Infinite Impulse Response (IIR) Filter. Ikhwannul Kholis, ST., MT. Universitas 17 Agustus 1945 Jakarta

Transcription

1 Infinite Impulse Response (IIR) Filter Ihwannul Kholis, ST., MT. Universitas 17 Agustus 1945 Jaarta

2 The Outline 8.1 State-of-the-art 8.2 Coefficient Calculation Method for IIR Filter Pole-Zero Placement Method Impulse Invariant Method Matched-z-transform (MZT) Method Bilinear z-transform (BZT) Method 8.3 Classical Analog Filter Butterworth Filter Chebysev Filter Elliptic Filter 8.4 Review

3 State of the art The basic IIR filter is characterized by the follo-wing equation : Where h() is the impulse response of the filter which is theoretically infinite in duration M N n x a n x b n y n x h n y ) ( ) ( ) ( ) ( ) ( ) ( 8.1

4 State of the art (cont d) b and a are the coefficients of the filter x(n) and y(n) are the input and output to the filter Transfer function for the IIR filter is : M N M M N N z a z b z a z a a z b b z b z H ) ( 8.1

5 8.1 State of the art (cont d) The important thing is to find suitable values for the coefficients b and a Note that the current output y(n) is a function of the past outputs y(n-). So that it show the feedbac system of some sort The strength of IIR filters comes from the flexibility the feedbac arrangement provides Remember that the transfer function of IIR filter can be shown as the pole and zero equations

6 8.1 State of the art (cont d) Here is an example tolerance scheme for an IIR bandpass filter Figure 8.1

7 8.1 State of the art (cont d) ε 2 : passband ripple parameter δ p : passband deviation δ s : stopband deviation f p1 and f p2 : passband edge frequency f p1 and f p2 : stopband edge frequency A p : passband ripple 10. log 10 (1+ ε 2) 20. log 10 (1- δ p ) A s : stopband attenuation -20. log 10 (δ p )

8 8.2 Coefficient calculation methods for IIR filters There are 4 methods to calculate the coefficients : 1. Pole-zero placement 2. Impulse invariant 3. Matched z-transform 4. Bilinear z-transform Learn carefully

9 8.2.1 Pole-zero placement Method The idea is : when a zero is placed at a given point on the z-plane, the frequency response will be zero at the corresponding point while a pole produces a pea at the corresponding frequency point Note that for the coefficient of the filter to be real, the poles and zeros must either be real Watch the figure 8.2 below

10 8.2.1 Pole-zero placement Method (cont d) Figure 8.2

11 8.2.1 Pole-zero placement Method (cont d) Here is an example to mae a bandpass digital filter which is required to meet the following specifications : - complete signal rejection at dc and 250 Hz - a narrow passband centered at 125 Hz - a 3dB bandwidth of 10 Hz Fist we must determine where to place the poles and zeros on the z-plane. Watch the frequency on 250 Hz and 125 Hz

12 8.2.1 Pole-zero placement Method (cont d) These are at angles of 0 o and 360 o x 250/ o and the place poles at o x 125/ o The radius, r, of the poles is determined by the desired bandwidth. An approximate relationship between r, for r > 0.9 and bandwidth bw is given by : r 1 (bw/fs).π So that, by substituting the value of bw10 Hz and Fs500 Hz, giving r After it, we can draw the pole-zero diagram below :

13 8.2.1 Pole-zero placement Method (cont d) Figure 8.3

14 8.2.1 Pole-zero placement Method (cont d) From the pole-zero diagram, the transfer function can be written as follow : H ( z) ( z ( z re 1)( z )( z jπ / 2 + 1) re jπ / 2 ) 2 z 1 2 z z z 2

15 8.2.1 Pole-zero placement Method (cont d) So that, the difference equation is : y( n) y( n 2) + x( n) x( n 2) Loo at again the transfer function. It shows filter which is a second-order section, with coefficients : b 0 1 a 1 0 b 1 0 a b 2-1

16 8.2.2 Impulse Invariant Method First, consider these component : - H (s) : a suitable analog transfer function - h (t ) : the impulse response - h (nt) : z transforming with T sampling interval - H (z) : desired transfer function Those component are useful and obtained by using Laplace Transform and also z-transformation Loo at the example on DSP textboo

17 8.2.2 Impulse Invariant Method (cont d) Here are the steps in Impulse Invariant Method : 1. Determine a normalized analog filter H(s) that satisfies the specifications for the desired digital filter 2. If necessary, expand H(s) using partial fraction to simplify the next step 3. Obtain the z-transform of each partial fraction to obtain : M CK s p M K 1 K K 11 C e K p T z 1

