Digital Signal Processing
Moslem Amiri, Václav Přenosil
Masaryk University
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Finite Impulse Response Filters
2
Introduction
 Filtering
 Filtering is the processing of a time-domain signal
resulting in some change in that signal’s original
spectral content
 The change is usually the reduction, or filtering out, of
some unwanted input spectral components
 That is, filters allow certain frequencies to pass while
attenuating other frequencies
 Digital filter in Fig. 5-1(b)
 Can be a software program in a computer, a
programmable hardware processor, or a
dedicated integrated circuit
3
Introduction
4
An Introduction to FIR Filters
 FIR filter
 Given a finite duration of nonzero input values, an
FIR filter will always have a finite duration of
nonzero output values
 If FIR filter’s input is a sequence of all zeros,
filter’s output will be all zeros
 FIR filters use addition to calculate their outputs
 Averaging is a kind of FIR filter
5
An Introduction to FIR Filters
 Averaging example
 We’re counting the number of cars that pass over a
bridge every minute, and every minute we’ll calculate
average number of cars/minute over the last five minutes
Minute index
No. of cars/minute over the
last minute
No. of cars/minute averaged over
the last five minutes
1 10 -
2 22 -
3 24 -
4 42 -
5 37 27
6 77 40.4
7 89 53.8
8 22 53.4
9 63 57.6
10 9 52
6
An Introduction to FIR Filters
7
An Introduction to FIR Filters
 Fig. 5-2
 Sudden changes in input sequence of
cars/minute are flattened out by averager
 Since sudden transitions in a time sequence
represent high-frequency components, averager
is behaving like a lowpass filter
 Averager is an FIR filter
 No previous averager output value is used to
determine a current output value; only input values are
used to calculate output values
 In addition, when input goes to zero, averager’s output
approaches and settles to a value of zero
8
An Introduction to FIR Filters
∑−=
=+−+−+−+−=
++++=
n
nk
ave
ave
kxnxnxnxnxnxny
xxxxxy
4
)(
5
1
)]()1()2()3()4([
5
1
)(
)]5()4()3()2()1([
5
1
)5(
9
An Introduction to FIR Filters
 Fig. 5-3
 Referred to as filter structure
 Is a physical depiction of how to calculate
averaging filter outputs with the input sequence of
values shifted, in order, from left to right along the
top of filter as new output calculations are
performed
 Delay elements, called unit delays, merely
indicate a shift register arrangement where input
sample values are temporarily stored during an
output calculation
10
An Introduction to FIR Filters
11
An Introduction to FIR Filters
 Fig. 5-4
 In (a), each of the first five input values is multiplied
by 1/5, and the five products are summed to give
the fifth filter output value
 To calculate sixth output value, input sequence is
right-shifted, discarding the first input value of 10,
and the sixth input value, 77, is accepted on left
 The filter’s structure using this shifting process is
called a transversal filter
 Because we tap off five separate input sample
values to calculate an output value, the structure is
called a 5-tap tapped-delay line FIR filter
∑−=
=+−+−+−+−=
n
nk
ave kxnxnxnxnxnxny
4
)(
5
1
)(
5
1
)1(
5
1
)2(
5
1
)3(
5
1
)4(
5
1
)(
12
Convolution in FIR Filters
 Averaging filter convolution
 Convolution equation
 We can graphically depict this equation’s
calculations as shown in Fig. 5-5
 Input samples: x(0), x(1), x(2), …
 Filter coefficients: h(0) through h(4)
 In above equation, we use factor of 1/5 as filter
coefficients multiplied by averaging filter’s input
samples
 Time order of inputs in Fig. 5-5 is reversed
∑−=
=+−+−+−+−=
n
nk
ave kxnxnxnxnxnxny
4
)(
5
1
)(
5
1
)1(
5
1
)2(
5
1
)3(
5
1
)4(
5
1
)(
13
Convolution in FIR Filters
14
Convolution in FIR Filters
 FIR filter’s y(n)th output
 For a general M-tap FIR filter, nth output is
 This is convolution equation as it applies to digital
FIR filters
∑=
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−=
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+−+−+−+−=
4
0
)(
tscoefficienfilter
)()(
)()0()1()1()2()2()3()3()4()4()(
)(
5
