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DIGITAL FILTERS: DESIGN OF FIR FILTERS
Lecture 23-24 عرساڻي احمد احسان
Introduction to FIR filters
These have linear phase No feedback Output is function of the present and past
inputs only These are also called ‘all-zero’ and ‘non-
recursive’ filters These do not have any poles 1....1 110 Mnxbnxbnxbny M
1
0
M
kk knxbny
1
0
M
k
kzkhzH
Applications
Where: highly linear phase response is required Need to avoid complicated design
FIR Filter Design Methods
Windows Frequency-sampling
FIR Filter Design: Windows Method
0n
njdd enhH
Start from the desired frequency response Hd(ω)
Determine the unit (sample) pulse reponse
hd(n)=F-1{Hd(ω)} hd(n) is generally
infinite in length Truncate hd(n) to a
finite length M
deHnh njdd 2
1
Truncating hd(n)
Take only M terms N=0 to N=M-1
Remove all others
0 20 40 60 80 100-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
nhd
[n]
0 20 40 60 80 100-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
n
hd[n
]
Truncating hd(n)
Take only M terms N=0 to N=M-1
Remove all others Multiplying hd(n)
with a rectangular window
elsewhere
Mnnw
0
01
nwnhnh d
Determine H(ω)
Take Fourier transform of h(n)
Therefore, compute: Hd(ω) and W(ω)
Hd(ω) depends on the required response hd(n)
nwnhnh d
nwnhFnhF d
nwFnhFnhF d
WHH d
dvvWvHH d2
1
Computing W(ω)
W(ω)=F{w(n)} w(n) is a
rectangular pulse
0n
njenwW
1
0
M
n
njenwW
j
Mj
e
eW
1
1
2/sin
2/sin2/1
M
eW Mj
Example
otherwise
eH c
Mj
d0
02/1
A low-pass linear
phase FIR filter with the frequency response Hd(ω) is required
Hd(n) happens to be non-causal having infinite duration
deHnh njdd 2
1
2
1sinc
2
Mnnh cc
d
2
1
2121
sin
M
nM
n
Mnc
The impulse response hd(n)
0 20 40 60 80 100-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
n
hd[n
]
Windowing the hd(n)
0 20 40 60 80 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n
hd[n
]
The truncated hd(n)
0 20 40 60 80 100-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
n
h[n]
Example
A low-pass linear phase FIR filter with the frequency response Hd(ω) is required
2
1sinc
Mnnh ccd
2
110
M
nMn
Frequency of oscilation increases with M
Magnitude of oscillation doesn’t increase or decrease with M
Oscillations occur due to the Gibbs phenonmenon caused by the multipli-cation of the rectangular window with Hd(ω)
Other windows
Other windows
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
KaiserHammingHanningBartlettBlackmanTukeyLanczos
Spectrum of Kaiser window
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-140
-120
-100
-80
-60
-40
-20
0
Normalized frequency
Mag
nitu
de (
dB)
M=61M=31
(Cycles per sample)
Spectrum of Hanning window
Spectrum of Hamming Window
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-120
-100
-80
-60
-40
-20
0
Normalized frequency
Mag
nitu
de (
dB)
M=61M=31
(Cycles per sample)
Spectrum of Blackman Window
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-350
-300
-250
-200
-150
-100
-50
0
Normalized frequency
Mag
nitu
de (
dB)
M=61M=31
(Cycles per sample)
Spectrum of Tukey Window
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-350
-300
-250
-200
-150
-100
-50
0
Normalized frequency
Mag
nitu
de (
dB)
M=61M=31
(Cycles per sample)
Windows’ characteristics
The FIR filter’s response with Rectangular window
M=61
0 0.5 1 1.5 2 2.5 3
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Normalized frequency
Mag
nitu
de (
dB)
FIR filter’s response with Hamming window
M=61
0 0.5 1 1.5 2 2.5 3-120
-100
-80
-60
-40
-20
0
w
Mag
nitu
de (
dB)
FIR filter’s response with Blackman window
M=61
0 0.5 1 1.5 2 2.5 3
-150
-100
-50
0
wn
Mag
nitu
de (
dB)
FIR filter’s response with Kaiser window
M=61
0 0.5 1 1.5 2 2.5 3
-100
-80
-60
-40
-20
0
wn
Mag
nitu
de (
dB)
Using the FIR filter
-10 -8 -6 -4 -2 0 2 4 6 8 10
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
t
x(t)
Blackman’s filter output
-15 -10 -5 0 5 10 15-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
t
y
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