Signals and Systems

Fall 2003

Lecture #119 October 20031. DTFT Properties and Examples

2. Duality in FS & FT3. Magnitude/Phase of Transforms and Frequency Responses

Convolution Property Example

ratio of polynomials in

A, B – determined by partial fraction expansion

DT LTI System Described by LCCDE’s

— Rational function of e-jω, use PFE to get h[n]

From time-shifting property:

Example: First-order recursive system

with the condition of initial rest ⇔ causal

DTFT Multiplication Property

Periodic ConvolutionDerivation:

Calculating Periodic Convolutions

Suppose we integrate from –π to π:

where

otherwise

Example:

Duality in Fourier AnalysisFourier Transform is highly symmetric

CTFT: Both time and frequency are continuous and in general aperiodic

Same except for these differences

Suppose f() and g() are two functions related by

Then

Letт = t and r = w:Letт = -w and r = t:

Example of CTFT dualitySquare pulse in either time or frequency

domain

DTFS

Duality in DTFS

Then

Let m = n and r = -k:Let r = n and m = k:

Discrete & periodic in time periodic & discrete in frequency

Suppose are two functions related by

Duality between CTFS and DTFT

CTFS

DTFT

Periodic in time Discrete in frequency

Discrete in time Periodic in frequency

CTFS-DTFT Duality

Suppose is a CT signal and a DT sequence related by

Then

(periodic with period 2π)

Magnitude and Phase of FT, and Parseval Relation

CT:

Parseval Relation:

Energy density in ω

DT:

Parseval Relation:

Effects of Phase

• Not on signal energy distribution as a function of frequency

• Can have dramatic effect on signal shape/character

— Constructive/Destructive interference

• Is that important?

— Depends on the signal and the context

Demo: 1) Effect of phase on Fourier Series 2) Effect of phase on image processing

Log-Magnitude and Phase

Easy to add

Cascading:

or

and

Plotting Log-Magnitude and Phase

a) For real-valued signals and systems Plot for ω ≥ 0, often with a

logarithmic scale for frequency in CT

b) In DT, need only plot for 0 ≤ ω ≤ π (with linear scale)

c) For historical reasons, log-magnitude is usually plotted in units of decibels (dB):

power magnitude

So… 20 dB or 2 bels:= 10 amplitude gain= 100 power gain

10 decibels1 bel output powerinput power

A Typical Bode plot for a second-order CT system20 log|H(jω)| and ∠ H(jω) vs. log ω

40 dB/decade

Changes by -π

A typical plot of the magnitude and phase of a second- order DT frequency response20log|H(ejω)| and ∠ H(ejω) vs. ω

For real signals, 0 to π is enough