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5
1
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)3(
5
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)4(
5
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)(
k
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knxkh
nxhnxhnxhnxhnxhny
nxnxnxnxnxny
∑
−
=
−=
1
0
)()()(
M
k
knxkhny
15
Convolution in FIR Filters
 Impulse response of a filter
 Filter’s output time-domain sequence when input
is a single unity-valued sample (impulse)
preceded and followed by zero-valued samples
 Fig. 5-6
 FIR filter’s impulse response is identical to the
five filter coefficient values
 For this reason, the terms FIR filter coefficients and
impulse response are synonymous
 Because there are a finite number of coefficients,
impulse response will be finite in time duration
 Finite impulse response, FIR
16
Convolution in FIR Filters
17
Convolution in FIR Filters
 Process of convolution, as it applies to FIR
filters
 Two sequences resulting from h(k)*x(n) and
H(m)·X(m) are Fourier transform pairs
 Convolution in time domain is equivalent to
multiplication in frequency domain
)()()()()(
IDFT
DFT
mXmHnxkhny ⋅
 ←
 →
∗=
18
Convolution in FIR Filters
19
Convolution in FIR Filters
H(m) is sin(x)/x
function
20
Convolution in FIR Filters
Conversion of
discrete
frequency axis
in Fig. 5-8 to
continuous
m = N/2 = 32 = folding frequency = fs/2
ideal lowpass filter
poor lowpass filter
21
Convolution in FIR Filters
initial four output
samples (transient
response) are not
exactly sinusoidal:
the no. of filter unitdelay
elements = 4
averaging filter
attenuates higherfrequency
inputs
further
output is a
sinewave of same
frequency as input
filter’s coefficients
are symmetrical
 input/output
delay = half the
no. of unit-delay
elements = 2
samples (no
dependence on
frequency)
22
Convolution in FIR Filters
Averager
attenuates
higher-frequency
portion of input
spectrum
23
Convolution in FIR Filters
 Summary, so far
 FIR filters perform time-domain convolution by
summing the products of the shifted input samples
and a sequence of filter coefficients
 An FIR filter’s output sequence is equal to
convolution of input sequence and a filter’s impulse
response (coefficients)
 An FIR filter’s frequency response is DFT of filter’s
impulse response
 An FIR filter’s output spectrum is product of input
spectrum and filter’s frequency response
 Convolution in time domain and multiplication in
frequency domain are Fourier transform pairs
24
Convolution in FIR Filters
25
Convolution in FIR Filters
 Fig. 5-12
 Different sets of coefficients give us different
frequency magnitude responses
 A sudden change in values of coefficient
sequence, such as 0.2 to 0 transition in the first
coefficient set, causes ripples, or sidelobes, in
frequency response
 If we minimize suddenness of changes in
coefficient values, such as the third set of
coefficients in (a), we reduce sidelobe ripples in
frequency response
 However, reducing sidelobes results in increasing main
lobe width of lowpass filter
26
Convolution in FIR Filters
 We can have a filter with more than 5 taps
 But input signal sample shifting, multiplications by
constant coefficients, and summation are all there
is to it
27
Lowpass FIR Filter Design
 Design of a lowpass FIR filter
 Design procedure starts with determination of a
desired frequency response followed by
calculating filter coefficients that will give us that
response
 There are two predominant techniques used to
design FIR filters
 1) window method
 2) optimum method
28
Lowpass FIR Filter Design
 Window design method
 Begins with our deciding what frequency
response we want for our lowpass filter
 We can start by considering a continuous lowpass
filter, and simulating that filter with a digital filter
 We define the continuous frequency response H(f) to
be ideal, i.e., a lowpass filter with unity gain at low
frequencies and zero gain (infinite attenuation) beyond
some cutoff frequency, as shown in Fig. 5-14(a)
 Representing this H(f) response by a discrete
frequency response is the same—with one difference:
