Signals and Systems Fall 2003 Lecture #11 9 October 2003
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Transcript of Signals and Systems Fall 2003 Lecture #11 9 October 2003
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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
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Convolution Property Example
ratio of polynomials in
A, B – determined by partial fraction expansion
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DT LTI System Described by LCCDE’s
— Rational function of e-jω, use PFE to get h[n]
From time-shifting property:
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Example: First-order recursive system
with the condition of initial rest ⇔ causal
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DTFT Multiplication Property
Periodic ConvolutionDerivation:
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Calculating Periodic Convolutions
Suppose we integrate from –π to π:
where
otherwise
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Example:
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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:
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Example of CTFT dualitySquare pulse in either time or frequency
domain
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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
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Duality between CTFS and DTFT
CTFS
DTFT
Periodic in time Discrete in frequency
Discrete in time Periodic in frequency
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CTFS-DTFT Duality
Suppose is a CT signal and a DT sequence related by
Then
(periodic with period 2π)
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Magnitude and Phase of FT, and Parseval Relation
CT:
Parseval Relation:
Energy density in ω
DT:
Parseval Relation:
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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
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Demo: 1) Effect of phase on Fourier Series 2) Effect of phase on image processing
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Log-Magnitude and Phase
Easy to add
Cascading:
or
and
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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
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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
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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