Download - signal n system 4 chapter
-
7/30/2019 signal n system 4 chapter
1/13
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 forContinuous 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]
-
7/30/2019 signal n system 4 chapter
2/13
06EC44-Signals and System Chapter 4.2-2009
Krupa Rasane(KLE) Page 2
The fundamental period is the smallest positive integerNfor which the above holds and o= 2/N and k[n] = e
jk
on= e
jk(2/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
k[n] = ejkon = ejk(2/N)n , k = 0, 1, 2,. Etc.
All of these have fundamental frequencies that aremultiples of 2/Nand 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 of2 are identical. This differs from the situation in
continuous time in which the signals k[t] are all different fromone 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 of2 are identical.
This differs from the situation in continuous time in
which the signals k[t] are all different from one another.
The sequences k[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 =.
-
7/30/2019 signal n system 4 chapter
3/13
06EC44-Signals and System Chapter 4.2-2009
Krupa Rasane(KLE) Page 3
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.
-
7/30/2019 signal n system 4 chapter
4/13
06EC44-Signals and System Chapter 4.2-2009
Krupa Rasane(KLE) Page 4
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 asN1 n N1 and use theAnalysis Equation for Discrete time signals
-
7/30/2019 signal n system 4 chapter
5/13
06EC44-Signals and System Chapter 4.2-2009
Krupa Rasane(KLE) Page 5
Let m=n+N1 or n=m-N1, we get
-
7/30/2019 signal n system 4 chapter
6/13
06EC44-Signals and System Chapter 4.2-2009
Krupa Rasane(KLE) Page 6
whereSketches for different values of N are shown below. Fourier
series coefficients for the periodic square wave of example 2.
Plots for 2N1+1 = 5For 2N1+1 = 5 and N = 10
For N=20
For N = 40
-
7/30/2019 signal n system 4 chapter
7/13
06EC44-Signals and System Chapter 4.2-2009
Krupa Rasane(KLE) Page 7
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
-
7/30/2019 signal n system 4 chapter
8/13
06EC44-Signals and System Chapter 4.2-2009
Krupa Rasane(KLE) Page 8
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
-
7/30/2019 signal n system 4 chapter
9/13
06EC44-Signals and System Chapter 4.2-2009
Krupa Rasane(KLE) Page 9
-
7/30/2019 signal n system 4 chapter
10/13
06EC44-Signals and System Chapter 4.2-2009
Krupa Rasane(KLE) Page 10
Example
-
7/30/2019 signal n system 4 chapter
11/13
06EC44-Signals and System Chapter 4.2-2009
Krupa Rasane(KLE) Page 11
-
7/30/2019 signal n system 4 chapter
12/13
06EC44-Signals and System Chapter 4.2-2009
Krupa Rasane(KLE) Page 12
4.2.4 Summary
The Response of LTI Systems to Discrete ComplexExponentials.
Harmonically Related Discrete Complex Exponentials Convergence Issues of the DT/CT Fourier Series DTFourier Series Representation an Example
-
7/30/2019 signal n system 4 chapter
13/13
06EC44-Signals and System Chapter 4.2-2009
Krupa Rasane(KLE) Page 13
Properties of Fourier Representation in ContinuousTime 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 Feb07Jun07
Text and Reference Books have been referred during the notes preparation.