discrete frequency-domain representations are always
periodic with the period being fs
29
Lowpass FIR Filter Design
30
Lowpass FIR Filter Design
 Window design method
 We have two ways to determine lowpass filter’s
time-domain coefficients
 The first way is algebraic
 1. Develop an expression for discrete frequency
response H(m)
 2. Apply that expression to inverse DFT equation to get
time domain h(k)
 3. Evaluate that h(k) expression as a function of time
index k
 The second method is to define individual
frequency-domain samples representing H(m) and
then have a software routine perform inverse DFT of
those samples, giving FIR filter coefficients
31
Lowpass FIR Filter Design
 Window design method
 In either method, we need only define the
periodic H(m) over a single period of fs Hz
 Defining H(m) in Fig. 5-14(b) over frequency span −fs/2
to fs/2 is the easiest form to analyze algebraically
 Defining H(m) over frequency span 0 to fs is the best
representation if we use inverse DFT to obtain filter’s
coefficients
32
Lowpass FIR Filter Design
 Algebraic method
 We can define an arbitrary discrete frequency
response H(m) using N samples to cover −fs/2 to
fs/2 and establish K unity-valued samples for the
passband of lowpass filter as shown in Fig. 5-15
 A great deal of algebraic manipulation is required here
that digital filter designers avoid performing IDFT
algebraically
)/sin(
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/2
Nk
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emH
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N
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Nkmj
π
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⋅=
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33
Lowpass FIR Filter Design
34
Lowpass FIR Filter Design
 Software IDFT method of FIR filter design
35
Lowpass FIR Filter Design
 Software IDFT method of FIR filter design
 Using a 32-point inverse FFT to implement a 32point
inverse DFT of H(m) sequence in Fig. 5-
17(c), we get 32 h(k) values from k = −15 to k =
16 in Fig. 5-18(a)
 Because we want final 31-tap h(k) filter
coefficients to be symmetrical with their peak
value in center of coefficient sample set, we drop
k = 16 sample and shift k index to left from Fig. 5-
18(a), giving us the desired sin(x)/x form of h(k)
as shown in Fig. 5-18(b)
 This shift of index k will not change frequency
magnitude response of FIR filter
36
Lowpass FIR Filter Design
37
Lowpass FIR Filter Design
the more h(k)
terms we use as
filter coefficients,
the closer we’ll
approximate our
ideal lowpass filter
response
38
Lowpass FIR Filter Design
 Why are passband ripples in lowpass FIR
filter response in Fig. 5-19
 Replacing h(k) and x(n) with h∞(k) and w(k)
 h∞(k) represents an infinitely long sin(x)/x sequence of
ideal lowpass FIR filter coefficients
 w(k) represents a window sequence that we use to
truncate sin(x)/x terms
)()()()(
)()()()(
IDFT
DFT
IDFT
DFT
mXmHnxkh
mXmHnxkh
∗
 ←
 →
⋅
⋅
 ←
 →
∗
)()()()(
IDFT
DFT
mWmHkwkh ∗
 ←
 →
⋅ ∞∞
39
Lowpass FIR Filter Design
length of w(k) is the
no. of coefficients, or
taps, we intend to
use to implement our
lowpass FIR filter
40
Lowpass FIR Filter Design
FIR filter’s true
frequency response is
H(m) = H∞(m) * W(m)
we can view a
particular sample value
of H(m) = H∞(m) * W(m)
convolution as being
sum of products of
H∞(m) and W(m) for a
particular frequency
shift of W(m)
unity for all of H∞(m) 
a particular H(m) value
is sum of W(m)
samples that overlap
H∞(m) rectangle
with a W(m) frequency
shift of 0 Hz, sum of
W(m) samples that
overlap H∞(m)
rectangle is the value of
H(m) at 0 Hz
sum of positive and
negative W(m) samples
under H∞(m) rectangle
varies as W(m) is
shifted  ripples in
passband
peak of W(m)’s main
lobe is outside H∞(m)
rectangle  H(m)’s
passband begins to roll
off
as W(m) shift
continues, ripples
in H(m) beyond
the positive cutoff
frequency
ripples in H(m)
are caused by
sidelobes of
W(m)
41
Lowpass FIR Filter Design
 How many sin(x)/x coefficients do we have to
use (or how wide must w(k) be) to get nice
sharp falling edges and no ripples in H(m)
passband?
