Signals and Systems 講義 國立交通大學 電機系 陳永平 編 23. IIR Digital Filter Design A digital filter is an algorithm to convert a sequence of numbers representing an input signal into another sequence of numbers which changes the character of the input signal in some prescribed feature. Here, we will focus on the design of LTI digital filter to behave closely with a reference analog filter, especilly a lowpass filter. An nth-order discretized LTI filter can be described by the following difference equation yk an1 yk 1 a1 yk n 1 a0 yk n (1) bnuk bn1uk 1 b1uk n 1 b0uk n whose impulse response is h[k] and transfer function is given as n n1 k Yz bn z bn1z b1z b0 (2) Hz hkz n n1 k0 Uz z an1z a1z a0 The problem in digital filter design is to determine the set of coefficients ai and bi so that the filter performs the desired behavior. Based on the duration of impulse response h[k], the digital filters can be classified into two types, the infinite impulse response (IIR) and the finite impulse response (FIR). The impulse response h[k] of an IIR filter contains an infinite number of samples and the filter is often realized in a recursive structure. Below shows the structure of an 3-rd order IIR filter. b0 b1 b2 u[k] y[k] 1 1 1 + z z z b3 + a2 a1 a0 From (1), the example of 3-rd order IIR filter can be expressed as the following difference equation yk a2 yk 1 a1 yk n 1 a0 yk n (3) 1/5 Signals and Systems 講義 國立交通大學 電機系 陳永平 編 b3uk b2uk 1 b1uk 2 b0uk 3 Its transfer function is 3 2 b3 z b2 z b1z b0 Hz 3 2 z a2 z a1z a0 hkz k h0 h1z 1 hnz n (4) k0 and clearly h 0, i.e., the impulse response indeed consists of an infinite number of samples. The impulse response h[k] of an FIR filter contains a finite number of samples and the filter is often realized in a nonrecursive structure. Below shows the structure of an 3-rd order FIR filter. b0 b1 b2 u[k] y[k] 1 1 1 z z z b3 + From (1), the example of 3-rd order FIR filter can be expressed as the following difference equation yk b3ukb2uk 1b1uk 2b0uk 3 (5) Its transfer function is 1 2 3 Hz b3 b2 z b1z b0 z (6) h0z0 h1z1 h2z2 h3z3 and clearly hk 0 for k>3, i.e., the impulse response only consists of a finite number of samples. Next, we will discuss the design of the IIR digital lowpass filter based on the bilinear transform method. It is known that the z-transform H(z) and Laplace transform H(s) are related by the condition: z esT (7) which is not an easy work to implement the relation. Instead, the so-called bilinear transform H(p) is employed by introducing a variable p which is defined as 2/5 Signals and Systems 講義 國立交通大學 電機系 陳永平 編 sT sT 1 z 1 1 esT e 2 e 2 sT p C 1 C sT C sT sT Ctanh (8) 1 z 1 e 2 e 2 e 2 In frequency response, let s=j then jT T f p Ctanh jC tan jC tan jC tan v (9) 2 2 2 f0 2 1 f where 2f , f0 and v . Let the imaginary part of p be , then (9) can 2T f0 be written as p jC tan v j (10) 2 i.e., Ctan v (11) 2 which is periodic and implies only the frequency response in the range of 0ff0 or 0v1 is required for filter design. The relationship p,z and s-planes are illustrated in the following figure. A C z-plane s-plane D p-plane j B C B B 1 1 A A C A 0 0 D C D D j B A C From (11), if v is small or f f0 we have tan v v . That means (11) 2 2 can be approximate as f T C v C C (12) 2 2 f0 2 Then, the constant C can be chosen to satisfy and obttained as 3/5 Signals and Systems 講義 國立交通大學 電機系 陳永平 編 2 C 4 f (13) T 0 for small v. In addition, we can also choose C such that =p is a particular frequency of a prototype analog filter and =c is the desired cutoff frequency. Hence, cT C p cot (14) 2 Now, let’s use some examples of lowpass filter design for demonstration. Example Under sampling rate 2kHz and based on the bilinear transformation, derive a 1st order lowpass digital with cutoff frequency 200Hz and it is required that its low frequency response is closed to the analog filter. Sol: Choose the digital filter as below: 1 1 400 Hp p p 1 1 p 400 c 400 where the desired cutoff frequency is 200Hz or c=400 rad. To fit the requirement, we choose C based on (13) for low frequency response, i.e., 2 C 2 2000 4000 T From (8), we have 1 z 1 1 z 1 p C 4000 1 z 1 1 z 1 Hence, the digital filter is designed as 400 400 Hp p 400 1 z 1 1 z 1 4000 400 10 1 1 1 z 1 z 3.1416 1 z 1 0.23911 z 1 101 z 1 3.14161 z 1 1 0.5219z 1 4/5 Signals and Systems 講義 國立交通大學 電機系 陳永平 編 Example Under sampling rate 1kHz and based on the bilinear transformation, derive a lowpass digital filter from the 2nd order butterworth filter with cutoff frequency 100 Hz. Sol: Choose the prototype 2nd order butterworth filter below: 1 Hp 11.414 p p2 where p=1 is the normalized cutoff frequency. Since the desired cutoff frequency is 100Hz or c=200 rad. According to (14), we have cT 200 C p cot cot cot 3.0777 2 2000 10 which from (8) yields 1 z 1 1 z 1 p C 3.0777 1 z 1 1 z 1 Hence, the digital filter is designed as 1 1 Hp 2 2 11.414 p p 1 z 1 1 z 1 11.4143.0777 3.0777 1 1 1 z 1 z 0.0675 z 2 2z 1 z 2 1.143z 0.413 5/5 .