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7.26. The sampling theorem, as we have derived it, states that a signal x(t) must be sam- pled at a rate greater than its bandwidth (or equivalently, a rate greater than twice its highest frequency). This implies that if x(t) has a spectrum as indicated in Figure P7.26(a) then x(t) must be sampled at a rate greater than 2w2. However, since the signal has most of its energy concentrated in a narrow band, it would seem reason- able to expect that a sampling rate lower than twice the highest frequency could be used. A signal whose energy is concentrated in a frequency band is often referred to as a bandpass signal. There are a variety of techniques for sampling such signals, generally referred to as bandpass-sampling techniques. x(t) X(jw) 1 -02-01 (a) +00 p(t) 8(t-nT) Xp (t) W1 W2 W H(jw) x(t) p(t) H(jw) 1 T дід. Wa (b) Wa w Figure P7.26 Chap. 7 Problems 565 To examine the possibility of sampling a bandpass signal as a rate less than the total bandwidth, consider the system shown in Figure P7.26(b). Assuming that ww2w1, find the maximum value of T and the values of the constants A, wa, and wb such that x,(t) = x(t).

7.26. The sampling theorem, as we have derived it, states that a signal x(t) must be sam- pled at a rate greater than its bandwidth (or equivalently, a rate greater than twice its highest frequency). This implies that if x(t) has a spectrum as indicated in Figure P7.26(a) then x(t) must be sampled at a rate greater than 2w2. However, since the signal has most of its energy concentrated in a narrow band, it would seem reason- able to expect that a sampling rate lower than twice the highest frequency could be used. A signal whose energy is concentrated in a frequency band is often referred to as a bandpass signal. There are a variety of techniques for sampling such signals, generally referred to as bandpass-sampling techniques. x(t) X(jw) 1 -02-01 (a) +00 p(t) 8(t-nT) Xp (t) W1 W2 W H(jw) x(t) p(t) H(jw) 1 T дід. Wa (b) Wa w Figure P7.26 Chap. 7 Problems 565 To examine the possibility of sampling a bandpass signal as a rate less than the total bandwidth, consider the system shown in Figure P7.26(b). Assuming that ww2w1, find the maximum value of T and the values of the constants A, wa, and wb such that x,(t) = x(t).

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
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7.26. The sampling theorem, as we have derived it, states that a signal x(t) must be sam- pled at a rate greater than its bandwidth (or equivalently, a rate greater than twice its highest frequency). This implies that if x(t) has a spectrum as indicated in Figure P7.26(a) then x(t) must be sampled at a rate greater than 2w2. However, since the signal has most of its energy concentrated in a narrow band, it would seem reason- able to expect that a sampling rate lower than twice the highest frequency could be used. A signal whose energy is concentrated in a frequency band is often referred to as a bandpass signal. There are a variety of techniques for sampling such signals, generally referred to as bandpass-sampling techniques. x(t) X(jw) 1 -02-01 (a) +00 p(t) 8(t-nT) Xp (t) W1 W2 W H(jw) x(t) p(t) H(jw) 1 T дід. Wa (b) Wa w Figure P7.26 Chap. 7 Problems 565 To examine the possibility of sampling a bandpass signal as a rate less than the total bandwidth, consider the system shown in Figure P7.26(b). Assuming that ww2w1, find the maximum value of T and the values of the constants A, wa, and wb such that x,(t) = x(t).
Transcribed Image Text:7.26. The sampling theorem, as we have derived it, states that a signal x(t) must be sam- pled at a rate greater than its bandwidth (or equivalently, a rate greater than twice its highest frequency). This implies that if x(t) has a spectrum as indicated in Figure P7.26(a) then x(t) must be sampled at a rate greater than 2w2. However, since the signal has most of its energy concentrated in a narrow band, it would seem reason- able to expect that a sampling rate lower than twice the highest frequency could be used. A signal whose energy is concentrated in a frequency band is often referred to as a bandpass signal. There are a variety of techniques for sampling such signals, generally referred to as bandpass-sampling techniques. x(t) X(jw) 1 -02-01 (a) +00 p(t) 8(t-nT) Xp (t) W1 W2 W H(jw) x(t) p(t) H(jw) 1 T дід. Wa (b) Wa w Figure P7.26 Chap. 7 Problems 565 To examine the possibility of sampling a bandpass signal as a rate less than the total bandwidth, consider the system shown in Figure P7.26(b). Assuming that ww2w1, find the maximum value of T and the values of the constants A, wa, and wb such that x,(t) = x(t).
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