Design of Analog Filters Chapter 4B Part B .Analog Lowpass Filter Specifications ●●● ●●● ●●● Butterworth Approximation Digital Processing of Design of Other Types of Analog Filters Continuous-Time Signals Design of Analog Filters 1.Analog Lowpass Filter 1.Analog Lowpass Filter 1.Analog Lowpass Filter Specifications Specifications Specifications Typical magnitude response of an analog ·In the passband,.defined by0≤2s2。,we --passband edge frequency lowpass filter may be given as indicated require1-8n≤H(j2≤1+6，l2s2 ,--stopband edge frequency below 旧到 i.e..H()approximates unity within an 1+ error of t. .--peak ripple value in the passband --peak ripple value in the stopband 。in the stopband,defined by2,≤2≤o,we require|H.(j2≤d,2,≤Ω≤oi.e,lH.(U2 Peak passband ripple approximates zero within an error of a。=-20logo1-6,)dB Minimum stopband attenuation a,=-20logio()dB
Chapter 4B Digital Processing of Continuous-Time Signals Part B Design of Analog Filters Design of Analog Filters Analog Lowpass Lowpass Filter Specifications Filter Specifications Butterworth Approximation Design of Other Types of Analog Filters Design of Other Types of Analog Filters 1. Analog Lowpass Filter Specifications Typical magnitude response of an analog lowpass filter may be given as indicated below ( ) H j a 0 p s 1 p 1 p s c pass band stop band transition band 1. Analog Lowpass Filter Specifications In the passband, defined by , we require i.e., approximates unity within an error of In the stopband, defined by , we require i.e., approximates zero within an error of 0 p 1 ( )1 , p H j a pp ( ) H j a p s () , H j a ss ( ) H j a s 1. Analog Lowpass Filter Specifications -- passband edge frequency -- stopband edge frequency -- peak ripple value in the passband -- peak ripple value in the stopband Peak passband ripple Minimum stopband attenuation p s p s 10 p p 20log (1 ) dB 10 s s 20log ( ) dB
2.Butterworth Approximation 2.Butterworth Approximation 2.Butterworth Approximation The magnitude-square response of an N-th Typical magnitude responses with =1 order analog lowpass Butterworth Filter is .Gain in dB is G()=10logioH(j)dB given by ●AsG(0)=0 dB and H.0f=1+1 1 G2.)=101og1o(0.5)=-3.0103≈-3dB .First 2N-I derivatives of H(j)at =0 is called the 3-dB cutoff frequency are equal to zero The Butterworth lowpass filter thus is said to have a maximally-flat magnitude at =0 2.Butterworth Approximation 2.Butterworth Approximation 2.Butterworth Approximation Two parameters completely characterizing a According to the definition of the loss Butterworth lowpass filter are and N function ·Suppose 1 They are determined from the specified a=10l0gio 1010-1 H(2 k= bandedges andand minimum 1010-1 passband magnitude H(j )and 。We know that1 maximum stopband magnitude H(,) 1*Q） 。Then H( 2 and Nare thus determined from lg H,n,f=1+,1a严 H,a,f=1+a.1a,严 1+ -10品 2. =10品1+
2. Butterworth Approximation The magnitude-square response of an N-th order analog lowpass Butterworth Filter Butterworth Filter is given by First 2Nˉ1 derivatives of at are equal to zero The Butterworth lowpass filter thus is said to have a maximally-flat magnitude flat magnitude at 2 2 1 ( ) 1( / ) a N c H j 2 ( ) H j a 0 0 2. Butterworth Approximation Gain in dB is As G(0)=0 dB and is called the 3-dB cutoff frequency 10 G( ) 10lo c g (0.5) 3.0103 3 dB c 2 10 G Hj ( ) 10lo g a ( ) dB 2. Butterworth Approximation Typical magnitude responses with 1 c 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 Analog angular frequency in rad/s Magnitude N=2 N=4 0.7071 N=10 2. Butterworth Approximation Two parameters completely characterizing a Butterworth lowpass filter are and N They are determined from the specified bandedges bandedges and , and minimum passband magnitude magnitude , and maximum stopband magnitude magnitude and N are thus determined from c p s ( ) H j a p ( ) H j a s c 2 2 1 ( ) 1( / ) a p N p c H j 2 2 1 ( ) 1( / ) a s N s c H j 2. Butterworth Approximation According to the definition of the loss function We know that 10 2 1 10log H j ( ) 2 2 1 1 ( ) N H j c 2 10 1 10 p N p c 2 10 1 10 s N s c 2. Butterworth Approximation Suppose Then lg lg k N s p 10 10 10 1 10 1 p s k
