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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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