Filter Approximation Concepts

Home , Bessel filter, Chebyshev filter, Elliptic filter, Ripple (electrical), Stopband

622 (ESS)

1 Filter Approximation Concepts

How do you translate filter specifications into a mathematical expression which can be synthesized ? • Approximation Techniques

Why an ideal Brick Wall Filter can not be implemented ? • Causality: Ideal filter is non-causal • Rationality: No rational transfer function of finite degree (n) can have such abrupt transition

|H(jω)| h(t)

Passband Stopband

ω t -ωc ωc -Tc Tc 2 Filter Approximation Concepts

Practical Implementations are given via window specs.

|H(jω)|

Amax

Amin

ω ωc ωs

Amax = Ap is the maximum attenuation in the passband

Amin = As is the minimum attenuation in the stopband

ωs-ωc is the Transition Width 3 Approximation Types of Lowpass Filter

1 Definitions Ap

 Ripple = 1-Ap wc ws  Stopband attenuation =As Filter specs Maximally flat (Butterworth)  Passband (cutoff

frequency) = wc

 Stopband frequency = ws

Equal-ripple (Chebyshev) Elliptic

4 Approximation of the Ideal Lowpass Filter

Since the ideal LPF is unrealizable, we will accept a small error in the passband, a non-zero transition band, and a finite stopband attenuation 1 퐻 푗휔 2 = 1 + 퐾 푗휔 2 |H(jω)| • 퐻 푗휔 : filter’s transfer function • 퐾 푗휔 : Characteristic function (deviation of 푇 푗휔 from unity)

For 0 ≤ 휔 ≤ 휔푐 → 0 ≤ 퐾 푗휔 ≤ 1 For 휔 > 휔푐 → 퐾 푗휔 increases very fast Passband Stopband

ω ωc 5 Maximally Flat Approximation (Butterworth) Stephen Butterworth showed in 1930 that the gain of an nth order maximally flat magnitude filter is given by

1 퐻 푗휔 2 = 퐻 푗휔 퐻 −푗휔 = 1 + 휀2휔2푛

퐾 푗휔 ≅ 0 in the passband in a maximally flat sense

푑푘 퐾 푗휔 2 2 푘 = 0 for 푘 = 1,2, … , 2푛 − 1 푑 휔 휔=0

The corresponding pole locations (for 휀 = 1) can be determined as follows

1 1 2푛 퐻 푠 2 = = → 푠 = − −1 −푛 = 푒푗휋 2푘−1+푛 푠 2푛 푛 2푛 푝 1+ 1+ −1 푠 푗 휋 2푘−1+푛 푗 푠푝 = 푒 2 푛 푘 = 1, … , 2푛 6 Pole Locations: Maximally flat (ε=1)

The poles are located on the unity circle at equispaced angles 휋 2푘−1+푛 푠 = 푒푗휃푘 where 휃 = 푘 = 1,2, … , 2푛 푝 푘 2 푛 The real and imaginary parts are 2푘−1 휋 2푘−1 휋 푅푒 푆 = − sin Im 푆 = cos 푝 푛 2 푝 푛 2 Poles in the LHP are associated with 퐻 푠 and poles in the RHP are associated with 퐻 −s

휽풌 ퟎ° ±ퟏퟑퟓ° ±ퟏퟐퟎ°, ퟏퟖퟎ° ±ퟏퟏퟐ. ퟓ°, ±ퟏퟓퟕ. ퟓ° ±ퟏퟎퟖ°, ±ퟏퟒퟒ°, ퟏퟖퟎ° Magnitude Response of Butterworth Filter

8 Design Example

Design a 1-KHz maximally flat lowpass filter with: • Attenuation at 10 kHz ≥2000

 Normalized prototype: 휔푐 = 1 푟푎푑 푠 , 휔푠 = 10 푟푎푑 푠 2 2 1 1 2푛 6 퐻 푗휔푠 = 2푛 ≤ → 10 ≥ 4 × 10 or 푛 ≥ 3.3 1+휔푠 2000

Choose 푛 = 4  Pole locations 푗휃푘 푠푝 = 푒 where 휃푘 = ±112.5° , ±157.5° 푠푝1,2 = −0.383 ± 푗0.924 푠푝3,4 = −0.924 ± 푗0.383  Normalized transfer function 1 퐻 푠 = 푠2 + 0.765푠 + 1 푠2 + 1.848푠 + 1

