1. Unit 9 IIR Filter Design 1

2. Introduction The ideal filter Constant gain of at least unity in the pass band Constant gain of zero in the stop band The gain should increase from zero to the higher gain of the pass band at a single frequency (brick wall profile) Types of filter Low pass High pass Band pass Band stop 2

3. Introduction Types of filter 3

4. Introduction Digital filters can be divided broadly into two types: Finite Impulse Response – FIR Infinite Impulse Response – IIR FIR filters are never unstable and have a linear phase response IIR require fewer coefficients to achieve the same magnitude frequency response. The two types of filter are very different in their performance and design 4

5. IIR Filter Design Methods Direct Design – pole-zero placement Bilinear Transform Method Impulse Invariant Method Pole-zero Mapping Method 5

6. Pole-zero Placement Poles and zeros are placed on the z-plane to try and achieve the required frequency response. Often an element of trial and error is involved Tweaking can be small in trying to fine tune the filter response, tools like Matlab can speed up this process. Once the map is obtained we can then easily Obtain the system transfer function Convert the transfer function into a practical implementation 6

7. Pole-zero Placement Example 1 – Low Pass Filter A Low pass filter is required that has a dc gain of unity and a cut off frequency which is 0.25 the sampling frequency The filter is to have a transfer function of the form: 7

8. Pole-zero Placement Solution 8

9. Pole-zero Placement Example 2 – Band Pass Filter A band pass filter is required to meet the following specification Complete signal rejection at dc and 500Hz A narrow pass band centred at 250Hz A 3dB bandwidth of 20Hz Assume a sampling frequency of 1000Hz a) Find the transfer function using the zero pole placement method b) Find the difference equation c) Propose a diagrammatical solution 9

10. Pole-zero Placement Solution 10

11. Pole-zero Placement Solution 11

12. Pole-zero Placement Class Exercise An analogue signal has a spectrum that extends from dc to 9kHz. It is required to extract a particular frequency component at 2.25kHz. Using pole zero placement, design a band-pass filter that has the centre of its pass-band at 2.25kHz and a zero in its frequency response at dc and at 4.5kHz. Additionally, the filter magnitude response should be unity at 2.25kHz and at 2kHz. 12

13. Pole-zero Placement Class Exercise Solution 13

14. Pole-zero Placement Class Exercise Solution 14

15. IIR Filter Design using Analogue Filter Templates Pole zero placement is adequate for fairly simple designs Complex designs need an alternative approach There is a vast knowledge base of the design of analogue filters Butterworth Chebyshev Mathematical techniques have been developed to convert a filter described in the s-domain to that described in the z-domain These designs result in an approximation to the analogue design but an exact match is impossible…..a digital filter is not expected to operate above the Nyquist frequency Three techniques covered here Bilinear Transform Impulse Invariant Transform Pole-zero matching 15

16. Bilinear Transform The transform: Example: Consider a first order low pass analogue filter having a cut off frequency at 10 rads/s 16

17. Bilinear Transform The transform: Warping Effect 17

18. Bilinear Transform A change in sampling frequency will affect the problem of warping (try increasing the sampling frequency by a factor of 10 and then recalculate w’ for the previous example) Pre-warping only ensures that the gain is matched at the chosen frequency and 0Hz It the design has several break points then a decision has to be made as to which one to preserve It there is a difference of more than 1% between w and w’ then pre- warping is advised 18

19. Bilinear Transform Class Exercise An analogue filter has the transfer function: It is to be converted to its digital equivalent using the bilinear transform. It is necessary for the gain to be preserved at the upper breakpoint frequency of 2 rads s -1 with the sampling frequency of 2 Hz. Decide first if pre-warping is required. Produce the corresponding transfer function for the digital filter & plot the characteristic magnitude/phase responses for this filter. Has the gain been preserved at the required frequency ? 19

20. Impulse Invariant Method The approach in this method is to produce a discrete filter that as the same unit impulse response as the continuous prototype. The output from the digital filter will be a sampled response, but the envelope should match the response of the analogue prototype Consider the previous example: 20

21. Impulse Invariant Method Comparison between the analogue prototype and the digital representation from the IIT method This normalised digital filter gives a comparable magnitude response to the analogue prototype at low frequency but the discrepancy grows as the Nyquist frequency is approached. Ideally the signal should be sampled at a frequency which is much higher than the desired cut-off frequency. In this example, the sampling is at a factor x10 of the desired cut-off frequency i.e. F s =16 Hz with f c =1.6 Hz. 21

22. Impulse Invariant Method Class Exercise An analogue prototype has a transfer function given by: Convert this to its discrete normalised equivalent using the Impulse Invariant method using a sampling frequency of 20 Hz. Produce & compare the impulse responses of the two filters & produce & compare the corresponding frequency responses. 22

23. Pole-Zero Mapping Method This approach uses the relationship to convert the s-plane singularities (poles & zeros) of the CT prototype filter to equivalent z-plane singularities. Once these have been determined, then the corresponding transfer function can be constructed to describe the digital design. All that needs to be done further is to match the gain at some appropriate frequency (often 0 Hz). 23

24. Pole-Zero Mapping Method Consider again the earlier analogue LP example with a transfer function given by along with a sampling frequency of 16 Hz. This function has a single pole at s=-10. The corresponding pole in the z-plane is given by: Using Hence the discrete transfer function is given by: 24