06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009•...
Transcript of 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009•...
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Chapter 5.2 Fourier Transform for Discrete time Signal
DTFT
5.2.1 Fourier Transform for Discrete Time Signal Definition
Development of the Discrete-Time Fourier Transform
Fig 1: is a Finite duration signal x[n]; Fig 2 periodic signal
FT of an Aperiodic Signal • Define
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DTFT
It is convenient to think of ak as being defined for all
integers K. So:
1) ak+N = ak – Special property of DT Fourier coefficients
2) We only use N consecutive values of ak in the synthesis
equation
3) Since N is periodic, it is specified by a sequence of N
numbers, either in the
Time or in the Frequency domain.
4) It is convenient to think of ak as being defined for all
integers K. So:
a) ak+N = ak – Special property of DT Fourier coefficients
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b) We only use N consecutive values of ak in the synthesis
equation
c) Since N is periodic, it is specified by a sequence of N
numbers, either in the Time or in the Frequency domain
CT Verses FT an OverView
Periodicity of a DT Fourier Transform
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High Frequency Low Frequency Concept
Example 1 :
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Magnitude and Phase Spectra of the FT of the FT of above Example:
x[n], for a>1 and a<1
Example 2:
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Example 3:
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Convergence of DT Fourier Transform
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The Analysis Equation will converge: Either if the x[n] is absolutely summable, or
has finite Energy, that is,
DTFT Summarize
It is convenient to think of ak as being defined for all integers K.
So:
1) ak+N = ak – Special property of DT Fourier coefficients
2) We only use N consecutive values of ak in the synthesis
equation
3) Since N is periodic, it is specified by a sequence of N
numbers, either in the Time or in the Frequency domain
5.2.1 Properties of Discrete-Time Fourier Transform
• Linearity
• Time Shifting and Frequency Shifting
• Conjugation and Conjugate Symmetry
• Differencing and Accumulation
• Time Reversal
• Time Expansion
• Differentiation in frequency
• Parseval’s Relation
Discrete-Time Fourier Transform Pair
• Notation
• Synthesis Equation
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• Analysis Equation
• Also if
• Periodicity of DTFT
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References • Figures and images used in these lecture notes are adopted
from “Signals & Systems” by Alan V. Oppenheim and
Alan S. Willsky, 1997
• Feng-Li Lian, NTU-EE and Mark Fowler Signals and
Systems.
• Text and Reference Books have been referred during the
notes preparation.
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