Chapter 7 Finite Impulse Response(FIR) Filter Design

Properties of FIR filter FIR filter has characteristics: Always stable Exactly linear phase response Simple to implement The effect of finite wordlength in FIR is less than in IIR filters (7-1) (7-2)

Linear phase response Phase response of FIR filter Phase delay and group delay (7-3) where (7-4) (7-5) (7-6)

Conditions to be linear phase response (7-7) (7-8) where and are constants  Constant group and constant phase delay responses

Impulse response of the filter should have positive symmetry to satisfy Eq. (7-7), thus (7-9) (7-10)

For symmetry condition (7-11) (7-12) (7-13) (7-14)

For the condition given in Eq. (7-8) Constant group delay only Negative symmetric impulse response (7-15) (7-16)

Fig. 7-1.

Example 7-1 Symmetric impulse response for linear phase response No phase distortion Using the symmetry condition For

Using the symmetry condition Frequency response Using the symmetry condition Compact form where

Using the symmetry condition where

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

Zeros of FIR filters Transfer function for FIR filter Positive symmetry (types 1 and 2) (7-17)

This means that should be zero. has zero at (i.e., ) , then should have zero at This means that should be zero. If each complex zero does not exist on the unit circle, then have 4 conjugate reciprocal zeros : If is real and is complex number, should have conjugate zero at

If zero exists on unit circle, then , i.e., becomes If zeros are real and do not exist on the unit circle, then If zeros exist on , then

Impulse response of type 3 (even N) and type 4 (odd N) If zeros exist on , then For symmetry and odd N, Should have mandatory zero at Impulse response of type 3 (even N) and type 4 (odd N) For z = 1 Mandatory zero at For and even N (7-18)

Mandatory zero Mandatory zero Mandatory zero Mandatory zero Fig. 7-2.

FIR filter design Design of FIR Filter Need to decide : Type of filter Amplitude and/or phase responses Tolerances Sampling frequency Wordlength of the input data To calculate filter coefficients, select the method from: Window method, Optimal method, Frequency sampling method

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 commonly used methods for obtaining Window, optimal, and frequency sampling methods

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

Fig. 7-4.

Truncation for 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 cutoff frequencies

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.

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 Using the stopband attenuation requirements Using empirical relationships The number of filter coefficients N where is the stopband attenuation , since the passband and stopband ripples are nearly equal (7-25) where is the normalized transition width

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)

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 or Blackman 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.

Advantages and disadvantages Simplicity Lack of flexibility The passband and stopband edge frequencies cannot be precisely specified For a given attenuation specification, filter designer must find a suitable window

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.

The procedure of optimal method 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 Transfer function of lowpass filter Symmetric property gives (7-28) where where and ,

Let Normalized passband : Normalized stopband : Desired magnitude response Weighting function (7-30) (7-31)

Find with , (7-32) (7-33) where are and

Alternation theorem Let If has equiripple inside and exhibit at least m+2 alternations, then (7-34) where

From equation (7-33) and (7-34) Matrix form (7-35)

Summary Step 1. Select filter length as 2m+1 Step 2. Select 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. Calculate ,

Example 7-4 Specification of desired filter Normalized frequency Filter length : 3 , Normalized frequency

From and (7-37)

Transfer function of optimal filter Selection of new Transfer function of optimal filter (7-38) (7-39)

Fig. 7-13.

Optimal method using MATLAB Based on Park-McClellan and Remez algorithm Calculation of coefficient 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.

Fig.7-14.

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.

Fig. 7-15.

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