Introduction
In a previous module we have seen how we can describe a continuous-time signal in frequency domain with the Fourier Transform for Continuous-time signals (FTC). A similar approach is also possible for discrete-time signals if we take into account the specific property that such a signal is defined only at discrete instances of time and not in between. The resulting description in frequency domain is the so called Fourier Transform for Discrete-time signals (FTD). Most of the derivations are not formal but have an intuitive character based on physical insight.
Screencast video [⯈]
Module overview
This module will cover the following topics:
- From FTC to FTD - This section will describe the process of deriving the FTD starting from the FTC. From this derivation it is now possible to derive the relative frequency spectrum of a sampled discrete-time signal.
- FTD examples [⯈] - After a proper foundation of the involved mathematics, it is beneficial to apply the FTD in practice. Through several examples the FTD will be applied to get the reader more familiar with this transform.
- FTD properties [⯈] - Just like with the FTC, there are several properties which greatly simply calculations. These properties will be derived in this section.
Summary
$$ \boxed{ \begin{eqnarray*} X(e^{j\theta}) \overset{\mathcal{F}\left\{x[n]\right\}}{=\! =} \sum_{n=- \infty}^{\infty} x[n] e^{-jn\theta} &\circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ & x[n] \overset{\mathcal{F}^{-1}\{X(e^{j\theta})\}}{=\! =\! =\! =\! =\!} \frac{1}{2 \pi} \int_{-\pi}^{\pi} X(e^{j\theta}) e^{jn\theta} \mathrm{d} \theta \end{eqnarray*}} $$
Main differences with continuous-time Fourier transforms:
- The signal samples $x[n]$ are discrete in time and not periodic,
- The normalized frequency $\theta = \omega \cdot T_s$,
- The Fundamental Interval is usually chosen as: $-\pi \leq \theta < \pi$,
- The frequency representation $X(e^{j\theta})$ is a continuous function and periodic,
- The FTD is evaluated over an infinite summation of samples $x[n]$,
- The IFTD is evaluated by an integral over one period of the frequency representation $X(e^{j\theta})$.
Discrete phasor property: $$ \textbf{DPP:} \qquad \sum_{p=-\infty}^{\infty} e^{-jp \theta} = \sum_{k=-\infty}^{\infty} 2 \pi \delta(\theta + k \cdot 2 \pi) $$
Most important FTD pairs:
| $x[n]$ | $X(e^{j\theta})$ |
|---|---|
| $\delta[n]$ | $1$ |
| $1$ | $2\pi\delta(\theta)$ |
| $\cos(n\theta_0)$ | $\pi\delta(\theta+\theta_0) + \pi\delta(\theta-\theta_0)$ |
| $a^nu[n], \ |a|<1$ | $\frac{1}{1-ae^{-j\theta}}$ |
| Sinc: $\frac{\theta_c}{\pi}\cdot \frac{\sin(\theta_c n)}{\theta_c n}$ | Block: $\begin{cases} 1 & |\theta|<\theta_c \newline 0 & \text{otherwise}\end{cases}$ |
| Block: $u[n]-u[n-N]$ | Dirichlet: $e^{-j\frac{N-1}{2}\theta}\frac{\sin(N\frac{\theta}{2})}{\sin(\frac{\theta}{2})}$ |
Most important FTD properties: $$ \begin{eqnarray*} x^\ast[n] & \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ & X^\ast(e^{-j\theta}) \newline x[-n] & \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ & X(e^{-j\theta})\newline x[n-n_0] & \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ & e^{-jn_0 \theta} \cdot X(e^{j\theta})\newline e^{jn \theta_0} \cdot x[n] & \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ & X(e^{j(\theta - \theta_0)})\newline x[n] \star y[n] & \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ & X(e^{j\theta}) \cdot Y(e^{j\theta})\newline x[n] \cdot y[n] & \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ & X(e^{j\theta}) \star Y(e^{j\theta}) \newline \sum_{n=-\infty}^{\infty} |x[n]|^2 &=& \frac{1}{2 \pi} \int_{\theta = - \pi}^{\pi} |X(e^{j\theta}) |^2 \mathrm{d} \theta \end{eqnarray*} $$