EEM478-WEEK8 Finite Impulse Response (FIR) Filters

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1 EEM478-WEEK8 Finite Impulse Response (FIR) Filters

2 Learning Objectives Introduction to the theory behind FIR filters: Properties (including aliasing). Coefficient calculation. Structure selection. Implementation in Matlab, C, assembly and linear assembly. Chapter 14, Slide 2

3 Introduction Amongst all the obvious advantages that digital filters offer, the FIR filter can guarantee linear phase characteristics. Neither analogue or IIR filters can achieve this. There are many commercially available software packages for filter design. However, without basic theoretical knowledge of the FIR filter, it will be difficult to use them. Chapter 14, Slide 3

4 Properties of an FIR Filter Filter coefficients: x[n] b k y[n] N represents the filter input, represents the filter coefficients, represents the filter output, is the number of filter coefficients (order of the filter). Chapter 14, Slide 4

5 Properties of an FIR Filter Filter coefficients: FIR equation Filter structure Chapter 14, Slide 5

6 Properties of an FIR Filter Filter coefficients: If the signal x[n] is replaced by an impulse [n] then: Chapter 14, Slide 6

7 Properties of an FIR Filter Filter coefficients: If the signal x[n] is replaced by an impulse [n] then: Chapter 14, Slide 7

8 Properties of an FIR Filter Filter coefficients: If the signal x[n] is replaced by an impulse [n] then: Chapter 14, Slide 8

9 Properties of an FIR Filter Filter coefficients: Finally: Chapter 14, Slide 9

10 Properties of an FIR Filter Filter coefficients: With: The coefficients of a filter are the same as the impulse response samples of the filter. Chapter 14, Slide 10

11 Frequency Response of an FIR Filter By taking the z-transform of h[n], H(z): Replacing z by e -j in order to find the frequency response leads to: Chapter 14, Slide 11

12 Frequency Response of an FIR Filter Since e -j2k = 1 then: Therefore: FIR filters have a periodic frequency response and the period is 2. Chapter 14, Slide 12

13 Frequency Response of an FIR Filter Frequency response: x[n] FIR y[n] x[n] y[n] Freq F s /2 Freq F s /2 Chapter 14, Slide 13

14 Frequency Response of an FIR Filter Solution: Use an anti-aliasing filter. x(t) ADC x[n] FIR y[n] Analogue Anti-Aliasing x(t) y[n] F s /2 Freq Freq F s /2 Chapter 14, Slide 14

15 Phase Linearity of an FIR Filter A causal FIR filter whose impulse response is symmetrical is guaranteed to have a linear phase response. Even symmetry Odd symmetry Chapter 14, Slide 15

16 Phase Linearity of an FIR Filter A causal FIR filter whose impulse response is symmetrical (ie h[n] = h[n-1-n] for n = 0, 1,, N-1) is guaranteed to have a linear phase response. Chapter 14, Slide 16

17 Phase Linearity of an FIR Filter Application of 90 linear phase shift: I Signal separation Q 90 o delay delay 90 o IH QH Reverse Forward Chapter 14, Slide 17

18 Design Procedure To fully design and implement a filter five steps are required: (1) Filter specification. (2) Coefficient calculation. (3) Structure selection. (4) Simulation (optional). (5) Implementation. Chapter 14, Slide 18

19 Filter Specification - Step 1 Chapter 14, Slide 19

20 Coefficient Calculation - Step 2 There are several different methods available, the most popular are: Window method. Frequency sampling. Parks-McClellan. We will just consider the window method. Chapter 14, Slide 20

21 Window Method First stage of this method is to calculate the coefficients of the ideal filter. This is calculated as follows: Chapter 14, Slide 21

22 Window Method Second stage of this method is to select a window function based on the passband or attenuation specifications, then determine the filter length based on the required width of the transition band. Using the Hamming Window: Chapter 14, Slide 22

23 Window Method The third stage is to calculate the set of truncated or windowed impulse response coefficients, h[n]: for Where: for Chapter 14, Slide 23

24 close all; clear all; Window Method Matlab code for calculating coefficients: fc = 8000/44100; N = 133; n = -((N-1)/2):((N-1)/2); n = n+(n==0)*eps; [h] = sin(n*2*pi*fc)./(n*pi); [w] = *cos(2*pi*n/N); d = h.*w; [g,f] = freqz(d,1,512,44100); figure(1) plot(f,20*log10(abs(g))); axis([0 2*10^ ]); figure(2); stem(d); xlabel('coefficient number'); ylabel ('Value'); title('truncated Impulse Response'); figure(3) freqz(d,1,512,44100); axis([0 2*10^ ]); % cut-off frequency % number of taps % avoiding division by zero % generate sequence of ideal coefficients % generate window function % window the ideal coefficients % transform into frequency domain for plotting % plot transfer function % plot coefficient values % use freqz to plot magnitude and phase response Chapter 14, Slide 24

