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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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