1) A system transforms an input signal into an output signal through interconnections between components. Systems can be continuous-time or discrete-time depending on if the input and output signals are continuous or discrete. 2) Systems are linear if they satisfy superposition, meaning the response to a weighted sum of inputs equals the sum of responses. Nonlinear systems do not satisfy superposition. 3) A system is time-invariant if a time shift in the input results in the same time shift in the output. Otherwise, it is time-varying.

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Lecture 3
Systems and Classification of Systems
Systems
A system is an interconnection of components that transforms an input signal into an
output signal. We can therefore view a system as a mapping (or
transformation) from an input function onto an output function.
A. Continuous-time and Discrete-time systems:
If the input (x) and output (y) are continuous-time (CT)signals, then the systemis called
a continuous-time system. If the input (x) and output (y) are discrete-time (DT) signals
or sequence, then the system is called a discrete-time system. Referring to the above
Figs
y(t) = H x(t)= H u(t) for continuous-time system
y[n] = H x[n]= H u[n] for discrete-time system
Hybrid system may have CT and DT input and output signals.
B. Linear and NonlinearSystems
A system is linear if it satisfies the principle of superposition (additivity and
homogeneity)
That is, the response(y) of a linear system to a weighted sum of inputs (x) equal to the
same weighted sum of output signals.

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Example
Example

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Example
Example

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Any system that does not satisfy the superposition principle is classified as nonlinear
system. Example of nonlinear systems are
A system is nonlinear if it has
 Nonlinear elements
 Nonzero initial condition
 Internal sources
C. Time-invariant and Time-varying Systems
A system is called time-invariant or fixed, if a time-shift (delay or advance) in the input
signal causes the same time shift in the out-put signal. That is its input-output
relationship does not change with time, i.e., if H [x(t)] = y(t), then
H [x(t-T)] = y(t-T) for any value of T
For example, the system described by y(t) =
x(t)+A x(t-T)

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is time-invariant if A and T are constants.
For a discrete-time system, the system is time-variant if
H [x(n-k)] = y(n- ) for any value of k
In the time-invariant systems the shape of the response y(t) depends only on the shape
of the input x(t) and not on the time when it is applied. When one more coefficients are
function of time, the system is called time-varying system.
Example
Example

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Example

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D. Linear Time-invariant Systems
If the system is linear and also time-invariant, then it is called a linear time-invariant
(LTI) system.
E. Systems with Memory and without Memory
A system is said to be memory-less if the output at any time depends on only the input
at that time. Otherwise, the system is said to have memory.
Example
(i) Memory-less system → resistive system
(ii) System with memory → capacitive or inductive
system given by
F. Instantaneous system and Dynamic system
Systems may be modeled by instantaneous (Non dynamic) linear relationships such as
y(t) = Ax(t)+B
Where A and B are constants, or a nonlinear relationship such as y(t)
= Ax2(t)+Bx(t)+C
Instantaneous systems are referred to as memory-less since the output y(t) depends only
on the instantaneous value of x(t).
The discrete-time signal described by y[n] = 2x[n] is memory-less, since the value of
y[n] at time n depends only on the present value of the input x[n].
Dynamic Systems
Continuous-time systems are often modeled by linear time-invariant differential
equation. In general the input-output relationship may be given by

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The coefficients of the above differential equation are the parameters of the physical
system. Dynamic systems are systems with memory. For example an inductor has
memory, since the current in an inductor given by
depends on all past values of the voltage v(t). The memory of an inductor extends into
the infinite past.
G. Causaland NoncausalSystems
A system is called causal if its output y(t) at an arbitrary time t = to, depends ononly the
input x(t) for t ≤ to,. That is, the output of a causal system at the present time depends
on only the present and/or past values of the input, not on its future values. Thus, in a
causal system, it is not possible to obtain an output before an input is applied to the
system. A system is called noncausal if it is not causal. Examples of noncausal systems
are
y(t) = x (t+1) y[n]=x[n]
Note that all memoryless systems are causal, but not vice versa.