BASIC (Common) DISCRETE-TIME SIGNALS
The unit step sequence u[n] is defined as
The unit impulse (or unit sample) sequence
d[n] is defined as
The last of these is especially important as it gives rise to the sifting
property of the unit sample function, which selects the value of a
function at a specific time and is especially important in studying the
relationship of an operation called convolution to time domain analysis
of linear time invariant systems. The sifting property is shown and
derived below
Its general discrete form is written as
Introduction to Systems
System Representation:
Ø A system is a mathematical model of a physical process that relates the input
(or excitation) signal to the output (or response) signal.
Ø Then the system is viewed as a transformation (or mapping) of x into y. This
transformation is represented by the mathematical notation
System Classifications and Properties
v Classification of Systems:
A. Continuous;Time and Discrete-Time Systems:
B. Causal vs. Noncausal systems
• A system is called causal if its output y ( t ) at an arbitrary time t = t0 depends on only
the input x ( t ) for t ≤ t0. 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.
• Similarly, an anticausal system is one in which the output depends only on current or
future inputs, but not past inputs. Finally, a noncausal system is one in which the
output depends on both past and future inputs.
• All "realtime" systems must be causal, since they can not have future inputs available
to them.
C. Linear Systems and Nonlinear Systems:
If the operator (T or H) satisfies the following two conditions, then T is called a linear
operator and the system represented by a linear operator T is called a linear system:
1. Additivity (superposition), 2. Homogeneity (or Scaling).
A nonlinear system is any system that does not have at least one (and/or) of these
properties.
1. Additivity (superposition),
To demonstrate that a system H obeys
the superposition property of linearity
is to show that
H [f1 (t) + f2 (t)] = H [f1 (t)] + H [f2 (t)]
2. Homogeneity (or Scaling)
for any signals x and any scalar a.
To show that a system H obeys the scaling property is to show that
H [kf (t)] = kH [f (t)]
It is possible to check a system for linearity in a single (though larger) step. To do this,
simply combine the first two steps to get
H {k1f1 (t) + k2f2 (t)} = k2H [f1 (t)] + k2H [f2 (t)]
D. Time-Invariant and Time-Varying Systems:
A system is called time-invariant if a time shift (delay or advance) in the input signal
causes the same time shift in the output signal. Thus, for a continuous-time system,
the system is time-invariant if ST [f (t)] = f (t -T) for all T,
HST = STH
Any system that does not have this property is said to be time varying.
E. Linear Time-Invariant Systems
a) Linear Systems:
Linear Scaling
Superposition
E. Linear Time-Invariant Systems
b) Time Invariant Systems:
E. Linear Time-Invariant Systems
Ø If the system is linear and also time-invariant, then it is called a linear rime-invariant
(LTI) system.
Scaling
LTI Systems in Series LTI Systems in Parallel
(Systems in series are also called
cascaded systems).
F. Stable Systems:
A system is bounded-input/bounded-output (BIBO) stable if for any bounded input x
defined by
the corresponding output y is also bounded defined by
where k1 and k2 are finite real constants.
G. Feedback Systems:
A special class of systems of great importance consists of systems having feedback.
In a feedback system, the output signal is fed back and added to the input to the
system:
Time Domain Analysis of Continuous Time (LTI) Systems