퐻1 퐻2 푄 = 1.306, 휔푐 = 1 푄 = 0.541, 휔푐 = 1 9

Design Example

푠  Denormalized transfer function 푠 = 휔푐 1.5585 × 1015 퐻 푠 = 푠2 + 4.8 × 103 푠 + 3.95 × 107 푠2 + 1.16 × 104 푠 + 3.95 × 107

퐻1 퐻2 3 3 푄 = 1.306 , 휔푐 = 2휋10 푄 = 0.541 , 휔푐 = 2휋10

10 Design Example -Discussion

 We can sharpen the transition in the previous example by increasing the quality factor of one of

the two cascaded filters (Q1=3.5 instead of 1.3)

 To alleviate the peaking problem in the previous ′ response, we can reduce 휔푐2 휔푐2 = 0.6휔푐1

 Passband ripples are now existing • They can be tolerated in some applications

 The resulting response has steeper transition than Butterworth response 11 Equiripple Filter Approximation (Chebyshev I) This type has a steeper transition than Butterworth filters of the same order but at the expense of higher passband ripples |H(jω)|2

Magnitude response of this type is given by 1 1/(1+ε2) 1 퐻 푗휔 2 = 1+ 퐾 푗휔 2 ω -1 1 퐾 푗휔 = 휀퐶 휔 푛 2 |K(jω)| −1 퐶푛 휔 = cos 푛 cos 휔 휔 ≤ 1 ε2 ω Going back and forth in ±1 range for -1 1 휔 ≤ 1 ε-1|K(jω)| 푪풏 is called Chebyshev’s polynomial 1 ω -1 -1 0 1 Defining the passband area of 푲 풋흎 Equiripple Filter Approximation (Chebyshev I) Chebyshev’s Polynomial e푗푛휙+e−푗푛휙 퐶 휔 = cos 푛 cos−1 휔 = 푛 2 풏 푪풏 For the stopband (휔 > 1) 0 1 1 휔 −1 휙 = cos 휔 is complex 2 2휔2 − 1 Thus, 퐶 > 1 푛 3 3 4휔 − 3휔 Since 4 8휔4 − 8휔2 + 1 cos 푛휙 = cosh 푛푗휙 푛 2휔퐶 − 퐶 and 푛−1 푛−2 푗휙 = cosh 휔 Then −1 퐶푛 휔 = cosh 푛 cosh 휔 for 휔 > 1

13 Properties of the Chebyshev polynomials

1 Cn cos(ncos (w))1 w1

Faster response in the stopband In the passband, w<1, Cn is For n>3 and w>1 limited to 1, then, the ripple is determined by e

Butterwort h Chebyshev 2 1  n1 N( jw)  C 2 2nw2 n1  23 w2 n1 2 2 n   1e C n w

2 2 The -3 dB frequency can be found as: e C n w3dB 1 w1    1 1 1  w3dB cosh cosh  w1  n e 

14 Pole Locations (Chebyshev Type I)

15 Magnitude Response (Chebyshev Type I)

16 Comparison of Design Steps for Maximally Flat and Chebyshev Cases

Step Maximally Flat Chebyshev Find n log 100.1훼푚푖푛 − 1 / 100.1훼푚푎푥 − 1 cosh−1 100.1훼푚푖푛 − 1 100.1훼푚푎푥 − 1 0.5 푛 = 푛 = −1 2 log 휔푠 2 cosh 휔푠 Round up to an integer Round up to an integer Find ε 100.1훼푚푎푥 − 1 1 2 100.1훼푚푎푥 − 1 1 2 180° Find pole If 푛 is odd 휃 = 0°, ±푘 Find 휃푘 as in Butterworth case 푘 푛 180° Find locations If 푛 is even 휃 = ±푘 푘 2푛 1 1 훼 = sinh−1 푛 휀 −1 푛 Radius = Ω0 = 휀 휔푝 Then, −휎 = Ω cos 휃 푘 0 푘 −휎푘 = sin 휃푘 sinh 훼 ±휔 = Ω sin 휃 푘 0 푘 ±휔푘 = cos 휃푘 cosh(훼)

