EELE503 Modern filter design Filter Design - Introduction A filter will modify the magnitude or phase of a signal to produce a desired frequency response or time response. One way to classify ideal filters is by frequency response Lowpass H(0<f<B =, H(f =0 elsewhere Highpass H(f>B =, H(f =0 elsewhere Bandpass H( f<f<f=, H(f =0 elsewhere B=f-f Bandstop H(f<f<f =0, H(f = elsewhere B=f=f also called a band-reject filter Allpass H(f =, argh(f=θ(f
Filter Design - Introduction An ideal filter will pass desired frequencies with no loss or phase distortion, and provide infinite attenuation to unwanted frequencies. It may be shown that an ideal rectangular filter response would require an infinite number of poles to realize. Modern analog filter design results in an approximation to the desired ideal response. 3 Ideal Filter x(t X(f h(t, H(f y(t Y(f H(f H(f Linear phase θ ( ω = d θ (0 τ ω 4
Ideal Filter-Magnitude, Phase/Delay For a transfer function H(s, at real frequencies, with s=jω, H ( jπf = H ( jπf e jθ ( jπf = G( ω e jθ ( ω Where G(ω and θ(ω are the gain and phase components. Phase Delay Pd(ω is defined as: Group Delay τ d (ω is defined as: θ ( ω Pd( ω = ω θ ( ω τ d ( ω = ω 5 Ideal Filter-Magnitude, Phase/Delay Linear Phase : θ ( ω = d θ (0 τ ω Linear Phase Distortion: θ ( ω = θ (0 τ ω τ ω d... The τ term is called the parabolic group delay distortion and has units of sec 6 3
Ideal Filter-Magnitude, Phase/Delay Both Pd(ω and τ d (ω are functions of frequency Phase delay Pd(ω is the absolute delay and is usually of little significance Group Delay τ d (ω is used as the criterion to evaluate phase nonlinearity. Group Delay is constant for all frequencies in the passband of an ideal filter. 7 Ideal Filter-Magnitude, Phase/Delay Linear phase variation with frequency (over a band of frequencies implies a constant Group Delay no phase distortion in that band of frequencies In order to preserve the integrity of a pulse x(t, it is mandatory that the Group Delay of the system be constant up to the maximum frequency component of the pulse. This implies equal time delay for all frequencies of interest. 8 4
Convolution of a rectangle and an exponential. w 3( t = w ( t w( t w ( λ w ( t λ dλ 9 Couch, Digital and Analog Communication Systems, Seventh Edition 007 Pearson Education, Inc. All rights reserved. 0-3-449-0 Power Signal Through a Filter: 0 5
6 Power Signal Through a Filter: time average or power autocorr: Periodic Signal Through a Filter: Time Average Autocorrelation a periodic signal for = = = = ( ( ( ( ( ( ( ( ( ( o o o o o o T T o x T T o k kt kt o k T T T x dt t x t x T R dt t x t x kt k Lim dt t x t x kt Lim dt t x t x T Lim R τ τ τ τ τ τ
Discreet Time Average Autocorrelation 3 Power Signal Through a Filter: 4 7
Discrete Time Average Autocorrelation Through a LTI System 5 Power Autocorrelation: R(0= power 6 8
Filters-Applications Modify the frequency spectrum of a signal remove out of band distortion Reduce the magnitude of unwanted signals, example 60 Hz hum reduce noise power by reducing bandwidth Waveform Shaping: y(t=x(t*h(t Matched signal detection 7 http://en.wikipedia.org/wiki/butterworth_filter Linear analog electronic filters Butterworth filter Chebyshev filter Elliptic (Cauer filter Bessel filter Gaussian filter Optimum "L" (Legendre filter 8 9
Filter Example: Hs ( Vx Vy Hs ( := + RC s ht ( t RC e RC t τ e τ 9 Let s=jπf=jω Filter Example: ( H ω Hf ( ( tan ( ω R C tan θω + + j ω ω o j f f o θ( f tan f f o ω o := RC f o := π R C ω ω o 0 0
