This module includes a table of properties of the DTFS.
Introduction
In this module we will discuss the basic properties of the Discrete-Time Fourier Series. We will begin by refreshing your memory of our basic
Fourier series equations:
Let
denote the transformation from
to the Fourier coefficients
maps complex valued functions to sequences of
complex numbers .
The rate of decay of the Fourier series determines if
has
finite energy .
Parsevalstheorem demonstration
Symmetry properties
Even signals
Even signals
Odd signals
Odd signals
*
Real signals
Real signals
*
*
Differentiation in fourier domain
Since
then
A differentiator
attenuates the low
frequencies in
and
accentuates the high frequencies. It
removes general trends and accentuates areas of sharpvariation.
A common way to mathematically measure the smoothness of a
function
is to see how many derivatives are finite energy.
This is done by looking at the Fourier coefficients of thesignal, specifically how fast they
decay as
.If
and
has the form
,
then
and has the form
.So for the
derivative to have finite energy, we need
thus
decays
faster than
which implies that
or
Thus the decay rate of the Fourier series dictates
smoothness.
Fourier differentiation demo
Integration in the fourier domain
If
then
If
, this expression doesn't make sense.
Integration accentuates low frequencies and attenuates high
frequencies. Integrators bring out the
general
trends in signals and suppress short term variation(which is noise in many cases). Integrators are
much nicer than differentiators.
Fourier integration demo
Signal multiplication
Given a signal
with Fourier coefficients
and a signal
with Fourier coefficients
,
we can define a new signal,
,
where
.
We find that the Fourier Series representation of
,
,
is such that
.
This is to say that signal multiplication in the time domainis equivalent to
discrete-time
circular convolution in the frequency domain.The proof of this is as follows
Conclusion
Like other Fourier transforms, the DTFS has many useful properties, including linearity, equal energy in the time and frequency domains, and analogs for shifting, differentation, and integration.
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