17-03-2014, 06:17 PM
Abstract
In this paper, a new method for the state space
representation of a system is proposed, which is based
on the companion form technique. It is very important
to have almost equal coefficients of state space
equations since their coefficients are proportional to
devices’ currents or voltages, e.g. transistors’ currents
for log domain filters. The method gives us the
opportunity to choose two arbitrary parameters (α, β)
to be able to obtain more balanced state space
equations. This method is applied a log domain filter,
which can be considered an adaptive filter since it can
be electronically tuned. It is particularly useful for
higher order log domain filters synthesized in the state
space.
1. Introduction
There are several methods to synthesize a filter or
more generally a system. The state space synthesis
method is one of these techniques. Regarding this
method, we need state space equations, i.e. differential
equations and output equations [1], [2]. In this paper,
we propose a new method to obtain the state space
equations of a system from the transfer function.
When a transfer function is given, there could be a
number of state space equations. Clearly, each
representation leads to a new topology although some
of them cannot be realized by known devices.
The state space approach is a general method to
define a system. This method is a valuable tool for
control applications and circuit synthesis. The method
can be applied for both nonlinear systems and timevariant
systems. The method can also be used for
current-mode circuits, systematic synthesis and
computer aided design [2], [3].
The properties discussed above are well suited for
log domain filters or more generally ELIN, Externally
Linear Internally Nonlinear, circuits. These circuits can
be considered as adaptive since they can be
electronically tunable. By changing the values of
current sources, one can tune the filter’s pole
frequency, band gain and quality factor automatically.
These kind of adaptive hardware systems are applied to
communication, electronic, and even biological signal
processing systems [4-6].
The first systematic synthesis method of log domain
filters, called Frey’s state space synthesis method, was
introduced in [7] through the state space
representation. Later, some researches implemented
this method to synthesize this kind of circuits [4], [8],
[9-11]. Frey’s theory uses existing state space
representation of a system, but does not discuss how to
obtain it. Other researchers synthesizing filters in the
log domain in state space have also used existing state
space representations. It seems there is a lack of
discussion on how to obtain state space representation
of a system used in Frey’s theory in the literature
[12-15].
Coefficients of the state variables and input of a
state space representation are proportional to
transistors’ currents in the log domain. In order to
make the log domain filter work, transistors’ currents
have to be relatively close to each others. Otherwise, it
may not be possible to reach an equilibrium state.
Depending on the transistors used in a design, their
currents’ level should be in a limited range. For
example, if a transistor has a DC 1mA, same type of
transistor cannot have 1 A or 1 μA. Note that there are
difficulties in determining this kind of state space
representation.
In this paper, a new method for the state space
representation of a system is proposed, which is based
on the companion form technique. It is very important
to have almost equal coefficients of state space
equations since their coefficients are proportional to
devices’ currents or voltages, e.g. transistors’ currents
for log domain filters. The method gives us the
opportunity to choose two arbitrary parameters (α, β)
to be able to obtain more balanced state space
equations. This method is applied a log domain filter,
which can be considered an adaptive filter since it can
be electronically tuned. It is particularly useful for
higher order log domain filters synthesized in the state
space.
1. Introduction
There are several methods to synthesize a filter or
more generally a system. The state space synthesis
method is one of these techniques. Regarding this
method, we need state space equations, i.e. differential
equations and output equations [1], [2]. In this paper,
we propose a new method to obtain the state space
equations of a system from the transfer function.
When a transfer function is given, there could be a
number of state space equations. Clearly, each
representation leads to a new topology although some
of them cannot be realized by known devices.
The state space approach is a general method to
define a system. This method is a valuable tool for
control applications and circuit synthesis. The method
can be applied for both nonlinear systems and timevariant
systems. The method can also be used for
current-mode circuits, systematic synthesis and
computer aided design [2], [3].
The properties discussed above are well suited for
log domain filters or more generally ELIN, Externally
Linear Internally Nonlinear, circuits. These circuits can
be considered as adaptive since they can be
electronically tunable. By changing the values of
current sources, one can tune the filter’s pole
frequency, band gain and quality factor automatically.
These kind of adaptive hardware systems are applied to
communication, electronic, and even biological signal
processing systems [4-6].
The first systematic synthesis method of log domain
filters, called Frey’s state space synthesis method, was
introduced in [7] through the state space
representation. Later, some researches implemented
this method to synthesize this kind of circuits [4], [8],
[9-11]. Frey’s theory uses existing state space
representation of a system, but does not discuss how to
obtain it. Other researchers synthesizing filters in the
log domain in state space have also used existing state
space representations. It seems there is a lack of
discussion on how to obtain state space representation
of a system used in Frey’s theory in the literature
[12-15].
Coefficients of the state variables and input of a
state space representation are proportional to
transistors’ currents in the log domain. In order to
make the log domain filter work, transistors’ currents
have to be relatively close to each others. Otherwise, it
may not be possible to reach an equilibrium state.
Depending on the transistors used in a design, their
currents’ level should be in a limited range. For
example, if a transistor has a DC 1mA, same type of
transistor cannot have 1 A or 1 μA. Note that there are
difficulties in determining this kind of state space
representation.