Nyquist Sampling Thm - Question

Thread starter cepheid
Start date Oct 1, 2006
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Sampling

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SUMMARY

The Nyquist sampling frequency for the signal x(t) + x(t-1) can be derived from the original signal's Nyquist frequency, denoted as ω_s. By applying the Fourier transform, represented as ℱ, the spectrum of the combined signal is expressed as ℱ{x(t) + x(t-1)} = (1 + e^{-jω})X(jω), where X(jω) is the Fourier transform of x(t). This relationship indicates that the Nyquist frequency for the new signal is influenced by the phase shift introduced by the term x(t-1).

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Oct 1, 2006

cepheid

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Here's my question:

Suppose a signal x(t) has a Nyquist sampling frequency \omega_s. Compute the Nyquist sampling frequency for the following signal in terms of \omega_s:

x(t) + x(t-1)

Well my first thought was, let's see how the spectrum of this new signal compares to that of the original signal. Computing the Fourier transform, an operation I've denoted by script F, I arrived at the result that:

\mathcal{F}\{x(t) + x(t-1)\} = (1 + e^{-j\omega})X(j\omega)

where X(jw) is the FT of x(t). I'm really not sure how to use this result to proceed.