17 Inverse Chebyshev Approximation (Chebyshev Type II) This type has • Steeper transition compared to Butterworth filters (but not as steep as type I) • No passband ripples |H(jω)|2 • Equal ripples in the stopband 1 Magnitude response of this type is given by 2 1 1/(1+ε-2) 퐻 푗휔 = ω 1+ 퐾 푗휔 2 -1 1 2 |K(jω)| 1 ∞ 퐾 푗휔 = 1 휀퐶푛 휔 ε-2 ω -1 1 1 −1 1 1 퐶 = cos 푛 cos ≤ 1 ε|K(jω)| 푛 휔 휔 휔 ∞ 휔 ≥ 1 1 Going back and forth in ±1 range for ω 0 휔 ≥ 1 -1 1 -1 푪 is Chebyshev’s polynomial 풏 ∞ Defining the stopband area of 푲 풋흎 Inverse Chebyshev Approximation (Chebyshev Type II) For the passband (휔 < 1)

1 1 퐶 = cosh 푛 cosh−1 for 휔 < 1 푛 휔 휔 1 1 푛 퐶 ≅ 2푛−1 for 휔 ≪ 1 푛 휔 휔 Attenuation 훼 1 훼 = 10 log 1 + 1 푑퐵 휀2퐶2 푛 휔 1 1 훼 = 10 log 1 + 훼 = 10 log 1 + 푚푎푥 2 2 1 푚푖푛 휀2 휀 퐶푛 휔푝

To find the required order for a certain filtering template 훼 10 1 2 2 1 −1 1 10 푚푖푛 −1 퐶푛 = cosh 푛 cosh = 훼 10 휔푝 휔푝 10 푚푎푥 −1

1 2 cosh−1 10훼푚푖푛 10 − 1 10훼푚푎푥 10 − 1 푛 = 1 cosh−1 19 휔푝 Pole/zero Locations (Inverse Chebyshev)

Pole/zero locations 2 2 1 1 휀 퐶푛 휔 퐻 푗휔 2 = = 1 2 2 1 1 + 휀−2퐶−2 1 + 휀 퐶푛 휔 푛 휔 We have imaginary zeros at ±휔푧,푘 where 2 1 퐶푛 = 0 휔푧,푘 푘휋 휔 = sec , 푘 = 1,3,5, … , 푛 푧,푘 2푛 If 푠푘 = 휎푘 + 푗휔푘 are the poles of Chebyshev filter

Then, 1 푝푘 = 훼푘 + 푗훽푘 = are the poles of inverse Chebyshev filter 푠푘

Magnitude and quality factor of imaginary poles 1 푝푘 = 푄푖퐶ℎ푒푏 = 푄퐶ℎ푒푏 푠푘 20 Pole/zero Locations (Inverse Chebyshev)

Poles of Chebyshev and inverse Chebyshev filters are reciprocal

Since the poles are on the radial line, they have the same pole Q

Imaginary zeros creates nulls in the stopband

n=2 n=3 n=4 n=5 21 Magnitude Response (Inverse Chebyshev)

22 Elliptic Filter Approximation

Elliptic filter Hjw • Equal ripple passband and stopband

• Nulls in the stopband w

• Sharpest transition band compared I to same-order Butterworth and m Chebyshev (Type I and II)

Ellipse Pole/zero Locations (Elliptic)

Imaginary zeros creates nulls in the stopband

n=2 n=3 n=4 n=5

24 Magnitude Response (Elliptic)

25 Design Example

Design a lowpass filter with: • 휔푝 = 1 푅푝 = 1 푑퐵 • 휔푠 = 1.5 푅푠 = 40 푑퐵

Matlab function buttord, cheb1ord, cheb2ord, and ellipord are used to find the least order filters that meet the given specs. 26 Design Example

Magnitude response

Filter approximation meeting the same specification yield Order (Butterworth)> Order (Chebyshev)> Order (Elliptic) 27 Design Example

Pole/zero locations

Note that Chebyshev and Eliptic approximations needs high-Q poles 28 Design example

Filter order for 푅푝 = 1 and 푅푠 = 40

Eliptic filter always yields the least order. Is it always the best choice ? 29 Step response of the design example

• Inverse Chebyshev filter has the least overshoot and ringing

• Ringing and overshoots can be problematic in some applications

• The pulse deformation is due to the fact that the filter introduces different time delay to the different frequency components (Phase distortion) 30 Phase Distortion

