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Q4 A transmitter generates message signals that are described by a baseband stationary random process x(t). The power spectral density (PSD) of x(t) is shown in Figure Q4-A. The transmission channel is distortionless, but the receiver is affected by additive noise. The noise signals can be described by a stationary random process n(t). The PSD of n(t) is flat over the entire frequency range and is shown in Figure Q4-B. a. Find the Fourier transform of Sx(t), i.e., the PSD of x(t) with its variable renamed as t. b. Use the duality of the Fourier transform and the result in Q4-a, find the autocorrelation of x(t). c. To limit the effect of noise, an ideal lowpass filter (LPF) is used at the receiver. What should be the bandwidth of the ideal LPF such that the average power of the noise is minimized while the message signal itself is not affected by the ideal LPF? d. What is the signal-to-noise ratio (SNR) at the output of the ideal LPF, where the bandwidth of the ideal LPF comes from Q4-c? e. Suppose the message signals are to be transmitted using a carrier signal that oscillates at a frequency of 106 Hz. With sufficient reasoning, specify the frequency response of the filter that you would use at the receiver side to limit the effect of noise. Sx(f) 10-3 -104 -f (Hz) 104 Figure Q4-A 10-9 Figure Q4-B Sn (f) -f (Hz)

Q4 A transmitter generates message signals that are described by a baseband stationary random process x(t). The power spectral density (PSD) of x(t) is shown in Figure Q4-A. The transmission channel is distortionless, but the receiver is affected by additive noise. The noise signals can be described by a stationary random process n(t). The PSD of n(t) is flat over the entire frequency range and is shown in Figure Q4-B. a. Find the Fourier transform of Sx(t), i.e., the PSD of x(t) with its variable renamed as t. b. Use the duality of the Fourier transform and the result in Q4-a, find the autocorrelation of x(t). c. To limit the effect of noise, an ideal lowpass filter (LPF) is used at the receiver. What should be the bandwidth of the ideal LPF such that the average power of the noise is minimized while the message signal itself is not affected by the ideal LPF? d. What is the signal-to-noise ratio (SNR) at the output of the ideal LPF, where the bandwidth of the ideal LPF comes from Q4-c? e. Suppose the message signals are to be transmitted using a carrier signal that oscillates at a frequency of 106 Hz. With sufficient reasoning, specify the frequency response of the filter that you would use at the receiver side to limit the effect of noise. Sx(f) 10-3 -104 -f (Hz) 104 Figure Q4-A 10-9 Figure Q4-B Sn (f) -f (Hz)

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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Q4 A transmitter generates message signals that are described by a baseband stationary random process x(t). The power spectral density (PSD) of x(t) is shown in Figure Q4-A. The transmission channel is distortionless, but the receiver is affected by additive noise. The noise signals can be described by a stationary random process n(t). The PSD of n(t) is flat over the entire frequency range and is shown in Figure Q4-B. a. Find the Fourier transform of Sx(t), i.e., the PSD of x(t) with its variable renamed as t. b. Use the duality of the Fourier transform and the result in Q4-a, find the autocorrelation of x(t). c. To limit the effect of noise, an ideal lowpass filter (LPF) is used at the receiver. What should be the bandwidth of the ideal LPF such that the average power of the noise is minimized while the message signal itself is not affected by the ideal LPF? d. What is the signal-to-noise ratio (SNR) at the output of the ideal LPF, where the bandwidth of the ideal LPF comes from Q4-c? e. Suppose the message signals are to be transmitted using a carrier signal that oscillates at a frequency of 106 Hz. With sufficient reasoning, specify the frequency response of the filter that you would use at the receiver side to limit the effect of noise. Sx(f) 10-3 -104 -f (Hz) 104 Figure Q4-A Sn (f) 10-9 Figure Q4-B -f (Hz)
Transcribed Image Text:Q4 A transmitter generates message signals that are described by a baseband stationary random process x(t). The power spectral density (PSD) of x(t) is shown in Figure Q4-A. The transmission channel is distortionless, but the receiver is affected by additive noise. The noise signals can be described by a stationary random process n(t). The PSD of n(t) is flat over the entire frequency range and is shown in Figure Q4-B. a. Find the Fourier transform of Sx(t), i.e., the PSD of x(t) with its variable renamed as t. b. Use the duality of the Fourier transform and the result in Q4-a, find the autocorrelation of x(t). c. To limit the effect of noise, an ideal lowpass filter (LPF) is used at the receiver. What should be the bandwidth of the ideal LPF such that the average power of the noise is minimized while the message signal itself is not affected by the ideal LPF? d. What is the signal-to-noise ratio (SNR) at the output of the ideal LPF, where the bandwidth of the ideal LPF comes from Q4-c? e. Suppose the message signals are to be transmitted using a carrier signal that oscillates at a frequency of 106 Hz. With sufficient reasoning, specify the frequency response of the filter that you would use at the receiver side to limit the effect of noise. Sx(f) 10-3 -104 -f (Hz) 104 Figure Q4-A Sn (f) 10-9 Figure Q4-B -f (Hz)
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