Nyquist Sampling Thm - Question

Thread starter cepheid
Start date Oct 1, 2006
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Sampling

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The discussion revolves around determining the Nyquist sampling frequency for the signal x(t) + x(t-1) based on the original signal's Nyquist frequency, ω_s. The user computes the Fourier transform of the new signal and finds that it can be expressed as (1 + e^{-jω})X(jω), where X(jω) is the Fourier transform of x(t). The challenge lies in understanding how this result affects the Nyquist sampling frequency. The conversation highlights the need for further analysis of the spectrum to derive the new sampling frequency. Ultimately, the relationship between the original signal and the modified signal is crucial for determining the appropriate sampling rate.

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.