Filter Notes. Terminology and Real Filter Concepts. Transition band and filter shape factor

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1 Filter Notes Dr. Bradley J. Bazuin Western Michigan University College of Engineering and Applied Sciences Department of Electrical and Computer Engineering 93 W. Michigan Ave. Kalamazoo MI,

2 Filter Notes Terminology and Real Filter Concepts Transition band and filter shape factor Filter Bandwidth Definitions Butterworth Filter Examples in Matlab Chebyshev Type I Examples in Matlab MATLAB Filtering Comparing classical analog filters Matlab functions to support analog filter generation Filter Notes 2

3 Real Filters: Terminology db -XX db Power (db) Passband BW PB f p BW SB SF Transition Band BW BW SB PB f s Freq. Stopband Low Pass Filter Passband Frequencies where signal is meant to pass Stopband Frequencies where some defined level of attenuation is desired Transition-band The transitions frequencies between the passband and the stopband Filter Shape Factor The ratio of the stopband bandwidth to the passband bandwidth Filter Notes 3

4 Real Filters: Terminology db Power (db) Stopband -XX db f sl f pl Transition Band SF Passband BW PB BW SB BW BW f pu Transition Band SB PB Freq. f su Stopband Band Pass Filter Passband Frequencies where signal is meant to pass Stopband Frequencies where some defined level of attenuation is desired Transition-band The transitions frequencies between the passband and the stopband Filter Shape Factor The ratio of the stopband bandwidth to the passband bandwidth Filter Notes 4

5 Passband/Stopband Ripple p db db p db Passband f p Freq. Power (db) BW PB Transition Band Stopband s db BW SB f s Based on the filter type, the passband and stopband may not be smooth The change in the band is defined as an allowable ripple Note: for passbands, a 3 db ripple may not be desirable; therefore, for real filters the 3 db point is often of no interest. Filter Notes 5

6 Bandwidths that are Used Notes and figures are based on or taken from materials in the course textbook: Filter Notes Bernard Sklar, Digital Communications, Fundamentals and Applications, 6 Prentice Hall PTR, Second Edition, 2.

7 Bandwidth Definitions (a) Half-power bandwidth. This is the interval between frequencies at which Gx(f ) has dropped to half-power, or 3 db below the peak value. (b) Equivalent rectangular or noise equivalent bandwidth. The noise equivalent bandwidth was originally conceived to permit rapid computation of output noise power from an amplifier with a wideband noise input; the concept can similarly be applied to a signal bandwidth. The noise equivalent bandwidth WN of a signal is defined by the relationship WN = Px/Gx(fc), where Px is the total signal power over all frequencies and Gx(fc) is the value of Gx(f ) at the band center (assumed to be the maximum value over all frequencies). (c) Null-to-null bandwidth. The most popular measure of bandwidth for digital communications is the width of the main spectral lobe, where most of the signal power is contained. This criterion lacks complete generality since some modulation formats lack well-defined lobes. Notes and figures are based on or taken from materials in the course textbook: Filter Notes Bernard Sklar, Digital Communications, Fundamentals and Applications, 7 Prentice Hall PTR, Second Edition, 2.

8 Bandwidth Definitions (2) (d) Fractional power containment bandwidth. This bandwidth criterion has been adopted by the Federal Communications Commission (FCC Rules and Regulations Section 2.22) and states that the occupied bandwidth is the band that leaves exactly.5% of the signal power above the upper band limit and exactly.5% of the signal power below the lower band limit. Thus 99% of the signal power is inside the occupied band. (e) Bounded power spectral density. A popular method of specifying bandwidth is to state that everywhere outside the specified band, Gx(f ) must have fallen at least to a certain stated level below that found at the band center. Typical attenuation levels might be 35 or 5 db. (f) Absolute bandwidth. This is the interval between frequencies, outside of which the spectrum is zero. This is a useful abstraction. However, for all realizable waveforms, the absolute bandwidth is infinite. Notes and figures are based on or taken from materials in the course textbook: Filter Notes Bernard Sklar, Digital Communications, Fundamentals and Applications, 8 Prentice Hall PTR, Second Edition, 2.

