1. 1 Fourier Representation of Signals and LTI Systems. CHAPTER 3 EKT 232

2. 2  Signals are represented as superposition's of complex sinusoids which leads to a useful expression for the system output and provide a characterization of signals and systems.  Example in music, the orchestra is a superposition of sounds generated by different equipment having different frequency range such as string, base, violin and ect. The same example applied the choir team.  Study of signals and systems using sinusoidal representation is termed as Fourier Analysis introduced by Joseph Fourier (1768-1830).  There are four distinct Fourier representations, each applicable to different class of signals. 3.1 Introduction.

3. 3 Fourier Series Discrete Time Fourier series (DTFS)

4. Fourier Series Notice that in the summation is over exactly one period, a finite summation. This is because of the periodicity of the complex sinusoid, This occurs because discrete time n is always an integer.

5. 5 Fourier Series

6. 6 CT Fourier Series Definition

7. 11/9/20157 CTFS Properties Linearity Dr. Abid Yahya

8. 11/9/20158 CTFS Properties Time Shifting

9. 11/9/20159 CTFS Properties Frequency Shifting (Harmonic Number Shifting) A shift in frequency (harmonic number) corresponds to multiplication of the time function by a complex exponential. Time Reversal

10. 11/9/201510 CTFS Properties Change of Representation Time (m is any positive integer) Dr. Abid Yahya

11. 11/9/201511 CTFS Properties Change of Representation Time

12. 11/9/201512 CTFS Properties Time Differentiation

13. 11/9/2015. J. Roberts - All Rights Reserved 13 Time Integration is not periodic CTFS Properties Case 1Case 2

14. 11/9/201514 CTFS Properties Multiplication-Convolution Duality

15. 11/9/201515 Fourier Series(DTFS)

16. 11/9/201516 Notice that in the summation is over exactly one period, a finite summation. This is because of the periodicity of the complex sinusoid, This occurs because discrete time n is always an integer. Fourier Series(DTFS)

17. 11/9/201517 Fourier Series(DTFS)

18. 11/9/201518 DTFS Properties Linearity

19. 11/9/201519 DTFS Properties Time Shifting

20. 11/9/201520 DTFS Properties Frequency Shifting (Harmonic Number Shifting)

21. 11/9/201521 DTFS Properties Time Scaling If a is not an integer, some values of z[n] are undefined and no DTFS can be found. If a is an integer (other than 1) then z[n] is a decimated version of x[n] with some values missing and there cannot be a unique relationship between their harmonic functions. However, if then

22. 11/9/201522 DTFS Properties Change of Representation Time (q is any positive integer)

23. 11/9/201523 DTFS Properties First Backward Difference Multiplication- Convolution Duality Dr. Abid Yahya

24. The Fourier Transform

25. 11/9/201525 Extending the CTFS The CTFS is a good analysis tool for systems with periodic excitation but the CTFS cannot represent an aperiodic signal for all time The continuous-time Fourier transform (CTFT) can represent an aperiodic signal for all time Dr. Abid Yahya

26. 11/9/201526 ForwardInverse f form  form ForwardInverse Definition of the CTFT or Commonly-used notation:

27. 11/9/201527 Some Remarkable Implications of the Fourier Transform The CTFT expresses a finite-amplitude, real-valued, aperiodic signal which can also, in general, be time-limited, as a summation (an integral) of an infinite continuum of weighted, infinitesimal-amplitude, complex sinusoids, each of which is unlimited in time. (Time limited means “having non-zero values only for a finite time.”)

28. The Discrete-Time Fourier Transform

29. 11/9/201529 Extending the DTFS Analogous to the CTFS, the DTFS is a good analysis tool for systems with periodic excitation but cannot represent an aperiodic signal for all time The discrete-time Fourier transform (DTFT) can represent an aperiodic signal for all time Dr. Abid Yahya

30. 11/9/201530 Definition of the DTFT F Form  Form ForwardInverse ForwardInverse

31. 11/9/201531 The Four Fourier Methods

32. 11/9/2015Dr. Abid Yahya32 Relations Among Fourier Methods Multiplication-Convolution Duality

33. 11/9/201533 Relations Among Fourier Methods Time and Frequency Shifting Dr. Abid Yahya

34. 11/9/201534 Tutorials 1. Compute the CTFS:,

35. 35 2. Find the frequency-domain representation of the signal in Figure 3.1 below. Figure 3.1: Time Domain Signal. Solution: Step 1: Determine N and  . The signal has period N=5, so   =2  /5. Also the signal has odd symmetry, so we sum over n = -2 to n = 2 from equation

36. 36 Step 2: Solve for the frequency-domain, X[k]. From step 1, we found the fundamental frequency, N =5, and we sum over n = -2 to n = 2.Cont’d…

37. 37 From the value of x{n} we get,Cont’d…