06EC44-Signals and System Chapter Fourier Representation for four Signal Classes

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1 Chapter 5.1 Fourier Representation for four Signal Classes 5.1.1Mathematical Development of Fourier Transform If the period is stretched without limit, the periodic signal no longer remains periodic but remains a single pulse x(t), corresponding to one period of the periodic pulse, x(t). It also represents a transition from a power signal to an energy signal. As, Also if we replace ωo by an infinitesimally small quantity dω 0, the discrete frequency kωo may be replaced by a continuous frequency ω. The factor1/t in the equation means that Krupa Rasane(KLE) Page 1

2 Continuous Time Fourier Transform Hence, we can extend the formula for continuous-time Fourier series coefficients for a periodic signal to periodic signals as well. The continuous-time Fourier series is not defined for aperiodic signals, but we call the formula Inverse Transforms the (continuous time) Fourier transform Replacing a k with Ta k and multiplying and dividing the RHS of FS Synthesis equation by 2π/ωo i.e Krupa Rasane(KLE) Page 2

3 Inverse Transforms If we have the full sequence of Fourier coefficients for a periodic signal, we can reconstruct it by multiplying the complex sinusoids of frequency ω 0 k by the weights X k and summing: We can perform a similar reconstruction for aperiodic signals: These are called the inverse transforms Comparison of FS and FT Fourier Series : Used for Periodic Signals Fourier Transform: Used for Non-periodic Signals FS coefficients are a complex-valued function of integer k, FT is a complex-valued function of the variable - < ω< Krupa Rasane(KLE) Page 3

4 5.1.3 Convergence Issues Dirichlets Conditions Condition 1: Over any period, x(t) must be absolutely integrable, i.e each coefficient to be finite Condition 2: In any finite interval, x(t) is of bounded variation; i.e., There are no more than a finite number of maxima and minima during any single period of the signal Condition 3: In any finite interval, x(t) has only finite number of discontinuities. Furthermore, each of these discontinuities is finite Convergence Issues Note that the above are sufficient conditions and not necessary conditions. Fourier transform for the analysis of many useful signals would be impossible if these were necessary conditions. FT Examples for typical signals 5.1.4FT Examples for typical signals Examples1 Find the Fourier transform of the following functions: (a) The unit Impulse (b) The rect function (c) The decaying exponential Ex 1(a) : The Unit Impulse Krupa Rasane(KLE) Page 4

5 All the coefficients will have the same value i.e 1 Krupa Rasane(KLE) Page 5

6 5.1.5 Definition of Sinc Function Krupa Rasane(KLE) Page 6

7 5.1.6 Band Limited and Time Concept Krupa Rasane(KLE) Page 7

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13 5.1.7 Magnitude and Phase Spectra The Fourier transform X(jω), in general, is a complex quantity and may be expressed in an exponential form as follows: For real signals This means as seen in above example, X(jω) versus ω display even Symmetry and the phase spectrum, that is plot of φ(ω) versus ω display odd symmetry. Krupa Rasane(KLE) Page 13

14 Important facts about Spectra Nonperiodic signals have continuous spectra. Effects of symmetry on the FT of real x(t) are: Even Symmetry in : The FT, is real and even symmetric. Odd Symmetry in : The FT, is purely imaginary and odd symmetric. No symmetry in : The real part of, is even symmetric and imaginary part of, is odd symmetric. Example 4.1 Find the Fourier Transform of the signal Also plot the magnitude and phase spectra. Soln: Since is real,, exhibits odd symmetry. Hence is purely imaginary. The magnitude spectrum displays even symmetry and phase spectrum exhibits odd symmetry. Magnitude spectra for Ex4.1 The magnitude spectrum displays even symmetry Krupa Rasane(KLE) Page 14

15 Phase spectra for Ex4.1 and phase spectrum exhibits odd symmetry Example 4.2 Find the Fourier Transform of the signal shown in fig below Also plot the magnitude and phase spectra. Soln: -Since is real, - is neither odd nor even. Hence has both real and imaginary parts. Krupa Rasane(KLE) Page 15

16 5.1.8 Properties of Fourier Transform Linearity Time Shift Frequency Shift Scaling Frequency Differentiation Time differentiation Convolution Integration or Accumulation Modulation Parsevals theorem or Rayleigh s theorem Duality or similarity theorem Symmetry Krupa Rasane(KLE) Page 16

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18 Example : Breaking the signal into two known pulse signals as below Krupa Rasane(KLE) Page 18

19 2. Time Shift Property Shift of Time Signal gives Linear Phase Shift of Frequency Components. If the time signal is Time Scaled by a. Then The FT is Freq. Scaled by 1/a. And Amplitude scaling by in the frequency domain.that means Compression of x(t) to x(at) in time domain leads to Expansion of X(jω) by a An interesting duality!!! Scaling effect on signal Krupa Rasane(KLE) Page 19

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21 This is called Folding Property. 3) Frequency Shift Property 4) Time Differentiation If Krupa Rasane(KLE) Page 21

22 5) Frequency Differentiation 6) Convolution Krupa Rasane(KLE) Page 22

23 7) Integration and Accumulation Krupa Rasane(KLE) Page 23

24 8) Modulation 10) Parsevals theorem or Rayleigh s theorem Krupa Rasane(KLE) Page 24

25 11) Duality or similarity theorem 12) Duality or similarity theorem Krupa Rasane(KLE) Page 25

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27 Example 1: 06EC44-Signals and System Chapter Find the Fourier Transform of the signal. What is the restriction on α for FT to exist? Given : For α<0, we see exponential growth and Dirichlets condition is violated. Hence FT does not exist for α <0. For α > 0, Example 2: What is the energy of the signal x(t) = e - α t u(t) and what is the energy in the frequency band ω 0.5 rad/sec? Soln :The energy is Krupa Rasane(KLE) Page 27

28 Example 3 Find FT for the following signal. Soln: Since Example 4 Find the inverse Fourier Transform of the following signals. Soln: Krupa Rasane(KLE) Page 28

29 Example 5 Find x(t), if Krupa Rasane(KLE) Page 29

30 Reference 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 and Mark Fowler Signals and Systems. Text and Reference Books have been referred during the notes preparation. Krupa Rasane(KLE) Page 30