EEE422
Signals and Systems Laboratory
Filters (FIR)
Kevin D. Donohue
Electrical and Computer Engineering
University of Kentucky
Filters
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
Filter implementations are broadly classified based on the
extent of their impulse responses:
 FIR = finite impulse response (impulse response goes to
zero and stays there after a finite number of time samples)
 IIR = infinite impulse response (impulse response
continues infinitely in time).
Filter Specifications
Example:Low-pass filter frequency response
1
Amplitude Scale
Cutoff Frequency
0.8
Passband
Stopband
0.6
0.4
0.2
Transitionband
0
0
0.2
Ripple
0.4
0.6
Normalize Frequency (fs/2 = 1)
0.8
1
Filter Specifications
Example Low-pass filter frequency response (in
dB)
10
Cutoff Frequency
0
Scale in dB
-10
-20
Stopband
Passband
-30
-40
Ripple
-50
-60
-70
0
Transitionband
0.2
0.4
0.6
0.8
Normalized Frequency (fs/2 = 1)
1
Filter Specifications
Example Low-pass filter frequency response (in dB) with
ripple in both bands
Filter Magnitude Response
0
Passband Ripple
Stopband Ripple
-10
dB
-20
-30
Transition
-40
-50
0
0.2
0.4
0.6
Normalize Hz
0.8
1
Useful Filter Functions
The sinc function and the rectangular pulse form a Fourier
transform pair.
f  
1
rect   
B 
0
for - B  f  B
2
2
for f elsewhere
sintB
B sinctB  B
tB
Ideal Low-Pass Filter
Response indicates the ideal low-pass filter is
non-causal and infinite. If using in a filter
design process need to approximate
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.
Example
Use impulse response method to compute a
24th order low-pass filter for fs = 1 and cut-off
frequency of .125 Hz.
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 since at 25 symmetric time points
btemp = 0.25*sinc(0.25*t)
Example
Delay response by
24/2 = 12 seconds.
Then apply tapering
window:
wtap = hamming(25);
plot(t+12,wtap,’g—’)
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.
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))
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))
Parks-Mcclellan Filter
To better control ripple over designated pass and stop bands,
an iterative algorithm was 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.
•
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 = [0 .12 .13 .5]/(fs/2) %
Need to normalize at Nyquest
Interval between pass and stop band – transition band
include specified cut-off (.125 Hz)
mag = [1 1 0 0]; % define pass and stopband amplitudes corresponding to vector f
bpm = firpm(24, f,mag); % create FIR filter coefficients
PM Filter Results
Examine magnitude and phase response of
resulting filter:
Parks-Mcclellan Filter Example
Try again with bigger transition band
fs = 1 % Sampling Frequency
f = [0 .10 .15 .5]/(fs/2) %
Need to normalize at Nyquest
Interval between pass and stop band – transition band
include specified cut-off (.125 Hz)
mag = [1 1 0 0]; % define pass and stopband amplitudes corresponding to vector f
bpm = firpm(24, f,mag); % create FIR filter coefficients
PM Filter Results
Examine and compare magnitude and phase
responses of resulting filters: