1. EEE422 Signals and Systems Laboratory
Filters (FIR)
Kevin D. Donohue Electrical and Computer Engineering University of Kentucky

2. Digital Filters – Difference Eqn
The x[k] be the input and y[k] be the output. The difference equation representing a digital filter is given by:

3. Digital filters – Transfer Function
Take Z-Transform of equation and compute transfer function:

4. Finite and Infinite Impulse Response
The output of a Finite Impulse Response (FIR) goes to zero after a finite number of increments (All zero filter)
The output of an Infinite Impulse Response (IIR) never goes to zero (pole and zero filter)

5. Filters spectral magnitude emphasis
Filter are designed based on specifications given by:
spectral magnitude emphasis
delay and phase properties through the group delay and phase spectrum
implementation and computational structures

6. Filter Specifications
Example:Low-pass filter frequency response

7. Filter Specifications
Example Low-pass filter frequency response (in dB)

8. Filter Specifications
Example Low-pass filter frequency response (in dB) with ripple in both bands

9. Useful Filter Functions
The sinc function and the rectangular pulse form a Fourier transform pair.

10. Ideal Low-Pass Filter
Response indicates the ideal low-pass filter is non-causal and infinite. If applied to design a real-time filter design process, need to approximate …

11. Impulse Response Method for FIR Filter Design
Approach:
Create ideal filter specifications for the magnitude response.
Generate impulse response through sinc function relationship.
Shift (delay) in time so sufficient response energy occurs after t=0 to make response closer to ideal in a causal implementation
Truncate sequence to make causal and finite: include 0 to N-1 to obtain coefficients for an Nth order filter. A tapering window is used to limit sharp transitions at the end points, which translate into ripple in the pass and stop bands.

12. Example FIR Design
Use impulse response method to compute a 24th order low-pass filter for fs = 1 and cut-off frequency of .125 Hz.

13. Example
Use impulse response method to compute a 24th order low-pass filter for fs = 1 and cut-off frequency of .125 Hz.
t = [24/2:1:24/2]; % evaluate at 25 symmetric time points btemp = 0.25*sinc(0.25*t)

14. Example
Delay response by 24/2 = 12 seconds. Then apply tapering window: wtap = hamming(25); plot(t+12,wtap,’g—’)

15. Example
Plot below are the coefficient values used to implement the FIR filter. Because is symmetric it will have a linear phase that accounts for the shift in time. All filtered outputs will undergo a 24/2 sample delay.

16. Example Use fir1()
Generate filter coefficient and check response by examining the Magnitude and Phase response.
fs = 1; % Sampling Frequency
b = fir1(24,.125/(fs/2), hamming(25)); % Compute filter coefficients, normalize frequency by Nyquest
[h,f] = freqz(b,1,1024,1); % Arguments: b is vector of FIR coefficients, 1 is the scaling on the output term, 1024 are number of points to evaluate frequency response at, 1 is the sampling frequency
plot(f,abs(h))
plot(f,phase(h))

17. Example Use fir1()
Generate filter coefficients without a tapering window and plot magnitude and phase response:
fs = 1; % Sampling Frequency
b = fir1(24,.125/(fs/2), boxcar(25)); % Compute filter coefficients, normalize frequency by Nyquest
[h,f] = freqz(b,1,1024,1); % Arguments: b is vector of FIR coefficients, 1 is the scaling on the output term, 1024 are number of points to evaluate frequency response at, 1 is the sampling frequency
plot(f,abs(h))
plot(f,phase(h))

18. Parks-Mcclellan Filter
To better control ripple over designated pass and stop bands, an iterative algorithm is applied to adjust filter coefficients to spread the ripple out, making it uniform over the bands of interest, thus minimizing the maximum deviation from the ideal flat pass/stop band for a given order.

19. Parks-Mcclellan Filter Example
Need to specify band range and amplitude over normalized frequency range. Can use as many bands as desired, but there should be space between defined bands for a transition to occur.
Example: apply previous design specs for PM filter
fs = 1 % Sampling Frequency
f = [ ]/(fs/2) % Need to normalize at Nyquest
mag = [ ]; % define pass and stopband amplitudes corresponding to vector f
bpm = firpm(24, f,mag); % create FIR filter coefficients
Interval between pass and stop band – transition band include specified cut-off (.125 Hz)

20. PM Filter Results
Examine magnitude and phase response of resulting filter:

21. Parks-Mcclellan Filter Example
Try again with bigger transition band
fs = 1 % Sampling Frequency
f = [ ]/(fs/2) % Need to normalize at Nyquest
mag = [ ]; % define pass and stopband amplitudes corresponding to vector f
bpm = firpm(24, f,mag); % create FIR filter coefficients
Interval between pass and stop band – transition band include specified cut-off (.125 Hz)

22. PM Filter Results
Examine and compare magnitude and phase responses of resulting filters: