Bài giảng Xử lý tín hiệu số - Fourier Transform
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- Xử lý tín hiệu số Fourier Transform Ngô Quốc Cường ngoquoccuong175@gmail.com sites.google.com/a/hcmute.edu.vn/Ngô Quốc Cường 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) 3
- 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 5
- 1.Frequency analysis of discrete time signal 1.1. Fourier series for periodic signals • Solution of example 1: 6
- 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 2 : 8
- 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 9
- 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 10
- 1.Frequency analysis of discrete time signal 1.2. Fourier transform of aperiodic signals • Example 2: 11
- 1.Frequency analysis of discrete time signal 1.2. Fourier transform of aperiodic signals • Solution of example 2: 12
- 1.Frequency analysis of discrete time signal 1.2. Fourier transform of aperiodic signals • Solution of example 2 (cont’d): 13
- 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 14
- 2. Properties of Fourier transform 15
- 2. Properties of Fourier transform 16
- 2. Properties of Fourier transform • Example 3: 17
- 2. Properties of Fourier transform • Solution of Example 3: 18
- 2. Properties of Fourier transform • Solution of Example 3: 19
- 2. Properties of Fourier transform • Solution of Example 3 (cont’d): a=0.8 20
- 2. Properties of Fourier transform • Example 4: 21
- 2. Properties of Fourier transform • Solution of Example 4: 22
- 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 23
- 3. Frequency domain characteristics of LTI systems • The Fourier transform of the unit sample response h(k) of the system • The function H(휔) exists if the system is BIBO stable • The response of the system to the complex exponential is 24
- 3. Frequency domain characteristics of LTI systems • Example 25
- 3. Frequency domain characteristics of LTI systems • Solution • First, • At 휔 = /2 yields • The output is 26
- 3. Frequency domain characteristics of LTI systems • In general, H(휔) is a complex function of 휔, hence • Expressed in terms of real and image components 27
- 3. Frequency domain characteristics of LTI systems • Example 28
- 3. Frequency domain characteristics of LTI systems • The impulse response • It follows that • Hence 29
- 3. Frequency domain characteristics of LTI systems 30
- 3. Frequency domain characteristics of LTI systems • Example • Determine the response of the system (with given impulse response h(n)) to the input signal x(n) 31
- 3. Frequency domain characteristics of LTI systems • Solution • The frequency response • With each term in the input signal • The response to the input signal 32
- 4. Discrete Fourier Transform 4.1. Frequency domain sampling • Aperiodic finite- energy signals have continuous spectra • Sample X(휔) periodically at a spacing of 훿휔 radians, take N equidistant samples in the interval 0 ≤ 휔 ≤ 2 with spacing 휔 = 2 / 33
- 4. Discrete Fourier Transform 4.1. Frequency domain sampling 34
- 4. Discrete Fourier Transform 4.2. Discrete Fourier Transform • A finite-duration sequence x(n) of length L has the DFT where N ≥ L • Recover x(n) from its DFT (inverse DFT) 35
- 4. Discrete Fourier Transform • 4.2. Discrete Fourier Transform • Exercise – Compute the 8-point DFT of 36
- 4. Discrete Fourier Transform • 4.2. Discrete Fourier Transform • Exercise – Compute the 6-point DFT of 37
- 4. Discrete Fourier Transform 4.2. Properties of the DFT Denote • Periodicity, Linearity, Circular symmetries • Time reversal • Circular time shift • Circular frequency shift • Complex conjugate properties • Circular correlation • Parseval ’s theorem 38
- 4. Discrete Fourier Transform 39
- 4. Discrete Fourier Transform 4.2. Properties of the DFT • Circular symmetries – The N-point DFT of x(n) is equivalent to the N-point DFT of xp(n), in which – Shift the periodic sequence xp(n) by k unit to the right, we obtain another periodic sequence 40
- 4. Discrete Fourier Transform 4.2. Properties of the DFT • Circular symmetries – x’(n) is related to x(n) by a circular shift. 41
- 4. Discrete Fourier Transform 4.2. Properties of the DFT • Circular symmetries – The circular shift can be represented as the index modulo N. – Example 45
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution – Suppose we have two finite-duration sequences of length N, x1(n) and x2(n). Their respective N-point DFTS are: – Multiply two sequences, the result is a DFT, say X3(k), of a sequence x3(n) 46
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution – The inverse of X3(k) is – Circular convolution 47
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution • Example 48
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution • Solution 49
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution 50
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution 51
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution 52
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution 53
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution • The circular of the two sequences yields the sequence 54
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution • Example • By means of DFT and IDFT, determine the sequence x3(n) corresponding to the circular convolution of x1(n) and x2(n) 55
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution • Solution • Compute the DFTs 56
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution • Solution • Thus, • Multiply the two DFTs • The IDFT of X3(k) 57
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution • Solution • Thus, 58
- 4. Discrete Fourier Transform 4.3. Multiplication of two DFTs and Circular convolution 59
- 4. Discrete Fourier Transform • Exercise 60
- 5. Fast Fourier Transform • The DFT can be written in the formula where • For each value of k, direct computation of X(k) involves – N complex multiplications – N-1 complex additions • For all N values, the DFT requires – N2 complex multiplications – N2-N complex additions 61
- 5. Fast Fourier Transform • Direct computation of DFT does not exploit the symmetry and periodicity properties 62
- 5. Fast Fourier Transform 5.1. Radix-2 FFT algorithms • Split N-point data sequence x(n) into two N/2-point sequence • The N-point DFT of x(n) can be expressed as 63
- 5. Fast Fourier Transform 5.1. Radix-2 FFT algorithms • Because , • We also have • Result in a reduction of the number of multiplication 64
- 5. Fast Fourier Transform 5.1. Radix-2 FFT algorithms • We can repeat the process for each of the sequences f1(n) and f2(n) 65
- 5. Fast Fourier Transform 5.1. Radix-2 FFT algorithms • Thus • The total number of multiplication is reduced again 66
- 5. Fast Fourier Transform 5.1. Radix-2 FFT algorithms • Comparison of computational complexity 67
- 5. Fast Fourier Transform 5.1. Radix-2 FFT algorithms • Perform 8-point DFT in 3 stages 68
- 5. Fast Fourier Transform 5.1. Radix-2 FFT algorithms • Basic butterfly computation 70
- 5. Fast Fourier Transform 5.1. Radix-2 FFT algorithms • Compute 8-point DFT (using FFT algorithm) of x(n)={1, 2, 3, 0, 0, 0, 0, 0} 71
- 5. Fast Fourier Transform 5.2. Radix-4 FFT algorithms 72