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Nptel Digital signal processing 6

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Signals in Natural DomainChapter 6 : Discrete fourier series and Discrete fourier transform
In the last chapter we studied fourier transform representation of aperiodic signal. Now we consider periodicand finite duration sequences
Discrete Fourier series Representation of a periodic signal
Suppose that is a periodic signal with period N, that is As is continues time periodic signal, we would like to represent in terms of discrete time complexexponential signals are given by
!. # All these signals have frequencies is that are multiples of the some fundamental frequency, , and thusharmonically related. $hese are two important distinction between continuous time and discrete time complex exponential. $he firstone is that harmonically related continuous time complex exponential are all distinct for different valuesof
k
, while there are only
N
different signals in the set.$he reason for this is that discrete time complex exponentials which differ in frequency by integer multiple of are identical. $husSo if two values of
k
differ by multiple of
N
, they represent the same signal. Another difference betweencontinuous time and discrete time complex exponential is that for different
k
have period whichchanges with
k
. In discrete time exponential, if
k
and
N
are relative prime than the period is
N
and not
N/k.
$husif
N
is a prime number, all the complex exponentials given by !. # will have period
N
. In a manner analogous tothe continuous time, we represent the periodic signal as !.%#where !.&#In equation !.%# and !.&# we can sum over any consecutive
N
values. $he equation !.%# is synthesis equationand equation !.&# is analysis equation. Some people use the faction
/N
in analysis equation. 'rom !.&# wecan see easily that$hus discrete 'ourier series coefficients are also periodic with the same period
N
.
Example 1:
So, and , since the signal is periodic with periodic with period 5, coefficients are also periodic with period 5, and.
Now we show that substituting equation !.&# into !.%# we indeed get.interchanging the order of summation we get
(6.4) Now the sum
if
n - m
multiple of
N
and for
n - m
# not a multiple of
N
this is a geometric series, so sum is As
m
varies from ( to
N -
, we have only one value of
m
namely
m
)
n
, for which the inner sum if non*+ero. Sowe set the -S of !.# as.
%&/!
Signals in Natural DomainChapter 6 : Discrete fourier series and Discrete fourier transform
Properties of Discrete-ime Fourier Series
-ere we use the notation similar to last chapter. 0et be periodic with period
N
and discrete 'ourier series coefficients be then the writewhere 0-S represents the signal and -S its 1'S coefficients

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