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E h, .__..............._.._.._...u u “t...._.—.—_.._.___._...w,....__.....__.m_. 5 Fig .‘r3 A "teqt'tbooig exarnple" wherethe nonzero 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
nonzero 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 zerosE Yet note well math is “not zeroexcept 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 sitdation _
7 here involves a sigh'al that is'hea'rly iaIWays‘ Zero (is nonzero only for three" discretesai‘ilples),? the actual signal is, in general, always non—zero, either as a Continuoustime 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).. 203 Bandwidth Assumptions We can not escape the issue of bandwidth which we have introduced above” First of all,
we heedtogdeal 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 .......mmiau_._...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 recenStructed '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 antialiasing“ (guard) filter at the input. '  " ' ' . I. .
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2 m. ___.,.....E.._I,... ..__..%..__
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1 ._.,...,_.._.€{_._ ._.. “jg..
! i 5 reconstru' " i
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.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 asampling replica (0.,5'to 0.85.) is . '
included. This is a failure of the output antiimaging (reconstruction, '
smoothing) filter. ' EN#200' (1.8) BEODSP SAMPLING 7(1 4) ...
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 Fall '05
 HEMAMI
 Digital Signal Processing, Signal Processing, 2 m, Whilewej, oomponent, jg.

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