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Signal and System Lecture 11

This document discusses properties of signals and systems including: 1. Properties of the discrete-time Fourier transform (DTFT) and examples of DTFT calculations. 2. Duality between the Fourier series and Fourier transform. 3. How to plot the magnitude and phase of transforms and frequency responses, including examples of Bode plots for continuous and discrete-time systems.

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346 views19 pages

Signal and System Lecture 11

This document discusses properties of signals and systems including: 1. Properties of the discrete-time Fourier transform (DTFT) and examples of DTFT calculations. 2. Duality between the Fourier series and Fourier transform. 3. How to plot the magnitude and phase of transforms and frequency responses, including examples of Bode plots for continuous and discrete-time systems.

Uploaded by

ali_rehman87
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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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 LCCDE’s

— Rational function of e-jω,


use PFE to get h[n]
Example: First-order recursive system

with the condition of initial rest ⇔ causal


DTFT Multiplication Property
Calculating Periodic Convolutions
Example:
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
Example of CTFT duality
Square pulse in either time or frequency domain
DTFS

Duality in DTFS

Then
Duality between CTFS and DTFT

CTFS

DTFT
CTFS-DTFT Duality
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
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
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
A typical plot of the magnitude and phase of a second-
order DT frequency response
20log|H(ejω)| and ∠ H(ejω) vs. ω

For real signals,


0 to π is enough

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