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In .26, we considered one procedure for bandpass sampling and reconstruction. Another
Chapter 7, Problem 7.27(choose chapter or problem)
In Problem 7.26, we considered one procedure for bandpass sampling and reconstruction. Another procedure, used when (\(x)t\) is real, consists of multiplying (\(x)t\) by a complex-exponential and then sampling the product. The sampling system is shown in Figure P7.27(a). With (\(x)t\) real and with \(X(j \omega)\) nonzero only for \(\omega_{1}<|\omega|<\omega_{2},\) the frequency is chosen to be\(\omega_{0}=(1 / 2)\left(\omega_{1}+\omega_{2}\right)\), and the lowpass filter \(H_{1}(j \omega)\) has cutoff frequency \((1 / 2)\left(\omega_{2}-\omega_{1}\right)\).
(a) For \(X(j \omega)\) as shown in Figure P7.27(b), sketch \(X_{p}(j \omega)\).
(b) Determine the maximum sampling period \(T\) such that \(x(t)\) is recoverable from \(x_{p}(t)\).
(c) Determine a system to recover \(x(t)\) from \(x_{p}(t)\).
Figure P7.27
Questions & Answers
QUESTION:
In Problem 7.26, we considered one procedure for bandpass sampling and reconstruction. Another procedure, used when (\(x)t\) is real, consists of multiplying (\(x)t\) by a complex-exponential and then sampling the product. The sampling system is shown in Figure P7.27(a). With (\(x)t\) real and with \(X(j \omega)\) nonzero only for \(\omega_{1}<|\omega|<\omega_{2},\) the frequency is chosen to be\(\omega_{0}=(1 / 2)\left(\omega_{1}+\omega_{2}\right)\), and the lowpass filter \(H_{1}(j \omega)\) has cutoff frequency \((1 / 2)\left(\omega_{2}-\omega_{1}\right)\).
(a) For \(X(j \omega)\) as shown in Figure P7.27(b), sketch \(X_{p}(j \omega)\).
(b) Determine the maximum sampling period \(T\) such that \(x(t)\) is recoverable from \(x_{p}(t)\).
(c) Determine a system to recover \(x(t)\) from \(x_{p}(t)\).
Figure P7.27
Step 1 of 9
(a) Redraw the sample system with signal \(X_{1}(j \omega)\) and \(X_{2}(j \omega)\) indication.
Let \(X_{1}(j \omega)\) be the Fourier transform of the signal \(x_{1}(t)\) obtained by multiplying \(x(t) with \(e^{-j \omega_{0} t}\).
Let \(X_{2}(j \omega)\) be the Fourier transform of the signal \(x_{2}(t)\) obtained at the output of the low pass filter \(H(j \omega)\).