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 1.Frequency analysis of discrete time signal 1.1 Fourier series for periodic signals – Given a periodic signal x(n) with period N (DTFS) 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 1.Frequency analysis of discrete time signal 1.1 Fourier series for periodic signals • Example 1: Determine the spectra of the signal 1.Frequency analysis of discrete time signal 1.1 Fourier series for periodic signals • Solution of example 1: 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: 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 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 Discrete Fourier Transform 4.3 Multiplication of two DFTs and Circular convolution • Solution • Thus, 58 Discrete Fourier Transform 4.3 Multiplication of two DFTs and Circular convolution 59 Discrete Fourier Transform • Exercise 60 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 Fast Fourier Transform • Direct computation of DFT does not exploit the symmetry and periodicity properties 62 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 Fast Fourier Transform 5.1 Radix-2 FFT algorithms • Because , • We also have • Result in a reduction of the number of multiplication 64 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 Fast Fourier Transform 5.1 Radix-2 FFT algorithms • Thus • The total number of multiplication is reduced again 66 Fast Fourier Transform 5.1 Radix-2 FFT algorithms • Comparison of computational complexity 67 Fast Fourier Transform 5.1 Radix-2 FFT algorithms • Perform 8-point DFT in stages 68 69 Fast Fourier Transform 5.1 Radix-2 FFT algorithms • Basic butterfly computation 70 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 Fast Fourier Transform 5.2 Radix-4 FFT algorithms 72 [...]... 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. ..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... input signal 32 4 Discrete Fourier Transform 4.1 Frequency domain sampling • Aperiodic finite- energy signals have continuous spectra • Sample X(𝜔) periodically at a spacing of 𝛿𝜔 radians, take N equidistant samples in the interval 0 ≤ 𝜔 ≤ 2𝜋 with spacing 𝜔 = 2𝜋/𝑁 33 4 Discrete Fourier Transform 4.1 Frequency domain sampling 34 4 Discrete Fourier Transform 4.2 Discrete Fourier Transform • A finite-duration... 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 ... • 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(𝜔) exists if the system is BIBO stable • The response of the system to the complex exponential is 24 3 Frequency domain characteristics ... of Fourier transform 15 Properties of Fourier transform 16 Properties of Fourier transform • Example 3: 17 Properties of Fourier transform • Solution of Example 3: 18 Properties of Fourier transform. .. (inverse DFT) 35 Discrete Fourier Transform • 4.2 Discrete Fourier Transform • Exercise – Compute the 8-point DFT of 36 Discrete Fourier Transform • 4.2 Discrete Fourier Transform • Exercise – Compute... Properties of Fourier transform Frequency domain characteristics of LTI systems Discrete Fourier Transform Fast Fourier Transform 1.Frequency analysis of discrete time signal 1.1 Fourier series