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Nonrecursive Digital Filters. Digital Filters & Filter Equation General Equation Transfer function Frequency response - FIR - Convolution.

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Presentation on theme: "Nonrecursive Digital Filters. Digital Filters & Filter Equation General Equation Transfer function Frequency response - FIR - Convolution."— Presentation transcript:

1 Nonrecursive Digital Filters

2 Digital Filters & Filter Equation General Equation Transfer function Frequency response - FIR - Convolution

3 Disadvantage : takes computation time Advantage : stable (zeros only) linear phase (no phase distortion) same phase shift to all frequencies Nonrecursive Filter

4  Smoothness of the signal  correlated to the increment of M  Width of mainlobe  negatively correlated to M  increment of M  narrow band lowpass filter 2M+1 coefficients, symmetric to n=0 Moving Average Filter Impulse response of moving average filter

5 5-point (M = 2) No zeros at z=0 since passband around 21-point (M = 10) Frequency Response of Moving Average Filter

6   = 0  peak value = 1  unwanted side lobe  first side lobe 22% of main lobe  5 terms  4 zeros missing zero at z = 1  21 terms  20 zeros passband contains at  = 0  Zeros lie actually on the unit circle  true nulls in the corresponding frequency ex) Frequency Response of Moving Average Filter

7 1.0 Ideal Lowpass Filter Method

8 Limit to 2M+1 terms, and start from n=0 (bandwidth : center frequency : ) Center frequency : bandwidth : Lowpass Filter Replace with Design of Highpass/Bandpass Filters using Lowpass Filter

9 Cutoff frequency : Sampling rate : Lowpass Filter Design

10 Cutoff frequency highpass filter Highpass Filter Design

11 Cutoff frequency : Sampling rate : Duration of impulse response : Center frequency Bandwidth Bandpass Filter Design

12 Frequency Transformation

13 Recursive Digital Filters

14  Recursive filter  powerful : separate control over the numerator and denominator of H(z)  If the magnitude of the denominator becomes small at the appropriate frequency  produce sharp response peaks by arranging General Form of Filters

15 Find the difference equation of Bandpass Filter (a) Center frequency :  =  /2, -3dB Bandwidth :  /40, Maximum gain : 1 (b) No frequency component at  = 0,  =  - Assume BC is straight line - d = 1 - r (r > 0.9) - 2d = 2 (1-r) - 2 (1-r) [rad] =  /40 = 3.14/40, r = No frequency component at  = 0 and  =  -  two zeroes at z = +1 and -1 origin Example #1

16 -3dB band-width :  /40 Maximum gain : (28.35dB) in equaiton ①, K = (6.15) -1 = ① The corresponding difference equaion is : y[n+2] y[n] = {x[n+2] - x[n]} subtracting 2 from each term in brackets y[n] = y[n-2] {x[n] - x[n-2]} Example #1

17 Design a band-reject filter which stops 60Hz powerline noise from ECG signal -10Hz cutoff bandwidth at -3dB point -Poles and zeros as in the picture (solution)- f s = 1.2 kHz - f max : 600Hz - 2  : 1200 =  o : 60 -  o (60Hz) = 0.1  y[n+2] y[n+1] y[n] = x[n+2] x[n+1] + x[n] y[n] = y[n-1] y[n-2] + x[n] x[n-1] + x[n-2] Example #2

18 Butterworth Chebyshev – 1 st order Elliptic Types of Filters Chebyshev – 2 nd order

19 Butterworth Chebyshev Elliptic analog digital ripple Butterworth, Chebyshev, Elliptic Filters

20 Find the minimum order of Filter Cutoff frequency  1 = 0.2  Frequency response of less than 30dB at  = 0.4  Example #3

21 H(s) H(z) Bilinear Transformation

22

23  Another method of deriving a digital filter from an analog filter  A sampled version of that of the reference analog filter Impulse-invariant Filters

24

25 Transfer function of analog filter Impulse-invariant filter Impulse-invariant Filters

26  A zero at the origin of the z-plane  A polse at  The impulse response of each analog subfilter takes a simple exponential form  For the i-th subfilter Impulse-invariant Filters

27 Design of Recursive Digital Filters

28 or Prototype : when frequency responses at N = 1, 2, 3 Butterworth LP Analog Filter Design (prototype)

29 When N : odd When N : even N = 1 ; N = 2 ; N = 3 ; only 3 effective terms Determination of Poles

30 1 st order2 nd order Determination of Poles

31 Order of filter Design a lowpass Butterworth filter : -3dB at 1 rad/sec (prototype filter) gain of less than 0.1 for the frequency greater than 2 rad/sec Example

32 or N : order, : cutoff frequency, r : ripple amplitude ( : ripple parameter) Chebyshev LP Analog Filter Design (prototype) Order of filter

33 Chebyshev Prototype Denominator Polynomials

34 Maximum passband ripple : 1dB, Cutoff frequency : less than 1.3 rad/sec Attenuation in stopband : 40dB for greater than 5 rad/sec ripple parameter cutoff frequency : -3dB point is half the magnitude 2 nd order Passband characteristic 3 rd order Example

35 form Analog Filter Frequency Transformation

36 Butterworth bandpass filter Prototype equivalent frequency Filter order 3 rd order Maximum attenuation of 0.2dB for Minimum attenuation of 50dB for Example


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