The purpose of this week is to understand sampling in the frequency domain, the discrete fourier transform (DFT), and time-domain aliasing.

1. Frequency Domain Sampling / DFT

The basic idea of the DFT is that we are sampling the DTFT of a signal so that we can store it in memory. Recall the DTFT:

$$X(\omega)=DTFT\{x[n]\}=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}$$

We sample $X(\omega)$ as such:

$$X_N(k)=X(\omega_k = \frac{2\pi k}{N})=\sum_{n=0}^{N-1}x[n]e^{-j\frac{2\pi k}{N} n}$$

Which is called the N-point DFT of $x$.

1.1. Various Notes

1. The DFT goes $\omega_k \in [0,2\pi)$

2. Time-Domain Aliasing

2.1. Formulation, Nyquist Sampling

Let

$$X_s(\omega)=X(\omega)\sum_{k=-\infty}^{\infty}\delta(\omega-k\frac{2\pi}{N})$$

Then

$$x_s[n]=x[n]*\sum_{k=-\infty}^{\infty}\frac{N}{2\pi}\delta[n-kN]=\frac{N}{2\pi}\sum_{k=-\infty}^{\infty}x[n-kN]$$

Thus

$$\boxed{x_s[n]=\frac{N}{2\pi}\sum_{k=-\infty}^{\infty}x[n-kN]}$$

Which shows that when we sample in the frequency domain, we get periodic replications of the time domain signal in the time domain.

In order to prevent aliasing, we require

$$\boxed{N>L}$$

Where $L$ is the length of $x$.