Chapter 5
THE APPLICATION OF THE Z TRANSFORM
5.6 Transfer Functions for Digital Filters
5.7 Amplitude and Delay Distortion
c 2005- Andreas Antoniou
Copyright Victoria, BC, Canada
Email: aantoniou@ieee.org
March 7, 2008
Frame # 1
Slide # 1
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Introduction
Previous presentations dealt with the frequency response
of discrete-time systems, which is obtained by using the
transfer function.
Frame # 2
Slide # 2
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Introduction
Previous presentations dealt with the frequency response
of discrete-time systems, which is obtained by using the
transfer function.
In this presentation, we examine some of the basic types
of transfer functions that characterize some typical firstand second-order filter types known as biquads.
Frame # 2
Slide # 3
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Introduction
Previous presentations dealt with the frequency response
of discrete-time systems, which is obtained by using the
transfer function.
In this presentation, we examine some of the basic types
of transfer functions that characterize some typical firstand second-order filter types known as biquads.
Biquads are often used as basic digital-filter blocks to
construct high-order filters.
Frame # 2
Slide # 4
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
First-Order Transfer Functions
A first-order transfer function can have only a real zero and
a real pole, i.e.,
z − z0
H(z) =
z − p0
Frame # 3
Slide # 5
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
First-Order Transfer Functions
A first-order transfer function can have only a real zero and
a real pole, i.e.,
z − z0
H(z) =
z − p0
To ensure that the system is stable, the pole must satisfy
the condition −1 < p0 < 1.
Frame # 3
Slide # 6
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
First-Order Transfer Functions
A first-order transfer function can have only a real zero and
a real pole, i.e.,
z − z0
H(z) =
z − p0
To ensure that the system is stable, the pole must satisfy
the condition −1 < p0 < 1.
The zero can be anywhere on the real axis of the z plane.
Frame # 3
Slide # 7
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
First-Order Transfer Functions Cont’d
2
2
1
1
jIm z
jIm z
If the pole is close to point (1, 0) and the zero is close to or
at point (−1, 0), then we have a lowpass filter.
0
−1
−2
−2
Frame # 4
Slide # 8
0
−1
−1
0
Re z
1
2
A. Antoniou
−2
−2
−1
0
Re z
1
Digital Signal Processing – Secs. 5.6 and 5.7
2
First-Order Transfer Functions Cont’d
2
2
1
1
jIm z
jIm z
If the pole is close to point (1, 0) and the zero is close to or
at point (−1, 0), then we have a lowpass filter.
If the zero and pole positions are interchanged, then we
get a highpass filter.
0
−1
−2
−2
Frame # 4
Slide # 9
0
−1
−1
0
Re z
1
2
A. Antoniou
−2
−2
−1
0
Re z
1
Digital Signal Processing – Secs. 5.6 and 5.7
2
First-Order Transfer Functions Cont’d
Certain applications require discrete-time systems that
have a constant amplitude response and a varying phase
response.
Such systems can be constructed by using allpass transfer
functions.
Frame # 5 Slide # 10
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
First-Order Transfer Functions Cont’d
Certain applications require discrete-time systems that
have a constant amplitude response and a varying phase
response.
Such systems can be constructed by using allpass transfer
functions.
A first-order allpass transfer function is of the form
H(z) =
z − 1/p0
p0 z − 1
= p0
z − p0
z − p0
where the zero is the reciprocal of the pole.
Frame # 5 Slide # 11
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
First-Order Transfer Functions Cont’d
Certain applications require discrete-time systems that
have a constant amplitude response and a varying phase
response.
Such systems can be constructed by using allpass transfer
functions.
A first-order allpass transfer function is of the form
H(z) =
z − 1/p0
p0 z − 1
= p0
z − p0
z − p0
where the zero is the reciprocal of the pole.
