Chapter 2: Bilinear transform
In digital signal processing and discrete-time control theory, the bilinear transform, which is sometimes referred to as Tustin's approach after Arnold Tustin, is utilized to convert continuous-time system representations into discrete-time representations and vice versa.
The bilinear transform is a specific instance of a conformal mapping, more specifically a Mobius transformation, which is frequently employed for the purpose of converting a given transfer function.
H.
a
()
s.
()
"H_a"(s)" is a display style.
the relationship between a linear, time-invariant (LTI) filter operating in the continuous-time domain (which is commonly referred to as an analog filter) and a transfer function
H.
d.
()
z
()
Displaying the function H_{d}(z)
in the discrete-time domain of a linear, shift-invariant filter (which is commonly referred to as a digital filter, but there are analog filters created using switched capacitors that are discrete-time filters). On the map, it depicts positions.
j
?
The display style used is j-omega.
the axis,
R.
e.
[[
s.
I]
=
0
The equation \Re} [s] = 0 is displayed in a display format.
with respect to the unit circle, in the s-plane
|
z
|
=
1.
The value of the variable "z" is equal to one.
when viewed from the z-plane. Additional bilinear transforms can be utilized for the purpose of warping the frequency response of any discrete-time linear system. For instance, they can be utilized to simulate the non-linear frequency resolution of the human auditory system. Furthermore, these transforms can be implemented in the discrete domain by effectively substituting the unit delays of a system.
()
z
--
1.
()
The display style is derived from the equation z raised to the power of -1.
when using all-pass filters of the first order.
In addition to maintaining stability, the transform maps each and every point of the frequency response of the continuous-time filter at the same time.
H.
a
()
j
?
a
()
The expression {\displaystyle H_{a}(j\omega _{a})
as a point in the frequency response of the discrete-time filter that corresponds to the point in question,
H.
d.
()
e.
j
?
d.
T.
()
The expression "H_{d}(e^{j\omega _{d}T})" looks like this:
nevertheless, to a frequency that is somewhat different, as will be demonstrated in the section on frequency warping that follows. This indicates that for each and every feature that is observed in the frequency response of the analog filter, there is a matching feature in the frequency response of the digital filter where the gain and phase shift are the same. However, the frequency of the digital filter may be somewhat different from the frequency of the analog filter. At low frequencies, the change in frequency is barely perceptible, but at frequencies that are quite close to the Nyquist frequency, it is quite noticeable.
A first-order Padé approximant of the natural logarithm function, the bilinear transform creates an exact mapping of the z-plane to the s-plane. This mapping is known as the bilinear transform. When the Laplace transform is applied to a discrete-time signal, the outcome is precisely the Z transform of the discrete-time sequence with the substitution of. This is because each and every element of the discrete-time sequence is associated to a unit impulse that is proportionally delayed.
where is the
T.
T is the display style.
represents the numerical integration step size of the trapezoidal rule that is utilized in the formulation of the bilinear transform; or, to put it another way, stands for the sample period. A solution can be found for the bilinear approximation described above.
s.
This is a display style.
or an approximation that is comparable for
s.
=
()
1.
I/
T.
()
"ln"
?
()
z
()
A display style of s equal to (1/T)ln(z)
able can be carried out?
The first-order bilinear approximation of this mapping, as well as the inverse of this mapping, is
In its most basic form, the bilinear transform makes use of this first-order approximation and incorporates it into the continuous-time transfer function.
H.
a
()
s.
()
"H_a"(s)" is a display style.
That is the case
If the poles of a continuous-time causal filter are located on the left half of the complex s-plane, then the filter is considered to be stable during the process. If the poles of a discrete-time causal filter are located within the unit circle in the complex z-plane, then the filter is considered to be stable. Through the use of the bilinear transform, the left half of the complex s-plane is transformed into the interior of the unit circle in the z-plane. It is therefore possible to transform filters that were built in the continuous-time domain and are stable into filters that are created in the discrete-time domain and that maintain that stability.
on a similar vein, a continuous-time filter is considered to be minimum-phase if the zeros of its transfer function are located on the left half of the complicated two-dimensional plane. If the zeros of a discrete-time filter's transfer function are located within the unit circle in the complex z-plane, then the filter is considered to meet the minimum-phase condition. Once this is accomplished, the same mapping property ensures that continuous-time filters that are minimum-phase are translated into discrete-time filters that maintain the property of being minimum-phase.
There is a transfer function present in a general LTI system.
H.
a
()
s.
()
=
b.
0
+1
b.
1.
s.
+1
b.
2.
s.
2.
+1
*
+1
b.
Q
s.
Q
a
0
+1
a
1.
s.
+1
a
2.
s.
2.
+1
*
+1
a
P
s.
P
{\displaystyle H_{a}(s)={\frac {b_{0}+b_{1}s+b_{2}s^{2}+\cdots +b_{Q}s^{Q}}{a_{0}+a_{1}s+a_{2}s^{2}+\cdots +a_{P}s^{P}}}}
In practice, the order of the transfer function N is the greater of P and Q; nevertheless, Q is more likely to be the greater of the two. This is because the transfer function must be appropriate for the system to be stable. Putting the bilinear transform into practice
s.
=
K.
z
--
1.
z
+1
1.
(s) equals K multiplied by the ratio of z-1 to z+1.
given that K is defined as either 2/T or, alternatively, if frequency warping is being used, given that
H.
d.
()
z
()
=
b.
0
+1
b.
1.
()
K.
z
--
1.
z
+1
1.
()
+1
b.
2.
()
K.
z
--
1.
z
+1
1.
()
2.
+1
*
+1
b.
Q
()
K.
z
--
1.
z
+1
1.
()
Q
a
0
+1
a
1.
()
K.
z
--
1.
z
+1
1.
()
+1
a
2.
()
K.
z
--
1.
z
+1
1.
()
2.
+1
*
+1
b.
P
()
K.
z
--
1.
z
+1
1.
()
P
{\displaystyle H_{d}(z)={\frac {b_{0}+b_{1}\left(K{\frac {z-1}{z+1}}\right)+b_{2}\left(K{\frac {z-1}{z+1}}\right)^{2}+\cdots +b_{Q}\left(K{\frac {z-1}{z+1}}\right)^{Q}}{a_{0}+a_{1}\left(K{\frac {z-1}{z+1}}\right)+a_{2}\left(K{\frac {z-1}{z+1}}\right)^{2}+\cdots +b_{P}\left(K{\frac {z-1}{z+1}}\right)^{P}}}}
When the numerator and denominator are multiplied by the biggest power of (z + 1)-1 that is present, which is denoted by (z + 1)-N, the result...
