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A signal limited in bandwidth to lw I < W can be recovered from nonuniformly spaced
Chapter 7, Problem 7.37(choose chapter or problem)
A signal limited in bandwidth to \(|\omega|<W\) can be recovered from nonuniformly
spaced samples as long as the average sample density is \(2(W / 2 \pi)\) samples per second. This problem illustrates a particular example of nonuniform sampling. Assume that in Figure P7.37(a):
1. \(x(t)\) is band limited; \(X(j \omega)=0,|\omega|>W\).
2. \(p(t)\) is a nonuniformly spaced periodic pulse train, as shown in Figure P7.37(b).
3. \(f(t)\) is a periodic waveform with period \(T=2 \pi / W\). Since \(f(t)\) multiplies an impulse train, only its values \(f(0)=a \text { and } f(\Delta)=b \text { at } t=0 \text { and } t=\Delta\), respectively, are significant.
4. \(H_{1} (jw)\) is a \(90^{\circ}\) phase shifter; that is,
\(H_{1}(j \omega)=\left\{\begin{array}{ll} j, & \omega>0 \\ -j, & \omega<0 \end{array} .\right.\)
5. \(H_{2}(j \omega)\) is an ideal lowpass filter; that is,
\(H_{2}(j \omega)=\left\{\begin{array}{ll} K, & 0<\omega<W \\ K^{*}, & -W<\omega<0, \\ 0, & |\omega|>W \end{array}\right.\)
where \(K\) is a (possibly complex) constant.
(a) Find the Fourier transforms of \(p(t), y_{1}(t), y_{2}(t) \text {, and } y_{3}(t) \text {. }\)
(b) Specify the values of \(a, b \text {, and } K\) as functions of \(\Delta\) such that \(z(t)=x(t)\) for any band-limited \(x(t)\) and any \(\Delta\) such that \(0<\Delta<\pi / W\).
Questions & Answers
QUESTION:
A signal limited in bandwidth to \(|\omega|<W\) can be recovered from nonuniformly
spaced samples as long as the average sample density is \(2(W / 2 \pi)\) samples per second. This problem illustrates a particular example of nonuniform sampling. Assume that in Figure P7.37(a):
1. \(x(t)\) is band limited; \(X(j \omega)=0,|\omega|>W\).
2. \(p(t)\) is a nonuniformly spaced periodic pulse train, as shown in Figure P7.37(b).
3. \(f(t)\) is a periodic waveform with period \(T=2 \pi / W\). Since \(f(t)\) multiplies an impulse train, only its values \(f(0)=a \text { and } f(\Delta)=b \text { at } t=0 \text { and } t=\Delta\), respectively, are significant.
4. \(H_{1} (jw)\) is a \(90^{\circ}\) phase shifter; that is,
\(H_{1}(j \omega)=\left\{\begin{array}{ll} j, & \omega>0 \\ -j, & \omega<0 \end{array} .\right.\)
5. \(H_{2}(j \omega)\) is an ideal lowpass filter; that is,
\(H_{2}(j \omega)=\left\{\begin{array}{ll} K, & 0<\omega<W \\ K^{*}, & -W<\omega<0, \\ 0, & |\omega|>W \end{array}\right.\)
where \(K\) is a (possibly complex) constant.
(a) Find the Fourier transforms of \(p(t), y_{1}(t), y_{2}(t) \text {, and } y_{3}(t) \text {. }\)
(b) Specify the values of \(a, b \text {, and } K\) as functions of \(\Delta\) such that \(z(t)=x(t)\) for any band-limited \(x(t)\) and any \(\Delta\) such that \(0<\Delta<\pi / W\).
ANSWER:
Step 1 of 8
A signal limited in bandwidth to \(|\omega|<W\) can be recovered from non-uniformly spaced samples as long as the average sample density is \(2\left(\frac{W}{2 \pi}\right)\) samples per second.
The system is shown in Figure 1.
