The power spectrum of a bandpass process X(t) is shown in...

The power spectrum of a bandpass process $X(t)$ is shown in Figure P7-72. $X(t)$ is applied to a product device where the second multiplying input is $3 \cos \left(\omega_0 t\right)$. Plot the power spectrum of the device's output $3 X(t) \cos \left(\omega_0 t\right)$.

Key Concepts

Scaling Effects in Spectra

The multiplication by a constant factor (here a factor of 3) scales the amplitude of themodulated signal's power spectrum. This scaling is important to account for when analyzing signal power after modulation since it adjusts the energy levels across the shifted spectral components.

Multiplicative Modulation

Multiplicative modulation involves multiplying a signal by a periodic function (in this case, a cosine) which results in the shifting of the original signal’s spectrum. This is a standard technique used to translate the frequency content of a signal, effectively moving its spectral components to new center frequencies.

Spectral Shifting and Replication

When a signal is multiplied by a cosine, the operation in the time domain corresponds to a convolution in the frequency domain. This causes the original spectrum to be shifted to the positive and negative frequencies corresponding to the cosine’s frequency, leading to spectral replicas of the original bandpass signal at new locations.

Power Spectrum

The power spectrum describes how the power of a signal is distributed across frequency components. It is the Fourier transform of the autocorrelation function of the process, showing the energy present at each frequency, which is essential for analyzing the frequency content of random signals.

Bandpass Process

A bandpass process confines its energy to a specific band of frequencies, meaning its power spectrum is nonzero only within a limited frequency range. This property allows the signal to be isolated in a particular frequency band, which is common in communication systems.

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