For example, plot( [0:49], x, '*') plots the samples as a star. Next, overplot as a continuous curve, the' 'cosine burst" part of x. This you will do by calculating the cosine on intervals of perhaps 0. 01. Does it seem as though you should be able to get the continuous curve from just the samples x? Next form the sinc interpolation sum for the samples x, using a bandwidth of 0.5, and overplot this curve as well. - How well does the sine sum approximate the cosine burst? To see this better, make additional plots and adjust the axis so that you are looking at the middle of the sequence and the ends of the sequence separately. BONUS Reproduce Fig. 4e of the sampling notes. You may start, if you wish, by estimating the input samples from the graph. However, the real trick is to calculate the input samples. Fig. 3 A' 'textbook example“ where the non-zero samples are convolved ' with a sine function (correspdnding to an ideal low-pass with“. cutoff equal to 0_. 5 M Here the sample points [(0,1),'(1,2), and (2,4)] are constructed witha low- ‘ pas’s With cutoff 0. 85, then sampled at 1, and reconstructed With a cutoff of ‘ _O 5. The recenstruction shows aliasing (lower frequency output) because ' the input bandwidth exceeds _0 5. This Is a failing of the anti-aliasing (guard) filter at the input. . -- Fig. 4d Here the samples [(0,1), (1,2), and (2,4)] are constructed with a bandwidth ' of 0:5, sampledat a rate 1- (no problem),:but thengr'e'oonstructed with azfilter with a cutoff of 0,85; The reconstruction shows higher frequency ' :combonents bebause _a ponion of a sampling replica (015't0'0'85) is , included This is a failure of the output anti—imaging (re_construbtion-,' smoothing) filter o —— —-—~~—i—~—-— — 3*ft;mmu " :liftmii _- i Fig .'-‘4e" There exist samples at t=O,1, and 2, which when passed through an ideal - * tow-pass with cutoff 0.3 will havean output that goes through (0,1), (1,2), énd (2,-1), and these are shown by the (*’s)‘ Finding these is just a matter ' of solving three éqtratio‘ns in three unknowns,- starting with the sinc- expansion. Compare to Fig. 4a. ' -- ' ...
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