Filter Example: The Group delay of the RC low pass is: τ d ( ω ω o ω + ω o τ d ( f π f o f f o + Figure 5 Characteristics of an RC low-pass filter. Couch, Digital and Analog Communication Systems, Seventh Edition 007 Pearson Education, Inc. All rights reserved. 0-3-449-0
Figure 6 Distortion caused by an RC low-pass filter. 3 Couch, Digital and Analog Communication Systems, Seventh Edition 007 Pearson Education, Inc. All rights reserved. 0-3-449-0 Figure 6 Distortion caused by an RC low-pass filter. 4 Couch, Digital and Analog Communication Systems, Seventh Edition 007 Pearson Education, Inc. All rights reserved. 0-3-449-0
Figure 6 Distortion caused by an RC low-pass filter. 5 Couch, Digital and Analog Communication Systems, Seventh Edition 007 Pearson Education, Inc. All rights reserved. 0-3-449-0 A LPF Distortion Problem: Assume we want the amplitude Linearity <% and the group delay variation (linearity <5% Find the usable bandwidth of the st order Butterworth filter if the 3dB bandwidth is MHz 6 3
A LPF Distortion Problem: Constraints: ( H0 ( Hf a H0 ( ( τ d ( 0 τ d f φ τ d ( 0 ε a 0.0 ε φ 0.05 % Voltage amplitude error 5% delay variation f o 0 6 := τ := π f o Hf ( := f + j f o τ d ( f := f o π f + f o 7 A LPF Distortion Problem: So the amplitude error will limit the usable bandwidth to 03 KHz V/V Hf ( 0.88 0.75 0.63 Filter Magnitude Response 0.5. 0 3. 0 4. 0 5. 0 6 f 8 4
A LPF Distortion Problem: Degrees 90 arg( H( f 80 π 90 90 45 0 45 Filter Phase Response 90 00. 0 3. 0 4. 0 5. 0 6 00 f Hz 0 6 9 A LPF Distortion Problem: usec 0.59 τ.d (0 f 6 0 0. 0.5 0. 0.05 Filter Group Delay Response 0 00. 0 3. 0 4. 0 5. 0 6. 0 7 00 f 0 7 Hz 30 5
A LPF Distortion Problem: Amplitude Error: Phase Error: ( ε a f a ( Hf a := ε H0 ( φ f p 0.0 0.95 f a + f o f o ( := f o τ d ( f p τ d ( 0 f p + f o f a := f o 0.98 f o f p := f 0.95 o f a =.03 0 5 Hz f p =.94 0 5 Hz 3. A LPF Distortion Problem: stop -3 Filter Amplitude and Phase Error.03 0 5 Error ε.a ( f ε.φ ( f 0.05.05 50 7 0. 0 3. 0 4. 0 5. 0 6 000 f.0 0 6 Hz Amplitude Error Group Delay Error 3 6
Filter Noise Equivalent Bandwidth We often equate the -3dB bandwidth of a real Filter to the bandwidth of an ideal filter that would Pass the same noise power. W 3dB -3dB W noise 33 Filter Noise Equivalent Bandwidth Real Filter 3 db White Noise S n (f Ideal Filter W Find W For Equal Powers From Filters S n (f is a White Noise Power Density N o Watts/Hz 34 7
Filter Noise Equivalent Bandwidth P ( S n ( f Hf ( df N o 0 ( Hf ( df P f S n (Π f W eq df W eq W eq N o df N o W eq W eq 0 ( Hf ( df 35 Butterworth lowpass filters ( Hf ( f + f o n Where n is the filter order (number of poles Note that n= is the RC lowpass 36 8
Butterworth lowpass filters 37 Butterworth lowpass filters Passband n x 6 db per octive n x 0 db per decade 38 9
Butterworth lowpass filters EQUIVELENT NOISE BANDWITH FOR BUTTERWORTH FILTERS B Noise_Power_From_Real_Filter_With_3db_Bandwidth_of_ Noise_Power_From_Ideal_Filter_With_Bandwidth_of_ n :=.. 6 filter orders from to 6 f ( f n d + 0 B n := Π ( f, df 0 First order filter with a 3dB bandwidth of passes 57% more noise power when compared with an ideal filter with a bandwidth of. A 3rd order filter only passes 4.7% more noise power when compared with an ideal filter. note that: 0 n = B n =.57. 3.047 4.06 5.07 6.0 f + ( f d π π =.57 39 0