Different Types of Systems
- Continuous and Discrete Time Systems
- Fixed and Time-varying Systems
- Linear and Non-linear Systems
- Lumped and Distributed Systems
- Instantaneous and Dynamic Systems
- Active and Passive Systems
- Causal and Non-causal Systems
- Stable and Unstable Systems
- Invertible and Non-invertible Systems
Continuous and Discrete Time Systems
Signals are represented mathematically as functions of one or more independent variables. We classify signals as being either continuous-time (functions of a real-valued variable) or discrete-time (functions of an integer-valued variable).
(a) Continuous-time signal (b) Discrete-time signal
The examples given below are common in our daily life.
Continuous-time systems
- Atmospheric pressure as a function of altitude
- Electric circuits composed of resistors, inductors, capacitors driven by continuous-time sources.
Discrete-time systems
- Weekly stock market index
- Balance in a bank account from month to month.
Time-Invariant (Fixed) and Time-Varying Systems
A system is time-invariant or fixed if the behaviour and characteristics of the system do not change with time. Otherwise, the system is time-varying.
Mathematically, if the input x(t) gives the output y(t), then the system is time-invariant if the input x(t – T) gives the output y(t – T) for any delay T. Hence, a time-shift of the input gives the same time-shift of the output.
Time-invariant system
Whether a system is time-invariant or time-varying can be seen in the differential equation (or difference equation) describing it. Time-invariant systems are modeled with constant coefficient equations. A constant coefficient differential (or difference) equation means that the parameters of the system are not changing over time and an input now will give the same result as the input later.
Linear and Non-Linear Systems
A system, in continuous-time or discrete-time, is said to be linear, if it obeys the properties of superposition, i.e., additivity and homogeneity (or scaling), while a system is non-linear if it does not obey at least any one of these properties.
The superposition principle says that the output to a linear combination of input signals is the same linear combination of the corresponding output signals. Mathematically, the linearity condition is based on two properties.
- Additivity If the input signals x_{1}(t) and x_{2}(t) correspond to the output signals y_{1}(t) and y_{2}(t), respectively, then the input signal {x_{1}(t) + x_{2}(t)} should correspond to the output signal {y_{1}(t) + y_{2}(t)}.
- Homogeneity If the input signal x_{1}(t) corresponds to the output signal y_{1}(t), then the input signal a_{1}x_{1}(t) should correspond to the output signal a_{1}y_{1}(t) for any constants a_{1}.
Lumped and Distributed Systems
All physical systems contain distributed parameters because of the physical size of the system components. For example, the resistance of a resistor is distributed throughout its volume.
However, if the size of the system components is very small with respect to the wavelength of the highest frequency present in the signals associated with it, then the system components behave as if it all were occurring at a point. This system is said to be lumped-parameter system.
Distributed parameter systems are modeled as given below.
- By partial differential equations if they are continuous-time systems
- By partial difference equations if they are discrete-time systems.
Lumped parameter systems are modeled with ordinary differential or difference equations.
Instantaneous (Static or Memoryless) and Dynamic Systems
An instantaneous or static or memoryless system is a system where the output at any specific time depends on the input at that time only. On the other hand, a dynamic system is one whose output depends on the past or future values of the input in addition to the present time.
A static system has no memory. Physically, it contains no energy-storage elements, whereas a dynamic system has one or more energy-storage element(s).
Active and Passive Systems
A system having no source of energy is known as a passive system, for example, electric circuits containing resistance, capacitance, inductance, diodes, etc.
A system having source of energy together with other passive elements is known as an active system, for example, electric circuits containing voltage source or current source or op-amp, etc.
Causal and Non-causal Systems
A system is said to be causal if the output of the system depends only on the input at the present time and/or in the past, but not the future value of the input. Thus, a causal system is nonanticipative, i.e., output cannot come before the input.
On the other hand, the output of a non-causal system depends on the future values of the input.
(a) Causal systems (b) Non-causal systems
Stable and Unstable Systems
A stable system is one where the output does not diverge as long as the input does not diverge. A bounded input produces a bounded output. For this reason, this type of system is known as bounded input-bounded output (BIBO) stable system.
Mathematically, a stable system must have the following property:
If x(t) be the input and y(t) be the output, then the output must satisfy the condition.
| y(t) | ≤ M_{y} < ∝; for all t
whenever the input satisfy the condition
| x(t) | ≤ M_{x} < ∝; for all t
where, M_{x} and M_{y}_{ }both represent a set of finite positive numbers.
If these conditions are not met, i.e., the output of the system grows without limit (diverges) from a bounded input, then the system is unstable.
Invertible and Non-invertible Systems
A system is referred to as an invertible system if
- distinct inputs lead to distinct output, and
- the input can be recovered from the output.
Invertible system
The property of invariability is important in the design of communication systems. When a transmitted signal propagates through a communication channel, it becomes distorted due to the physical characteristics of the channel. An equalizer is connected in cascade with the channel in the receiver to compensate this distortion. By designing the equalizer to be inverse of the channel, the transmitted signal is restored.