IUST-EE Chapter 7: IIR Filter Design Techniques Contents Performance Specifications Pole-Zero Placement Method Impulse Invariant Method Bilinear Transformation Classical Analog Filters DSP-Shokouhi
Advantages (Digital vs. Analog Filters) Unique Characteristics Independent to Environment Automatic Adjustable Response Multi Inputs or Channels VLSI Technology (Low Power & Low Cost) High Precision Repeatability Wide Range of Frequency (from DC) DSP-Shokouhi 3 Disadvantages (Digital vs. Analog Filters) Speed Limitation Finite Wordlength Effect Long Design Development Time DSP-Shokouhi 4
Digital Filter Design Several design procedures are available for computing IIR and FIR filter coefficients. IIR Pole-zero placement Impulse invariance Bilinear transformation FIR Window Optimal Frequency Sampling DSP-Shokouhi 5 Filter Design Steps Recalculation DSP-Shokouhi 6
Filter Specifications 1 or p :Passband deviation Ap As or s f p :Stopband deviation :Passband edge frequency 0log :Stopband attenuation s 0log(1 p ):Passband ripple F :Transition band f s : Stopband edge frequency DSP-Shokouhi 7 Filter Specifications DSP-Shokouhi 8
Filter Specifications DSP-Shokouhi 9 IIR Filter Design Methods Pole-Zero Placement Method The poles and zeroes of the transfer function are placed in such a way to obtain the desired frequency response. Impulse Invariant Method The digital filter is designed using the well known analog filter design techniques and sampling from the analog impulse response. Bilinear z-transform The digital filter is designed using the well known analog filter design techniques and an algebraic DSP-Shokouhi transformation between the variables s and z. 10
Pole-Zero Placement Method Example 1: Design a Bandpass filter with the following specifications: Sampling frequency is 500 Hz Complete signals rejection are at dc and 50 Hz Pass band center is at 15 Hz 3 db bandwidth at 15 Hz is 10 Hz DSP-Shokouhi 11 Pole-Zero Placement Method Solution: Zeros Poles f z z 1 f s 0 z f H ( z ) p1, f s rp 1 0.937 Q z 1 / z 0.877969 DSP-Shokouhi 1
Pole-Zero Placement Method DSP-Shokouhi 13 Pole-Zero Placement Method Example : Bandstop filter Design a Bandstop filter with the following specifications Notch frequency is at 50 Hz 3 db width of notch is 5 Hz Sampling frequency is 500 Hz DSP-Shokouhi 14
Pole-Zero Placement Method Solution f z z 1 f 5 s Q 10 z 1.618z 1 H ( z ) p z z 1.5663z 0.9370 rp 1 0.968 Q DSP-Shokouhi 15 Pole-Zero Placement Method DSP-Shokouhi 16
Design of Discrete-time IIR Filters From Continuous-time Filters DSP-Shokouhi 17 Impulse Invariant Method A method for obtaining discrete time system from frequency response of a continuous time system. The impulse response of the continuous time filter is sampled to achieve discrete time filter. h n] T h ( nt ) [ d c d The discrete & continuous time frequency responses are related linearly. DSP-Shokouhi 18
Impulse Invariant Method DSP-Shokouhi 19 Impulse Invariant Method DSP-Shokouhi 0
Impulse Invariant Method DSP-Shokouhi 1 Impulse Invariant Method DSP-Shokouhi
Impulse Invariance with a Butterworth Filter (1) () (3) DSP-Shokouhi 3 Impulse Invariance with a Butterworth Filter (3) (4) DSP-Shokouhi 4
Impulse Invariance with a Butterworth Filter (4) DSP-Shokouhi 5 Impulse Invariance with a Butterworth Filter (5) (6) DSP-Shokouhi 6
Impulse Invariant Method Example: Find the discrete equivalent system of the following transfer function: DSP-Shokouhi 7 f c =150Hz, f s =1.8kHz Impulse Invariant Method Example: N k N k s T k d k k c z e A T z H s s A s H k d 1 1 1 1 ) ( ) ( 1 1 1 1 0.3530 1.0308 1 0.3078 ) (, z z z z H j A j A j s j s p p DSP-Shokouhi 8
Impulse Invariant Method Example: x[n] y[n] DSP-Shokouhi 9 Bilinear Transformation An Algebraic transformation between variables s and z Maps onto Avoids the problem of aliasing A nonlinear transformation that causes frequency warping s T d 1 z 1 z 1 1 DSP-Shokouhi 30
Bilinear Transformation DSP-Shokouhi 31 Bilinear Transformation DSP-Shokouhi 3
DSP-Shokouhi 33 DSP-Shokouhi 34
DSP-Shokouhi 35 Bilinear Transformation The frequency axis becomes nonlinear. Higher frequencies become more compressed. DSP-Shokouhi 36
Bilinear Transformation of a Butterworth Filter (1) () DSP-Shokouhi 37 Bilinear Transformation of a Butterworth Filter (3) DSP-Shokouhi 38
Bilinear Transformation of a Butterworth Filter (4) DSP-Shokouhi 39 Bilinear Transformation of a Butterworth Filter (5) DSP-Shokouhi 40
Bilinear Transformation of a Butterworth Filter (6) DSP-Shokouhi 41 Bilinear Transformation- Example Example f c =150Hz, f s =1.8kHz DSP-Shokouhi 4
BZT- Example Solution: 1 H( s ) 1 RCs 1 1 z s ( 1 ) BZT T 1 z d T, d 0.408 z H(z) 1 0.1584z -1 1 DSP-Shokouhi 43 Continuous-Time Filters DSP-Shokouhi 44
DSP-Shokouhi 45 DSP-Shokouhi 46
DSP-Shokouhi 47 DSP-Shokouhi 48
DSP-Shokouhi 49 Designing Other Filter Types from Lowpass Filter DSP-Shokouhi 50
FREQUENCY TRANSFORMATIONS OF LOWPASS IIR FILTERS Tolerance scheme for a multiband filter DSP-Shokouhi 51 DSP-Shokouhi 5