Signal and System

Signals and systems are frequently discussed without knowing the exact parameters being represented. This is the same as using x and y in algebra, without assigning a physical meaning to the variables. This brings in a fourth rule for naming signals. If a more descriptive name is not available, the input signal to a discrete system is usually called: x[n], and the output signal: y[n]. For continuous systems, the signals: x(t) and y(t) are used.

There are many reasons for wanting to understand a system. For example, you may want to design a system to remove noise in an electrocardiogram, sharpen an out-of-focus image, or remove echoes in an audio recording. In other cases, the system might have a distortion or interfering effect that you need to characterize or measure. For instance, when you speak into a telephone, you expect the other person to hear something that resembles your voice. Unfortunately, the input signal to a transmission line is seldom identical to the output signal. If you understand how the transmission line (the system) is changing the signal, maybe you can compensate for its effect. In still other cases, the system may represent some physical process that you want to study or analyze. Radar and sonar are good examples of this. These methods operate by comparing the transmitted and reflected signals to find the characteristics of a remote object. In terms of system theory, the problem is to find the system that changes the transmitted signal into the received signal.
At first glance, it may seem an overwhelming task to understand all of the possible systems in the world. Fortunately, most useful systems fall into a category called linear systems. This fact is extremely important. Without the linear system concept, we would be forced to examine the individual characteristics of many unrelated systems. With this approach, we can focus on the traits of the linear system category as a whole. Our first task is to identify what properties make a system linear, and how they fit into the everyday notion of electronics, software, and other signal processing systems.

Signal

A signal is a function representing a physical quantity or variable, and typically it
contains information about the behavior or nature of the phenomenon. For instance, in a
RC circuit the signal may represent the voltage across the capacitor or the current flowing
in the resistor. Mathematically, a signal is represented as a function of an independent
variable t. Usually t represents time. Thus, a signal is denoted by x(t).

System

A system is a mathematical model of a physical process that relates the input (or
excitation) signal to the output (or response) signal.
Let x and y be the input and output signals, respectively, of a system. Then the system
is viewed as a transformation (or mapping) of x into y. This transformation is represented
by the mathematical notation

y = Tx

where T is the operator representing some well-defined rule by which x is transformed
into y.

Convolution

Convolution is a powerful way of characterizing the input-output relationship of time invarient linear systems.
There are two convolution theorems

1) Time Convolution
Time convolution theorem is for time domain

2) Frequency Convolution
this theorem is for frequency domain

Energy and Power Signals

x(t) (or x[n]) is said to be an energy signal (or sequence) if and only if 0 < E < infinity, and
so P = 0.

x(t) (or x[n]) is said to be a power signal (or sequence) if and only if 0 < P < infinity, thus
implying that E = infinity.

Signals that satisfy neither property are referred to as neither energy signals nor power
signals.

Note that a periodic signal is a power signal if its energy content per period is finite, and
then the average power of this signal need only be calculated over a period

Real and Complex Signals

Real and Complex Signals:
A signal x(t) is a real signal if its value is a real number, and a signal x(t) is a complex
signal if its value is a complex number. A general complex signal x( t) is a function of the form

x ( t ) = x , ( t ) + i x 2 ( t )


where x,( t ) and x2( t ) are real signals and j = m.
Note that in Eq. t represents either a continuous or a discrete variable.

Analog and Digital Signals

If a continuous-time signal x(t) can take on any value in the continuous interval (a, b),
where a may be - infinity and b may be + infinity, then the continuous-time signal x(t) is called an
analog signal. If a discrete-time signal x[n] can take on only a finite number of distinct
values, then we call this signal a digital signal

Continuous-Time and DiscreteContinuous-Time and Discrete-Time Signals

A signal x(t) is a continuous-time signal if t is a continuous variable. If t is a discrete
variable, that is, x(t) is defined at discrete times, then x(t) is a discrete-time signal. Since a
discrete-time signal is defined at discrete times, a discrete-time signal is often identified as
a sequence of numbers, denoted by {x,) or x[n], where n = integer. Illustrations of a
continuous-time signal x(t) and of a discrete-time signal x[n] are shown in Fig.


A discrete-time signal x[n] may represent a phenomenon for which the independent
variable is inherently discrete. For instance, the daily closing stock market average is by its
nature a signal that evolves at discrete points in time (that is, at the close of each day). On
the other hand a discrete-time signal x[n] may be obtained by sampling a continuous-time signal x(t) such as

x(to), +,)' . 7 ~ ( t , )., . *

or in a shorter form as
x[O], x[l], ..., x[n], . ..
or xo, x ~ ,. .. , x,, . . .
where we understand that
x, =x[n] =x(t,)
and x,'s are called samples and the time interval between them is called the sampling
interval. When the sampling intervals are equal (uniform sampling), then
x,, =x[n] =x(nT,)
where the constant T, is the sampling interval.
A discrete-time signal x[n] can be defined in two ways:
1. We can specify a rule for calculating the nth value of the sequence.