The frequency response of a system characterized by H(z)
is given by
H(ejωT ) =
Frame # 5 Slide # 12
p0 ejωT − 1
p0 cos ωT + jp0 sin ωT − 1
=
jωT
e
− p0
cos ωT + j sin ωT − p0
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
First-Order Transfer Functions Cont’d
···
H(ejωT ) =
p0 ejωT − 1
p0 cos ωT + jp0 sin ωT − 1
=
cos ωT + j sin ωT − p0
ejωT − p0
The amplitude and phase responses are given by
p0 cos ωT − 1 + jp0 sin ωT M(ω) = cos ωT − p0 + j sin ωT 1
(p0 cos ωT − 1)2 + (p0 sin ωT )2 2
=1
=
(cos ωT − p0 )2 + (sin ωT )2
and
θ (ω) = tan−1
p0 sin ωT
sin ωT
− tan−1
p0 cos ωT − 1
cos ωT − p0
respectively.
Frame # 6 Slide # 13
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Lowpass Biquad
A lowpass second-order transfer function can be
constructed by placing a complex-conjugate pair of poles
anywhere inside the unit circle and a pair of zeros at the
Nyquist point:
2
jIm z
1
0
−1
−2
−2
Frame # 7 Slide # 14
−1
A. Antoniou
0
Re z
1
2
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Lowpass Biquad Cont’d
The transfer function of the lowpass biquad assumes the
form:
HLP (z) =
z 2 + 2z + 1
(z + 1)2
=
(z − rejφ )(z − re−jφ )
z 2 − 2r (cos φ)z + r 2
where 0 < r < 1.
Frame # 8 Slide # 15
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Lowpass Biquad Cont’d
As the poles move closer to the unit circle, the amplitude
response develops a peak at frequency ω = φ/T while the
slope of the phase response tends to become steeper and
steeper at that frequency.
50
0
40
− 0.5
r = 0.99
r = 0.99
30
Phase shift, rad
−1.0
Gain, dB
20
10
r = 0.50
0
−2.0
−3.0
−20
Frame # 9 Slide # 16
−1.5
−2.5
−10
−30
r = 0.50
0
5
Frequency, rad/s
10
A. Antoniou
−3.5
0
5
Frequency, rad/s
10
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Highpass Biquad
A highpass second-order transfer function can be
constructed by placing a complex-conjugate pair of poles
anywhere inside the unit circle and a pair of zeros at point
(1, 0):
2
jIm z
1
0
−1
−2
−2
Frame # 10 Slide # 17
−1
A. Antoniou
0
Re z
1
2
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Highpass Biquad Cont’d
The transfer function of the highpass biquad assumes the
form:
HHP (z) =
(z − 1)2
(z 2 − 2z + 1)
=
z 2 − 2r (cos φ)z + r 2
z 2 − 2r (cos φ)z + r 2
where 0 < r < 1.
Frame # 11 Slide # 18
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Highpass Biquad Cont’d
As the poles move closer to the unit circle, the amplitude
response develops a peak at frequency ω = φ/T while the slope
of the phase response tends to become steeper and steeper at
that frequency.
50
3.5
40
3.0
30
Phase shift, rad
2.5
Gain, dB
20
10
r = 0.50
0
1.5
0.5
−20
Frame # 12 Slide # 19
r = 0.50
2.0
1.0
−10
−30
r = 0.99
r = 0.99
0
0
5
Frequency, rad/s
10
A. Antoniou
0
5
Frequency, rad/s
10
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Bandpass Biquad
A bandpass second-order transfer function can be
constructed by placing a complex-conjugate pair of poles
anywhere inside the unit circle, zeros at points (-1, 0) and
(1, 0):
2
jIm z
1
0
−1
−2
−2
Frame # 13 Slide # 20
−1
A. Antoniou
0
Re z
1
2
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Bandpass Biquad Cont’d
The transfer function of the bandpass biquad assumes the
form:
HBP (z) =
z2
(z + 1)(z − 1)
− 2r (cos φ)z + r 2
where 0 < r < 1.
