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 FIR filter are very simple to implement. Nonrecursive FIR filters suffer less from the effect of finite wordlength than IIR filters

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

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

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

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

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)

Center of symmetry Fig. 7-1.

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

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

Frequency response where

(3) where

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

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

If zero on unit circle If zero not exist on the unit circle If zeros on

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

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

ILPF Satisfies spec’s Fig. 7-3.

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

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

Fig. 7-4.

Truncation to FIR Rectangular Window

Fig. 7-5.

Fig. 7-6.

Fig. 7-7.

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

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

Characteristics of common window functions Fig. 7-8.

Table 7.3 summary of important features of common window functions

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

Kaiser Formulas – for LPF design

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

Using Hamming window

Fig. 7-9.

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

Using Kaiser window

Fig. 7-10.

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

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

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)

Practical response Ideal response Fig. 7-11.

Fig. 7-12.

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

Optimal FIR filer design where where and , Let This weighting function permits different peak error in the two band

where are and Find

Alternation theorem Let If has equiripple inside bands and more than m+2 extremal point then where

From equation (7-33) and (7-34) Equation (7-35) is substituted equation (7-32) Matrix form

Summary Step 1. Select filter length as 2m+1 Step 2. Select m+2 point in F Step 3. Calculate and e using equation (7) Step 4. Calculate using equation (5). If in some of f , go to step 5, otherwise go to step 6 Step 5. Determine m local minma or maxma points Step 6. Calculate when where

Example 7-4 Specification of desired filter Ideal low pass filter Filter length : 3 Normalized frequency

From Cutoff frequency : not the optimal filter

: has the minimum (N=3)

Fig. 7-13.

Optimization using MATLAB Park-McClellan Remez where N is the filter order (N+1 is the filter length) F is the normalized frequency of border of pass band M is the magnitude of frequency response WT is the weight between ripples

Example 7-5 Specification of desired filter Band pass region : 0 – 1000Hz Transition region : 500Hz Filter length : 45 Sampling frequency : 10,000Hz Normalized frequency of border of passband Magnitude of frequency response

Table 7-4.

Fig.7-14.

Example 7-6 Specification of desired filter Band pass region : 3kHz – 4kHz Transition region : 500Hz Pass band ripple : 1dB Rejection region : 25dB Sampling frequency : 20kHz Frequency of border of passband

Transform dB to normal value Filter length Remezord (MATLAB command) where and ripple value(dB) of pass band and rejection band

Table 7-5.

Fig. 7-15.

7. Frequency sampling method Frequency sampling filters Taking N samples of the frequency response at intervals of Filter coefficients where are samples of the ideal or target frequency response

For linear phase filters (for N even) For N odd Upper limit in summation is where

Fig. 7-16.

Example 7-7 (1) Show the Expanding the equation is real value

Sampling frequency : 18kHz Filter length : 9 (2) Design of FIR filter Band pass region : 0 – 5kHz Sampling frequency : 18kHz Filter length : 9 Fig. 7-17.

Samples of magnitude in frequency Table 7-6.

8. Comparison of most commonly method Window method The easiest, but lacks flexibility especially when passband and stopband ripples are different Frequency sampling method Well suited to recursive implementation of FIR filters Optimal method Most powerful and flexible