• Consider a filter with a transfer function 퐻 푗휔 = 퐻 푗휔 푒푗휙 휔 • Let us apply two sine waves at different frequencies 푣푖푛 푡 = 퐴1 sin 휔1푡 + 퐴2 sin 휔2푡 • The filter output is 휙1 휙2 푣표푢푡 푡 = 퐴1 퐻 푗휔1 sin 휔1 푡 + + 퐴2 퐻 푗휔2 sin 휔2 푡 + 휔1 휔2

• Assuming that the difference between 퐻 푗휔1 and 퐻 푗휔2 is small, the shape of the time domain output signal will be preserved if the two signals are delays by the same amount of time 휙 휔 휙 휔 1 = 2 휔1 휔2 • This condition is satisfied for

휙 휔 = 푡0휔 푡0 = 푐표푛푠푡푎푛푡

• A filter with this characteristic is called “linear phase” 31 Linear Phase Filters

• For this type of filters: The magnitude of the signals is scaled equally, and they are delayed by the same amount of time v (t)  Kv (t  t ) out in 0

Filter Vout Vin  jw(t0 ) Vout(s)  Vin (s)Ke 

• The filter transfer function is 퐻 푠 = 퐾푒−푗휔푡0 퐻 푗휔 = 퐾 휙 푗휔 = 푡0휔

휙 푗휔 • In this types of filters the phase delay 휏 = − , and the group delay 휏 = 푃퐷 휔 퐺퐷 푑휙 푗휔 − are constant and equal 휔

32 Linear Phase Filter Approximation

• For a typical lowpass filter 퐾 퐾 퐻 푠 = 2 = 2 3 1 + 푎1푠 + 푎2푠 + ⋯ 1 − 푎2휔 + ⋯ + 푗휔 푎1 − 푎3휔 + ⋯

Thus the phase shift is given by 3 −1 푎1 − 푎3휔 + ⋯ 휙 휔 = arg 퐻 푗휔 = − tan 2 1 − 푎2휔 + ⋯

푥3 푥5 • Using power-series expansion tan−1 푥 = 푥 − + − ⋯ 3 5 The condition for linear phase is satisfied if

휕 휕 푥3 푥5 tan−1 푥 = 푥 − + − ⋯ = 푐표푛푠푡푎푛푡 휕휔 휕휔 3 5

These are called Bessel polynomials, and the resulting networks are called Thomson filters 33 Linear Phase Filter Approximation

• Typically the phase behavior of these filters shows some deviation

휏퐺퐷 Ideal response • Errors are measured in time t0 • 휏퐺퐷 is the Group Delay 휕휙 휏 = − Actual response 퐺퐷 휕휔

34 Bessel (Thomson) Filter Approximation

• All poles

• Poles are relatively low Q

• Maximally flat group delay (Maximally linear phase response)

• Poor stopband attenuation

http://www.rfcafe.com/references/electrical/bessel-poles.htm

35 Bessel Filter Approximation

36 Comparison of various LPF Group delay

Ref: A. Zverev, Handbook of filter synthesis, Wiley, 1967.

37 Filter Design Conventional Procedure

Transform your filter specs into a normalized LPF

Filter order, zeros, poles and/or values for the passive elements can be obtained from tables or from a software package like FIESTA or Matlab

If you use biquadratic sections, you need poles and zeros matching

For ladder filters, the networks can be obtained from tables

Transform the normalized transfer function to your filter by using Filter transformation (LP to BP, HP, BR) Frequency transformation Impedance denormalization

You obtain the transfer function or your passive network

38 Frequency Transformations

Lowpass to Highpass n 1 H0 H0 H0p s  then Hlp (s)   Hhp (p)   p n n i n a si  1  a pni i ai   i i0 i0  p  i0

H(s) H(p)

w 0.66 1 1.5 2 0.5 1 1.5 2 p  2 1 0.66 0.5

39  Lowpass to Highpass H H pn H (s)  0  H (p)  0 lp n hp n i ni ais aip i0 i0

N zeros at  are translated to zero Poles are not the same!!! The main characteristics of the lowpass filter are maintained

for a highpass filter with cutoff frequency at w0, then

w0 H(p) s  This transformation scheme p translates w=1 to p=w0 p 0.66 1 1.5 2

p/w0 1 40 Frequency Transformations

. The Lowpass to Highpass transformation can also be applied to the elements

Element impedance transforme d Resistor R R  The resistors are not affected w L inductor sL 0 p  L is transformed in a capacitor Ceq=1/w0L 1 p capacitor sC w C 0  C is transformed in an inductor Leq=1/w0C

Example: Design a 1KHz HP-filter from a LP prototype.

