A
real, Nperiodic, discretetime
signal x[n] can be represented as a discretetime Fourier series
_{}
where
_{}, and the discretetime 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 discretetime 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 discretetime 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,…, N1 
Amplitude Scale 
c x[n] 
_{} 
Time Shift 
x[n – M] 
_{} 
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 _{} 
_{} 
* For this time scaling, the period of y[n]
is 2N.
** For the running sum,
y[n] is periodic if and only if _{}.
Applet by Lan Ma. 