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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]

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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> .

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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.

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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

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Let m=n+N1 or n=m-N1, we get

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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

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Example

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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.