A real, N-periodic, discrete-time signal x[n] can be represented as a discrete-time Fourier series

where , and the discrete-time fundamental frequency is .

The complex coefficients  can be calculated from the expression

The  are called the spectral coefficients of the signal x[n].  A plot of  vs k is called the magnitude spectrum of x[n], and a plot of  vs k is called the phase spectrum of x[n]. These plots, particularly the magnitude spectrum, provide a picture of the frequency composition of the signal. Recall that the spectral coefficients for a periodic signal repeat according to .

The applet below illustrates properties of the discrete-time Fourier series. You can enter a value of  N up to 15 and specify x[n] by sketching the signal or by sketching magnitude and phase spectra with the mouse. Then standard operations can be performed on x[n] and the effects on the signal and its spectra are displayed. (Click the Reset button between operations.)

Let y[n]  be the signal resulting from an operation on x[n], and let the discrete-time Fourier series coefficients of y[n] be specified by . The table below lists the operations available, where M and  are integers and c is a real constant. For the ideal low pass filter,  specifies the highest frequency, , that is passed by the filter. For the ideal low pass filter,  specifies the lowest frequency passed by the filter.

## Operation

Resulting signal y[n]

Spectral coefficients of y[n]

k=0, 1,…, N-1

Amplitude Scale

c x[n]

Time Shift

x[n – M]

Time Scale*

Time Reverse

x[- n]

First Difference

x[n] -  x[n-1]

Running Sum**

x[n]+ x[n-1]+ x[n-2]+¼

Ideal Low Pass

filtered version of x[n],

cutoff frequency

Ideal High Pass

filtered version of x[n],

cutoff frequency

*    For this time scaling, the period of y[n] is 2N.

**  For the running sum, y[n] is periodic if and only if .