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Nonrecursive Digital Filters
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Digital Filters & Filter Equation
General Equation - FIR - Convolution Frequency response Transfer function
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Nonrecursive Filter • Disadvantage : takes computation time
• Advantage : stable (zeros only) linear phase (no phase distortion) same phase shift to all frequencies
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Impulse response of moving average filter
2M+1 coefficients, symmetric to n=0 Smoothness of the signal correlated to the increment of M Width of mainlobe negatively correlated to M increment of M narrow band lowpass filter
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Frequency Response of Moving Average Filter
5-point (M = 2) 21-point (M = 10) No zeros at z=0 since passband around
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Frequency Response of Moving Average Filter
= 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)
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Ideal Lowpass Filter Method
1.0
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Design of Highpass/Bandpass Filters using Lowpass Filter
(bandwidth : center frequency : ) Lowpass Filter Replace with Limit to 2M+1 terms, and start from n=0 Center frequency : bandwidth :
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Lowpass Filter Design Cutoff frequency : Sampling rate :
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Highpass Filter Design
Cutoff frequency highpass filter
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Bandpass Filter Design
Cutoff frequency : Sampling rate : Duration of impulse response : Center frequency Bandwidth
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Frequency Transformation
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Recursive Digital Filters
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General Form of Filters
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
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Example #1 Find the difference equation of Bandpass Filter
(a) Center frequency : = /2, dB Bandwidth : /40, Maximum gain : 1 (b) No frequency component at = 0, = origin - Assume BC is straight line - d = 1 - r (r > 0.9) - 2d = 2 (1-r) - 2 (1-r) [rad] = /40 = 3.14/40, r = 0.961 - No frequency component at = 0 and = - two zeroes at z = +1 and -1
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Example #1 ① -3dB band-width : /40 Maximum gain : 26.15 (28.35dB)
in equaiton ①, K = (6.15)-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]}
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Example #2 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) - fs = 1.2 kHz - fmax : 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]
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Types of Filters Butterworth Chebyshev – 1st order
Chebyshev – 2nd order Elliptic
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Butterworth, Chebyshev, Elliptic Filters
analog digital Butterworth Chebyshev Elliptic ripple
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Example #3 Find the minimum order of Filter Cutoff frequency 1= 0.2
Frequency response of less than 30dB at = 0.4
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Bilinear Transformation
H(s) H(z)
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Bilinear Transformation
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Impulse-invariant Filters
Another method of deriving a digital filter from an analog filter A sampled version of that of the reference analog filter
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Impulse-invariant Filters
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Impulse-invariant Filters
Transfer function of analog filter Impulse-invariant filter
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Impulse-invariant Filters
The impulse response of each analog subfilter takes a simple exponential form For the i-th subfilter A zero at the origin of the z-plane A polse at
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Design of Recursive Digital Filters
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Butterworth LP Analog Filter Design (prototype)
Prototype : when or frequency responses at N = 1, 2, 3
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Determination of Poles
When N : odd When N : even N = 1 ; N = 2 ; N = 3 ; only 3 effective terms
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Determination of Poles
1st order 2nd order
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Example 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 Order of filter
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Chebyshev LP Analog Filter Design (prototype)
or N : order, : cutoff frequency, r : ripple amplitude ( : ripple parameter) Order of filter
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Chebyshev Prototype Denominator Polynomials
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Example 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 2nd order Passband characteristic 3rd order
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Analog Filter Frequency Transformation
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Example Butterworth bandpass filter Maximum attenuation of 0.2dB for
Minimum attenuation of 50dB for Prototype equivalent frequency Filter order 3rd order
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