2.Butterworth Approximation 2.Butterworth Approximation 2.Butterworth Approximation .Since order Nmust be an integer,value Since H(s)H (-s)evaluated at s=j is Hence,the transfer function of an analog obtained is rounded up to the next highest simply equal to H(),it follows that Butterworth lowpass filter has the form of integer 1 H.(s)H(-s)= C H.(s)= n Q This value of Nis used next to determine by satisfying either the stopband edge or the 1+(←s212 D6g+∑dF'T6-m passband edge specification exactly The poles of this expression occur on a circle Where If the stopband edge specification is satisfied, of radius at equally spaced points B=2.es2M,1=l,2,N then the passband edge specification is Because of the stability and causality.the Denominator D(s)is known as the exceeded providing a safety margin transfer function itself will be specified by just Butterworth polynomial of order N the poles in the negative real half-plane ofs 3.Design of Other Types of 3.Design of Other Types of 2.Butterworth Approximation Analog Filters Analog Filters Example- Highpass Filter Lets denote the Laplace transform variable of Determine the lowest order of a Butterworth Stepl Develop of specifications of a prototype analog lowpass filter H(s)and s lowpass filter with a I dB cutoff frequency at prototype analog lowpass filter Hs)from denote the Laplace transform variable of 1 kHz and a minimum attenuation of 40 dB at specifications of desired analog filter Hp(s) desired analog filter Hp(3) 5 kHz 1 using a frequency(spectrum)transformation. The mapping froms-domain to $-domain is 1+(2000x12 Step 2 Design the prototype analog lowpass given by the invertible transformations=F(s) filter 1=-40 Then 1+(10000r/2.)2 Step 3 Determine the transfer function Hps) HD向=Hn(slFo N=logo 10-1 10805=3.2811 of desired analog filter by applying the Hi(s)=Hp()-) 10-1 inverse frequency transformation to H(s)
2. Butterworth Approximation Since order N must be an integer integer, value obtained is rounded up to the next highest integer This value of N is used next to determine by satisfying either the stopband edge or the passband edge specification exactly If the stopband edge specification is satisfied, then the passband edge specification is exceeded providing a safety margin 2. Butterworth Approximation Since evaluated at is simply equal to , it follows that The poles of this expression occur on a circle of radius at equally spaced points Because of the stability and causality, the transfer function itself will be specified by just the poles in the negative real half negative real half-plane of s H sH s a a () ( ) s j 2 ( ) H j a 2 2 1 () ( ) 1 / a a N c H sH s s c 2. Butterworth Approximation Hence, the transfer function of an analog Butterworth lowpass filter has the form of Where Denominator is known as the Butterworth polynomial Butterworth polynomial of order N 1 0 1 ( ) ( ) ( ) N N c c a N l N N N l l l l C H s D s s ds s p [ ( 2 1)/ 2 ], 1, 2, , j Nl N l c p e lN ( ) DN s 2. Butterworth Approximation Example Example – Determine the lowest order of a Butterworth lowpass filter with a 1 dB cutoff frequency at 1 kHz and a minimum attenuation of 40 dB at 5 kHz 10 2 1 10log 1 1 (2000 / ) N c 10 2 1 10log 40 1 (10000 / ) N c 0.1 10 10 4 10 1 1 log log 3.2811 10 1 5 N ! " # 3. Design of Other Types of Analog Filters Highpass Highpass Filter Step1 Develop of specifications specifications of a prototype analog lowpass filter HLP(s) from specifications of desired analog filter HD(s) using a frequency (spectrum) frequency (spectrum) transformation. Step 2 Design the prototype analog lowpass filter Step 3 Determine the transfer function HD(s) of desired analog filter by applying the inverse frequency transformation to HLP(s) 3. Design of Other Types of Analog Filters Let s denote the Laplace transform variable of prototype analog lowpass filter HLP(s) and denote the Laplace transform variable of desired analog filter The mapping from s -domain to -domain is given by the invertible transformation Then s ˆ s ˆ ( )ˆ H s D s Fs ( )ˆ 1 ( )ˆ ˆ ( ) () () ˆ () ()ˆ D LP sFs LP D sF s Hs H s H s Hs
3.Design of Other Types of Analog Filters Spectral Transformation (Lowpass to Highpass) s=22 where is the passband edge frequency of H(s)and is the passband edge frequency of H(s) On the imaginary axis the transformation is 2=-92 Ω
3. Design of Other Types of Analog Filters Spectral Transformation (Lowpass Lowpass to Highpass Highpass) where is the passband edge frequency of dddd and is the passband edge frequency of On the imaginary axis the transformation is ˆ ˆ P P s s P ( ) H s LP ˆ P ( )ˆ H s HP ˆ ˆ P P