 As long as w(k) is a finite number of unity values
(i.e., a rectangular window of finite width), there
will be sidelobe ripples in W(m), and this will
induce passband ripples in final H(m) frequency
response
42
Lowpass FIR Filter Design
we can make filter’s
transition region
narrower using
additional h(k) filter
coefficients, but we
cannot eliminate
passband ripple
a wider w(k) does
not even reduce
peak-to-peak ripple
magnitudes, as long
as w(k) has sudden
discontinuities
43
Lowpass FIR Filter Design
 Gibbs’s phenomenon
 The ripple is known as Gibbs’s phenomenon,
which manifests itself anytime a function (w(k) in
this case) with an instantaneous discontinuity is
represented by a Fourier series
 No finite set of sinusoids will be able to change
fast enough to be exactly equal to an
instantaneous discontinuity
 No matter how wide w(k) window is, W(m) will
always have sidelobe ripples
44
Lowpass FIR Filter Design
 Windows used in FIR filter design
 We can minimize FIR passband ripple with
window functions the same way we minimized
DFT leakage
 Window FIR design method is the technique of
reducing w(k)’s discontinuities by using window
functions other than rectangular window
45
Lowpass FIR Filter Design
1,...,2,1,0for
1
4
cos08.0
1
2
cos5.042.0)(
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−=






−
+





−
−=
Nk
N
k
N
k
kw
ππ
smoothly tapered
h(k) coefficients
passband ripples
are greatly reduced
the price we paid for reduced
passband ripple is a wider
H(m) transition region
we can get a steeper filter response rolloff
by increasing the no. of taps in FIR
filter (we should use a 63-coefficient
Blackman window for a 63-tap FIR filter)
46
Lowpass FIR Filter Design
greatly reduced
sidelobe levels of
Blackman window
Blackman window’s
main lobe is almost
three times as wide
as rectangular
window’s main lobe
47
Lowpass FIR Filter Design
 Summary of window method of FIR filter
design
 We pick a window function and multiply it by
sin(x)/x values from h∞(k) to get our final h(k) filter
coefficients
48
Lowpass FIR Filter Design
 Window functions with more control over their
frequency responses
 There are two window functions with more
flexibility in trading off the window’s main lobe
width and sidelobe levels
 Chebyshev and Kaiser window functions
49
Lowpass FIR Filter Design
[ ]
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γ and β control
parameters give us
control over
Chebyshev and
Kaiser windows’
main lobe widths
and sidelobe levels
50
Lowpass FIR Filter Design
unlike the constant
sidelobe peak levels
of Chebyshev window,
Kaiser window’s
sidelobes decrease
with increased
frequency
Kaiser sidelobes
are higher than
Chebyshev
window’s
sidelobes near
main lobe
51
Lowpass FIR Filter Design
Selecting different
values for γ
enables us to
adjust sidelobe
levels and see
what effect those
values have on
main lobe width
Chebyshev
window function’s
stopband
attenuation, in dB
AttenCheb = -20 γ
52
Lowpass FIR Filter Design
53
Lowpass FIR Filter Design
 Chebyshev or Kaiser, which is the best?
 Depends on the application
 Digital filter designers typically experiment with
various values of γ and β for Chebyshev and
Kaiser windows to get the optimum WdB(m) for a
particular application
 Blackman window’s very low sidelobe levels
outweigh its wide main lobe in many applications
54
Bandpass FIR Filter Design
 Bandpass FIR filter design
 Window method of lowpass FIR filter design can
be used as the first step
 Let’s say we want a 31-tap FIR filter with the
frequency response shown in Fig. 5-22(a), but
instead of being centered about zero Hz, we want
filter’s passband to be centered about fs/4 Hz
 If we define a lowpass FIR filter’s coefficients as
hlp(k), to find hbp(k) coefficients of a bandpass FIR
filter, we can shift Hlp(m)’s frequency response by
multiplying hlp(k) lowpass coefficients by a
sinusoid of fs/4 Hz (sshift(k))
55
Bandpass FIR Filter Design
actual magnitude of
|Hbp(m)| is half that
of the original
|Hlp(m)| because half
the values in hbp(k)
are zero when
sshift(k) corresponds
exactly to fs/4
when we design an
N-tap bandpass FIR
filter centered at a
frequency of fs/4 Hz,
we only need to
perform N/2
multiplications for
each filter output
sample
hlp(k) lowpass
coefficients have not
been multiplied by
any window function.
