EECS 247 Lecture 2: Filters © 2010 Page 1EE247 - Lecture 2Filters• 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– BiquadsEECS 247 Lecture 2: Filters © 2010 Page 2NomenclatureFilter Types wrt Frequency Range Selectivity jH jHLowpass HighpassBandpass Band-reject(Notch)Provide frequency selectivity jH jHAll-pass jHPhase shaping or equalizationEECS 247 Lecture 2: Filters © 2010 Page 3Filter 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 output-referred third order intercept point)• Area/pole & Power/poleEECS 247 Lecture 2: Filters © 2010 Page 40x 10Frequency (Hz)Filter Magnitude versus Frequency CharacteristicsExample: Lowpass jH jH 0HPassband Ripple (Rpass)Transition BandcfPassbandPassband GainstopfStopband FrequencyStopband Rejectionf H j [dB]3dBfdB3EECS 247 Lecture 2: Filters © 2010 Page 5Filters• 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– BiquadsEECS 247 Lecture 2: Filters © 2010 Page 6Quality 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 eachEECS 247 Lecture 2: Filters © 2010 Page 7Component Quality Factor (Q)• For any component with a transfer function:• Quality factor is defined as: Energy Storedper unit timeAverage Power Dissipation1HjR jXXQREECS 247 Lecture 2: Filters © 2010 Page 8Component Quality Factor (Q) Inductor & Capacitor Quality Factor• Inductor Q :Rs series parasitic resistance• Capacitor Q :Rp parallel parasitic resistanceRsLssLL1LYQR j L RRpCpCCp1Z Q CR1jRCEECS 247 Lecture 2: Filters © 2010 Page 9Pole Quality FactorxxjPpPolexQ 2s-Plane• Typically filter singularities include pairs of complex conjugate poles.• Quality factor of complex conjugate poles are defined as:EECS 247 Lecture 2: Filters © 2010 Page 10Bandpass Filter Quality Factor (Q) 0.1 1 10f1fcenterf20-3dBDf = f2 - f1 H jfFrequencyMagnitude [dB]Q fcenter/DfEECS 247 Lecture 2: Filters © 2010 Page 11Filters• 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– BiquadsEECS 247 Lecture 2: Filters © 2010 Page 12• Consider a continuous-time filter with s-domain transfer function G(s):• Let us apply a signal to the filter input composed of sum of two sine waves at slightly different frequencies (D):• The filter output is:What is Group Delay?vIN(t) = A1sin(t) + A2sin[(+D) t]G(j) G(j)ej()vOUT(t) = A1 G(j) sin[t+()] + A2 G[ j(+D)] sin[(+D)t+ (+D)]EECS 247 Lecture 2: Filters © 2010 Page 13What is Group Delay?{ ]}[vOUT(t) = A1 G(j) sint + ()+{ ]}[+ A2 G[ j(+D)] sin(+D)t +(+D)+D(+D)+D ()+d()dD[ ][1)( ]1 -Dd()d()+()-()DD<<1Since then D0[ ]2EECS 247 Lecture 2: Filters © 2010 Page 14What is Group Delay?Signal Magnitude and Phase Impairment{ ]}[vOUT(t) = A1 G(j) sint + ()+{ ]}[+ A2 G[ j(+D)]sin(+D)t +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-zerothe filter’s output at frequency +D is time-shifted differently than the filter’s output at frequency “Phase distortion”dEECS 247 Lecture 2: Filters © 2010 Page 15• Phase distortion is avoided only if:• 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=PDimplies linear phase• Note: Filters with ()=k+c are also called linear phase filters, but they’re not free of phase distortionWhat is Group Delay?Signal Magnitude and Phase Impairmentd()d()-= 0EECS 247 Lecture 2: Filters © 2010 Page 16What is Group Delay?Signal Magnitude and Phase Impairment• If GR=PD No phase distortion[ )](vOUT(t) = A1 G(j) sint - GR+[+ A2 G[ j(+D)] sin(+D))](t - GR• If alsoG( j)=G[ j(+D)] for all input frequencies within the signal-band, vOUTis a scaled, time-shifted replica of the input, with no “signal magnitude distortion” • In most cases neither of these conditions are exactly realizableEECS 247 Lecture 2: Filters © 2010 Page 17• 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=-kSummaryGroup DelayEECS 247 Lecture 2: Filters © 2010 Page 18Filters• Filters: – Nomenclature– Specifications • Magnitude/phase response
View Full Document
Unlocking...