Class 24 (The Discrete Fourier Transform)

The table below illustrates when you want to use each of the Fourier Transforms. For example, if your signal is aperiodic and discrete in time, you can get the frequency representation from the Discrete-Time Fourier Transform. If your signal is aperiodic and discrete in frequency, you can get the time representation from the inverse [continuous-time] Fourier Series.

The Discrete-Time Fourier Series

As described in the table above, we use the Discrete-Time Fourier Series when we need to find the frequency representation of a signal that is both periodic and discrete in time. We can derive the Discrete-Time Fourier Series very much like we can derive the Discrete-Time Fourier Transform -- by sampling our signal $x(t)$ and plugging it into the Fourier Series equations.

The result is a pair of direct and inverse transforms defined by $$ X[k] = \frac{1}{N_0} \sum_{n = (N_0 \textrm{ period})} x[n] e^{-j n \frac{2 \pi}{N_0}k }$$ $$ x[n] = \sum_{k = (N_0 \textrm{ period})} X[k] e^{j n \frac{2 \pi}{N_0} k }$$ In the above equations, $\Omega_0 = 2 \pi / N_0$ is the fundamental frequency of the periodic signal in time and $N_0$ is the period of the signal. The above equations are the direct and inverse Discrete-Time Fourier Series equations.

The Discrete Fourier Transform (DFT)

When we compute the "Fourier Transform" on the computer, we are actually using the Discrete Fourier Transform. Since on computers cannot hold continuous information, the time domain and frequency domain representations need to both be discrete. Hence, the Discrete Fourier Transform (DFT) is very similar to the Discrete-Time Fourier Series. The difference is that we assume the signal starts at $n=0$ and the fundamental period is the length of the finite signal $N_s = N_0$. That is, the sum goes from $n=0$ to $n=N_0$ and the fundamental angular frequency is $\Omega_0 = 2 \pi / N_0$. We also change where the $1/N_0$ factor is (in the inverse transform rather than forward transform).

Given this, the Discrete Fourier Transform is defined by $$ X[k] = \sum_{n = 0}^{N_0-1} x[n] e^{-j n \frac{2 \pi}{N_0} k} $$ $$ x[n] = \frac{1}{N_0} \sum_{k = 0}^{N_0-1} X[k] e^{j n \frac{2 \pi}{N_0} k} $$ This implies that to get discrete signals in time and frequency, we essentially assume the signals are also periodic in both time and frequency (but only ever compute one period).