1.4142 -4 1 1 1.12x10

1.4142 1 1.12x10-4 1

41 Frequency Transformations

Lowpass to Bandpass transformation

n p2 1 H H p s  then H (s)  0  H (p)  0 lp n bp 2n p i i ais bi p i0 i0 • n zeros at w=0 and n zeros at  •even number of poles •The bandwidth of the BP is equal to the bandwidth of the LP

•In the p-domain 2 2 s  s  w  w  p  jv    1 or v     1 2  2  2  2 

v1  0.5 1.25 v2  0.5 1.25

w v -1 0 1 -1 0 1 Bandwidth=1

42 Frequency Transformations

Note that v2 v1 1 and v2 v1 1

w v -1 1 v1 v2

Lowpass prototype Bandpass filter

43 Frequency Transformations

2 2 General transformation 1  p  w  s   0  BW p    2 BW  BW  2 1  v1       w0 2 2  2  BWw  BWw  2 v     w0 0  w0 2  2  2 BW  BW  2 1  v2       w0 2  2  BW

w v -1 1 v1 w0 v2 Lowpass prototype Bandpass filter

44 Frequency Transformations

The Lowpass to Bandpass transformation can also be applied to the elements

Element impedance transforme d L BW 2 Resistor R R BW Lw0 L w 2L 1 inductor sL p  0 BW BW p BW 1 1 2 capacitor Cw0 sC C w 2C 1 p  0 BW BW p

C BW  Note that for w=w0

 for the inductor Zeq=0  for the capacitor Yeq=0 (Zeq=)

45 Frequency Transformations

• In general, for double-resistance terminated ladder filters

 around w=w0 Rin Lossless

ladder R 1 H s  L   ww 0 R  R R L RL in L 1 Rin

 In the passband, the transfer function can be very well controlled

Low-sensitivy

46 Frequency Transformations

 Lowpass to Bandreject transformation

1 s   2 2  1  p  w0    BW p 

• Lowpass to Highpass transformation (notch at w=0)

•Shifting the frequency to w0 and adjusting the bandwidth to BW . Bandpass transformation!!! BW

w p 1 1 w0

47 Transformation Methods • Transformation methods have been developed where a low pass filter can be converted to another type of filter by simply transforming the complex variable s.

• Matlab lp2lp, lp2hp, lp2bp, and lp2bs functions can be used to transform a low pass filter with normalized cutoff frequency, to another low-pass filter with any other specified frequency, or to a high pass filter, or to a band-pass filter, or to a band elimination filter, respectively.

48 LPF with normalized cutoff frequency, to another LPF with any other specified frequency • Use the MATLAB buttap and lp2lp functions to find the transfer function of a third-order Butterworth low-pass filter with cutoff frequency fc=2kHz.

% Design 3 pole Butterworth low-pass filter (wcn=1 rad/s) [z,p,k]=buttap(3); [b,a]=zp2tf(z,p,k); % Compute num, den coefficients of this filter (wcn=1rad/s) f=1000:1500/50:10000; % Define frequency range to plot w=2*pi*f; % Convert to rads/sec fc=2000; % Define actual cutoff frequency at 2 KHz wc=2*pi*fc; % Convert desired cutoff frequency to rads/sec [bn,an]=lp2lp(b,a,wc); % Compute num, den of filter with fc = 2 kHz Gsn=freqs(bn,an,w); % Compute transfer function of filter with fc = 2 kHz semilogx(w,abs(Gsn)); grid; xlabel('Radian Frequency w (rad/sec)') ylabel('Magnitude of Transfer Function') title('3-pole Butterworth low-pass filter with fc=2 kHz or wc = 12.57 kr/s')

49 LPF with normalized cutoff frequency, to another LPF with any other specified frequency 3-pole Butterworth low-pass filter with fc=2 kHz or wc = 12.57 kr/s 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3 Magnitude of FunctionTransfer Magnitude 0.2

0.1

0 3 4 5 10 10 10 Radian Frequency w (rad/sec) 50 High-Pass Filter • Use the MATLAB commands cheb1ap and lp2hp to find the transfer function of a 3-pole Chebyshev high-pass analog filter with cutoff frequency fc = 5KHz.

