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Digital Filter Specifications

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Digital Filter Specifications

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  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 1

  2. Digital Filter Specifications • These filters are unealizable becausetheir 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 2

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

  4. Digital Filter Specifications • In the passband we require that with a deviation • In the stopband we require that with a deviation 4

  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 5

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

  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 7

  8. Digital Filter Specifications • Example - Let kHz, kHz, and kHz • Then 8

  9. Selection of Filter Type • 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 9

  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: 10

  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 11

  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 12

  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 13

  14. FIR: Linear phase • Linear Phase:  The impulse response is required to be • so that for N even: 14

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

  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 16

  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] 17

  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 18

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

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

  21. FIR: Linear phase • Let where • Thus 21

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

  23. FIR: Linear phase • So that • Thus • where 23

  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 24

  25. Fundamental Relationships • hence (note Fourier form) 25

  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 26

  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 27

  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 28

  29. Windows Commonly used windows • Rectangular 1 • Bartlett • Hann • Hamming • Blackman • Kaiser 29

  30. Kaiser window • Kaiser window 30

  31. Example • Lowpass filter of length 51 and 31

  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 32

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

  34. Linear-Phase FIR Filter Design by Optimisation • Modified form of weighted error function where 34

  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] 35

  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. 36

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

  38. FIR Digital Filter Order Estimation • Hermann-Rabiner-Chan’s Formula: where with 38

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

  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 40