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Signals and Systems Fall 2003 Lecture #11 9 October 2003

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. ratio of polynomials in.

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Signals and Systems Fall 2003 Lecture #11 9 October 2003

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  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

  2. Convolution Property Example ratio of polynomials in A, B – determined by partial fraction expansion

  3. DT LTI System Described by LCCDE’s From time-shifting property: • — Rational function of e-jω, use PFE to get h[n]

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

  5. DTFT Multiplication Property Periodic Convolution Derivation:

  6. Calculating Periodic Convolutions Suppose we integrate from –π to π: where otherwise

  7. Example:

  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 Letт = t and r = w: Letт = -w and r = t:

  9. Example of CTFT duality • Square pulse in either time or frequency domain

  10. DTFS Discrete & periodic in time periodic & discrete in frequency Duality in DTFS Suppose are two functions related by Then Let m = n and r = -k: Let r = n and m = k:

  11. Duality between CTFS and DTFT CTFS Periodic in time Discrete in frequency DTFT Discrete in time Periodic in frequency

  12. CTFS-DTFT Duality Suppose is a CT signal and a DT sequence related by Then (periodic with period 2π)

  13. Magnitude and Phase of FT, and Parseval Relation CT: Parseval Relation: Energy density in ω DT: Parseval Relation:

  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

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

  16. Log-Magnitude and Phase or and Cascading: Easy to add

  17. 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): output power 1 bel 10 decibels input power power magnitude • So… 20 dB or 2 bels: • = 10 amplitude gain • = 100 power gain

  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

  19. 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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