1. Professor A G Constantinides 1 Digital Filter Specifications We discuss in this course only the magnitude approximation problem There are four basic types of ideal filters with magnitude responses as shown below

2. Professor A G Constantinides 2 Digital Filter Specifications These filters are unealizable because their impulse responses infinitely long non- causal In practice the magnitude response specifications of a digital filter in the passband and in the stopband are given with some acceptable tolerances In addition, a transition band is specified between the passband and stopband

3. Professor A G Constantinides 3 Digital Filter Specifications For example the magnitude response of a digital lowpass filter may be given as indicated below

4. Professor A G Constantinides 4 Digital Filter Specifications In the passband we require that with a deviation In the stopband we require that with a deviation

5. Professor A G Constantinides 5 Digital Filter Specifications Filter specification parameters - passband edge frequency - stopband edge frequency - peak ripple value in the passband - peak ripple value in the stopband

6. Professor A G Constantinides 6 Digital Filter Specifications Practical specifications are often given in terms of loss function (in dB) Peak passband ripple dB Minimum stopband attenuation dB

7. Professor A G Constantinides 7 Digital Filter Specifications In practice, passband edge frequency and stopband edge frequency are specified in Hz For digital filter design, normalized bandedge frequencies need to be computed from specifications in Hz using

8. Professor A G Constantinides 8 Digital Filter Specifications Example - Let kHz, kHz, and kHz Then

9. Professor A G Constantinides 9 The transfer function H(z) meeting the specifications must be a causal transfer function For IIR real digital filter the transfer function is a real rational function of H(z) must be stable and of lowest order N for reduced computational complexity Selection of Filter Type

10. Professor A G Constantinides 10 Selection of Filter Type For FIR real digital filter the transfer function is a polynomial in with real coefficients For reduced computational complexity, degree N of H(z) must be as small as possible If a linear phase is desired, the filter coefficients must satisfy the constraint:

11. Professor A G Constantinides 11 Selection of Filter Type Advantages in using an FIR filter - (1) Can be designed with exact linear phase, (2) Filter structure always stable with quantised coefficients Disadvantages in using an FIR filter - Order of an FIR filter, in most cases, is considerably higher than the order of an equivalent IIR filter meeting the same specifications, and FIR filter has thus higher computational complexity

12. Professor A G Constantinides 12 FIR Design FIR Digital Filter Design Three commonly used approaches to FIR filter design - (1) Windowed Fourier series approach (2) Frequency sampling approach (3) Computer-based optimization methods

13. Professor A G Constantinides 13 Finite Impulse Response Filters The transfer function is given by The length of Impulse Response is N All poles are at. Zeros can be placed anywhere on the z- plane

14. Professor A G Constantinides 14 FIR: Linear phase Linear Phase: The impulse response is required to be so that for N even:

15. Professor A G Constantinides 15 FIR: Linear phase for N odd: I) On we have for N even, and +ve sign

16. Professor A G Constantinides 16 FIR: Linear phase II) While for –ve sign [Note: antisymmetric case adds rads to phase, with discontinuity at ] III) For N odd with +ve sign

17. Professor A G Constantinides 17 FIR: Linear phase IV) While with a –ve sign [Notice that for the antisymmetric case to have linear phase we require The phase discontinuity is as for N even]

18. Professor A G Constantinides 18 FIR: Linear phase The cases most commonly used in filter design are (I) and (III), for which the amplitude characteristic can be written as a polynomial in

19. Professor A G Constantinides 19 FIR: Linear phase For phase linearity the FIR transfer function must have zeros outside the unit circle

20. Professor A G Constantinides 20 FIR: Linear phase To develop expression for phase response set transfer function In factored form Where, is real & zeros occur in conjugates

21. Professor A G Constantinides 21 FIR: Linear phase Let where Thus

22. Professor A G Constantinides 22 FIR: Linear phase Expand in a Laurent Series convergent within the unit circle To do so modify the second sum as

23. Professor A G Constantinides 23 FIR: Linear phase So that Thus where

24. Professor A G Constantinides 24 FIR: Linear phase are the root moments of the minimum phase component are the inverse root moments of the maximum phase component Now on the unit circle we have and

25. Professor A G Constantinides 25 Fundamental Relationships hence (note Fourier form)

26. Professor A G Constantinides 26 FIR: Linear phase Thus for linear phase the second term in the fundamental phase relationship must be identically zero for all index values. Hence 1) the maximum phase factor has zeros which are the inverses of the those of the minimum phase factor 2) the phase response is linear with group delay equal to the number of zeros outside the unit circle

27. Professor A G Constantinides 27 FIR: Linear phase It follows that zeros of linear phase FIR trasfer functions not on the circumference of the unit circle occur in the form

28. Professor A G Constantinides 28 Design of FIR filters: Windows (i) Start with ideal infinite duration (ii) Truncate to finite length. (This produces unwanted ripples increasing in height near discontinuity.) (iii) Modify to Weight w(n) is the window

29. Professor A G Constantinides 29 Windows Commonly used windows Rectangular 1 Bartlett Hann Hamming Blackman Kaiser

30. Professor A G Constantinides 30 Kaiser window βTransition width (Hz) Min. stop attn dB 2.121.5/N30 4.542.9/N50 6.764.3/N70 8.965.7/N90

31. Professor A G Constantinides 31 Example Lowpass filter of length 51 and

32. Professor A G Constantinides 32 Frequency Sampling Method In this approach we are given and need to find This is an interpolation problem and the solution is given in the DFT part of the course It has similar problems to the windowing approach

33. Professor A G Constantinides 33 Linear-Phase FIR Filter Design by Optimisation Amplitude response for all 4 types of linear- phase FIR filters can be expressed as where

34. Professor A G Constantinides 34 Linear-Phase FIR Filter Design by Optimisation Modified form of weighted error function where

35. Professor A G Constantinides 35 Linear-Phase FIR Filter Design by Optimisation Optimisation Problem - Determine which minimise the peak absolute value of over the specified frequency bands After has been determined, construct the original and hence h[n]

36. Professor A G Constantinides 36 Linear-Phase FIR Filter Design by Optimisation Solution is obtained via the Alternation Theorem The optimal solution has equiripple behaviour consistent with the total number of available parameters. Parks and McClellan used the Remez algorithm to develop a procedure for designing linear FIR digital filters.

37. Professor A G Constantinides 37 FIR Digital Filter Order Estimation Kaiser’s Formula: ie N is inversely proportional to transition band width and not on transition band location

38. Professor A G Constantinides 38 FIR Digital Filter Order Estimation Hermann-Rabiner-Chan’s Formula: where with

39. Professor A G Constantinides 39 FIR Digital Filter Order Estimation Fred Harris’ guide: where A is the attenuation in dB Then add about 10% to it

40. Professor A G Constantinides 40 FIR Digital Filter Order Estimation Formula valid for For, formula to be used is obtained by interchanging and Both formulae provide only an estimate of the required filter order N If specifications are not met, increase filter order until they are met