Filter Notes - Homepages at WMUbazuinb/ECE6560/FilterNotes.pdfFilter Notes 15 Chebyshev Type I Filter PSD (2) 10-1 10 0-9-8-7-6-5-4-3-2-1 0 1 Chebyshev Type I Filter Family Frequency

Filter Notes

Dr. Bradley J. BazuinWestern Michigan University

College of Engineering and Applied SciencesDepartment of Electrical and Computer Engineering

1903 W. Michigan Ave.Kalamazoo MI, 49008-5329

Filter Notes 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 3

Real Filters: TerminologyLow 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 bandwidthPB

SB

BWBWSF

0 dB

-XX dBStopband

Passband

TransitionBand

PBBW

SBBW

Freq.

Pow

er (d

B)

pf

sf

Filter Notes 4

Real Filters: TerminologyBand 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 bandwidthPB

SB

BWBWSF

0 dB

-XX dBStopband

Passband

TransitionBand

PBBW

SBBW

Freq.

Pow

er (d

B)

TransitionBandStopband

plf

slf

puf

suf

Filter Notes 5

Passband/Stopband Ripple

• 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.

Stopband

Passband

TransitionBand

PBBW

SBBW

Freq.Po

wer

(dB)

dBpdB0

dBp

dBs

pf

sf

Filter Notes 6

Bandwidths that are Used

Notes and figures are based on or taken from materials in the course textbook: Bernard Sklar, Digital Communications, Fundamentals and Applications,

Prentice Hall PTR, Second Edition, 2001.

Filter Notes 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.



Filter Notes 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 2.202) and states that the occupied bandwidth is the band that leaves exactly 0.5% of the signal power above the upper band limit and exactly 0.5% 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 50 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.



Filter Notes 9

Other Filter Types (Bandwidth and Center Frequency)

Vectron International, General technical information, http://www.vectron.com/products/saw/pdf_mqf/TECHINFO.pdf

BW

typi

cally

a

band

-pas

s filt

er

Op-Amp max freq.

Resonant electro-mechanical devices used for electrical filtering.

When op-amps, Rs and Cs don’t work anymore.

Filter Notes 10

Butterworth Low Pass Filter

• Maximally Flat, Smooth Roll-off, identical 3dB point for all filter orders

n2

0ww1

1jwHjwH

n2

0

n

n2

0

n2

n2

0

2

ws11

1w

sj1

1wj

s1

1sH

M.E. Van Valkenburg, Analog Filter Design, Oxford Univ. Press, 1982. SBN: 0-19-510734-9

10-1 100 101 102 103-120

-100

-80

-60

-40

-20

0

Butterworth Filter Family

Frequency (normalized)

Atte

nuat

ion

(dB

)

1st order2nd order3rd order4th order5th order

Filter Notes 11

Butterworth Filter PSD

10-1 100 101 102 103-120

-100

-80

-60

-40

-20

0

Butterworth Filter Family


Atte

nuat

ion

(dB

)


Filter Notes 12

Butterworth Filter PSD (2)

10-1 100-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1Butterworth Filter Family


Atte

nuat

ion

(dB

)


Filter Notes 13

Matlab Script: ButterPlot.m%% Butterworth filter plots%

freqrange = logspace(-1,3,1024)';wrange=2*pi*freqrange;

[B1,A1]=butter(1,2*pi,'s');[H1] = freqs(B1,A1,wrange);





Hmatrix=[H1 H2 H3 H4 H5];

figure(1)semilogx(freqrange,dB(psdg(Hmatrix)));gridtitle('Butterworth Filter Family');xlabel('Frequency (normalized)');ylabel('Attenuation (dB)');legend('1st order','2nd order','3rd order','4th order','5th order','Location','SouthWest');axis([10^-1 10^3 -120 3]);

figure(2)semilogx(freqrange,dB(psdg(Hmatrix)));gridtitle('Butterworth Filter Family');xlabel('Frequency (normalized)');ylabel('Attenuation (dB)');legend('1st order','2nd order','3rd order','4th order','5th order','Location','SouthWest');axis([10^-1 3 -9 1]);

Filter Notes 14

Chebyshev Type IFilter PSD (Cheby1Plot.m)

10-1 100 101 102 103-120

-100

-80

-60

-40

-20

0

Chebyshev Type I Filter Family


Atte

nuat

ion

(dB

)


Filter Notes 15

Chebyshev Type IFilter PSD (2)

10-1 100-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1Chebyshev Type I Filter Family


Atte

nuat

ion

(dB

)


Filter Notes 16

Available MATLAB Filters(Signal Proc. TB)

• http://www.mathworks.com/help/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