In practice, we’d use
an hlp(k) that has
been windowed to
reduce passband
ripple
If we wanted to
center bandpass
filter’s response at
some frequency
other than fs/4, we
need to modify
sshift(k) to represent
sampled values of a
sinusoid whose
frequency is equal to
the desired
bandpass center
frequency
56
Highpass FIR Filter Design
 Highpass FIR filter design
 We can use the bandpass FIR filter design
technique to design a highpass FIR filter
 To obtain coefficients for a highpass filter, we
need only modify the shifting sequence sshift(k) to
make it represent a sampled sinusoid whose
frequency is fs/2
 hhp(k) = hlp(k) . sshift(k) = hlp(k) . (1,-1,1,-1,etc.)
57
Highpass FIR Filter Design
hhp(k) is merely hlp(k)
with the sign
changed for every
other coefficient
|Hhp(m)| response
has the same
amplitude as the
original |Hlp(m)|
hlp(k) lowpass
coefficients have not
been modified by
any window function.
In practice, we’d use
a windowed hlp(k) to
reduce passband
ripple
58
Parks-McClellan Exchange FIR Filter Design Method
 Parks-McClellan FIR filter design method
 Also called Remez Exchange, or Optimal method
 A popular technique used to design highperformance
FIR filters
to use this method,
we have to visualize
a desired frequency
response Hd(m): we
have to establish
desired fpass, fstop, δp
and δs
59
Parks-McClellan Exchange FIR Filter Design Method
 Parks-McClellan FIR filter design method
 Passband and stopband ripples, in decibels, are
related to δp and δs by
 Next, we apply these parameters to a software
routine that generates the filter’s N time-domain
h(k) coefficients where N is the minimum number
of filter taps to achieve the desired filter response
)(log20rippleStopband
)1(log20ripplePassband
10
10
s
p
δ
δ
⋅−=
+⋅=
60
Parks-McClellan Exchange FIR Filter Design Method
some software
Parks-McClellan
routines assume that
we want δp and δs to
be as small as
possible and require
us only to define the
desired values of
Hd(m) response as
shown by solid black
dots
filter designer has
the option to define
some of Hd(m)
values in transition
band, and software
calculates remaining
undefined Hd(m)
transition band
values
61
Parks-McClellan Exchange FIR Filter Design Method
the three filters have
roughly the same
stopband sidelobe
levels, near main
lobe, but ParksMcClellan
filter has
the more desirable
(steeper) transition
band roll-off
62
Half-band FIR Filters
 Half-band FIR filter
 Very useful in signal decimation and interpolation
applications
 Its frequency magnitude response is symmetrical
about fs/4 point
 fpass + fstop = fs/2
 When filter has an odd number of taps, filter’s timedomain
impulse response has every other filter
coefficient being zero, except center coefficient
 This enables us to avoid approximately half the
number of multiplications when implementing this filter
 For an N-tap half-band FIR filter, we’ll only need to
perform (N + 1)/2 + 1 multiplications per output sample
63
Half-band FIR Filters
coefficients for a
31-tap half-band
filter where Δf is
defined to be
approximately
fs/32 using ParksMcClellan
FIR
filter design
method
alternating h(k)
coefficients are
zero, so we
perform 17
multiplications
per output
sample instead of
the expected 31
multiplications
h(1) and h(5)
multipliers are
absent
64
Half-band FIR Filters
 Half-band FIR filter
 To build linear-phase N-tap half-band FIR filters,
having alternating zero-valued coefficients, N + 1
must be an integer multiple of four
 If this restriction is not met, e.g. when N = 9, the first
and last coefficients will both be equal to zero and can
be discarded, yielding a 7-tap half-band filter
 When designing a half-band filter, assuming that
the modeled filter has a passband gain of unity,
ensure that filter has a gain of 0.5 (−6 dB) at fs/4
 Numerical computation errors yield alternate filter
coefficients that are not exactly zero-valued 
force those values to zero
65
Phase Response of FIR Filters
the 25 h(k) sequence
is padded with 103
zeros to take a 128point
DFT, resulting in
H(m) sample values
at m = 17, H(m)
experiences a polarity
change of its real part
while its imaginary
part remains
negative—this
induces a true phaseangle
discontinuity (in
Fig. 5-35(c)) that
really is a constituent
of H(m) at m = 17.
