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Chapter 7 Finite Impulse Response(FIR) Filter Design

Chapter 7 Finite Impulse Response(FIR) Filter Design. 1. Features of FIR filter. Characteristic of FIR filter FIR filter is always stable FIR filter can have an exactly linear phase response

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Chapter 7 Finite Impulse Response(FIR) Filter Design

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  1. Chapter 7 Finite Impulse Response(FIR) Filter Design

  2. 1. Features of FIR filter • Characteristic of FIR filter • FIR filter is always stable • FIR filter can have an exactly linear phase response • FIR filter are very simple to implement. Nonrecursive FIR filters suffer less from the effect of finite wordlength than IIR filters

  3. 2. Linear phase response • Phase response of FIR filter • Phase delay and group delay (1) where (2)

  4. Condition of linear phase response (3) (4) Where and is constant Constant group delay and phase delay response

  5. If a filter satisfies the condition given in equation (3) • From equation (1) and (2) thus

  6. It is represented in Fig 7.1 (a),(b)

  7. When the condition given in equation (4) only • The filter will have a constant group delay only • It is represented in Fig 7.1 (c),(d)

  8. Center of symmetry Fig. 7-1.

  9. Table 7.1 A summary of the key point about the four types of linear phase FIR filters

  10. Example 7-1 • Symmetric impulse response for linear phase response. No phase distortion

  11. Frequency response where

  12. (3) where

  13. 3. Zero distribution of FIR filters • Transfer function for FIR filter

  14. Four types of linear phase FIR filters • have zero at is real and is imaginary

  15. If zero on unit circle • If zero not exist on the unit circle • If zeros on

  16. Necessary zero Necessary zero Necessary zero Necessary zero Fig. 7-2.

  17. 4. FIR filter specifications • Filter specifications peak passband deviation (or ripples) stopband deviation passband edge frequency stopband edge frequency sampling frequency

  18. ILPF Satisfies spec’s Fig. 7-3.

  19. Characterization of FIR filter • Most commonly methods for obtaining • Window, optimal and frequency sampling methods

  20. 5. Window method • FIR filter • Frequency response of filter • Corresponding impulse response • Ideal lowpass response

  21. Fig. 7-4.

  22. Truncation to FIR • Rectangular Window

  23. Fig. 7-5.

  24. Fig. 7-6.

  25. Fig. 7-7.

  26. Table 7.2 summary of ideal impulse responses for standard frequency selective filters and are the normalized passband or stopband edge frequencies; N is the length of filter

  27. Common window function types • Hamming window where N is filter length and is normalized transition width

  28. Characteristics of common window functions Fig. 7-8.

  29. Table 7.3 summary of important features of common window functions

  30. Kaiser window where is the zero-order modified Bessel function of the first kind where typically

  31. Kaiser Formulas – for LPF design

  32. Example 7-2 • Obtain coefficients of FIR lowpass using hamming window • Lowpass filter Passband cutoff frequency Transition width Stopband attenuation Sampling frequency

  33. Using Hamming window

  34. Fig. 7-9.

  35. Example 7-3 • Obtain coefficients using Kaiser or Blackman window Stopband attenuation passband attenuation Transition region Sampling frequency Passband cutoff frequency

  36. Using Kaiser window

  37. Fig. 7-10.

  38. Summary of window method 1. Specify the ‘ideal’ or desired frequency response of filter, 2. Obtain the impulse response, , of the desired filter by evaluating the inverse Fourier transform 3. Select a window function that satisfies the passband or attenuation specifications and then determine the number of filter coefficients 4. Obtain values of for the chosen window function and the values of the actual FIR coefficients, , by multiplying by

  39. Advantages and disadvantages • Simplicity • Lack of flexibility • The passband and stopband edge frequencies cannot be precisely specified • For a given window(except the Kaiser), the maximum ripple amplitude in filter response is fixed regardless of how large we make N

  40. 6. The optimal method • Basic concepts • Equiripple passband and stopband • For linear phase lowpass filters • m+1 or m+2 extrema(minima and maxima) • Weighted • Approx. error Weighting function Ideal desired response Practical response where m=(N+1)/2 (for type1 filters) or m =N/2 (for type2 filters)

  41. Practical response Ideal response Fig. 7-11.

  42. Fig. 7-12.

  43. Optimal method involves the following steps • Use the Remez exchange algorithm to find the optimum set of extremal frequencies • Determine the frequency response using the extremal frequencies • Obtain the impulse response coefficients

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