9 Other Filter Types (Bandwidth and Center Frequency) When op-amps, Rs and Cs don t work anymore. Op-Amp max freq. BW typically a band-pass filter Resonant electromechanical devices used for electrical filtering. Vectron International, General technical information, Filter Notes 9

10 Butterworth Low Pass Filter H jw H jw 2n H s 2 s j w j 2n n w 2n s w s w w 2n 2n Attenuation (db) st order 2nd order 3rd order 4th order Butterworth Filter Family 5th order Frequency (normalized) Maximally Flat, Smooth Roll-off, identical 3dB point for all filter orders Filter Notes M.E. Van Valkenburg, Analog Filter Design, Oxford Univ. Press, 982. SBN:

11 Butterworth Filter PSD Butterworth Filter Family -2 Attenuation (db) st order 2nd order 3rd order 4th order 5th order -2 Filter Notes Frequency (normalized)

12 Butterworth Filter PSD (2) Butterworth Filter Family - -2 Attenuation (db) st order 2nd order 3rd order 4th order 5th order -9 Filter Notes - Frequency (normalized) 2

13 Matlab Script: ButterPlot.m % % Butterworth filter plots % freqrange = logspace(-,3,24)'; wrange=2*pi*freqrange; [B,A]=butter(,2*pi,'s'); [H] = freqs(b,a,wrange); [B2,A2]=butter(2,2*pi,'s'); [H2] = freqs(b2,a2,wrange); [B3,A3]=butter(3,2*pi,'s'); [H3] = freqs(b3,a3,wrange); [B4,A4]=butter(4,2*pi,'s'); [H4] = freqs(b4,a4,wrange); figure() semilogx(freqrange,db(psdg(hmatrix))); grid title('butterworth Filter Family'); xlabel('frequency (normalized)'); ylabel('attenuation (db)'); legend('st order','2nd order','3rd order','4th order','5th order','location','southwest'); axis([^- ^3-2 3]); figure(2) semilogx(freqrange,db(psdg(hmatrix))); grid title('butterworth Filter Family'); xlabel('frequency (normalized)'); ylabel('attenuation (db)'); legend('st order','2nd order','3rd order','4th order','5th order','location','southwest'); axis([^- 3-9 ]); [B5,A5]=butter(5,2*pi,'s'); [H5] = freqs(b5,a5,wrange); Hmatrix=[H H2 H3 H4 H5]; Filter Notes 3

14 Chebyshev Type I Filter PSD (ChebyPlot.m) Chebyshev Type I Filter Family -2 Attenuation (db) st order 2nd order 3rd order 4th order 5th order Filter Notes Frequency (normalized) 4

15 Chebyshev Type I Filter PSD (2) Chebyshev Type I Filter Family - -2 Attenuation (db) st order 2nd order 3rd order 4th order 5th order -9 Filter Notes - Frequency (normalized) 5

16 Available MATLAB Filters (Signal Proc. TB) /signal/index.html Analog (s-domain) or Digital (z-domain) Butterworth Chebyshev Type I Chebyshev Type II Elliptic or Cauer Bessel Digital Windows barthannwin bartlett blackman blackmanharris bohmanwin chebwin flattopwin gausswin hamming hann kaiser nuttallwin parzenwin rectwin triang tukeywin s.html Filter Notes 6

17 Analog Lowpass Filter Design Butterworth Monotonic Decreasing Magnitude All poles Chebyshev (Cheby Type ) Passband Ripple All poles Inverse Chebyshev (Cheby Type2) Stopband Ripple Elliptical or Cauer Filter Passband Ripple Stopband Ripple Bessel Filter Linear Phase Maximized Monotonic Filter Comparison: Magnitude Butter Bessel Cheby Cheby2 Ellip Spec Butterworth Order Predication Filter Order = 4 3dB BW = Hz Bessel Order Predication Filter Order = 4 3dB BW = Hz Chebyshev Type II Order Predication Filter Order = 3 3dB BW = Hz Elliptical or Cauer Order Predication Filter Order = 3 3dB BW = Hz Chebyshev Type I Order Predication Filter Notes Filter Order = 3 3dB BW = Hz 7

18 Matlab Filter Generation () Passband Stopband Passband Ripple (db) Stopband Ripple (db) fpass=; fstop=; AlphaPass=.5; AlphaStop=6; w#### = 2 x pi x f#### Filter Order and other design parameters [Nbutter, Wnbutter] = buttord(wpass, wstop, AlphaPass, AlphaStop,'s'); [Ncheby, Wncheby] = chebord(wpass, wstop, AlphaPass, AlphaStop,'s'); [Ncheby2, Wncheby2] = cheb2ord(wpass, wstop, AlphaPass, AlphaStop,'s'); [Nellip, Wnellip] = ellipord(wpass, wstop, AlphaPass, AlphaStop,'s'); Filter Notes 8

