EE247 Lecture 2 Material covered today Nomenclature Filter specifications Quality factor Frequency characteristics Group delay Filter types Butterworth Chebyshev I Chebyshev II Elliptic Bessel Group delay comparison example EECS 247 Lecture 2 Filters 2004 H K Page 1 Nomenclature Filter Types Lowpass Highpass Bandpass H j H j H j Band reject Notch Provide frequency selectivity EECS 247 Lecture 21 Filters H j H j All pass Phase shaping or equalization 2004 H K Page 2 Filter Specifications Frequency characteristics lowpass filter 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 Power pole Area pole EECS 247 Lecture 2 Filters 2004 H K Page 3 Lowpass Filter Frequency Characteristics H j Passband Ripple Rpass f 3dB H 0 3dB Passband Gain Transition Band H j 0 H j Passband EECS 247 Lecture 2 Filters fc fs t o p Frequency Hz Stopband Rejection x 10 f Stopband Frequency 2004 H K Page 4 Quality Factor Q The term Quality Factor Q has different definitions Component quality factor inductor capacitor Q Pole quality factor Bandpass filter quality factor Next 3 slides clarifies each EECS 247 Lecture 2 Filters 2004 H K Page 5 Component Quality Factor Q For any component with a transfer function H j 1 R jX Quality factor is defined as Q X Energy S t o r e d A v e r a g e Power D i s s i p a t i o n R EECS 247 Lecture 2 Filters perunittime 2004 H K Page 6 Inductor Capacitor Quality Factor Inductor Q YL Rs 1j L L QL Rs L Rs Capacitor Q ZC Rp QC CRp 1 1 j C Rp C EECS 247 Lecture 2 Filters 2004 H K Page 7 Pole Quality Factor j s Plane P x x QP o l e x 2 x EECS 247 Lecture 2 Filters 2004 H K Page 8 Bandpass Filter Quality Factor Q H jf Q fcenter f 0 Magnitude dB 0 5 3dB f 10 15 20 1 fcenter 0 1 EECS 247 Lecture 2 Filters 10 Frequency 2004 H K Page 9 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 sinewaves at slightly different frequencies vIN t A1sin t A2 sin t The filter output is vOUT t A1 G j sin t A2 G j sin t EECS 247 Lecture 2 Filters 2004 H K Page 10 What is Group Delay A2 G j sin Since then 1 2 t vOUT t A1 G j sin t 0 d d d d 1 1 EECS 247 Lecture 2 Filters 2004 H K Page 11 What is Group Delay Signal Magnitude and Phase Impairment vOUT t A1 G j sin t A2 G j sin d t d If the second term in the phase of the 2 nd sinwave is non zero then the filter s output at frequency is time shifted differently than the filter s output at frequency Phase distortion If the second term is zero then the filter s output at frequency and the output at frequency are each delayed in time by PD is called the phase delay and has units of time EECS 247 Lecture 2 Filters 2004 H K Page 12 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 2004 H K Page 13 What is Group Delay Signal Magnitude and Phase Impairment If GR PD No phase distortion A G j sin t vOUT t A1 G j sin t GR 2 GR If also G j G j 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 realizable exactly EECS 247 Lecture 2 Filters 2004 H K Page 14 Summary Group Delay Phase delay is defined as PD time Group delay is defined as 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 2004 H K Page 15 Filter Types Butterworth Lowpass Filter d N H j d 0 0 Moderate phase distortion 20 40 60 0 5 200 3 1 400 0 1 2 Normalized Group Delay Maximally flat amplitude within the filter passband Phase degrees Magnitude dB 0 Normalized Frequency Example 5th Order Butterworth filter EECS 247 Lecture 2 Filters 2004 H K Page 16 Butterworth Lowpass Filter j All poles Poles located on the unit circle with equal angles s plane Example 5th Order Butterworth filter EECS 247 Lecture 2 Filters 2004 H K Page 17 Filter Types Chebyshev I Lowpass Filter 20 40 35 0 200 400 0 0 1 Normalized Group Delay Chebyshev I filter Equal ripple passband Sharper transition band compared to Butterworth Poorer group delay Phase degrees Magnitude dB 0 2 Normalized Frequency Example 5th Order Chebyshev filter EECS 247 Lecture 2 Filters 2004 H K Page 18 Chebyshev I Lowpass Filter Characteristics j s plane All poles Poles located on an ellipse inside the unit circle Allowing more ripple in the passband Narrower transition band Sharper cut off Higher pole Q Chebyshev I LPF 3dB passband ripple Chebyshev I LPF 0 1dB passband ripple Example 5th Order Chebyshev I Filter EECS 247 Lecture 2 Filters 2004 H K Page 19 Filter Types Cheybshev II Lowpass 0 20 40 60 0 Phase deg Chebyshev II filter Ripple in stopband Sharper transition band compared to Butterworth Passband group delay superior to Chebyshev I Magnitude dB Bode Diagram 90 180 270 360 0 0 5 1 1 5 2 Frequency Hz Example 5th Order Chebyshev II filter EECS 247 Lecture 2 Filters 2004 H K Page 20 Filter Types Cheybshev II Lowpass j Both poles zeros No of poles n No of zeros n 1 s plane Poles located both inside outside of the unit circle Zeros located on j axis Ripple in the stopband only Example 5th Order Chebyshev II Filter poles zeros EECS 247 Lecture 2 Filters 2004 H K Page 21 Filter Types Elliptic Lowpass Filter Elliptic filter Ripple in passband Ripple in the stopband Sharper transition band compared to Butterworth both Chebyshevs Poorer group delay 20 40 60 0 Phase degrees Magnitude dB 0 200 400 0 1 2 Normalized Frequency Example 5th Order Elliptic …
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