Introduction to Signal Processing - Michigan State University
Transcript of Introduction to Signal Processing - Michigan State University
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Introduction to Signal ProcessingSummer 2007
1. DTFT Properties and Examples2. Duality in FS & FT3. Magnitude/Phase of Transforms
and Frequency Responses
Chap. 5 Chap. 6
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Convolution Property Example
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DT LTI System Described by LCCDE’s
— Rational function of e-jω, use PFE to get h[n]
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Example: First-order recursive system
with the condition of initial rest , causal
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DTFT Multiplication Property
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Calculating Periodic Convolutions
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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
Suppose f(•) and g(•) are two functions related by
Then
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Example of CTFT dualitySquare pulse in either time or frequency domain
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DTFS
Duality in DTFS
Then
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Duality between CTFS and DTFT
CTFS
DTFT
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CTFS-DTFT Duality
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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 Series2) Effect of phase on image processing
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Log-Magnitude and Phase
Easy to add
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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):
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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 response 20log|H(ejω)| and ∠ H(ejω) vs. ω