19 Matlab Filter Generation (2) Filter Transfer Function Generation [numbutter,denbutter] = butter(nbutter,wnbutter,'low','s') [numbesself,denbesself] = besself(nbutter,wnbutter) [numcheby,dencheby] = cheby(ncheby,alphapass, Wncheby,'low','s') [numcheby2,dencheby2] = cheby2(ncheby2,alphastop, Wncheby2,'low','s') [numellip,denellip] = ellip(nellip,alphapass,alphastop, Wnellip,'low','s'); Spectral Response from Transfer Function [Specbutter]=freqs(numbutter,denbutter,wspace); [Specbesself]=freqs(numbesself,denbesself,wspace); [Speccheby]=freqs(numcheby,dencheby,wspace); [Speccheby2]=freqs(numcheby2,dencheby2,wspace); [Specellip]=freqs(numellip,denellip,wspace); Filter Notes 9

20 Matlab Filter Generation (3) figure() semilogx((fspace),db(psdg([specbutter Specbesself Speccheby Speccheby2 Specellip])),... specfreq,specmag,'k-.',specfreq2,specmag2,'k-.',specfreq3,specmag3,'k-.'); title('filter Comparison: Magnitude') legend('butter','bessel','cheby','cheby2','ellip','spec') Filter Comparison: Magnitude -5 - Butter Bessel Cheby Cheby2 Ellip Spec -5 Filter Notes

21 Generating Digital Filters MATLAB loves digital filter. Leave off the s from the design programs Select the passband and stopband in terms of the sample rate/2 from ( to ) IIR filters of defined order are generated. For the spectrum, use freqz instead of freqs Filter Notes 2

22 Matlab Digital Filter Generation () Sample Rate Fs=8; Passband fpass=; Stopband fstop=375; Passband Ripple (db) AlphaPass=.; Stopband Ripple (db) AlphaStop=8; w#### = f####/(fs/2) Filter Order and other design parameters [Nbutter, Wnbutter] = buttord(wpass, wstop, AlphaPass, AlphaStop); [Ncheby, Wncheby] = chebord(wpass, wstop, AlphaPass, AlphaStop); [Ncheby2, Wncheby2] = cheb2ord(wpass, wstop, AlphaPass, AlphaStop); [Nellip, Wnellip] = ellipord(wpass, wstop, AlphaPass, AlphaStop); Filter Notes 22

23 Matlab Digital Filter Generation (2) Filter Transfer Function Generation [numbutter,denbutter] = butter(nbutter,wnbutter,'low') [numcheby,dencheby] = cheby(ncheby,alphapass, Wncheby,'low') [numcheby2,dencheby2] = cheby2(ncheby2,alphastop, Wncheby2,'low') [numellip,denellip] = ellip(nellip,alphapass,alphastop, Wnellip,'low') Spectral Response from Transfer Function [Specbutter,wspace]=freqz(numbutter,denbutter,fftsize,'whole') [Speccheby,wspace]=freqz(numcheby,dencheby,fftsize,'whole'); [Speccheby2,wspace]=freqz(numcheby2,dencheby2,fftsize,'whole'); [Specellip,wspace]=freqz(numellip,denellip,fftsize,'whole'); Filter Notes 23

24 Matlab Digital Filter Generation (3) plot((wspace)/(2*pi),db(psdg([specbutter Speccheby Speccheby2 Specellip Specfirpm])),... specfreq/(2*pi),specmag,'k-.',specfreq2/(2*pi),specmag2,'k-.',specfreq3/(2*pi),specmag3,'k-.',... specfreq4/(2*pi),specmag4,'k-.',specfreq5/(2*pi),specmag5,'k-.') legend('butter','cheby','cheby2','ellip','firpm','spec.','location','southeast') title('digital Filter Comparison: Magnitude') 2 Digital Filter Comparison: Magnitude -2 Magnitude (db) Filter Notes /2 Butter Cheby Cheby2 Ellip FIRPM Spec.

