ENEE 322: Continuous-Time Fourier Transform (Chapter 4) Richard J. La Spring 2016

Motivation Recall: Continuous-time Fourier series representation allows us to write a large class of periodic signals as a linear combination of complex exponentials with harmonic frequencies Most of real signals are not periodic Usefulness of the idea below will be severely limited if aperiodic signals could not be written in terms of complex exponentials LTI with IR h(t)

Continuous-time Fourier Transform (1) It turns out a large class of aperiodic signals can also be written as linear combination of complex exponentials Suppose that a continuous-time signal x(t) is absolutely integrable, i.e., Fourier transform: Inverse Fourier transform:

Continuous-time Fourier Transform (2) * Interpretation of inverse Fourier transform Let denote the set of all complex-valued continuous-time signals with finite energy Inner product of two signals is given by Fourier transform: where

Continuous-time Fourier Transform (3) * Substituting last expression in the inverse Fourier transform In turns out is an orthogonal basis for Fourier transform of a signal x(t) (with finite energy) tells us how the energy of the signal is spread in the frequency domain In the same way the Fourier series representation tells us how the average power of a periodic signal is spread in the frequency domain

Continuous-time Fourier Transform (4) Fourier transform pair: There are many ways to represent a signal Signal is the representation in time domain Fourier transform is just another representation of the same signal in frequency domain Allows us to go back and forth between time domain and frequency domain

Fourier Transform of periodic signals Fourier transform of a periodic signal with period T Example #1:

Examples (2) Example #2: Example #3: Example #4:

Properties of continuous-time Fourier transform (1) Linearity: Time shifting: Time reversal: Complex conjugation: Special case: x(t) is real – What does it say about real and imaginary parts of the Fourier transform?

Properties of continuous-time Fourier transform (2) Parseval’s relation: Differentiation: Integration: Time & frequency scaling:

Properties of continuous-time Fourier transform (3) Duality (*) Oftentimes we know and We can compute y(t) quite easily without performing inverse Fourier transform

Properties of continuous-time Fourier transform (4) Example #1: What is the signal y(t) that has Fourier transform given below?

Properties of continuous-time Fourier transform (5) Example #2: Differentiation in frequency domain Example #3: Shifting in frequency domain (often used in communication systems due to regulation)

Properties of continuous-time Fourier transform (6) Example #5: Integration in frequency domain

Convolution & Multiplication Property (1) Recall: Take a complex exponential and pass it through an LTI system with impulse response h(t) where Guess what would be the output when the input is given by LTI with IR h(t) Fourier transform of h(t) evaluated at !!!!

Convolution & Multiplication Property (2) Earlier with weighted sum of complex exponentials where Now LTI with IR h(t) LTI with IR h(t) ????

Convolution & Multiplication Property (3) Answer: Drum roll please …. Look at the expression for the output carefully LTI with IR h(t) inverse Fourier transform

Convolution & Multiplication Property (4) Second equality tells us …. Fourier transform of the output is given by the product of the Fourier transform of the input signal and that of the impulse response Gives us another method for computing the output (in time domain) given an input signal and the impulse response Compute the Fourier transform of the input signal and the impulse response (called “frequency response”) Multiply the Fourier transforms to obtain the Fourier transform of the output Take the inverse Fourier transform to obtain the output in time domain

Convolution & Multiplication Property (5) Convolution in time domain Multiplication in frequency domain The Fourier transform of the impulse response called the “frequency response” of the system Tells us how each component with different frequency is changed by the system in magnitude and phase Caveat: Since the Fourier transform is not defined for all signals, the frequency response is not defined for all LTI systems. We will assume that the frequency response of the LTI systems we deal with exists

Convolution & Multiplication Property (6) Implication for cascade systems The frequency response of a cascade of LTI systems is given by the product of the frequency responses of the LTI systems S1 S2

Convolution & Multiplication Property (5) What if we have a multiplication in time domain? Multiplication in time domain becomes convolution in frequency domain

Examples (1) Example : Suppose that we have an LTI system with impulse response Find the output when the input x(t) is given by (a) (b)