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Unformatted text preview: ns an impulse at the highest frequency
B, the sampling rate must be Fs > 2B Hz. Such is the case when f (t) = sin 27rBt. This signal
is bandlimited to B Hz, but all of its samples are zero when taken a t a rate F . = 2B (starting at
t = 0), and f (t) cannot be recovered from its Nyquist samples. 322 5 Sampling 5.1 T he Sampling T heorem Irhh(t~ ~ (b) 323
J (t) Sampled signal 7,n i '1 Reconstructed signal i r······~ yet) kT
(a) -5 5 (b) . 'J(Hz)- IH(w)1
(c) (d) -21tB - 20 -5 o 21tB 411B 00- Fig. 5.3 Simple interpolation using a zero-order hold circuit. 20 . 'J(Hz)- l(t) Answer: T he N yquist interval is 0.01 s econd a nd t he N yquist sampling r ate is 100 Hz for b oth t he
signals. \ l (e) 5.1-1 ,
- 0.3 -0.1 \ . .......
0 0.2 " ".t .... -49Jt
- 20 '. -5 5 20 .'J(Hz) 1(1) (g) Fig. 5.2 Effect of undersampling and oversampling.
(Fig. 5.2h). Hence, F(w) can be recovered from F(w) using an ideallowpass filter or even
a practicallowpass filter (shown dotted in Fig. 5.2h).t •
6 Exercise E 5.1
F ind t he N yquist r ate a nd t he N yquist interval for t he signals (a) sine ( 100m) a nd
(b) sine (1001Tt)+ sinc(501Tt).
t The filter s hould h ave a c onstant g ain between 0 t o 5 Hz, a nd zero gain beyond 10 Hz. I n practice,
t he g ain b eyond 10 Hz can be made negligibly small, b ut n ot zero. -+ Signal Reconstruction: T he Interpolation Formula T he process of reconstructing a continuous-time signal f (t) from its samples is
also known as i nterpolation. I n Sec. 5.1, we saw t hat a signal f (t) b and limited
to B Hz c an b e reconstructed (interpolated) exactly from its samples. This reconstruction is accomplished by passing t he s ampled signal t hrough a n ideal lowpass
filter of bandwidth B Hz. As seen from Eq. (5.3) [or Fig. 5.1e], t he s ampled signal
contains a component ~f(t) a nd t o recover f (t) (or F (w)), t he s ampled signal must
be passed through a n i deallowpass filter of b andwidth B Hz a nd gain T . T hus, the
reconstruction (or interpolating) filter transfer function is H(w) = T rect (~)
T he i nterpolation process here is expressed in t he frequency-domain as a filtering
operation. Now, we shall examine this process from a different viewpoint, t hat of
To begin with, let us consider a very simple interpolating filter whose impulse
response is r ect ( +), d epicted in Fig. 5.3a. This is a g ate pulse centered a t t he origin,
having u nit height, a nd w idth T ( the sampling interval). We shall find t he o utput
of this filter when t he i nput is t he s ampled signal f (t). E ach sample in f (t), being
an impulse, produces a t t he o utput a g ate pulse of height equal t o t he s trength of
the sample. For instance, t he k th s ample is a n impulse of s trength f (kT) l ocated a t
t - kT, a nd c an be expressed as f (kT)6(t - kT). W hen t his impulse passes through
the filter, it produces a t t he o utput a g ate pulse o f height f ( kT), c entered a t t = k T
(shown shaded in Fig. 5.3b). Each sample in f (...
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