Frame # 14 Slide # 21
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Bandpass Biquad Cont’d
As the poles move closer to the unit circle, the amplitude
response develops a peak at frequency ω = φ/T while the slope
of the phase response tends to become steeper and steeper at
that frequency.
40
2
r = 0.99
30
r = 0.99
1
Phase shift, rad
Gain, dB
20
10
r = 0.50
0
−10
r = 0.50
0
−1
−20
−30
Frame # 15 Slide # 22
0
5
Frequency, rad/s
10
A. Antoniou
−2
0
5
Frequency, rad/s
10
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Notch Biquad
A notch second-order transfer function can be constructed
by placing a complex-conjugate pair of poles anywhere
inside the unit circle, and a complex-conjugate pair of
zeros on the unit circle.
jIm z
There are three possibilities:
1
1
1
0
0
0
−1
−1
−1
−1
0
1
−1
0
1
−1
0
Re z
Frame # 16 Slide # 23
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
1
Second-Order Notch Biquad Cont’d
The transfer function of the bandpass biquad assumes the
form:
HN (z) =
z 2 − 2(cos ψ)z + 1
z 2 − 2r (cos φ)z + r 2
where 0 < r < 1.
Frame # 17 Slide # 24
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Notch Biquad Cont’d
If ψ = π/4, ψ = π/2, or ψ = 3π/4, the notch filter behaves as a
highpass, bandstop, or lowpass filter.
30
4
ψ = π/4
ψ = π/2
ψ = 3π/4
3
20
2
Phase shift, rad
Gain, dB
10
0
−10
1
0
−1
−2
−20
−30
Frame # 18 Slide # 25
−3
0
5
Frequency, rad/s
10
A. Antoniou
−4
0
5
Frequency, rad/s
10
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Allpas Biquad
An allpass second-order transfer function can be
constructed by placing a complex-conjugate pair of poles
anywhere inside the unit circle and a complex-conjugate
pair of zeros that are the reciprocals of the poles outside
the unit circle.
2
jIm z
1
0
−1
−2
−2
Frame # 19 Slide # 26
−1
A. Antoniou
0
Re z
1
2
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Allpass Biquad Cont’d
The transfer function of the bandpass biquad assumes the
form:
HAP (z) =
r 2 z 2 − 2r (cos φ)z + 1
z 2 − 2r (cos φ)z + r 2
where 0 < r < 1.
Frame # 20 Slide # 27
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Allpass Biquad Cont’d
The transfer function of the bandpass biquad assumes the
form:
HAP (z) =
r 2 z 2 − 2r (cos φ)z + 1
z 2 − 2r (cos φ)z + r 2
where 0 < r < 1.
We note that the numerator coefficients are the same as
the denominator coefficients but in the reverse order.
Frame # 20 Slide # 28
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Allpass Biquad Cont’d
The transfer function of the bandpass biquad assumes the
form:
HAP (z) =
r 2 z 2 − 2r (cos φ)z + 1
z 2 − 2r (cos φ)z + r 2
where 0 < r < 1.
We note that the numerator coefficients are the same as
the denominator coefficients but in the reverse order.
The above is a general property, that is, an arbitrary
transfer function with the above coefficient symmetry is an
allpass transfer function independently of the order.
Frame # 20 Slide # 29
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Second-Order Allpass Biquad Cont’d
MAP (ω)
= |HAP (e
jωT
)| = HAP (e
jωT
1
∗
jωT 2
) · HAP (e )
1
= HAP (ejωT ) · HAP (e−jωT ) 2
=
HAP (z) · HAP (z −1 )
=
=
1
2
z=ejωT
r 2 z 2 + 2r (cos φ)z + 1 r 2 z −2 + 2r (cos φ)z −1 + 1
· −2
z 2 + 2r (cos φ)z + r 2
z + 2r (cos φ)z −1 + r 2
r 2 z 2 + 2r (cos φ)z + 1 r 2 + 2r (cos φ)z + z 2
·
z 2 + 2r (cos φ)z + r 2 1 + 2r (cos φ)z + z 2 r 2
Frame # 21 Slide # 30
A. Antoniou
1
2
z=ejωT
1
2
=1
z=ejωT
Digital Signal Processing – Secs. 5.6 and 5.7
High-Order Filters Cont’d
Higher-order transfer functions can be obtained by forming
products or sums of first- and/or second-order transfer
functions.
