EE247 Lecture 2 Filters Filters Nomenclature Specifications Quality factor Magnitude phase response versus frequency characteristics Group delay Filter types Butterworth Chebyshev I II Elliptic Bessel Group delay comparison example Biquads EECS 247 Lecture 2 Filters 2010 Page 1 Nomenclature Filter Types wrt Frequency Range Selectivity Lowpass Highpass Bandpass Band reject Notch H j H j H j H j Provide frequency selectivity EECS 247 Lecture 2 Filters All pass H j Phase shaping or equalization 2010 Page 2 Filter Specifications Magnitude response versus frequency characteristics Passband ripple Rpass Cutoff frequency or 3dB frequency Stopband rejection Passband gain Phase characteristics Group delay SNR Dynamic range SNDR Signal to Noise Distortion ratio Linearity measures IM3 intermodulation distortion HD3 harmonic distortion IIP3 or OIP3 Input referred or outputreferred third order intercept point Area pole Power pole EECS 247 Lecture 2 Filters 2010 Page 3 Filter Magnitude versus Frequency Characteristics Example Lowpass H j dB Passband Ripple Rpass f 3dB H 0 3dB Passband Gain Transition Band H j 0 H j fc fstop Frequency Hz Passband EECS 247 Lecture 2 Filters Stopband Frequency Stopband Rejection x 10 f 2010 Page 4 Filters Filters Nomenclature Specifications Magnitude phase response versus frequency characteristics Quality factor Group delay Filter types Butterworth Chebyshev I II Elliptic Bessel Group delay comparison example Biquads EECS 247 Lecture 2 Filters 2010 Page 5 Quality Factor Q The term quality factor Q has different definitions in different contexts Component quality factor inductor capacitor Q Pole quality factor Bandpass filter quality factor Next 3 slides clarifies each EECS 247 Lecture 2 Filters 2010 Page 6 Component Quality Factor Q For any component with a transfer function H j 1 R jX Quality factor is defined as Q X Energy Stored Average Power Dissipation R EECS 247 per unit time Lecture 2 Filters 2010 Page 7 Component Quality Factor Q Inductor Capacitor Quality Factor Inductor Q Rs series parasitic resistance YL 1 Rs j L QL L Rs Rs L Capacitor Q Rp parallel parasitic resistance Rp 1 ZC 1 j C Rp EECS 247 QC CR p C Lecture 2 Filters 2010 Page 8 Pole Quality Factor j Typically filter singularities include pairs of complex conjugate poles s Plane Quality factor of complex conjugate poles are defined as QP o l e EECS 247 P x x p 2 x Lecture 2 Filters 2010 Page 9 Bandpass Filter Quality Factor Q H jf Q fcenter Df Magnitude dB 0 3dB D f f2 f 1 0 1 EECS 247 f1 1 fcenter Lecture 2 Filters f2 10 Frequency 2010 Page 10 Filters Filters Nomenclature Specifications Magnitude phase response versus frequency characteristics Quality factor Group delay Filter types Butterworth Chebyshev I II Elliptic Bessel Group delay comparison example Biquads EECS 247 Lecture 2 Filters 2010 Page 11 What is Group Delay Consider a continuous time filter with s domain transfer function G s G j G j e j Let us apply a signal to the filter input composed of sum of two sine waves at slightly different frequencies D vIN t A1sin t A2sin D t The filter output is vOUT t A1 G j sin t A2 G j D sin D t D EECS 247 Lecture 2 Filters 2010 Page 12 What is Group Delay vOUT t A1 G j sin t A2 G j D sin D t D D D then D 2 0 1 Since D D EECS 247 d D d d d 1 1 D D Lecture 2 Filters 2010 Page 13 What is Group Delay Signal Magnitude and Phase Impairment vOUT t A1 G j sin t A2 G j D sin D t d d D d PD is called the phase delay and has units of time If the delay term d is zero the filter s output at frequency D and the output at frequency are each delayed in time by If the term d is non zero the filter s output at frequency D is timeshifted differently than the filter s output at frequency Phase distortion EECS 247 Lecture 2 Filters 2010 Page 14 What is Group Delay Signal Magnitude and Phase Impairment Phase distortion is avoided only if d d 0 Clearly if k k a constant no phase distortion This type of filter phase response is called linear phase Phase shift varies linearly with frequency GR d d is called the group delay and also has units of time For a linear phase filter GR PD k GR PD implies linear phase Note Filters with k c are also called linear phase filters but they re not free of phase distortion EECS 247 Lecture 2 Filters 2010 Page 15 What is Group Delay Signal Magnitude and Phase Impairment If GR PD No phase distortion A G j D sin D t vOUT t A1 G j sin t GR 2 GR If also G j G j D for all input frequencies within the signal band vOUT is a scaled time shifted replica of the input with no signal magnitude distortion In most cases neither of these conditions are exactly realizable EECS 247 Lecture 2 Filters 2010 Page 16 Summary Group Delay Phase delay is defined as PD Group delay is defined as time GR d d time If k k a constant no phase distortion For a linear phase filter GR PD k EECS 247 Lecture 2 Filters 2010 Page 17 Filters Filters Nomenclature Specifications Magnitude phase response versus frequency characteristics Quality factor Group delay Filter types examples considered all lowpass the highpass and bandpass versions similar characteristics Butterworth Chebyshev I II Elliptic Bessel Group delay comparison example Biquads EECS 247 Lecture 2 Filters 2010 Page 18 Filter Types wrt Frequency Response Lowpass Butterworth Filter H j d 0 40 60 0 Phase degrees d N 20 Moderate phase distortion 0 5 200 3 1 400 0 1 2 Normalized Group Delay Maximally flat amplitude within the filter passband Magnitude dB 0 Normalized Frequency Example 5th Order Butterworth filter EECS 247 Lecture 2 Filters 2010 Page 19 Lowpass Butterworth Filter j All poles s plane Number of poles equal to filter order Poles located on the unit circle with equal angles pole Example 5th Order Butterworth Filter EECS 247 Lecture 2 Filters 2010 Page 20 Filter Types Chebyshev I Lowpass Filter Chebyshev I filter Ripple in the passband Sharper transition band compared to Butterworth for the same number of poles 20 40 35 Phase degrees 0 Poorer group delay compared to Butterworth More ripple in passband poorer phase response 200 400 0 0 1 Normalized Group Delay Magnitude dB 0 2 Normalized Frequency Example 5th Order Chebyshev filter EECS 247 Lecture 2 Filters 2010 Page 21 Chebyshev I Lowpass Filter Characteristics j All poles Poles located on an ellipse inside the unit circle s plane Allowing more ripple in the passband Narrower transition band Sharper cut off Higher pole Q Poorer phase response Chebyshev I LPF 3dB passband ripple Chebyshev I LPF 0 1dB
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