25 Window Method Chapter 14, Slide 25

26 Realisation Structure Selection - Step 3 Direct form structure for an FIR filter: Chapter 14, Slide 26

27 Realisation Structure Selection - Step 3 Direct form structure for an FIR filter: Linear phase structures: N even: N Odd: Chapter 14, Slide 27

28 Realisation Structure Selection - Step 3 (a) N even. (b) N odd. Chapter 14, Slide 28

29 Realisation Structure Selection - Step 3 Direct form structure for an FIR filter: Cascade structures: Chapter 14, Slide 29

30 Realisation Structure Selection - Step 3 Direct form structure for an FIR filter: Cascade structures: Chapter 14, Slide 30

31 Implementation - Step 5 Implementation procedure in C with fixed-point: Set up the codec (\Links\CodecSetup.pdf). Transform: to C code. (\Links\FIRFixed.pdf) Configure timer 1 to generate an interrupt at 8000Hz (\Links\TimerSetup.pdf). Set the interrupt generator to generate an interrupt to invoke the Interrupt Service Routine (ISR) (\Links\InterruptSetup.pdf). Chapter 14, Slide 31

32 Implementation - Step 5 Implementation procedure in C with floating-point: Same set up as fixed-point plus: Convert the input signal to floating-point format. Convert the coefficients to floating-point format. With floating-point multiplications there is no need for the shift required when using Q15 format. See \Links\FIRFloat.pdf Chapter 14, Slide 32

33 Implementation - Step 5 Implementation procedure in assembly: Same set up as fixed-point, however: is written in assembly. (\Links\FIRFixedAsm.pdf) The ISR is now declared as external. Chapter 14, Slide 33

34 Implementation - Step 5 Implementation procedure in assembly: The filter implementation in assembly is now using circular addressing and therefore: The circular pointers and block size register are selected and initialised by setting the appropriate values of the AMR bit fields. The data is now aligned using: #pragma DATA_ALIGN (symbol, constant (bytes)) Set the initial value of the circular pointers, see \Links\FIRFixedAsm.pdf. Chapter 14, Slide 34

35 Implementation - Step 5 *B5 *B5 *B5 *B5 b 0 b 1 b 2 b 3 *A5 *A5 *A5 *A5 x 0 x 1 x 2 x 3 y0 = b0*x0 + b1*x1 + b2*x2 + b3*x3 y[n] time Circular addressing link slide. Chapter 14, Slide 35

36 Implementation - Step 5 *B5 *B5 *B5 *B5 b 0 b 1 b 2 b 3 *A5 *A5 *A5 *A5 x 4 x 1 x 2 x 3 y0 = b0*x0 + b1*x1 + b2*x2 + b3*x3 y1 = b0*x1 + b1*x2 + b2*x3 + b3*x4 y[n] time Circular addressing link slide. Chapter 14, Slide 36

37 Implementation - Step 5 *B5 *B5 *B5 *B5 b 0 b 1 b 2 b 3 *A5 *A5 *A5 *A5 x 4 x 5 x 2 x 3 y0 = b0*x0 + b1*x1 + b2*x2 + b3*x3 y1 = b0*x1 + b1*x2 + b2*x3 + b3*x4 y2 = b0*x2 + b1*x3 + b2*x4 + b3*x5 y[n] time Circular addressing link slide. Chapter 14, Slide 37

38 FIR Code Code location: Code\Chapter 14 - Finite Impulse Response Filters Projects: Fixed Point in C: \FIR_C_Fixed\ Floating Point in C: \FIR_C_Float\ Fixed Point in Assembly: \FIR_Asm_Fixed\ Floating Point in Assembly: \FIR_Asm_Float\ Chapter 14, Slide 38

39 WEEK 8 Finite Impulse Response (FIR) Filters - End -

Digital Filtering: Realization

Digital Filtering: Realization Digital Filtering: Matlab Implementation: 3-tap (2 nd order) IIR filter 1 Transfer Function Differential Equation: z- Transform: Transfer Function: 2 Example: Transfer Function