System
A system can be viewed as a process for which cause-and effect relations exist.
An example of a physical system is an electric heater where the input signal is the ac voltage,
v(t), and we consider the output signal to be the temperature, of a certain point in space that is
close to the heater.
One representation of this system is the block diagram shown in Fig 2.1.
Fig. 2.1 System representation 1.
Another example is a system with input voltage v (t) and output voltage v0 (t) shown in Fig 2.2.
Fig. 2.2 System representation
Types of system
A continuous-time
A continuous-time system is a system in which continuous time input signals are applied and
results in continuous-time output signals.
Fig. 2.3 A continuous-time System representation.
A discrete-time
A discrete-time system is a system in which discrete-time input signals are applied and results
in discrete-time output signals.
..
Fig. 2.4 A discrete-time system representation
Classification of Systems
Systems are classified into the following categories:
1. Linear and Non-linear Systems
2. Time Variant and Time Invariant Systems
3. Linear Time variant and Linear Time invariant systems
4. Static and Dynamic Systems
5. Causal and Non-causal Systems
6. Invertible and Non-Invertible Systems
7. Stable and Unstable Systems
Linear and Non-linear Systems
A system is said to be linear when it satisfies the superposition and homogenate principles. By
linearity, one can describe the effects of a system by separating the input signal into simple parts
and using superposition at the output to restore the overall system output.
In other words, the system is linear if:
Consider two systems with inputs as x1 (t), x2 (t), and outputs as y1 (t), y2 (t) respectively.
Then, according to the superposition and homogenate principles, this condition must hold true:
From the above expression, is clear that the response of the overall system is equal to response
of individual system.
Time Variant and Time Invariant Systems
A system is said to be time variant if its input and output characteristics vary with time.
Otherwise, it is considered as time invariant.
A system is time invariant if a time shift in the input signal results in an identical time shift in
the output signal.
Linear Time Variant and Linear Time Invariant Systems
If a system is both linear and time variant, then it is called a linear time variant (LTV) system.
On the other hand, if a system is both linear and time invariant, then it is called a linear time
invariant (LTI) system.
Static and Dynamic Systems
A static system is a memory-less system whereas a dynamic system is a system with memory.
A system is memory-less if its output at a given time is dependent only on the input at that same
time. A memory-less system does not have memory to store any input values because it just
operates on the current input.
EXAMPLE: y(t) = 2x(t).
For the present value t=0, the system output is y(0) = 2x(0). Here, the output is only dependent
upon the present input. Hence the system is memory less or static is also memory-less.
Causal and Non-Causal Systems
A system is said to be causal if its output depends upon present and past inputs, but does not
depend upon future input.
Such a system is often referred to as being non-anticipative, as the system output does not
anticipate future values of the input.
EXAMPLE :y(t)=2x(t)+3x(t-3)
For the present value t=1, the system output is y(1) = 2x(1) + 3x(-2). Here, the system output
only depends upon present and past inputs. Hence, the system is causal.
For a non-causal system, the output depends upon the future inputs also.
EXAMPLE: y(t) = 2x(t) + 3x(t-3) + 6x(t+3).
For the present value t=1, the system output is y(1) = 2x(1) + 3x(-2) + 6x(4). Here, the system
output depends upon the future input. Hence the system is a non-causal system.
Invertible And Non-Invertible Systems
A system is said to invertible if the input of the system appears at the output.
In other words, a system is invertible if distinct inputs lead to distinct outputs, or if an inverse
system exists.
Stable and Unstable Systems
A stable system is one in which small inputs leads to responses that do not diverge. More
formally, if the input to a stable system is bounded, then the output must be also bounded and
therefore cannot diverge.
More Examples: Check whether or not the following system is:
