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Frequency Response & Digital Filters
S Wongsa
11
S Wongsa
Dept. of Control Systems and Instrumentation Engineering,
KMUTT
Today’s goalsToday’s goals
LTI Digital Filters
Digital filter representations and structures
Frequency response analysis of digital filters
Ideal filters
IIR & FIR filters
22
IIR & FIR filters
Relationship between ZT & DTFT
Frequency Response of Discrete-Time Systems
33
• In going from the DTFT to the ZT we replace
by . Ωje z
• This evaluation is equivalent to evaluating the z-transform on the unit circle in
the complex plane.
• Replacing with , ZT will become DTFT. z Ωje
Frequency Response of Discrete-Time Systems
• Frequency Response Analysis
Consider a DT transfer function H(z), the discrete frequency response function
(FRF) is
44
Ω=
Ω ==Ω jez
j zHeHH |)()()(
• is the magnitude or gain of the FRF. |)(| ΩH
• is the phase of the FRF. )(Ω∠H
where Ω is the discrete frequency in rad/sample.
Frequency Response of Discrete-Time Systems
System Response to Sampled Sinusoids
55
If a DT system is stable with transfer function H(z), then in steady-state
Anx =][ )0(][ AHny =
nAnx 0sin][ Ω= ))(sin(|)(|][ 000 Ω∠+ΩΩ= HnHAny
nAnx 0cos][ Ω= ))(cos(|)(|][ 000 Ω∠+ΩΩ= HnHAny
Frequency Response of Discrete-Time Systems
nAnx 0sin][ Ω= ))(sin(|)(|][ 000 Ω∠+ΩΩ= HnHAny
nAnx 0cos][ Ω= ))(cos(|)(|][ 000 Ω∠+ΩΩ= HnHAny
66
When , the system response is nznx =][
∑
∑∑∞
−∞=
−
∞
−∞=
−∞
−∞=
==
=−=
k
nkn
k
kn
k
zHzzkhz
zkhknxkhny
)(][
][][][][
)(cos()()cos( Ω∠+ΩΩ⇒Ω HnHn
Proof:
Frequency Response of Discrete-Time Systems
)(zHzz nn ⇒
Setting in this relationship yields Ω±= jez
)2( )(
)1( )(Ω−Ω−Ω−
ΩΩΩ
⇒
⇒jnjnj
jnjnj
eHee
eHee
77
Adding (1) and (2) yields
)(Re2)()()cos(2 ΩΩΩ−Ω−ΩΩ =+⇒Ω jnjjnjjnj eHeeHeeHen
With , we get )()()(
Ω∠ΩΩ =jeHjjj eeHeH
)(cos()()cos( ΩΩ ∠+Ω⇒Ω jj eHneHn
EXAMPLE
5.0)(
−=z
zzH
5.0)(
−=Ω Ω
Ω
j
j
e
eH
Frequency Response of Discrete-Time Systems
88
5.0−Ωje
Ω+−ΩΩ+Ω
=Ωsin)5.0(cos
sincos)(
j
jH
EXAMPLE
Ω+−ΩΩ+Ω
=Ωsin)5.0(cos
sincos)(
j
jH
If a sinusoidal input is appliednnx3
sin][π
=
1.1547,|)3/(| =πH o30)3/( −=∠ πH
Frequency Response of Discrete-Time Systems
99
)303
1.1547sin(][ o−= nnyπ
H = tf([1 0],[1 -0.5],-1);
[mag,phase]=bode(H,pi/3)
mag =
1.1547
phase =
-30.0000
5.0)(
−=z
zzHNB:
Bode Plot
A Bode plot in the discrete time is a graph of |H(Ω)| and ∠H(Ω) plotted
as a function of Ω, where Ω is usually ranging from 0 to π.
Ω+−ΩΩ+Ω
=Ωsin)5.0(cos
sincos)(
j
jH
5.0)(
−=z
zzH
Frequency Response of Discrete-Time Systems
MATLAB: freqz
1010
num = 1; % numerator
den = [1 -0.5]; % denominator
freqz(num,den); % DT frequency response
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-30
-20
-10
0
Normalized Frequency (×π rad/sample)
Phase (degrees)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5
0
5
10
Normalized Frequency (×π rad/sample)
Magnitude (dB)
MATLAB: freqz
NB: The Bode plot can also be
obtained from the ‘bode’ command of
MATLAB.
