Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms Topic three Fourier Transforms of Periodic Signals Properties of the CT Fourier Transform The Convolution Property of the CTFT Frequency Response and LTI Systems Revisited Multiplication Property and Parseval’s Relation The DT Fourier Transform

Fourier’s Derivation of the CT Fourier Transform  x(t) - an aperiodic signal view it as the limit of a periodic signal as T → ∞  For a periodic signal, the harmonic components are spaced ω 0 = 2π/T apart...  As T → ∞, ω 0 → 0, and harmonic components are spaced closer and closer in frequency Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 2

Motivating Example: Square wave Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 3 increases kept fixed Discrete frequency points become denser in ω as T increases

So, on with the derivation... Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 4 For simplicity, assume x(t) has a finite duration.

Derivation (continued) Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 5

Derivation (continued) Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 6

For what kinds of signals can we do this? (1) It works also even if x(t) is infinite duration, but satisfies: a) Finite energy In this case, there is zero energy in the error b) Dirichlet conditions c) By allowing impulses in x(t)or in X(j ω ), we can represent even more Signals E.g. It allows us to consider FT for periodic signals Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 7

Example #1 Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 8

Example #2: Exponential function Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 9 Even symmetry Odd symmetry

Example #3: A square pulse in the time-domain Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 10 Note the inverse relation between the two widths ⇒ Uncertainty principle

Useful facts about CTFT’s Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 11

Example #4: Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 12 Also a Gaussian! (Pulse width in t)(Pulse width in ω) ⇒ ∆t∆ ω ~ (1/a 1/2 )(a 1/2 ) = 1 Uncertainty Principle! Cannot make both ∆t and ∆ω arbitrarily small.

CT Fourier Transforms of Periodic Signals Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 13

Example #5: Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 14 “Line Spectrum”

Example #6: Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 15 x(t) Same function in the frequency-domain! Note: (period in t) T ⇔ (period in ω) 2π/T Inverse relationship again!

Properties of the CT Fourier Transform Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 16

Properties (continued) Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 17

The Properties Keep on Coming... Book Chapter4: Section1 Computer Engineering Department, Signals and Systems 18

The CT Fourier Transform Pair Book Chapter4: Section2 Computer Engineering Department, Signals and Systems 19 ─ FT (Analysis Equation) ─ Inverse FT (Synthesis Equation) Last lecture:some properties Today: further exploration

Book Chapter4: Section2 Computer Engineering Department, Signal and Systems 20 Convolution Property A consequence of the eigenfunction property : h(t) H(jω).a Synthesis equation for y(t)

The Frequency Response Revisited Book Chapter4: Section2 Computer Engineering Department, Signal and Systems 21 h(t) impulse response frequency response The frequency response of a CT LTI system is simply the Fourier transform of its impulse response Example #1: H(jω) Recall inverse FT

Example #2 A differentiator Book Chapter4: Section2 Computer Engineering Department, Signal and Systems 22 - an LTI system Differentiation property: 1) Amplifies high frequencies (enhances sharp edges) Larger at high ω o phase shift 2) +π/2 phase shift ( j = e jπ/2 )

Example #3: Impulse Response of an Ideal Lowpass Filter Book Chapter4: Section2 Computer Engineering Department, Signal and Systems 23 Define: sinc(θ) Questions: 1) Is this a causal system? No. 2) What is h(0)? 3) What is the steady-state value of the step response, i.e. s(∞)?

Example #4: Cascading filtering operations Book Chapter4: Section2 Computer Engineering Department, Signal and Systems 24

Book Chapter4: Section2 Computer Engineering Department, Signal and Systems 25 h(t)x(t) Example #6: Gaussian × Gaussian = Gaussian ⇒ Gaussian ∗ Gaussian = Gaussian

Example #2 from last lecture Book Chapter4: Section2 Computer Engineering Department, Signal and Systems 26

Book Chapter4: Section2 Computer Engineering Department, Signal and Systems 27 - a rational function of jω, ratio of polynomials of jω Partial fraction expansion Inverse FT

Example #8: LTI Systems Described by LCCDE’s (Linear-constant-coefficient differential equations) Book Chapter4: Section2 Computer Engineering Department, Signal and Systems 28 Using the Differentiation Property Transform both sides of the equation 1) Rational, can use PFE to get h(t) 2) If X(j ω ) is rational e.g. x(t)=Σcie -at u(t) then Y(j ω ) is also rational H(jω)

Parseval’s Relation Book Chapter4: Section2 Computer Engineering Department, Signal and Systems 29 Total energy in the time-domain Total energy in the time-domain - Spectral density Multiplication Property FT is highly symmetric, We already know that: Then it isn’t a surprise that: Convolution in ω — A consequence of Duality

Examples of the Multiplication Property Book Chapter4: Section2 Computer Engineering Department, Signal and Systems 30 For any s(t)...

Example (continued) Book Chapter4: Section2 Computer Engineering Department, Signal and Systems 31 Amplitude modulation (AM) Drawn assuming: