Signal Processing and Linear Systems-B.P.Lathi copy

441 t his r epresents s pectrum f w s hifted t o ws a

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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) 2"" B 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(w) is h (t) = 2 BT sine (211" B t) ( 5.9a) 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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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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