Frequency Response from Pole-Zero Location
Transfer function:
∏∏
∑
∑
=
−
=
−
−
=
−
=
−
−=
+=
N
k k
M
k k
kN
k
k
kM
k
k
zd
zcb
za
zb
zH
1
1
1
1
0
1
0
)1(
)1(
1
)(
Frequency response:
∏∏
∏∏
=
ΩΩ−
=
ΩΩ−
=
Ω−
=
Ω−
−
−=
−
−=Ω
N
k k
jjN
M
k k
jjM
N
k
j
k
M
k
j
k
dee
ceeb
ed
ecbH
1
10
1
10
)(
)(
)1(
)1()(
1111
The magnitude:
∏∏
∏∏
=
Ω
=
Ω
=
Ω
=
Ω
=−
−=Ω
N
k
j
k
M
k
j
k
N
k k
j
M
k k
j
ed
ecb
de
cebH
1
10
1
1
0
to from distance
to from distance)(
and phase:
( ) ( )∑∑ =
Ω
=
Ω −∠−−∠+−Ω=Ω∠N
k k
jM
k k
j deceMNH11
)()(
Linear phase term Sum of the angles from the zeros/poles to unit circle
Frequency Response of LTI – A Graphical View
Transfer function:
∏∏
=
−
=
−
−
−=
N
k k
M
k k
zd
zcbzH
1
1
1
1
0
)1(
)1()(
Ω= jezWe are going around the circle with
1212
∏ =−
k k zd1)1(
Frequency response:
Ω==Ω jez
zHH )()(
Adopted from Elena Punskaya, Basics of Digital Filters.
Frequency Response of LTI – A Graphical View
The magnitude of the frequency response is given by b0
times the product of the distances from the zeros to
divided by the product of the distances from the poles
Ω= jez
∏∏
∏∏
=
Ω
=
Ω
=
Ω
=
Ω
=−
−=Ω
N
k
j
k
M
k
j
k
N
k k
j
M
k k
j
ed
ecb
de
cebH
1
10
1
1
0
to from distance
to from distance)(
1313Adopted from Elena Punskaya, Basics of Digital Filters.
( ) ( )∑∑ =
Ω
=
Ω −∠−−∠+−Ω=Ω∠N
k k
jM
k k
j deceMNH11
)()(
to Ω= jez
The phase response is given by the sum of the angles from
the zeros to minus the sum of the angles from the
poles to plus a linear phase term
Ω= jez
Ω= jez )( MN −Ω
Frequency Response of LTI – A Graphical View
when is close to a pole, the
magnitude of the response rises
(resonance).
Ω= jez
when is close to a zero, the Ω= jez
1414Adopted from Elena Punskaya, Basics of Digital Filters.
magnitude of the response falls (a null).
Frequency Response of LTI – Example
1515Source: Ashok Ambadar, Digital Signal Processing: A Modern Introduction.
Example – Filters and Pole-Zero Plots
1616
Today’s goalsToday’s goals
LTI Digital Filters
Digital filter representations and structures
Frequency response analysis of digital filters
Ideal filters
IIR & FIR filters
1717
IIR & FIR filters
What is Digital Filter?
Digital filter is a system that performs mathematical operations on a discrete-time
signal and transforms it into another sequence that has some more desirable
properties, e.g.
Digital filter x[n] y[n]
1818
In this course, we limit ourselves to LTI digital filters only.
Example Applications of Digital Filters
Noise RemovalElectrocardiogram
(ECG)
1919
https://youtu.be/v3b-YhZmQu8?t=45
Example Applications of Digital Filters
Noise RemovalElectrocardiogram
(ECG)
2020
Low-pass filtered ECG
https://youtu.be/v3b-YhZmQu8?t=45
Example Applications of Digital Filters
Audio Processing
2121
Digital Filter Representations
][][
][...]1[][
...][...]2[]1[][
01
10
21
knxbknya
Mnxbn-xbnxb
Nnyanyanyany
M
k
k
N
k
k
M
N
−+−−=
−+++
+−−−−−−−=
∑∑==
The linear time-invariant digital filter can be described by the linear difference
equation
The order of the filter is the larger of M or N
2222
The order of the filter is the larger of M or N
kN
k
k
kM
k
k
za
zb
zX
zYzH
−
=
−
=
∑
∑
+==
1
0
1)(
)()(
Transfer function of the filter is
Adders:
Multipliers:
Digital Filter Structures: Common Elements
2323
Delays:
Digital Filter Structures
EXAMPLE: Echo Generation
y[n] = x[n] + αx[n − D], |α| < 1
Echoes are delayed signals and can be generated by the following difference equation:
where D is the delay in samples.
Direct sound Reflected sound
2424
where D is the delay in samples.
Ideal Filters – Magnitude Response
Ideal filters let frequency components over the passband pass through undistorted
(gain = 1), while components at the stopband are completely cut off (gain = 0).
2525
Ideal Filters – Phase Response
Ideal filter: admits a linear phase response
)(|)(|)( ΩΩ=Ω θjeHH
where
Ω−=Ω0
)( Nθ
Linear phase Nonlinear phase
2626
)(|)(|)( 0 ΩΩ=Ω Ω−XeHY
jNFourier transform of
][ 0Nnx −
Linear phase Nonlinear phase
Output is merely a delayed version of input.