25 Coherent Gain of a Digital Filter FIR: Sum the coefficients For x(k)= for all k, y( ) for a LPF settles to a constant To adjust for unity gain, divide the numerator coefficients by the coherent gain Filter Notes z a z a z b z b b z H n x b n x b n x b n y a n y a n y 2 2 b b b a a y 2 2 a a b b b y

26 Pulse Response and Risetime Low Pass Filters cause sharp signal edges to be smoothed. The amount of smoothing is based on the bandwidth of the filter More smoothing smaller bandwidth Fourier relationship: a narrow rect function in time results in a broad (wide bandwidth) sinc function in frequency a wide rect function in time results in a narrow (small bandwidth) sinc function in frequency Filter Notes 26

27 Filter Step Response Hz and Hz 4 th order Butterworth LPF Filters (s-domain) The step response can be used to help define the bandwidth required for pulse signals. Butterworth Filters.4 Step Response Attenuation (db) -6-8 Amplitude Hz Hz Frequency (normalized).2 Hz Hz Time (sec) Filter Notes 27

28 Filter Bandwidth for Pulses Pulse of length T rect t T T sin c f T 2.5 Null-to-null BW of null to null Single Sided BW T B/2 may be acceptable in some cases 2 T.5 B Filter Notes 28

29 Pulse Filtering Attenuation (db) Butterworth Filters 2.5 Hz 5. Hz. Hz 2. Hz Frequency (fs = Hz).6 PulseTest.m (Butterworth digital filters) Amplitude (db) Four one-sided BW filters. sec pulse responses B=/T = Hz 2B=2/T = 2 Hz B/2 = /2T = 5 Hz Fs = Hz Time (fs=hz) Filter Notes Butterworth Filters Test Signal 2.5 Hz 5. Hz. Hz 2. Hz

30 Matlab Code Example so far AnalogFilterCompare.m DigitalFilterCompare.m PulseTest.m Filter Notes 3

31 Design Parameters Signal Type Signal bandwidth Spectral filtering requirements Special requirements Signal to Noise Ratio (SNR) Bit Error Rate (BER) Intersymbol interference (ISI) Multipath considerations (reflected signal paths?) Filter Notes 3

32 Example Bandwidth Considerations Analog communications signals have definable bandwidths based on the modulation type Amplitude Modulation Phase Modulation Frequency Modulation Digital communications based on analog are concerned with symbol detection and ISI Amplitude Shift Keying (ASK or OOK) Phase Shift Keying (PSK) Frequency Shift Keying (FSK) Quadrature Amplitude Modulation (QAM) Filter Notes 32

33 Bandwidth Estimates for Common Communication Signals AM Signals: st A m t cos2 f t For both AM and DSB signals, for a signal with maximum frequency, f max W the bandwidths used should be: Bandpass Signal: f c f max f c f f c f max BPF bandwidth = Baseband Signal: LPF bandwidth = 2 f max f f max f max f max Filter Notes 33

34 Bandwidth Estimates for Common Communication Signals PM Signals: t s Acos 2 f t p m2 For both PM signals, for a signal with maximum frequency, f max W the bandwidth is based on Carson s rule and should be: BPF Bandwidth: BT 2 PM f max LPF bandwidth B pred PM f max t These equations wok with the following restrictions; abs(m(t)), A =, and PM After PM demodulation, a post-demodulation LPF with bandwidth W should be used to limit noise power contributions to the output. Filter Notes 34

35 Bandwidth Estimates for Common Communication Signals FM Signals: s t Acos2 f t 2 f m3 d where For both FM signals, for a signal with maximum frequency, f max W the bandwidth is based on the Deviation ratio (D) which typically uses Carson s rule for D>> or D<< and should be: BPF Bandwidth D>> or D<<: B T 2 D f max LPF Bandwidth D>> or D<<: B pred D f max BPF Bandwidth 2<D<: B T 2 D 2 f max LPF Bandwidth 2<D<: B pred D 2 f max These equations wok with the following restrictions; abs(m(t)) and A =. After FM demodulation, a post-demodulation LPF with bandwidth W should be used to limit noise power contributions to the output. Filter Notes 35 t D f W

36 Symbol Period PSK/FSK For a symbol period T, a 2/T LPF can be used. PulseTest2.m PulseTest3.m Note: in many communication systems, the goal for demodulation is one sample per symbol. If time alignment is required 2 or more samples per symbol may be required or desired for receiver design considerations. Filter Notes 36

Filter Notes. Terminology and Real Filter Concepts. Transition band and filter shape factor

Filter Notes. Terminology and Real Filter Concepts. Transition band and filter shape factor Filter Notes Dr. Bradley J. Bazuin Western Michigan University College of Engineering and Applied Sciences Department of Electrical and Computer Engineering 93 W. Michigan Ave. Kalamazoo MI, 498-5329 Filter