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.




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]+

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 .

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Applet by Lan Ma.