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06EC44-Signals and System –Chapter 4.2-2009•
Krupa Rasane(KLE) Page 1
Chapter 4.2
Fourier Representation for four Signal Classes
Fourier Representation for Continuous Time Signals
4.2.1Introduction Fourier Representation for
Continuous Time Vs Discrete Time Signals
Some Important Differences • DTFS is a finite series while FS is an infinite series
representation. Hence mathematical convergence issues
are not there in DTFS.
• Discrete-time signal x[n] is periodic with period N. i.e
x[n] = x[n+N]
06EC44-Signals and System –Chapter 4.2-2009•
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• The fundamental period is the smallest positive integer N
for which the above holds and ωo= 2π/N and φk[n] = ejk
ωon = e
jk(2π/N)n , k = 0, ±1, ±2,…. Etc.
Harmonically Related complex Exponentials
The DT Complex exponential signals that are
periodic with period N is given by
φk[n] = ejk ωon
= ejk(2π/N)n ,
k = 0, ±1, ±2,…. Etc. All of these have fundamental frequencies that are
multiples of 2π/N and are harmonically related.
As mentioned there are only N distinct signals in the set
given above. This is a consequence of the fact that discrete time
complex exponentials which differ in frequency by a
multiple of 2π are identical. This differs from the situation in
continuous time in which the signals φk[t] are all different from
one another.
As mentioned there are only N distinct signals in the set
given above.
This is a consequence of the fact that discrete time
complex exponentials which differ in frequency by a
multiple of 2π are identical.
This differs from the situation in continuous time in
which the signals φk[t] are all different from one another.
The sequences φk[n] are distinct only over a range of N
successive values of k. Thus the summation is on k, as k varies
over a range of N successive integers. Hence the limits of the
summation is expressed as k =<N> .
06EC44-Signals and System –Chapter 4.2-2009•
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Discrete time Fourier Series
These Equations play the same role for discrete time
periodic signals as the Synthesis and Analysis Equations
for Continuous time signals.
ak are referred to as the spectral coefficient of
x[n]. These coefficients specify a decomposition of x[n]
into a sum of N harmonically related complex
exponentials.
We also observe that the graph nature both in Time
domain and frequency domain are both discrete unlike in
Fourier Series for continuous times
Example 1:Find the Fourier Representation for the following.
Solution:
We can expand x[n] directly in terms of complex exponential
using the Eulers Formula.
06EC44-Signals and System –Chapter 4.2-2009•
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We get,
The Fourier Series Coefficient for the above Example .
Example 2: Find the Fourier Coefficient for the given
waveform.
where
Solution :
Select the range conveniently as –N1 ≤ n ≤ N1 and use the
Analysis Equation for Discrete time signals
06EC44-Signals and System –Chapter 4.2-2009•
Krupa Rasane(KLE) Page 5
Let m=n+N1 or n=m-N1, we get
06EC44-Signals and System –Chapter 4.2-2009•
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where
Sketches for different values of N are shown below. Fourier
series coefficients for the periodic square wave of example 2.
Plots for 2N1+1 = 5
For 2N1+1 = 5 and N = 10
For N=20
For N = 40
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Example 3:
Convergence Issues and comparisons for CT and DT
We Observed the Gibbs Phenomenon at the
discontinuity CT, whereby as the number of terms
increased, the ripples in the partial sum as in eg 3 became
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compressed towards the discontinuity, with the peak
amplitude of the ripples remaining constant
independently of the number of terms in the partial sum.
In DT eg3 with N=9, 2N1+1=5, and for several
values of M. For M=4, the partial sum exactly
equals x[n].
In contrast to the CT there is no Gibbs
phenomenon and no convergence issue in DTFS
4.2.2Properties for DTFS
06EC44-Signals and System –Chapter 4.2-2009•
Krupa Rasane(KLE) Page 9
06EC44-Signals and System –Chapter 4.2-2009•
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Example
06EC44-Signals and System –Chapter 4.2-2009•
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4.2.4 Summary
• The Response of LTI Systems to Discrete Complex
Exponentials.
• Harmonically Related Discrete Complex Exponentials
• Convergence Issues of the DT/CT Fourier Series
• DTFourier Series Representation an Example
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• Properties of Fourier Representation in Continuous
Time Domain
References
Figures and images used in these lecture notes are adopted from “Signals & Systems”
by Alan V. Oppenheim and Alan S. Willsky, 1997
Feng-Li Lian, NTU-EE, Signals and Systems Feb07 – Jun07
Text and Reference Books have been referred during the notes preparation.