Frame # 22 Slide # 31
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
High-Order Filters Cont’d
Higher-order transfer functions can be obtained by forming
products or sums of first- and/or second-order transfer
functions.
Corresponding high-order filters can be constructed by
connecting several biquads in cascade or in parallel.
Frame # 22 Slide # 32
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
High-Order Filters Cont’d
Higher-order transfer functions can be obtained by forming
products or sums of first- and/or second-order transfer
functions.
Corresponding high-order filters can be constructed by
connecting several biquads in cascade or in parallel.
Methods for obtaining transfer functions that will yield
specified frequency responses will be explored in later
chapters.
Frame # 22 Slide # 33
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion
In practice, a discrete-time system can distort the
information content of a signal to be processed.
Frame # 23 Slide # 34
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion
In practice, a discrete-time system can distort the
information content of a signal to be processed.
Two types of distortion can be introduced as follows:
Frame # 23 Slide # 35
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion
In practice, a discrete-time system can distort the
information content of a signal to be processed.
Two types of distortion can be introduced as follows:
– Amplitude distortion
Frame # 23 Slide # 36
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion
In practice, a discrete-time system can distort the
information content of a signal to be processed.
Two types of distortion can be introduced as follows:
– Amplitude distortion
– Delay (or phase) distortion
Frame # 23 Slide # 37
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
Consider an application where a digital filter characterized by a
transfer function H(z) is to be used to select a specific signal
xk (nT ) from a sum of signals
x (nT ) =
m
xi (nT )
i=1
Frame # 24 Slide # 38
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
Consider an application where a digital filter characterized by a
transfer function H(z) is to be used to select a specific signal
xk (nT ) from a sum of signals
x (nT ) =
m
xi (nT )
i=1
Let the amplitude and phase responses of the filter be M(ω) and
θ (ω), respectively.
Frame # 24 Slide # 39
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
Consider an application where a digital filter characterized by a
transfer function H(z) is to be used to select a specific signal
xk (nT ) from a sum of signals
x (nT ) =
m
xi (nT )
i=1
Let the amplitude and phase responses of the filter be M(ω) and
θ (ω), respectively.
Two parameters associated with the phase response are the
absolute delay τa (ω) and the group delay τg (ω) which are
defined as
τa (ω) = −
Frame # 24 Slide # 40
θ (ω)
ω
A. Antoniou
and τg (ω) = −
dθ (ω)
dω
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
Consider an application where a digital filter characterized by a
transfer function H(z) is to be used to select a specific signal
xk (nT ) from a sum of signals
x (nT ) =
m
xi (nT )
i=1
Let the amplitude and phase responses of the filter be M(ω) and
θ (ω), respectively.
Two parameters associated with the phase response are the
absolute delay τa (ω) and the group delay τg (ω) which are
defined as
τa (ω) = −
θ (ω)
ω
and τg (ω) = −
dθ (ω)
dω
As functions of frequency, τa (ω) and τg (ω) are known as the
absolute-delay and group-delay characteristics.
Frame # 24 Slide # 41
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
Now assume that the amplitude spectrum of signal xk (nT ) is
concentrated in frequency band B given by
B = {ω : ωL ≤ ω ≤ ωH }
as shown.
|Xk(ejωT)|
|X(ejωT)|
ωL
Frame # 25 Slide # 42
A. Antoniou
B
ωH
ω
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
Now assume that the amplitude spectrum of signal xk (nT ) is
concentrated in frequency band B given by
B = {ω : ωL ≤ ω ≤ ωH }
as shown.
