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Nonrecursive Digital Filters

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

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

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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

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

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= 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

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1.0 Ideal Lowpass Filter Method

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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

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Cutoff frequency : Sampling rate : Lowpass Filter Design

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Cutoff frequency highpass filter Highpass Filter Design

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

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Frequency Transformation

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Recursive Digital Filters

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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

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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

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-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

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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

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

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

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

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H(s) H(z) Bilinear Transformation

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

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

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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

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Design of Recursive Digital Filters

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

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

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1 st order2 nd order Determination of Poles

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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

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

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Chebyshev Prototype Denominator Polynomials

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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

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form Analog Filter Frequency Transformation

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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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