% Design 3 pole Type 1 Chebyshev low-pass filter, wcn=1 rad/s [z,p,k]=cheb1ap(3,3); [b,a]=zp2tf(z,p,k); % Compute num, den coef. with wcn=1 rad/s f=1000:100:100000; % Define frequency range to plot fc=5000; % Define actual cutoff frequency at 5 KHz wc=2*pi*fc; % Convert desired cutoff frequency to rads/sec [bn,an]=lp2hp(b,a,wc); % Compute num, den of high-pass filter with fc =5KHz Gsn=freqs(bn,an,2*pi*f); % Compute and plot transfer function of filter with fc = 5 KHz semilogx(f,abs(Gsn)); grid; xlabel('Frequency (Hz)'); ylabel('Magnitude of Transfer Function') title('3-pole Type 1 Chebyshev high-pass filter with fc=5 KHz ')

51 High-Pass Filter 3-pole Chebyshev high-pass filter with fc=5 KHz 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3 Magnitude of FunctionTransfer Magnitude 0.2

0.1

0 3 4 5 10 10 10 Frequency (Hz)

52 Band-Pass Filter • Use the MATLAB functions buttap and lp2bp to find the transfer function of a 3-pole Butterworth analog band-pass filter with the pass band frequency centered at fo = 4kHz , and bandwidth BW =2KHz.

[z,p,k]=buttap(3); % Design 3 pole Butterworth low-pass filter with wcn=1 rad/s [b,a]=zp2tf(z,p,k); % Compute numerator and denominator coefficients for wcn=1 rad/s f=100:100:100000; % Define frequency range to plot f0=4000; % Define centered frequency at 4 KHz W0=2*pi*f0; % Convert desired centered frequency to rads/s fbw=2000; % Define bandwidth Bw=2*pi*fbw; % Convert desired bandwidth to rads/s [bn,an]=lp2bp(b,a,W0,Bw); % Compute num, den of band-pass filter % Compute and plot the magnitude of the transfer function of the band-pass filter Gsn=freqs(bn,an,2*pi*f); semilogx(f,abs(Gsn)); grid; xlabel('Frequency f (Hz)'); ylabel('Magnitude of Transfer Function'); title('3-pole Butterworth band-pass filter with f0 = 4 KHz, BW = 2KHz') 53 Band-Pass Filter

3-pole Butterworth band-pass filter with f0 = 4 KHz, BW = 2KHz 1.4

1.2

0.8

0.6

0.4 Magnitude of FunctionTransfer Magnitude

0.2

0 2 3 4 5 10 10 10 10 Frequency f (Hz) 54 Band-Elimination (band-stop) Filter • Use the MATLAB functions buttap and lp2bs to find the transfer function of a 3-pole Butterworth band-elimination (band-stop) filter with the stop band frequency centered at fo = 5 kHz , and bandwidth BW = 2kHz.

[z,p,k]=buttap(3); % Design 3-pole Butterworth low-pass filter, wcn = 1 r/s [b,a]=zp2tf(z,p,k); % Compute num, den coefficients of this filter, wcn=1 r/s f=100:100:100000; % Define frequency range to plot f0=5000; % Define centered frequency at 5 kHz W0=2*pi*f0; % Convert centered frequency to r/s fbw=2000; % Define bandwidth Bw=2*pi*fbw; % Convert bandwidth to r/s % Compute numerator and denominator coefficients of desired band stop filter [bn,an]=lp2bs(b,a,W0,Bw); % Compute and plot magnitude of the transfer function of the band stop filter Gsn=freqs(bn,an,2*pi*f); semilogx(f,abs(Gsn)); grid; xlabel('Frequency in Hz'); ylabel('Magnitude of Transfer Function'); title('3-pole Butterworth band-elimination filter with f0=5 KHz, BW = 2 KHz') 55 Band-Elimination (band-stop) Filter 3-pole Butterworth band-elimination filter with f0=5 KHz, BW = 2 KHz 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3 Magnitude of FunctionTransfer Magnitude 0.2

0.1

0 2 3 4 5 10 10 10 10 Frequency in Hz 56 How to find the minimum order to meet the filter specifications ? The following functions in Matlab can help you to find the minimum order required to meet the filter specifications:

• Buttord for butterworth • Cheb1ord for chebyshev • Ellipord for elliptic • Cheb2ord for inverse chebyshev

57 Calculating the order and cutoff frequency of a inverse chebyshev filter • Design a 4MHz Inverse Chebyshev approximation with Ap gain at passband corner. The stop band is 5.75MHz with -50dB gain at stop band.

