Bài giảng xử lý tín hiệu số fourier transform ngô quốc cường

Xử lý tín hiệu số
Fourier Transform

Ngô Quốc Cường

Ngô Quốc Cường
sites.google.com/a/hcmute.edu.vn/ngoquoccuong


CONTENTS
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Frequency analysis of discrete time signal
Properties of Fourier transform
Frequency domain characteristics of LTI systems
Discrete Fourier Transform
Fast Fourier Transform

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1.Frequency analysis of discrete time signal
1.1. Fourier series for periodic signals
– Given a periodic signal x(n) with period N.

(DTFS)

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1.Frequency analysis of discrete time signal
1.1. Fourier series for periodic signals

• The spectrum of a signal x(n) which is periodic with period N,
is a periodic sequence with period N.

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1.Frequency analysis of discrete time signal
1.1. Fourier series for periodic signals
• Example 1: Determine the spectra of the signal

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1.Frequency analysis of discrete time signal
1.1. Fourier series for periodic signals
• Solution of example 1:

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1.Frequency analysis of discrete time signal

1.2. Fourier transform of aperiodic signals
• The Fourier transform of a finite energy signal x(n) is defined
as

• X(w) is periodic with period 2𝜋:

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1.Frequency analysis of discrete time signal
1.2. Fourier transform of aperiodic signals
• In summary, the Fourier transform pair of a discrete time is as
follows

• Uniform convergence is guaranteed if x(n) is absolutely
summable

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1.Frequency analysis of discrete time signal
1.2. Fourier transform of aperiodic signals
• The spectrum X(w) is, in general, a complex valued function
of frequency

• The energy density spectrum of x(n) is

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1.Frequency analysis of discrete time signal
1.2. Fourier transform of aperiodic signals
• Example 2:

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1.Frequency analysis of discrete time signal
1.2. Fourier transform of aperiodic signals
• Solution of example 2:

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1.Frequency analysis of discrete time signal
1.2. Fourier transform of aperiodic signals
• Solution of example 2 (cont’d):

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2. Properties of Fourier transform
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Symmetry
Linearity
Time shifting
Time reversal
Convolution theorem
Correlation theorem
Frequency shifting
Modulation theorem
Windowing theorem
Differentiation in frequency domain

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2. Properties of Fourier transform

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2. Properties of Fourier transform

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2. Properties of Fourier transform
• Example 3:

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2. Properties of Fourier transform
• Solution of Example 3:

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2. Properties of Fourier transform
• Solution of Example 3:

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2. Properties of Fourier transform
• Solution of Example 3 (cont’d):

a=0.8

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2. Properties of Fourier transform
• Example 4:

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2. Properties of Fourier transform

• Solution of Example 4:

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3. Frequency domain characteristics of LTI
systems
• The response of any relaxed-system to arbitrary input signal
is:

• Excite the system with the complex exponential

• Obtain the response

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3. Frequency domain characteristics of LTI
systems
• The Fourier transform of the unit sample response h(k) of the
system

• The function H(𝜔) exists if the system is BIBO stable

• The response of the system to the complex exponential is

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3. Frequency domain characteristics of LTI

systems
• Example

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