Additional phase
discontinuities occur
each time the real
part of H(m) reverses
polarity
66
Phase Response of FIR Filters
software adds 360° to
any negative angles
in the range of
−180° > ø ≥ −360°
one of the dominant
features of FIR filters
is their linear phase
response. Hø(m) is
linear over the
passband of H(m)
67
Phase Response of FIR Filters
 Group delay
 G = −dø/df
 For FIR filters, group delay is slope of Hø(m) response
curve
 When group delay is constant, as it is over passband of
all FIR filters having symmetrical coefficients, all
frequency components of filter input signal are delayed
by an equal amount of time G before they reach filter’s
output
 Crucial in communications signals
 For amplitude modulation (AM) signals, constant group
delay preserves time waveform shape of signal’s
modulation envelope
 Important because modulation portion of an AM signal
contains signal’s information
68
Phase Response of FIR Filters
 Group delay
 Over passband frequency range for a linearphase,
S-tap FIR filter, group delay is
 D = S−1 is the number of unit-delay elements
 ts is sample period (1/fs)
 Eliminating ts factor changes its dimensions to samples
 Passband phase-angle resolution
 N = the number of points in DFT
seconds
2
stD
G
⋅
=
N
G 
360⋅−
=∆φ
69
Phase Response of FIR Filters
128-point DFT was
used to obtain
frequency responses
in Figs. 5-34 and 5-
35; we could use N =
32-point or N = 64point
DFTs
phase-angle
resolution is much
finer here
S = 25-tap filter in Fig.
5-34(a)  G = 12 
Δø = −12 · 360°/32 =
−135°
S = 25-tap filter in Fig.
5-34(a)  G = 12 
Δø = −12 · 360°/128 =
−33.75°
70
Phase Response of FIR Filters
 For FIR filters, output phase shift (in degrees)
for passband frequency f = mfs/N, is
 Relationship between phase responses in Fig. 5-
36 considering the phase delay associated with
frequency of fs/32
N
Gm
mNfmH s

360
)/(delayphase
⋅⋅−
=∆⋅== φφ
DFT size, N Index m Hø(mfs / N)
32 1 −135°
64 2 −135°
128 4 −135°
71
Analyzing FIR Filters
 Analyzing tapped-delay line, nonrecursive
FIR filters
 Means determining FIR filter’s frequency
response based on known filter coefficients
 Two ways to analyze
 Using continuous-time Fourier algebra
 Using discrete Fourier transform
72
Analyzing FIR Filters
 Algebraic analysis of FIR filters
 Uses DTFT equation
 DTFT of an FIR filter having N coefficients
(impulse response) represented by h(k), where
k = 0, 1, 2, ..., N−1
 H(ω) is an (N−1)th-order polynomial
 ω is continuous and ranges from 0 to 2π rad/samp,
corresponding to a continuous-time frequency range of 0
to fs Hz
∑
∞
−∞=
−
=
n
jn
enxX ω
ω )()(
ωωωω
ω
ω
)1(20
1
0
)1(...)2()1()0(
)()(
−−−−−
−
=
−
−++++=
= ∑
Njjjj
N
k
jk
eNheheheh
ekhH
73
Analyzing FIR Filters
 Example
 A 4-tap FIR filter whose coefficients are h(k) =
[0.2, 0.4, 0.4, 0.2]
 Magnitude and phase (Hø(ω) = arctangent of ratio
of imaginary part over real part of H(ω)) are
plotted in Fig. 5-46
)]3sin(2.0)2sin(4.0)sin(4.0[
)3cos(2.0)2cos(4.0)cos(4.02.0
2.04.04.02.0
)3()2()1()0()()(
32order-3rdcomplex
320
3
0
ωωω
ωωω
ω
ωωω
ωωωωω
++−
+++=
+++= →
+++==
−−−
−−−−
=
−
∑
j
eee
ehehehehekhH
jjj
jjjj
k
jk
74
Analyzing FIR Filters
75
Analyzing FIR Filters
 DFT analysis of FIR filters
 The most convenient way to determine an FIR
filter’s frequency response is to perform DFT of
filter’s coefficients
∑
−
=
−
=
1
0
/2
)()(
N
k
Nkmj
ekhmH π
76
Analyzing FIR Filters
We need more |H(m)| frequency-domain
information. That is, we need improved
frequency resolution.
