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lectset3

# lectset3 - ~.L_f o E h.u u “t.—.—.w.m 5 Fig.‘r3...

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Unformatted text preview: ; ~...L.._f o.....____....- E h, .__..............._.._.._...u u “t...._.—.—_.._.___._...w-,....__....._-_.m_. 5 Fig .‘r3 A "teqt'tbooig exarnple" where-the non-zero samples arebonvOiveg' ' " with 'a sine 'fuh'ction (corresponding to an ideal low—pass With ' ' - cutoff equal to 0..5..-7_ ' __ ‘ ' ' ' " Fig. 3 shows an. example .of this reconstruction. Here we aSSUme that there are ohly three non-zero samples, x(0)=1, x(1)=_0,.6, and x(2)=-0.l4.: These are shown in terms of the individual Weighted sinc contributions, and the Sum of the three; Note that for this Special case, the reconstructed curye goes exactly through each of the original points, This statement includes the points hot spe'Ciﬁed'ﬁhus zero by default) — it goes eXactlylthi‘ough these zeros-E Yet note well math is “not zero-except at'these integer_points.. The 'rec'bvered ‘ curve is of couise a perfectly “geod Waveform bandlimited to' 0;.5f3.. Accordingly, wefcould 7 resample this curve at the same sampling frequency ts:t witha different initial Starting point, The resampling could be, done at (integers-+114)ffor example, in which case wewould get non—‘- ' zero samples for ﬂ (1., "Thus Whilewej may 'g‘étth‘e imprjession'(fro'rh Fig, 3)_ that the s-itdation _ 7 here involves a sigh'al that is'hea'rly iaIWays‘ Zero (is non-zero only for three" discretesai‘ilples),? the actual signal is, in general, always non—zero, either as a Continuous-time signal, or as "‘ '_ samples, Accordingly, Fig“ 3 is a "textbookexample" but must be understood to be a douny special case (special bandwidth, special initial timing).. 20-3 Bandwidth Assumptions We can not escape the issue of bandwidth which we have introduced above” First of all, we heed-togdeal with signals thathaveabandwidths lessthan 13/2, since we are goingto use _r_e_aj filters in association with our sampling, in fact, it is this issue of using real filters that ' provides us with a simple (perhaps even sufficient) manner of dealing with bandwidth ambiguities, EN#200 (14) BEoDSP : SAMPUNG (10) ' . i m .......mm-iau_._...wwi-._.__.__ Fig. .49 Here “the sample points [(0,1),(1,2), and (23)] are constructed with a'loiiv- . _ ' pass with c'utotf 0..85,-then sampled at 1, and recen-Structed 'with a cutoff of . 0.5. The 'reconstructiOn' 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. ' - " ' ' . I. . f. l i z 3 - swim, __.Mw.§..... . g . l 2 m. ___.,.....E.._I,... -..__..%..__ i l t 1 ._.,...,_.._.€{_._ ._.. “jg..- ! i -5 reconstru' " i ' l l i -3 ' ‘ _ .4 o -2 Eig. 4d Here the sarnples [(0,1), (1,2), and (2,4)] are constructed with a bandwidth - of 0.5, sampled at a irate 1 (no problem); but thenie’constructed with ajfilt'er - with a cutoff of 0.85:. The reconstruction shows higher frequency ' ' ' 1 : oomponent's because a partion of a-sampling replica (0.,5'to 0.85.) is . ' included. This is a failure of the output anti-imaging (reconstruction, -' smoothing) filter. ' EN#200' (1.8) BEODSP SAMPLING 7(1 4) ...
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lectset3 - ~.L_f o E h.u u “t.—.—.w.m 5 Fig.‘r3...

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