– http://www.mathworks.com/help/signal/windows.html

Filter Notes 17

Analog Lowpass Filter Design • Butterworth

– Monotonic Decreasing Magnitude

– All poles• Chebyshev (Cheby Type 1)

– Passband Ripple– All poles

• Inverse Chebyshev (Cheby Type2) – Stopband Ripple

• Elliptical or Cauer Filter – Passband Ripple– Stopband Ripple

• Bessel Filter– Linear Phase Maximized– Monotonic

101 102 103 104 105 106 107-160

-140

-120

-100

-80

-60

-40

-20

0

20Filter Comparison: Magnitude

ButterBesselCheby1Cheby2EllipSpec

Butterworth Order PredicationFilter Order = 4 3dB BW = 1778.28 Hz

Bessel Order PredicationFilter Order = 4 3dB BW = 1778.28 Hz

Chebyshev Type I Order PredicationFilter Order = 3 3dB BW = 1000 Hz

Chebyshev Type II Order PredicationFilter Order = 3 3dB BW = 8972.85 Hz

Elliptical or Cauer Order PredicationFilter Order = 3 3dB BW = 1000 Hz

Filter Notes 18

Matlab Filter Generation (1)• Passband • Stopband• Passband Ripple (dB)• Stopband Ripple (dB)

• fpass=1000;• fstop=10000;• AlphaPass=0.5;• AlphaStop=60;• w#### = 2 x pi x f####

[Nbutter, Wnbutter] = buttord(wpass, wstop, AlphaPass, AlphaStop,'s');

[Ncheby1, Wncheby1] = cheb1ord(wpass, wstop, AlphaPass, AlphaStop,'s');

[Ncheby2, Wncheby2] = cheb2ord(wpass, wstop, AlphaPass, AlphaStop,'s');

[Nellip, Wnellip] = ellipord(wpass, wstop, AlphaPass, AlphaStop,'s');

Filter Order and other design parameters

Filter Notes 19

Matlab Filter Generation (2)Filter Transfer Function Generation

[numbutter,denbutter] = butter(Nbutter,Wnbutter,'low','s')[numbesself,denbesself] = besself(Nbutter,Wnbutter)[numcheby1,dencheby1] = cheby1(Ncheby1,AlphaPass, Wncheby1,'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);[Speccheby1]=freqs(numcheby1,dencheby1,wspace);[Speccheby2]=freqs(numcheby2,dencheby2,wspace);[Specellip]=freqs(numellip,denellip,wspace);

Filter Notes 20

Matlab Filter Generation (3)figure(10)semilogx((fspace),dB(psdg([Specbutter Specbesself Speccheby1 Speccheby2 Specellip])), ...

specfreq1,specmag1,'k-.',specfreq2,specmag2,'k-.',specfreq3,specmag3,'k-.');title('Filter Comparison: Magnitude')legend('Butter','Bessel','Cheby1','Cheby2','Ellip','Spec')

101 102 103 104 105 106 107-150

-100

-50

0

Filter Comparison: Magnitude

ButterBesselCheby1Cheby2EllipSpec

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 (0 to 1)– IIR filters of defined order are generated.– For the spectrum, use freqz instead of freqs

Filter Notes 21

Filter Notes 22

Matlab Digital Filter Generation (1)• Sample Rate • Passband • Stopband• Passband Ripple (dB)• Stopband Ripple (dB)

• Fs=8000;• fpass=1000;• fstop=3750;• AlphaPass=0.1;• AlphaStop=80;• w#### = f####/(Fs/2)

[Nbutter, Wnbutter] = buttord(wpass, wstop, AlphaPass, AlphaStop);

[Ncheby1, Wncheby1] = cheb1ord(wpass, wstop, AlphaPass, AlphaStop);

[Ncheby2, Wncheby2] = cheb2ord(wpass, wstop, AlphaPass, AlphaStop);

[Nellip, Wnellip] = ellipord(wpass, wstop, AlphaPass, AlphaStop);

Filter Order and other design parameters

Filter Notes 23

Matlab Digital Filter Generation (2)Filter Transfer Function Generation

[numbutter,denbutter] = butter(Nbutter,Wnbutter,'low')[numcheby1,dencheby1] = cheby1(Ncheby1,AlphaPass, Wncheby1,'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')[Speccheby1,wspace]=freqz(numcheby1,dencheby1,fftsize,'whole');[Speccheby2,wspace]=freqz(numcheby2,dencheby2,fftsize,'whole');[Specellip,wspace]=freqz(numellip,denellip,fftsize,'whole');

Filter Notes 24

Matlab Digital Filter Generation (3)plot((wspace)/(2*pi),dB(psdg([Specbutter Speccheby1 Speccheby2 Specellip Specfirpm])), ...

specfreq1/(2*pi),specmag1,'k-.',specfreq2/(2*pi),specmag2,'k-.',specfreq3/(2*pi),specmag3,'k-.',...specfreq4/(2*pi),specmag4,'k-.',specfreq5/(2*pi),specmag5,'k-.')

legend('Butter','Cheby1','Cheby2','Ellip','FIRPM','Spec.','Location','SouthEast')title('Digital Filter Comparison: Magnitude')

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-160

-140

-120

-100

-80

-60

-40

-20

0

20

/2

Mag

nitu

de (d

B)

Digital Filter Comparison: Magnitude

ButterCheby1Cheby2EllipFIRPMSpec.

