The bilinear transform (also known as Tustin's method, after
Arnold Tustin) is used in
digital signal processing and discrete-time
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
to transform continuous-time system representations to discrete-time and vice versa.
The bilinear transform is a special case of a
conformal mapping (namely, a
Möbius transformation), often used to convert a
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
of a
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
,
time-invariant
In control theory, a time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is ...
(
LTI) filter in the
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
-time domain (often called an
analog filter
Analogue filters are a basic building block of signal processing much used in electronics. Amongst their many applications are the separation of an audio signal before application to bass, mid-range, and tweeter loudspeakers; the combining and ...
) to a transfer function
of a linear, shift-invariant filter in the
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a g ...
-time domain (often called a
digital filter
In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, t ...
although there are analog filters constructed with
switched capacitors that are discrete-time filters). It maps positions on the
axis,
, in the
s-plane
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the com ...
to the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
,
, in the
z-plane. Other bilinear transforms can be used to warp the
frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...
of any discrete-time linear system (for example to approximate the non-linear frequency resolution of the human auditory system) and are implementable in the discrete domain by replacing a system's unit delays
with first order
all-pass filter
An all-pass filter is a signal processing filter that passes all frequencies equally in gain, but changes the phase relationship among various frequencies. Most types of filter reduce the amplitude (i.e. the magnitude) of the signal applied to i ...
s.
The transform preserves
stability
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
** Asymptotic stability
** Linear stability
** Lyapunov stability
** Orbital stability
** Structural sta ...
and maps every point of the
frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...
of the continuous-time filter,
to a corresponding point in the frequency response of the discrete-time filter,
although to a somewhat different frequency, as shown in the
Frequency warping section below. This means that for every feature that one sees in the frequency response of the analog filter, there is a corresponding feature, with identical gain and phase shift, in the frequency response of the digital filter but, perhaps, at a somewhat different frequency. This is barely noticeable at low frequencies but is quite evident at frequencies close to the
Nyquist frequency
In signal processing, the Nyquist frequency (or folding frequency), named after Harry Nyquist, is a characteristic of a sampler, which converts a continuous function or signal into a discrete sequence. In units of cycles per second ( Hz), it ...
.
Discrete-time approximation
The bilinear transform is a first-order
Padé approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is ap ...
of the natural logarithm function that is an exact mapping of the ''z''-plane to the ''s''-plane. When the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
is performed on a discrete-time signal (with each element of the discrete-time sequence attached to a correspondingly delayed
unit impulse
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (al ...
), the result is precisely the
Z transform of the discrete-time sequence with the substitution of
:
where
is the
numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
step size of the
trapezoidal rule
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule; see Trapezoid for more information on terminology) is a technique for approximating the definite integral.
\int_a^b f(x) \, dx.
The trapezoidal rule works by ...
used in the bilinear transform derivation; or, in other words, the sampling period. The above bilinear approximation can be solved for
or a similar approximation for
can be performed.
The inverse of this mapping (and its first-order bilinear
approximation) is
:
The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function,
:
That is
:
Stability and minimum-phase property preserved
A continuous-time causal filter is
stable if the
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in C ...
of its transfer function fall in the left half of the
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
s-plane
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the com ...
. A discrete-time causal filter is stable if the poles of its transfer function fall inside the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
in the
complex z-plane. The bilinear transform maps the left half of the complex s-plane to the interior of the unit circle in the z-plane. Thus, filters designed in the continuous-time domain that are stable are converted to filters in the discrete-time domain that preserve that stability.
Likewise, a continuous-time filter is
minimum-phase if the
zeros of its transfer function fall in the left half of the complex s-plane. A discrete-time filter is minimum-phase if the zeros of its transfer function fall inside the unit circle in the complex z-plane. Then the same mapping property assures that continuous-time filters that are minimum-phase are converted to discrete-time filters that preserve that property of being minimum-phase.
Transformation of a General LTI System
A general
LTI system
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defin ...
has the transfer function
The order of the transfer function is the greater of and (in practice this is most likely as the transfer function must be
proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
for the system to be stable). Applying the bilinear transform
where is defined as either or otherwise if using
frequency warping, gives
Multiplying the numerator and denominator by the largest power of present, , gives
It can be seen here that after the transformation, the degree of the numerator and denominator are both .
Consider then the pole-zero form of the continuous-time transfer function
The roots of the numerator and denominator polynomials, and , are the
zeros and poles
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if ...
of the system. The bilinear transform is a
one-to-one mapping
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
, hence these can be transformed to the z-domain using
yielding some of the discretized transfer function's zeros and poles and
As described above, the degree of the numerator and denominator are now both , in other words there is now an equal number of zeros and poles. The multiplication by means the additional zeros or poles are
Given the full set of zeros and poles, the z-domain transfer function is then
Example
As an example take a simple
low-pass
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filte ...