77
Analyzing FIR Filters
a finer-granularity
version of H(m)
obtained by zero
padding h(k)
coefficients
circular white
dots correspond
to square dots in
Fig. 5-47(b)
78
Analyzing FIR Filters
 A filter’s complex H(m) frequency response
sequence








=
=
−
)(
)(
tan)(
)()(
1
)(
mH
mH
mH
emHmH
real
imag
mHj
φ
φ
79
Analyzing FIR Filters
 FIR filter (constant) group delay
 A filter has a linear phase response over its
passband and will induce no phase distortion in
its output signals
 ω is continuous and ranges from −π to π
radians/sample, corresponding to a continuous-time
frequency range of −fs/2 to fs/2 Hz
 Hø(ω) in radians
ω in radians/sample
G(ω) are time measured in samples
samples
)(
)(
)()()(
)(
ω
ω
ωωω
φωφ
d
dH
GeHH
Hj −
=→=
80
Analyzing FIR Filters
81
Analyzing FIR Filters
 FIR filter group delay
 Example: complex-valued frequency response of
a K-tap moving average filter is
 For symmetrical-coefficient FIR filters
samples2
)(
)2(
)(
)(
)(
22/)15()(
)2/sin(
)2/sin(
)(
5,
5,
filteraveragemovingtap-5=K
aofdelaygroup
5,
filteraveragemovingtap-5=K
aofresponsephase
2/)1(
=
−−
=
−
= →
−=−−= →
=
=
=
=
−−
ω
ω
ω
ω
ω
ωωω
ω
ω
ω
φ
φ
ω
d
d
d
dH
G
H
K
eH
K
Kma
K
Kj
ma
seconds
22
samples
2
seconds
samples
s
s
f
DDt
G
D
G
==
=D = the no. of unitdelay
elements in
filter’s delay line
82
Analyzing FIR Filters
 FIR filter group delay
 In general, group delay of a tapped-delay line FIR
digital filter, whose impulse response is
symmetric, is
 If a tapped-delay line (FIR) network has an
antisymmetrical impulse response, it also has a
linear phase response and its group delay is also
described by above equation
 Antisymmetrical impulse response: h(k)=−h(N−k−1)
where 0≤k≤(N−1)/2 when N is odd and 0≤k≤(N/2)−1
when N is even
samples
2
1lengthresponseimpulse
samples
−
=G
83
Analyzing FIR Filters
samples
2
samples
D
G = fractional delay
84
Analyzing FIR Filters
 FIR filter passband gain
 Passband gain is filter’s passband magnitude
response level around which the passband ripple
fluctuates
 In practice we design filters to have very small
passband ripple, so a lowpass filter’s passband
gain is roughly equal to its DC gain (gain at 0 Hz)
 DC gain is sum of filter’s impulse response sequence,
i.e., sum of FIR filter’s coefficients
 Most commercial FIR filter design software
packages compute filter coefficients such that
their passband gain is unity
85
Analyzing FIR Filters
passband gain
equals unity here
86
Analyzing FIR Filters
 Estimating the number of FIR filter taps
 How do we estimate the number of filter taps
(coefficients), N, that can satisfy a given
frequency magnitude response of an FIR filter?
 A simple expression proposed by Prof. Fred Harris for
N, for passband ripple values near 0.1 dB, is
 Atten = desired stopband attenuation measured in dB
 fpass and fstop are frequencies normalized to fs sample
rate in Hz
E.g., fpass = 0.2 means that continuous-time frequency
of fpass is 0.2fs Hz
)(22 passstop
FIR
ff
Atten
N
−
≈
87
Analyzing FIR Filters
8.21
)1000/2501000/350(22
48
)(22
25.035.0
FIR
dB48attenstopband
Hz350=
Hz250=
Hz1000=
Assuming
passstop
FIR =
−
≈ →
−
≈ =

N
ff
Atten
N
stop
pass
s
f
f
f
the lowpass filter
can be built using
a 22-tap FIR filter