Coherent Gain of a Digital Filter

• FIR: Sum the coefficients

• For x(k)=1 for all k, y(∞) for a LPF settles to a constant

• To adjust for unity gain, divide the sum of the numerator coefficients by the sum of the coherent gain

Filter Notes 25

22

11

22

110

1 zazazbzbbzH

2121 21021 nxbnxbnxbnyanyany

210211 bbbaay

21

210

1 aabbby

Filter Notes 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 27

Filter Step Response

• 1 Hz and 10 Hz 4th order Butterworth LPF Filters (s-domain)• The step response can be used to help define the bandwidth

required for pulse signals.

10-1 100 101 102 103 104-120

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0

Butterworth Filters


Atte

nuat

ion

(dB

)

1 Hz10 Hz

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

Ampl

itude

1 Hz10 Hz

Filter Notes 28

Filter Bandwidth for Pulses• Pulse of length T

• Null-to-null BW of

• Single Sided BW

• B/2 may be acceptable in some cases

TfcsinTTtrect

T2nulltonull

T1B -3 -2 -1 0 1 2 3

0

0.5

1

1.5

2

Filter Notes 29

Pulse Filtering

• Four one-sided BW filters• 0.1 sec pulse responses

– B=1/T = 10 Hz– 2B=2/T = 20 Hz– B/2 = 1/2T = 5 Hz– Fs = 100 Hz

0 10 20 30 40 50 60 70 80 90 100-160

-140

-120

-100

-80

-60

-40

-20

0

20Butterworth Filters

Frequency (fs = 100 Hz)

Atte

nuat

ion

(dB

)

2.5 Hz5.0 Hz10. Hz20. Hz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2Butterworth Filters

Time (fs=100Hz)

Am

plitu

de (d

B)

Test Signal2.5 Hz5.0 Hz10. Hz20. Hz

PulseTest1.m

(Butterworthdigital filters)

Filter Notes 30

Matlab Code Example so far

• AnalogFilterCompare.m• DigitalFilterCompare.m• PulseTest1.m

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 31

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

Filter Notes 33

Bandwidth Estimates for Common Communication Signals

AM Signals:• For both AM and DSB signals, for a signal with maximum

frequency , the bandwidths used should be:• Bandpass Signal:

– BPF bandwidth =

• Baseband Signal:– LPF bandwidth =

Wf max

maxmax ffffff ccc

max2 f

maxmax fff

maxf

tftmAts 01 2cos1

Filter Notes 34


PM Signals:• For both PM signals, for a signal with maximum frequency

, the bandwidth is based on Carson’s rule and should be:

• BPF Bandwidth:• LPF bandwidth

– These equations wok with the following restrictions; abs(m(t)) 1, A = 1, and

– After PM demodulation, a post-demodulation LPF with bandwidth W should be used to limit noise power contributions to the output.

Wf max

max12 fB PMT

max1 fB PMDpre

PM

tmtfAts p 202cos

Filter Notes 35


FM Signals:• For both FM signals, for a signal with maximum frequency

, the bandwidth is based on the Deviation ratio (D) which typically uses Carson’s rule for D>>1 or D<<1 and should be:

• BPF Bandwidth D>>1 or D<<1:• LPF Bandwidth D>>1 or D<<1:• BPF Bandwidth 2<D<10:• LPF Bandwidth 2<D<10:

– These equations wok with the following restrictions; abs(m(t)) 1 and A = 1.

– After FM demodulation, a post-demodulation LPF with bandwidth W should be used to limit noise power contributions to the output.

Wf max

max12 fDBT

max1 fDB Dpre

max22 fDBT

max2 fDB Dpre

t

f dmtfAts 30 22cosW

Dwhere f

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

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Filter Notes - Homepages at WMUbazuinb/ECE6560/FilterNotes.pdfFilter Notes 15 Chebyshev Type I Filter PSD (2) 10-1 10 0-9-8-7-6-5-4-3-2-1 0 1 Chebyshev Type I Filter Family Frequency