Deterministic and Random Signals

Deterministic and Random Signals:

Deterministic signals are those signals whose values are completely specified for any
given time. Thus, a deterministic signal can be modeled by a known function of time I .

Random signals are also called non deterministic signals are those signals that take random values at any given time and must be characterized statistically.

Even and Odd Signals

signal x ( t ) or x[n] is referred to as an even signal if

x ( - t ) = x ( r )
x [ - n ] = x [ n ]

A signal x ( t ) or x[n] is referred to as an odd signal if

Examples of even x ( - t ) = - x ( t ) , x [ - n ] = - x [ n ] and odd signals are shown in pictures

Periodic and Nonperiodic Signals

A continuous-time signal x ( t ) is said to be periodic with period T if there is a positive
nonzero value of T for which

x(t + T ) = x ( t ) all t

An example of such a signal is given in fig. From Eq it follows
that

x (t+ mT) = x(t)

for all t and any integer m. The fundamental period T, of x ( t ) is the smallest positive
value of T for which Eq holds. Note that this definition does not work for a constant

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

Stable Systems

A system is bounded-input/bounded-output (BIBO) stable if for any bounded input x
defined by

x < k1


the corresponding output y is also bounded defined by

x < k2


where k1 , and k2, are finite real constants. Note that there are many other definitions of
stability.

Linear Time-Invariant Systems

If the system is linear and also time-invariant, then it is called a linear rime-invariant
(LTI) system.

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
for any real value of T. For a discrete-time system, the system is time-invariant (or
shift-invariant ) if

~ { x [-nk ] ) =y[n - k ] (1.72)

for any integer k. A system which does not satisfy (continuous-time system) or(discrete-time system) is called a time-varying system. To check a system for
time-invariance, we can compare the shifted output with the output produced by the
shifted input

Linear Systems and Nonlinear Systems

If the operator T 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:

Given that Tx, = y, and Tx, = y,, then
T{x, +x2) =y, +Y,

for any signals x, and x2.

2. Homogeneity (or Scaling):

for any signals x and any scalar a.

Any system that does not satisfy T is classified as a
nonlinear system. Equation can be combined into a single condition as


T { ~+ wI 2 ) = ~ I Y+I a 2 Y z (1.68)

where a, and a, are arbitrary scalars. Equation (1.68) is known as the superposition
property. Examples of linear systems are the resistor and the capacitor.

Examples of nonlinear systems are
y =x 2 (1.69)
y = cos x (1.70)

Note that a consequence of the homogeneity (or scaling) property of linear
systems is that a zero input yields a zero output. This follows readily by setting a = 0 . This is another important property of linear systems.

Continuous;Time and Discrete-Time Systems

If the input and output signals x and p are continuous-time signals, then the system is
called a continuous-time system. If the input and output signals are discrete-time
signals or sequences, then the system is called a discrete-time system

Systems with Memory and without Memory

A system is said to be memoryless if the output at any time depends on only the input
at that same time. Otherwise, the system is said to have memory. An example of a
memoryless system is a resistor R with the input x ( t ) taken as the current and the voltage
taken as the output y ( t ) .
The input-output relationship (Ohm's law) of a resistor is
An example of a system with memory is a capacitor C with the current as the input x( t )
and the voltage as the output y ( 0 ; then

y (t)

A second example of a system with memory is a discrete-time system whose input and
output sequences are related by


y (n)

Causal and Noncausal Systems

A system is called causal if its output y ( t ) at an arbitrary time t = t,, depends on only
the input x ( t ) for t It o. 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


Note that all memoryless systems are causal, but not vice versa.

Unit - impulse Function

The unit impulse function 6(t), also known as the Dirac delta function, plays a central
role in system analysis. Traditionally, 6(t) is often defined as the limit of a suitably chosen
conventional function having unity area over an infinitesimal time interval as shown in the picture.

Unit Step Function

The integral of impulse function give Unit Step Function.

The integral of impulse function is a singularity function and called the Unit Step Function represented by u (t)

u (t) = 0 t <> 0


and for discrete time signals it is represented by u(n)

Unit Ramp Function

The Unit Ramp Function r(t) can be obtained by integrating the unit impulse function twice or integrating the unit step function once

r(t) = 0 t <> 0

Unit Pulse Function

The Unit Pulse Function is obtained from unit step signals as

unit pulse = u(t + 1/2) - u(t - 1/2)

the u(t + 1/2) and u(t - 1/2) are the unit step signals shifted by 1/2 units in the time axis towards the left and right respectively