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

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 ( ).  There are four distinct Fourier representations, each applicable to different class of signals. 3.1 Introduction.

3 Fourier Series Discrete Time Fourier series (DTFS)

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 Fourier Series

6 CT Fourier Series Definition

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

11/9/20158 CTFS Properties Time Shifting

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

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

11/9/ CTFS Properties Change of Representation Time

11/9/ CTFS Properties Time Differentiation

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

11/9/ CTFS Properties Multiplication-Convolution Duality

11/9/ Fourier Series(DTFS)

11/9/ 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)

11/9/ Fourier Series(DTFS)

11/9/ DTFS Properties Linearity

11/9/ DTFS Properties Time Shifting

11/9/ DTFS Properties Frequency Shifting (Harmonic Number Shifting)

11/9/ 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

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

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

The Fourier Transform

11/9/ 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

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

11/9/ 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.”)

The Discrete-Time Fourier Transform

11/9/ 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

11/9/ Definition of the DTFT F Form  Form ForwardInverse ForwardInverse

11/9/ The Four Fourier Methods

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

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

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

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