The
discretetime Fourier transform (DTFT) of a real, discretetime
signal x[n] is a complexvalued 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 2pperiodic
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 2pperiodic,
the magnitude and phase spectra need only be displayed for a 2p range
in w,
typically _{}.
The
applet below illustrates properties of the discretetime Fourier transform. You
can sketch x[n] or select from the provided
signals: a rectangular pulse and two onesided 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 discretetime
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 cutoff frequency _{} specifies the highest
frequency passed by the filter. For the ideal high pass filter, _{} specifies the lowest
frequency passed by the filter.
Operation on x[n]

Resulting signal y[n] 
Y(w), p < w £ p 
Amplitude Scale 
bx[n] 
bX(w) 
Time Shift 
x[n – N] 
_{} 
Time Scale 
_{} 
_{} 
Time Reverse 
x[ n] 
_{} 
First Difference 
x[n]  x[n1] 
_{} 
Running Sum 
x[n]+ x[n1]+ x[n2]+¼ 
_{} 
Ideal Low Pass 
filtered version of x[n], cutoff frequency _{} 
_{} 
Ideal High Pass 
filtered version of x[n], cutoff frequency _{} 
_{} 
Applet by Lan Ma. 