dtft properties

 

The discrete-time Fourier transform (DTFT) of a real, discrete-time signal x[n] �is a complex-valued function defined by

 

where w is a real variable (frequency) and . We assume x[n] is such that the sum converges for all w.

 

An important mathematical property is that X(w) is 2p-periodic in w, , since

 

for any (integer) value of n.

 

A plot of �vs w is called the magnitude spectrum of x[n], and a plot of �vs w is called the phase spectrum of x[n]. These plots, particularly the magnitude spectrum, provide a picture of the frequency composition of the signal. Since X(w) is 2p-periodic, the magnitude and phase spectra need only be displayed for a 2p range in w, typically .

 

The applet below illustrates properties of the discrete-time Fourier transform. You can sketch x[n] or select from the provided signals: a rectangular pulse and two one-sided exponential signals, , where u[n] is the unit step signal. (The exponentials continue for all n, that is, they are nonzero for all positive n. Sketched signals are assumed to be zero for all n outside the range .)

 

Sketch or select x[n] and click �show� in the top panel to display the corresponding spectra. Then choose an operation and click the corresponding �show� button to display the effects on the signal and its spectra.

 

Suppose y[n] �is the signal resulting from an operation on x[n], and let the discrete-time Fourier transform of y[n] be Y(w).The table below describes the operations available in the applet. For the ideal low pass filter, the cut-off frequency �specifies the highest frequency passed by the filter. For the ideal high pass filter, �specifies the lowest frequency passed by the filter.