21-07-2012, 05:03 PM
Discrete time Fourier Series
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A discrete time signal of fundamental period N can consists of frequency
components separated by 2п/N radians or f=1/N cycles.
Discrete time Fourier Series
But the frequency range for discrete time signals is unique over the
interval (-п, п) or (0, 2п).
* Consequently, the Fourier series representation of the discrete time
periodic signal will contain at most ‘N’ frequency components.
* This is the basic difference between F.S. representations for Continuous
and Discrete time signals.
Discrete Fourier Transform
Discrete Fourier Transform of a discrete time signal x(n)
is obtained by sampling one period of the Fourier
Transform X(ω) of the signal x(n) at a finite number of
frequency points.
The sampling is performed at N equally spaced points in
the period 0 to at ω
k=
The FT of a discrete time signal is a continuous function
of ω and so cannot be processed by any digital system.
The DFT converts the continuous function of ω in to a
discrete function of ω
Thus DFT allows us to perform frequency analysis on
digital systems.
Linear and circular convolutions
For performing the circular convolution both the sequences
should be of the same length, say,N.
If one of the sequences has length less than N, it should be
padded with zeroes so that both sequences has same length N.
The circular convolution then produces a sequence of length N.
But the linear convolution produces a sequence of length
N=N1+N2-1.
It is possible to employ circular convolution to find the output
response of systems by appending zeroes to both the
sequences such that the length of sequences are both
N=N1+N2-1.
Append N-N1 and N-N2 zeroes to the sequences to make the
lengths equal to N1+N2-1 and then perform circular convolution.