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]+� 
_{} 
Ideal Low Pass 
filtered version of x[n], cutoff frequency _{} 
_{} 
Ideal High Pass 
filtered version of x[n], cutoff frequency _{} 
_{} 
Applet by Lan Ma. 