Also assume that the filter has amplitude and phase responses
G0 for ω ∈ B
M(ω) =
and θ (ω) = −τg ω + θ0 for ω ∈ B
0
otherwise
respectively, where G0 and τg are constants.
|Xk(ejωT)|
|X(ejωT)|
ωL
Frame # 25 Slide # 43
A. Antoniou
B
ωH
ω
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
The z transform of the output of the filter is given by
Y (z) = H(z)X (z) = H(z)
m
i=1
Frame # 26 Slide # 44
A. Antoniou
Xi (z) =
m
H(z)Xi (z)
i=1
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
The z transform of the output of the filter is given by
Y (z) = H(z)X (z) = H(z)
m
i=1
Xi (z) =
m
H(z)Xi (z)
i=1
Thus the frequency spectrum of the output signal is
obtained as
Y (ejωT ) =
m
H(ejωT )Xi (ejωT )
i=1
=
m
M(ω)ejθ(ω) Xi (ejωT )
i=1
Frame # 26 Slide # 45
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
···
Y (ejωT ) =
m
M(ω)ejθ(ω) Xi (ejωT )
i=1
We have assumed that
G0 for ω ∈ B
M(ω) =
0
otherwise
and θ(ω) = −τg ω+θ0
for ω ∈ B
and hence we get
Y (ejωT ) = G0 e−jωτg +jθ0 Xk (ejωT )
since all signal spectrums except Xk (ejωT ) will be multiplied
by zero.
Frame # 27 Slide # 46
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
···
Y (ejωT ) = G0 e−jωτg +jθ0 Xk (ejωT )
If we now let τg = mT where m is a constant, we can write
Y (z) = G0 ejθ0 z −m Xk (z)
Frame # 28 Slide # 47
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
···
Y (ejωT ) = G0 e−jωτg +jθ0 Xk (ejωT )
If we now let τg = mT where m is a constant, we can write
Y (z) = G0 ejθ0 z −m Xk (z)
Therefore, from the time-shifting theorem of the z transform, we
deduce the output of the filter as
y (nT ) = G0 ejθ0 xk (nT − mT )
Frame # 28 Slide # 48
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
···
Y (ejωT ) = G0 e−jωτg +jθ0 Xk (ejωT )
If we now let τg = mT where m is a constant, we can write
Y (z) = G0 ejθ0 z −m Xk (z)
Therefore, from the time-shifting theorem of the z transform, we
deduce the output of the filter as
y (nT ) = G0 ejθ0 xk (nT − mT )
In effect, if the amplitude response of the filter is constant with
respect to frequency band B and zero elsewhere and its phase
response is a linear function of ω, that is, the group delay is
constant in frequency band B, then the output signal is a
delayed replica of signal xk (nT ) except that a constant multiplier
G0 ejθ0 is introduced.
Frame # 28 Slide # 49
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
If the amplitude response of the system is not constant in
frequency band B, then so-called amplitude distortion will
be introduced since different frequency components of the
signal will be amplified by different amounts.
Frame # 29 Slide # 50
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
If the amplitude response of the system is not constant in
frequency band B, then so-called amplitude distortion will
be introduced since different frequency components of the
signal will be amplified by different amounts.
If the group delay is not constant in band B, different
frequency components will be delayed by different
amounts, and delay (or phase) distortion will be introduced.
Frame # 29 Slide # 51
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
Amplitude and Delay Distortion Cont’d
Amplitude distortion can be quite objectionable in practice.
Consequently, the amplitude response is required to be flat
to within a prescribed tolerance in each frequency band
that carries information.
If the ultimate receiver of the signal is the human ear, e.g.,
when a speech or music signal is to be processed, delay
distortion turns out to be quite tolerable.
In other applications where images are involved, e.g.,
transmission of video signals, delay distortion can be as
objectionable as amplitude distortion, and the delay
characteristic is required to be fairly flat.
Frame # 30 Slide # 52
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7
This slide concludes the presentation.
Thank you for your attention.
Frame # 31 Slide # 53
A. Antoniou
Digital Signal Processing – Secs. 5.6 and 5.7