Nonrecursive Digital Filters

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

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

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

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

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)

Ideal Lowpass Filter Method 1.0

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 :

Lowpass Filter Design Cutoff frequency : Sampling rate :

Highpass Filter Design Cutoff frequency highpass filter

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

Frequency Transformation

Recursive Digital Filters

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

Example #1 Find the difference equation of Bandpass Filter (a) Center frequency :  = /2, -3dB 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

Example #1 ① -3dB band-width : /40 Maximum gain : 26.15 (28.35dB) in equaiton ①, K = (6.15)-1 = 0.03824 The corresponding difference equaion is : y[n+2] + 0.9235y[n] = 0.03824{x[n+2] - x[n]} subtracting 2 from each term in brackets y[n] = -0.9235y[n-2] + 0.03824{x[n] - x[n-2]}

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] - 1.8523 y[n+1] + 0.94833 y[n] = x[n+2] - 1.9021 x[n+1] + x[n] y[n] = 1.8523 y[n-1] - 0.94833 y[n-2] + x[n] - 1.9021 x[n-1] + x[n-2]

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

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

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

Bilinear Transformation H(s) H(z)

Bilinear Transformation

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

Impulse-invariant Filters

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

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

Design of Recursive Digital Filters

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

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

Determination of Poles 1st order 2nd order

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

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

Chebyshev Prototype Denominator Polynomials

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

Analog Filter Frequency Transformation

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