Bài giảng Digital Signal Processing - Chapter 4: FIR filtering and Convolution - Nguyen Thanh Tuan

Content
 Block processing methods
 Convolution: direct form, convolution table
 Convolution: LTI form, LTI table
 Matrix form
 Flip-and-slide form
 Overlap-add block convolution method
 Sample processing methods
 FIR filtering in direct form 
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  1. Chapter 4 FIR filtering and Convolution Nguyen Thanh Tuan, ClickM.Eng. to edit Master subtitle style Department of Telecommunications (113B3) Ho Chi Minh City University of Technology Email: nttbk97@yahoo.com
  2. Introduction  Block processing methods: data are collected and processed in blocks.  FIR filtering of finite-duration signals by convolution  Fast convolution of long signals which are broken up in short segments  DFT/FFT spectrum computations  Speech analysis and synthesis  Image processing  Sample processing methods: the data are processed one at a time- with each input sample being subject to a DSP algorithm which transforms it into an output sample.  Real-time applications  Digital audio effects processing  Digital control systems  Adaptive signal processing Digital Signal Processing 3 FIR Filtering and Convolution
  3. 11.1. Direct form  The convolution in the direct form: y()()() n  h m x n m m  For DSP implementation, we must determine  The range of values of the output index n  The precise range of summation in m  Find index n: index of h(m) 0≤m≤M index of x(n-m) 0≤n-m≤L-1 0 ≤ m ≤ n ≤m+L-1 ≤ M+L-1 0 n M L 1  Lx=L input samples which is processed by the filter with order M yield the output signal y(n) of length Ly L M=Lx M Digital Signal Processing 5 FIR Filtering and Convolution
  4. Example 1  Consider the case of an order-3 filter and a length of 5-input signal. Find the output ? h=[h0, h1, h2, h3] x=[x0, x1, x2, x3, x4 ] y=h*x=[y0, y1, y2, y3, y4 , y5, y6, y7 ] Digital Signal Processing 7 FIR Filtering and Convolution
  5. Example 2  Calculate the convolution of the following filter and input signals? h=[1, 2, -1, 1], x=[1, 1, 2, 1, 2, 2, 1, 1]  Solution : sum of the values along anti-diagonal line yields the output y: y=[1, 3, 3, 5, 3, 7, 4, 3, 3, 0, 1] Note that there are Ly=L+M=8+3=11 output samples. Digital Signal Processing 9 FIR Filtering and Convolution
  6. 1.3. LTI Form  LTI form of convolution:  LTI form of convolution provides a more intuitive way to under stand the linearity and time-invariance properties of the filter. Digital Signal Processing 11 FIR Filtering and Convolution
  7. 1.4. Matrix Form  Based on the convolution equations we can write y Hx  x is the column vector of the Lx input samples.  y is the column vector of the Ly =Lx+M put samples.  H is a rectangular matrix with dimensions (Lx+M)xLx . Digital Signal Processing 13 FIR Filtering and Convolution
  8. Example 4  Using the matrix form to calculate the convolution of the following filter and input signals? h=[1, 2, -1, 1], x=[1, 1, 2, 1, 2, 2, 1, 1]  Solution : since Lx=8, M=3 Ly=Lx+M=11, the filter matrix is 11x8 dimensional Digital Signal Processing 15 FIR Filtering and Convolution
  9. 1.6. Transient and steady-state behavior M  From LTI convolution: yn( )  hmxnm ( ) ( ) hxhx0n 1 n 1 hx M n M m 0  The output is divided into 3 subranges:  Transient and steady-state filter outputs: Digital Signal Processing 17 FIR Filtering and Convolution
  10. Example 5  Using the overlap-add method of block convolution with each bock length L=3, calculate the convolution of the following filter and input signals? h=[1, 2, -1, 1], x=[1, 1, 2, 1, 2, 2, 1, 1]  Solution : The input is divided into block of length L=3 The output of each block is found by the convolution table: Digital Signal Processing 19 FIR Filtering and Convolution
  11. 2. Sample processing methods  The direct form convolution for an FIR filter of order M is given by  Introduce the internal states Sample processing algorithm Fig: Direct form realization  Sample processing methods are of Mth order filter convenient for real-time applications Digital Signal Processing 21 FIR Filtering and Convolution
  12. Example 6 Digital Signal Processing 23 FIR Filtering and Convolution
  13. Hardware realizations  The FIR filtering algorithm can be realized in hardware using DSP chips, for example the Texas Instrument TMS320C25  MAC: Multiplier Accumulator Digital Signal Processing 25 FIR Filtering and Convolution
  14. Example 7  What is the longest FIR filter that can be implemented with a 50 nsec per instruction DSP chip for digital audio applications with sampling frequency fs=44.1 kHz ? Solution: Digital Signal Processing 27 FIR Filtering and Convolution
  15. Homework 2 Digital Signal Processing 29 FIR Filtering and Convolution
  16. Homework 4  Compute the output y(n) of the filter h(n) = {1, -1, 1, -1} and input x(n) = {1, 2, 3, 4, @, -3, 2, -1} Digital Signal Processing 31 FIR Filtering and Convolution