Chapter 7 Finite Impulse Response(FIR) Filter Design

- The design of a digital filter involves five steps: Filter design steps - The design of a digital filter involves five steps: (1) Specifications of the filter requirements (2) calculation of suitable filter coefficients (3) Representation of the filter by a suitable structure (realization) (4) Analysis of the effects of finite wordlength on the filter performance (5) Implementation of filter in software and/or hardware

Filter design Need to decide : Type of filter Amplitude and/or phase responses Tolerances Sampling frequency Wordlength of the input data

Filter specifications Important parameters Another important parameter peak passband deviation (or ripples) stopband deviation passband edge frequency stopband edge frequency sampling frequency Filter order N

ILPF Fig. 7-3.

FIR coefficient calculation Most common methods used for calculating Window method Optimal method

The window method of calculating FIR filter coefficients Step 1 : specify the desired frequency response of filter, Step 2 : obtain the impulse response, , of desired filter by evaluating the inverse Fourier transform Step 3 : select a window function and then determine the number of coefficients using the appropriate relationship between the filter length and the transition width, Step 4 : obtain values of for chosen window function and the values of the actual FIR coefficient, , by multiplying by (7-26)

Window method Design of FIR filter using window method Frequency response of filter, Impulse response, Ideal lowpass response (7-19) (7-20)

Table 7.2 Summary of ideal impulse responses for standard frequency selective filters and are the normalized passband or stopband cutoff frequencies

Fig. 7-4. (The frequency scale is normalized by T = 1)

Truncation for FIR Rectangular window

Fig. 7-5.

Fig. 7-6.

Fig. 7-7.

Some common window functions Hamming window Appropriate relationship between transition width and filter length (7-21) (7-22) where N is filter order and is normalized transition width

Properties of common window functions Fig. 7-8. Window functions: (a) Rectangular, (b) Hamming, (c) Blackman

Table 7.3 Summary of important features of common window functions

Kaiser window Trade-off transition width against ripple using a ripple control parameter, (7-23) where is zero-order modified Bessel function of the first kind where typically

Determination of parameter by using the stopband attenuation requirement through empirical relationships below The number of filter coefficients N where is the stopband attenuation value and since the passband and stopband ripples are nearly equal (7-25) where is the normalized transition width

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

Using Hamming window Considering the smearing effect of the window function

Symmetrical function Calculation of Using the symmetry property to obtain the other coefficients

Fig. 7-9.

Example 7-3 Obtain coefficients using Kaiser window From filter specifications Stopband attenuation passband attenuation Transition region Sampling frequency Passband cutoff frequency

Using Kaiser window The number of filter order N The ripple parameter Normalized cutoff frequency

Calculation of FIR coefficients

Symmetrical function Calculation of Using the symmetry property to obtain the other coefficients

Fig. 7-10.

 

The optimal method Basic concepts Equiripple passband and stopband Mathematically expressed as over the passbands and stopbands Weighted Approx. error Weighting function Ideal desired response Practical response

Practical response Ideal response Fig. 7-11.

 equiripple passband and stopband of the resulting filter response, with the ripple alternating in sign between equal amplitude levels The minima and maxima are known as extrema. For linear lowpass filters, for example, there are either m+1 or m+2 extrema, m=(N+1)/2, type 1 filter m= N/2, type 2 filter

Fig. 7-12. Extremal frequencies

The procedure of optimal method Use the alternation theorem (Parks and McClellan) to find the optimum set of extremal frequencies Determine the frequency response using the extremal frequencies Obtain the impulse response coefficients

Optimal FIR filer design Transfer function of lowpass filter Symmetric property gives (7-28) where where and ,

Let be defined by normalized frequency, then Normalize as -> Normalized passband : -> Normalized stopband : Desired magnitude response Weighting function (7-30) (7-31)

  (7-32) (7-33) for in and

Alternation theorem Let is the unique best approximation if and only if is equiripple at and has at least m+2 extremal points in , that is, there exists such that (7-34) where

From equations (7-33) and (7-34) Equation (7-32) into equation (7-35) yields Matrix form (7-35)

then the optimal filter is given by Summary Step 1. Choose filter length as 2m+1 Step 2. Choose m+2 points of in F Step 3. Calculate and e using equation (7-36) Step 4. Calculate using equation (7-29). If , go to step 5, otherwise go to step 6 Step 5. Determine m local minima or maxima points Step 6. Obtain , , then the optimal filter is given by

Example 7-4 Specification of desired filter Filter length : 3 , Choose three frequencies and normalize them, two of them are cutoff frequencies, the third one arbitrarily

The error at and is 2x0.196= 0.392, and the error at is 0.196. From and The error at and is 2x0.196= 0.392, and the error at is 0.196. => not have => Not the characteristic of the optimal filter (7-37)

For this filter for all f in Choose a new set of For this filter for all f in Transfer of the optimal function (7-38) (7-39)

Fig. 7-13. Characteristics of filter H(f)

Optimal method using MATLAB Based on Parks-McClellan and Remez algorithm Calculation of coefficients for FIR filter using Remez where N is the filter length F is the normalized frequency band edges M is the magnitude response WT is the relative weight between ripples

Example 7-5 Specification of desired filter Pass band : 0 – 1000Hz Transition band : 500Hz Filter length : 45 Sampling frequency : 10,000Hz Normalized frequency band edges Magnitude response

Table 7-4. Impulse response coefficients

Fig.7-14. Frequency response of filter

Example 7-6 Specification of desired filter Pass band : 3kHz – 4kHz Transition band : 500Hz Pass band ripple : 1dB Rejection band attenuation : 25dB Sampling frequency : 20kHz Frequency band edges and magnitude response

Estimation of filter length Pass and rejection band ripples Using Remezord in MATLAB where and are ripples of dB scale in pass and rejection band

Table 7-5. Impulse response coefficients of filter

Fig. 7-15. Frequency response of filter

Frequency sampling method Design of FIR filter Taking N samples of the frequency response at intervals of , Filter coefficients (7-40) where , are samples of desired frequency response

Linear phase filters with positive symmetrical impulse response For N even For N odd Upper limit in the summation is (7-41) where

Fig. 7-16.

Example 7-7 (1) Show the From equation (7-40) is symmetry is real value (7-42)

(2) Design of FIR filter Specification of desired filter Pass band : 0 – 5kHz Sampling frequency : 18kHz Filter length : 9 Selection of frequency samples at intervals of Fig. 7-17.

Coefficient of FIR filter using equation (7-42) Table 7-6.

Comparison of the window, optimum and frequency sampling methods Optimal method Easy and efficient way of computing FIR filter coefficients Making filter with good amplitude response characteristics for reasonable values of N Window method In the absence of the optimal software or when the passband and stopband ripples are equal, the window method represents a good choice Particularly simple method to apply and conceptually easy to understand Frequency sampling method Filters with arbitrary amplitude-phase response can be easily designed Lack of precise control for the location of the bandedge frequencies or the passband ripples