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
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
of decibels (dB):
power magnitude
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(e
jω
)| and ∠ H(e
jω
) vs,ω
For real signals,
0 to π is enough