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Unformatted text preview: t) will generate a corresponding gate 324 5 Sampling 5.1 T he Sampling Theorem pulse resulting i n t he filter o utput t hat is a s taircase approximation of J (t), shown
d otted in Fig. 5.3b. This filter thus gives a crude form of interpolation.
T he t ransfer function of this filter H (w) is t he Fourier transform o f t he impulse
response rect (~). Assuming t he Nyquist sampling rate; t hat is, T = 1 /2B
h (t) H (oo)
T = rect (~) = r ect(2Bt) a nd - 2lt8 H(w) 325 . (WT) = 21 . (W)
4B = T Slllc S IllC 00- (5.8) T he a mplitude response IH (w)1 for this filter, illustrated in Fig. 5.3c, explains the
reason for t he c rudeness of this interpolation. This filter, also known as t he z eroorder h old filter, is a poor form o f t he ideal lowpass filter (shown shaded in Fig.
5.3c) required f or e xact interpolation.t
We can improve on t he zero-order hold filter by using the f irst-order h old
filter, which r esults in a linear interpolation instead of the staircase interpolation.
T he linear i nterpolator, whose impulse response is a triangle pulse 6.(2~)' results
in a n i nterpolation in which successive sample tops are connected by s traight line
segments (see P rob. 5.1-5).
T he ideal interpolation filter transfer function obtained in Eq. (5.7) is illustrated
in Fig. 5.4a. T he impulse response of this filter, t he inverse Fourier transform of
h (t) = 2 BT sine (211" B t)
Assuming t he N yquist sampling rate; t hat is, 2 BT = 1, t hen
h (t) = sinc (211"Bt) 21tB
(a) ( b) Sampled signal t_ Reconstructed signal
f (1) 7 (I) ...... , ..
....: ' .. ~ .., t_
(c) F ig. 5 .4 I deal i nterpolation. E quation (5.10) is t he i nterpolation f ormula, which yields values of J (t) between
samples as a weighted sum of all t he sample values.
• E xample 5 .2 F ind a s ignal J (t) t hat is b andlimited t o B Hz, a nd whose s amples a re (5.9b) This h (t) is d epicted in Fig. 5.4b. Observe t he very interesting fact t hat h (t) = 0
a t all Nyquist sampling instants (t = ± 28) e xcept a t t = O. W hen t he sampled
signal j (t) is a pplied a t t he i nput of this filter, t he o utput is J (t). E ach sample in
/ (t), being a n i mpulse, generates a sine pulse o f height equal t o t he s trength of the
sample, as i llustrated in Fig. 5.4c. T he process is identical to t hat d epicted in Fig.
5.3b, except t hat h (t) is a sinc pulse instead of a gate pulse. Addition of t he ~nc
pulses generated by all t he samples results in J (t). T he k th sample of t he i nput J (t)
is t he impulse J ( kT)8(t - kT); t he filter o utput of this impulse is J (kT)h(t - kT).
Hence, t he filter o utput t o J (t), which is J (t), c an now be expressed as a sum
J (t) = L J(kT)h(t - kT) J(O) = 1 a nd J (±T) = f (±2T) = f (±3T) = . .. = 0 where t he s ampling i nterval T is t he N yquist interval for f it); t hat is, T = 1 /2B.
We use t he i nterpolation f ormula (5.lOb) t o c onstruct f it) from its samples. Since
all b ut o ne o f t he N yquist s amples are zero,...
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