Linear Phase Response
Linear-phase filters delay all frequencies by the same amount, thereby maximally
preserving waveshape.
2727Adapted from Elena Punskaya, Basics of Digital Filters.
Ω+Ω−=Ω 3150
)( 2
2 πθNonlinear phase:
Ω−=Ω 5)(1θ
Ideal Filters – Linear Phase Response
Group Delay: A measure of linearity of the phase is obtained from the group delay
function, which is defined as
ΩΩ
−=Ωd
d )()(
θτ
The group delay is constant when the phase is linear.
2828
0
Example – A simple ideal lowpass filter
≤Ω<Ω
Ω≤Ω=Ω
if ,0
if ,1)(
πc
c
dH
The impulse response is given by
n
nnh
c
ccd Ω
ΩΩ=
sin][
π
2929
Non-causal and infinite in duration.
Clearly cannot be implemented in
real-time!.
FIR & IIR Filters
Finite Impulse Response (FIR) Filters: N = 0, no feedback. The FIR filter has no poles,
only zeros, i.e.
kN
k
k
kM
k
k
za
zb
zX
zYzH
−
=
−
=
∑
∑
+==
1
0
1)(
)()(
3030
Infinite Impulse Response (IIR) Filters: 0≠ka
( )]2[]1[][3
1][ −+−+= nxnxnxny
][]1[6.0][ nxnyny +−=
FIR & IIR Filters
FIR Filers : the impulse response of a FIR filter lasts only a finite time
( )]2[]1[][3
1][ −+−+= nxnxnxny
0
0.1
0.2
0.3
0.4
=
=elsewhere 0
2,1,0,3/1][
nnh
3131
IIR Filters: the impulse response function of a IIR filter is non-zero over an infinite
length of time
][]1[6.0][ nxnyny +−=
0 1 2 3 4 50
n
FIR & IIR Filters
Stability: FIR filters are always stable . IIR filters can be unstable (because the filter
have poles in their transfer functions. ) if not designed properly.
Order: IIR filters are computationally more efficient than FIR filters as they require
fewer coefficients due to the fact that they use feedback or poles.
3232
fewer coefficients due to the fact that they use feedback or poles.
Phase: FIR filters can be guaranteed to have linear phase. IIR filters in general do
not have linear phase.
Symmetry conditions for linear phase response
MM
n
n zMhzhhznhzH −−
=
− +++==∑ ][...]1[]0[][)( 1
0
Consider an FIR filter of order M. By definition H(z) is the z-transform of h[n]
Sufficient conditions for the phase linearity of an FIR filter:
If h[n] is either symmetric or antisymmetric about its center point, the filter phase
response is a linear function of Ω
3333
response is a linear function of Ω
][][ nMhnh −=
Symmetrical impulse response:
e.g. M=4
][][ nMhnh −−=
Antisymmetrical impulse response:
e.g. M=4
Symmetry conditions for linear phase response
][][ nMhnh −=
Symmetrical impulse response:
EXAMPLE: for M=4,
)]4[]3[]2[]1[]0[(
]4[]3[]2[]1[]0[)(
2122
4321
−−−
−−−−
++++=
++++=
zhzhhzhzhz
zhzhzhzhhzH
The frequency response is
3434
The frequency response is
)]4[]3[]2[]1[]0[()( 222 Ω−Ω−ΩΩΩ− ++++=Ω jjjjj ehehheheheH
))2cos(]0[2)cos(]1[2]2[(
])][1[]2[]][0[()(
2
222
Ω+Ω+=
++++=ΩΩ−
Ω−ΩΩ−ΩΩ−
hhhe
eehheeheH
j
jjjjj
If h[n] is symmetric about its center point, then h[0] = h[4] and h[1] = h[3]:
Hence, the phase response is given by
Ω−=Ω∠ 2)(H
Symmetry conditions for linear phase response
][][ nMhnh −−=
Antisymmetrical impulse response:
EXAMPLE: for M=4,
)]4[]3[]2[]1[]0[(
]4[]3[]2[]1[]0[)(
2122
4321
−−−
−−−−
++++=
++++=
zhzhhzhzhz
zhzhzhzhhzH
The frequency response is
3535
The frequency response is
)]4[]3[]2[]1[]0[()( 222 Ω−Ω−ΩΩΩ− ++++=Ω jjjjj ehehheheheH
))2sin(]0[)sin(]1[(2
])][1[]][0[()(
2
222
Ω+Ω=
−+−=ΩΩ−
Ω−ΩΩ−ΩΩ−
hhje
eeheeheH
j
jjjjj
If h[n] is antisymmetric about its center point, then h[2]=0, h[0] = -h[4] and h[1] = -h[3]:
Hence, the phase response is given by
Ω−=Ω∠ 22
)(π
H