clear all; Fp = 4e6; Wp=2*pi*Fp; Fs=1.4375*Fp; Ws=2*pi*Fs; Fplot = 20*Fs; f = 1e6:Fplot/2e3:Fplot ; w = 2*pi*f; Ap = 1; As = 50; % Cheb2ord helps you find the order and wn (n and Wn) that %you can pass to cheby2 command. [n, Wn] = cheb2ord(Wp, Ws, Ap, As, 's'); [z, p, k] = cheby2(n, As, Wn, 'low', 's'); [num, den] = cheby2(n, As, Wn, 'low', 's'); bode(num, den) 58 Bode Plot

Bode Diagram 0

-50

-100

Magnitude (dB) Magnitude -150

-200 1260

1080

900

Phase (deg) Phase 720

540 5 6 7 8 9 10 10 10 10 10 10 10 Frequency (rad/sec)

59 References [1] S. T. Karris, “Signals and Systems with Matlab Computing and Simulink Modeling,” Fifth Edition. Orchard Publications [2] Matlab Help Files

60 Ladder Filters

.The ladder filter realization can be found in tables and/or can be obtained from FIESTA

.The elements must be transformed according to the frequency and impedance normalizations 0

-40

)

d (

-80

a g

-120

-160 4.0E+6 8.0E+6 1.2E+7 1.6E+7 2.0E+7 Frequency (Hz) 61 Sensitivity

y x y  Definition S  Y= transfer function and x = variable or element x y x

Some properties: ky y y x x Sx Skx Sx Skx Sx 1 1/ y y y yn y Sx S1/ x  Sx Sx  nSx y 1 y y y x2 S  Sx Sx S Sx xn n x2 n n n yi yi n yi  yiSx Si1  S yi Si1  i1 x  x x n i1  yi i1 For a typical H(s) H i H(s) 0 i ai jw ai jw S j a jwi a j a jjw S  i     a j i i ai jw ai jw

62 Sensitivity

• Sensitivity is a measure of the change in the performance of the system due to a change in the nominal value of a certain element.

y x y y x Dy Dy y Dx Sx  Sx  or  Sx  y x y Dx y x

Normalized variations at the output are determined by the sensitivity function and the normalized variations of the parameter Example: If the senstivity function is 10, then variations of Dx/x=0.01(1%) produce Dy/y=0.1(10%)

For a good design, the sensitivity functions should be < 5. Effects of the partial positive feedback (negative resistors)?

63 Sensitivity

gm . For a typical amplifier Av  g0

SAv 1, SAv  1 gm g0

•Sometimes the dc gain is enhanced by using a negative resistor

gm gm 1 Av   g g g g For large dc gain g02=g0 0 02 0 1 02 g0 g0 g0Sg 1 SAv 1, SAv   0   gm g0 g g g 0 02 1 02 g0

The larger the gain improvment the larger the sensitivity!!!!

64 Properties of Stable Network Functions

• Typically the transfer function A(s) 1b sb s 2 .. presents the form of a ratio of two N(s) K 1 2 polynomials 2 B(s) 1a1sa 2s .. • For non-negative elements the coefficients are real and positive • The poles are located in the left side of the s-plane

1 H(s) • System is stable sa • BOUNDED OUTPUT FOR BOUNDED INPUT h(t)e at t0

65 Properties of Network Functions: Frequency Transformation

 Most of the filter approximations are normalized to 1 rad/sec. Hence, it is necessary to denormalize the transfer function.

 Using the frequency transformation p w For inductors and  L   n jp L jw   capacitors:     n   1 rad/sec is translated to 1/  C   n rad/sec jp C jw      n 

66 Impedance denormalization

. Typically the network elements are normalized to 1 . Hence an impedance denormalization scheme must be used

Z  n Z . or

R   n R

L   L n C  C  n . Note that the transfer function is invariant with the impedance denormalization (RC and LC products remain constant!!!!)

. In general both frequency and impedance denormalizations are used

• There is a number of conventional filter magnitude approximations

• The choice of a particular approximation is application dependent

• Besides the magnitude specifications, there exists also a phase (group delay) specification. For this the Thompson (Bessel) approximation is used

• There are a host of Filter approximation software programs, including Matlab, Filsyn, and Fiesta2 developed at TAMU

Acknowledgment: Thanks to my colleague Dr. Silva-Martínez for providing some of the material for this presentation