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
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. 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 • 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
Truncation to FIR • Rectangular Window
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
Kaiser window where is the zero-order modified Bessel function of the first kind where typically
Example 7-2 • Obtain coefficients of FIR lowpass using hamming window • Lowpass filter Passband cutoff frequency Transition width Stopband attenuation Sampling frequency
Example 7-3 • Obtain coefficients using Kaiser or Blackman window Stopband attenuation passband attenuation Transition region Sampling frequency Passband cutoff frequency
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.
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