Nyquist Sampling Thm - Question

[cepheid]

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.

 

[LightHero]

The Nyquist sampling frequency for a signal is given by the highest frequency component of the signal, so if we take the Fourier transform of the new signal x(t) + x(t-1) and find the highest frequency component, then we can calculate the Nyquist sampling frequency in terms of \omega_s. To calculate the highest frequency component, we can take the magnitude of the Fourier transform and find the frequency at which it is highest. Therefore, the Nyquist sampling frequency for x(t) + x(t-1) is given by:\omega_{Nyq} = argmax(|X(j\omega)|) * \omega_s

 

[piercas]

Your approach of using the Fourier transform is a good start. The Nyquist sampling theorem states that in order to accurately reconstruct a signal, the sampling frequency must be at least twice the highest frequency component in the signal. In other words, the sampling frequency must be greater than or equal to 2\omega_{max}.

In this case, the original signal x(t) has a Nyquist sampling frequency of \omega_s. But when we add a delayed version of the signal, x(t-1), the highest frequency component in the new signal becomes 2\omega_{max} + \omega_s. This means that in order to accurately sample and reconstruct this new signal, the Nyquist sampling frequency would need to be at least 2(2\omega_{max} + \omega_s) = 4\omega_{max} + 2\omega_s.

So the Nyquist sampling frequency for the new signal, in terms of \omega_s, would be 4\omega_{max} + 2\omega_s. This is a generalization, as it assumes that the original signal x(t) has a maximum frequency component of \omega_{max}. If we know the specific frequencies in x(t), we can calculate the exact Nyquist sampling frequency for the new signal.