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Unformatted text preview: ECE425 ' LAB 3 FALL 2006 PERFECT RECONSTRUCTION FILTERS Week of Sept 18—23, 2006 INTRODUCTION In class, we studied the simplest possible example of a Perfect Reconstruction Filter (PRF), the sum/difference filters as shown in the figure below, We observed that it was easy to see why this worked if we looked at the problem in' the time domain, and harder if we looked at it in the frequency domain. (More advanced PRF’s are not, however, obvious in the time domain!) in this lab we will briefly verify the time-domain performance, and-then move to the frequency domain, Fig .. 1 High-Pass 1. TIME DOMAIN You receive an email that states that you have won $1 million and that all you need to do to claim the prize is to return your studentlD number by PRF, Zero-pad your student number (or similar) with a few zeros on each end, and run it through the PRF above. That is, simulate the run in Matlab, Does it work? Much to your surprise, you receive another email stating that the number you returned does not match the winning number. You suspect fraud — likely the originators of the contest never took ECE425 and have mixed up the two synthesis filters. You accuse them of this error and they reply that if they had done this, they would have known immediately upon receiving your reply. Why would they have known? 2. FREQUENCY DOMAIN Next, verify that the PRF works in the frequency domain. Specifically, show that the resampiers inside the PRF do create aliasing, but that the aliasing cancels in the final summation. Essentially you will need to recreate the set of nine figures shown below. These same figures have also been discussed in lecture. The top part of each figure shows the time samples (all real) while the bottom part shows the magnitude of the (generally complex) FFT of the samples. NOTES: Fig. 2a shows the test input. it is a length-16 sequence consisting of frequencies 3/16 and 6/16. “Eyeball” these from the time domain, and identify them in the FFT. Fig. 2b is n_ot present in the PRF. It is the resampling of the input —just to remind you how spectral replicas occur with resampling by 2. ' Fig. 3a and Fig. 3b show the filtering of the input in the analysis section. These may be obtained using Matlab’s filter. However, we want to avoid transients (must — to get “clean FFT’s) so we suggest you first add the first time sampie to the end of the input sequence (making it length 17) filter it, and keep samples 2 to 17.. Do you see the expected frequency shaping in the time domain and in the frequency domain? Keep in mind that we are not displaying the phase here, which the filters do change of course. ._ in Fig. 4a and Fig. 4b, we resample the now filtered input. Notice the new frequencies, the expected aliases that result from resampling by _2. Fig. 5a and Fig. 5b shows the next filtering (the synthesis part). Of course, the aliased terms are not removed by the filters, although their magnitudes change (and of course the phases — not shown). Given that we want the aliased terms to cancel at the output, it must be at least true that they have the same magnitude at the filter outputs. is this true? Fig. 6 shows the reconstruction. Did the aliases really cancel? Why is the reconstruction delayed, and is this expected? 2a INPUT) 33 (d of Fig. 1) (i of Fig (k of Fig ORIGNALJHS 915 —— ~¥p—~—v—r fir—_———'— s_.,_._|___ 90 I2 14 16 1a —————~—w—————~u—-—- u—~——.-——v—-— T ’ 1 n U—‘ l 1‘ ‘l—V—y—TJfi—IH ‘ . I 2L i l _J_ | |__ I l ..I—L... ‘ \—L.. o ‘ U 10 12 I4 15 6 ID I2 H 16 woman 20 —- —1 —-— 15 w 5 Le Let—a 9—9 3 9 9 WO— 5 . __ . _A_. J ... - ... 0 2 4 5 8 i0 ‘2 Ti ‘6 '-I “I a _s_. 4 .__.‘ _._# Lw_._ _. ‘ 6 3 ‘0 I1 l4 ‘0 ) LPORIGINALSAMP‘LED (e of Flg wmmmngn 2o m'T——-r r‘" w 20 — .—— r“—"'-*1-— ‘5 I5 m _ 5 i I T J. a s a ? a TL—Aa- e a T T e 6 _;____ 2 ‘ 6 I I 12 H II . ----1 2 1 n -‘ a _ _»_ a 2 4 e o «o 12 14 1s )m LFOKIG‘NALSAMPlEfllP (| of Fig 1) m "- wommmsmmm I5 II 1“ 10 o u e e o e e »_._ é o 2 4 5 I m ‘2 ‘4 '5 '5 o 2 a u 4': 13‘ 1: 94—“ 1'5 ---— orrgmal recovered OUTPUT ...
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