18 8.2.2 Impulse Invariant Method (cont d) 4. Obtain H(z) by combining the z-transforms of the partial fractions into second-order terms and possibly one-first-order term. If the actual sampling frequency is used then multiply H(z) by T

19 8.2.3 Matched z-transform (MZT) method It provides a simple way to convert an analog filter into an equivalent digital filter The idea is : each of the poles and zeros of the analog filter is mapped directly from the s-plane to the z-plane using the following equation : ( s a) (1 1 at It maps a pole or zero at the location sa in the s- plane, onto a pole or zero in the z-plane at ze at z e )

20 8.2.3 Matched z-transform (MZT) method (cont d) Here is an example for having a filter with a 3 db cutoff frequency of 150 Hz in sampling frequency of 1.28 Hz. The normalized of transfer function of an analog filter is given by : H ( s) s s + 1 To obtain the transfer function, watch the answer below

21 8.2.3 Matched z-transform (MZT) method (cont d) The cutoff frequency may be expressed as ω c 2π x rad/s. The transfer function of the denormalized analog filter is obtained by replacing s by s/ω c : H '( s) H ( s) s s 2 + s ωc 2 c ω 2ω s c + ω 2 c Find poles by abc formula

22 8.2.3 Matched z-transform (MZT) method (cont d) Remember : so that : We have the real and imaginary poles : p p r i b 2 s12 ± b 4ac 2 2ωc ωc j j 2

23 8.2.3 Matched z-transform (MZT) method (cont d) Then, p r T cos (p i T) p i T e prt The transfer function become : H ( z) z z

24 8.2.4 Bilinear z-transform (BZT) Method It is the most important method The idea is: to convert an analog filter H(s) into an equivalent digital filter is to replace s as follow: s z z 1, or 2 T That transformation maps the analog transfer function, H(s), from the s-plane into the discrete transfer function, H(z), in the z-plane

25 8.2.4 BZT Method (cont d) Loo at the figure below. It shows the transforma-tion using BZT method Figure 8.4 S-plane Z-plane

26 8.2.4 BZT Method (cont d) Here are the steps for using BZT 1. Use the digital filter specifications to find suitable normalized, prototype, analog low pass filter H(s) 2. Determine and prewarpe the bandedge or critical frequencies of the desired filter when : ω p ω p specified cutoff frequency prewarped cutoff frequency

27 8.2.4 BZT Method (cont d) Remember that in bandpass and bandstop filter, there are the lower and upper passband edge frequencies or we can say ω p1 and ω p2. ω 3. Denormalize the analog prototype filter by replacing s in the transfer function, H(s), using these following transformation : p ' tan ω pt 2

28 8.2.4 BZT Method (cont d) s s ω ' p lowpass to lowpass s s s ω ' s s s 2 2 p + ω Ws Ws + ω lowpass to highpass lowpass to bandpass lowpass to bandstop

29 8.2.4 BZT Method (cont d) where : ω 4. Apply the BZT to obtain the desired digital filter transfer function, H(z), by replacing s in the frequencyscaled (i.e. denormalized) transfer function, H (s) as follows 2 0 ω' p2 ω' p1 W ω' p2 ω' p1 s z 1 z + 1

30 8.2.4 Example of BZT Method Learn in DSP textboo [Ifeachor and Jervis] pages Its very urgent! Lowpass filter Highpass filter Bandpass filter

31 8.3 Classical Analog Filter There are four types of Classical Analog filter : 1. Butterworth filter 2. Chebysev type I 3. Chebysev type II 4. Elliptic All types of filter are derived from lowpass prototype filter

32 8.3.1 Butterworth Filter Here is setch of frequency response on Butterworth filter Figure 8.5

33 8.3.1 Butterworth Filter (cont d) The important equations on Butterworth filter are : H ( ω) 2 1 ω 1+ p ω p 2N As log Ap N p ωs 2log p ω p Magnitude square Frequency response Filter order

34 8.3.2 Chebysev Filter Chebysev Type I : equal ripple in the passband, monotonic in the stopband Chebysev Type II : equal ripple in the stopband, N monotonic in the passband As cosh Ap p 1 ωs cosh p ω p

35 8.3.2 Chebysev Filter Here is setch of frequency response on Chebysev Type 1 Type 2 Figure 8.6

36 8.3.3 Elliptic Filter The elliptic filter exhibits equiripple behavior in both the passband and the stopband This is the following magnitude-squared response: H ( 2 K ω') ε ( ω 2 ) G N G N (ω ) is a Chebysev rational function

37 8.3.3 Elliptic Filter Here is setch of frequency response on Elliptics Figure 8.7