Signals and Systems Fall 2003 Lecture #11 9 October 2003 1. DTFT - - PowerPoint PPT Presentation

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Signals and Systems Fall 2003 Lecture #11 9 October 2003 1. DTFT Properties and Examples 2. Duality in FS & FT 3. Magnitude/Phase of Transforms and Frequency Responses Convolution Property Example DT LTI System Described by LCCDEs

SLIDE 1

Signals and Systems

Fall 2003 Lecture #11

9 October 2003

1. DTFT Properties and Examples 2. Duality in FS & FT 3. Magnitude/Phase of Transforms and Frequency Responses

SLIDE 2

Convolution Property Example

SLIDE 3

DT LTI System Described by LCCDE’s

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

SLIDE 4

Example: First-order recursive system with the condition of initial rest ⇔ causal

SLIDE 5

DTFT Multiplication Property

SLIDE 6

Calculating Periodic Convolutions

SLIDE 7

Example:

SLIDE 8

Duality in Fourier Analysis

Fourier 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

SLIDE 9

Example of CTFT duality

Square pulse in either time or frequency domain

SLIDE 10

DTFS Duality in DTFS

Then

SLIDE 11

Duality between CTFS and DTFT

CTFS DTFT

SLIDE 12

CTFS-DTFT Duality

SLIDE 13

Magnitude and Phase of FT, and Parseval Relation

CT: Parseval Relation:

Energy density in ω

DT: Parseval Relation:

SLIDE 14

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

SLIDE 15

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

SLIDE 16

Log-Magnitude and Phase

Easy to add

SLIDE 17

Plotting Log-Magnitude and Phase

Plot for ω ≥ 0, often with a logarithmic scale for frequency in CT

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

b) In DT, need only plot for 0 ≤ ω ≤ π (with linear scale) a) For real-valued signals and systems c) For historical reasons, log-magnitude is usually plotted in units

f decibels (dB):

power magnitude

SLIDE 18

A Typical Bode plot for a second-order CT system

20 log|H(jω)| and ∠ H(jω) vs. log ω 40 dB/decade Changes by -π

SLIDE 19

A typical plot of the magnitude and phase of a second-

rder DT frequency response

20log|H(ejω)| and ∠ H(ejω) vs. ω

For real signals, 0 to π is enough