Bài giảng "Xử lý tín hiệu số: Fourier Transform" cung cấp cho người học các kiến thức: Frequency analysis of discrete time signal, properties of Fourier transform, frequency domain characteristics of LTI systems, discrete Fourier Transform, fast Fourier Transform. Mời các bạn cùng tham khảo nội dung chi tiết.
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Nội dung Text: Bài giảng Xử lý tín hiệu số: Fourier Transform - Ngô Quốc Cường (ĐH Sư phạm Kỹ thuật)
- Xử lý tín hiệu số
Fourier Transform
Ngô Quốc Cường
ngoquoccuong175@gmail.com
Ngô Quốc Cường
sites.google.com/a/hcmute.edu.vn/ngoquoccuong
- CONTENTS
• Frequency analysis of discrete time signal
• Properties of Fourier transform
• Frequency domain characteristics of LTI systems
• Discrete Fourier Transform
• Fast Fourier Transform
2
- 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.
4
- 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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- 7
- 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
• 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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