RC filter
A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC ci ...
. This continuous-time filter has a transfer function
:
If we wish to implement this filter as a digital filter, we can apply the bilinear transform by substituting for
the formula above; after some reworking, we get the following filter representation:
:
The coefficients of the denominator are the 'feed-backward' coefficients and the coefficients of the numerator are the 'feed-forward' coefficients used to implement a real-time
digital filter
In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, t ...
.
Transformation for a general first-order continuous-time filter
It is possible to relate the coefficients of a continuous-time, analog filter with those of a similar discrete-time digital filter created through the bilinear transform process. Transforming a general, first-order continuous-time filter with the given transfer function
:
using the bilinear transform (without prewarping any frequency specification) requires the substitution of
:
where
:
.
However, if the frequency warping compensation as described below is used in the bilinear transform, so that both analog and digital filter gain and phase agree at frequency
, then
:
.
This results in a discrete-time digital filter with coefficients expressed in terms of the coefficients of the original continuous time filter:
:
Normally the constant term in the denominator must be normalized to 1 before deriving the corresponding
difference equation. This results in
:
The difference equation (using the
Direct form I) is
:
General second-order biquad transformation
A similar process can be used for a general second-order filter with the given transfer function
:
This results in a discrete-time
digital biquad filter with coefficients expressed in terms of the coefficients of the original continuous time filter:
:
Again, the constant term in the denominator is generally normalized to 1 before deriving the corresponding
difference equation. This results in
:
The difference equation (using the
Direct form I) is
:
Frequency warping
To determine the frequency response of a continuous-time filter, the
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
is evaluated at
which is on the
axis. Likewise, to determine the frequency response of a discrete-time filter, the transfer function
is evaluated at
which is on the unit circle,
. The bilinear transform maps the
axis of the ''s''-plane (of which is the domain of
) to the unit circle of the ''z''-plane,
(which is the domain of
), but it is not the same mapping
which also maps the
axis to the unit circle. When the actual frequency of
is input to the discrete-time filter designed by use of the bilinear transform, then it is desired to know at what frequency,
, for the continuous-time filter that this
is mapped to.
:
:
This shows that every point on the unit circle in the discrete-time filter z-plane,
is mapped to a point on the
axis on the continuous-time filter s-plane,
. That is, the discrete-time to continuous-time frequency mapping of the bilinear transform is
:
and the inverse mapping is
:
The discrete-time filter behaves at frequency
the same way that the continuous-time filter behaves at frequency
. Specifically, the gain and phase shift that the discrete-time filter has at frequency
is the same gain and phase shift that the continuous-time filter has at frequency
. This means that every feature, every "bump" that is visible in the frequency response of the continuous-time filter is also visible in the discrete-time filter, but at a different frequency. For low frequencies (that is, when
or
), then the features are mapped to a ''slightly'' different frequency;
.
One can see that the entire continuous frequency range
:
is mapped onto the fundamental frequency interval
:
The continuous-time filter frequency
corresponds to the discrete-time filter frequency
and the continuous-time filter frequency
correspond to the discrete-time filter frequency
One can also see that there is a nonlinear relationship between
and
This effect of the bilinear transform is called frequency warping. The continuous-time filter can be designed to compensate for this frequency warping by setting
for every frequency specification that the designer has control over (such as corner frequency or center frequency). This is called pre-warping the filter design.
It is possible, however, to compensate for the frequency warping by pre-warping a frequency specification
(usually a resonant frequency or the frequency of the most significant feature of the frequency response) of the continuous-time system. These pre-warped specifications may then be used in the bilinear transform to obtain the desired discrete-time system. When designing a digital filter as an approximation of a continuous time filter, the frequency response (both amplitude and phase) of the digital filter can be made to match the frequency response of the continuous filter at a specified frequency
, as well as matching at DC, if the following transform is substituted into the continuous filter transfer function.
This is a modified version of Tustin's transform shown above.
:
However, note that this transform becomes the original transform
:
as
.
The main advantage of the warping phenomenon is the absence of aliasing distortion of the frequency response characteristic, such as observed with
Impulse invariance
Impulse invariance is a technique for designing discrete-time infinite-impulse-response (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of the discre ...
.
See also
*
Impulse invariance
Impulse invariance is a technique for designing discrete-time infinite-impulse-response (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of the discre ...
*
Matched Z-transform method
References
External links
MIT OpenCourseWare Signal Processing: Continuous to Discrete Filter DesignLecture Notes on Discrete EquivalentsThe Art of VA Filter Design
{{DEFAULTSORT:Bilinear Transform
Digital signal processing
Transforms
Control theory