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

Time propertyPeriodicNonperiodic Continuous (t)Fourier series (FS) Fourier Transform (FT) Discrete [n]Discrete-Time Fourier Series (DTFS) Discrete-Time Fourier Transform (DTFT) 2 Relationship between Time Properties of a Signal and the Appropriate Fourier Representation

Periodic signal:FS Representations 3

Notice that in; where, Ω 0 =2π/N fundamental frequency of x[n]. Similarly for continous x(t) fundamental period T, where, ω 0 =2π/T fundamental frequency of x(t). Show the relationship… (3.1) (3.2)

10/8/20155 Freq of the kth sinusoid is kω 0 and each sinusoid has a common period T, A sinusoid whose freq is an integer multiple of a fundamental freq is said to be a harmonic of the sinusoid at the fundamental freq. Thus, is the kth harmonic of

NonPeriodic Signal:FT Representations 6

10/8/20157 (3.3) (3.4)

10/8/20158 Identify the fourier representation for the following signals; a.x[n]=(1/2) n u[n] b. x(t)=e-t cos (2 πt) u(t) c. x(t)=1-cos (7 πt )+sin (6 πt)

9 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series (DTFS) (3.5) (3.6)

10 Fourier Series Are the DTFS coefficients of the signal x[n]. We can denote that x[n] and X[k] are a DTFS pair and the relationship as;

Exercise 10/8/201511

12 1. 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

13 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.

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

15Cont’d…

16 CT Fourier Series Definition

10/8/ CTFS Properties Linearity Dr. Abid Yahya

10/8/ CTFS Properties Time Shifting

10/8/ 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

10/8/ CTFS Properties Change of Representation Time (m is any positive integer) Dr. Abid Yahya

10/8/ CTFS Properties Change of Representation Time

10/8/ CTFS Properties Time Differentiation

10/8/2015. J. Roberts - All Rights Reserved 23 Time Integration is not periodic CTFS Properties Case 1Case 2

10/8/ CTFS Properties Multiplication-Convolution Duality

10/8/ Fourier Series(DTFS)

10/8/ 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)

10/8/ Fourier Series(DTFS)

10/8/ DTFS Properties Linearity

10/8/ DTFS Properties Time Shifting

10/8/ DTFS Properties Frequency Shifting (Harmonic Number Shifting)

10/8/ 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

10/8/ DTFS Properties Change of Representation Time (q is any positive integer)

10/8/ DTFS Properties First Backward Difference Multiplication- Convolution Duality

The Fourier Transform

10/8/ 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

10/8/ ForwardInverse f form  form ForwardInverse Definition of the CTFT or Commonly-used notation:

10/8/ 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

10/8/ 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

10/8/ Definition of the DTFT F Form  Form ForwardInverse ForwardInverse

10/8/ The Four Fourier Methods

10/8/ Relations Among Fourier Methods Multiplication-Convolution Duality

10/8/ Relations Among Fourier Methods Time and Frequency Shifting

10/8/ Tutorials 1. Compute the CTFS:,