Introduction to signals and systems Operations on it

UNIT-I
SOME USEFUL OPERATIONS ON SIGNALS
 Time shifting, Time scaling, Time inversion.
 Signal models: Impulse function, Unit step function
 Exponential function, Even and odd signals.
 Linear and Non-linear systems
 Constant parameter and time varying parameter systems,
 Static and dynamic systems, Causal and Non-causal systems,
 Lumped Parameter and distributed parameter systems
 Continuous-time and discrete-time systems, Analog and digital systems
MECS, Dept. of CME Page 1

UNIT-I
SOME USEFUL OPERATIONS ON SIGNALS
SIGNAL:
A signal is a function that conveys information about a phenomenon. In electronics and
telecommunications, it refers to any time varying voltage, current or electromagnetic wave that carries
information. A signal may also be defined as an observable change in a quality such as quantity.
Signal is a function of one or more independent variables, which contain some information.
Example: voice signal, video signal, signals on telephone wires etc.
Note: Noise is also a signal, but the information conveyed by noise is unwanted hence it is
considered as undesirable.
SYSTEM:
System is a device or combination of devices, which can operate on signals and produces
corresponding response. Input to a system is called as excitation and output from it is called
as response.
For one or more inputs, the system can have one or more outputs.
Example: Communication System
Some of the basic signals are
Unit Step Function:
Unit step function is denoted by u(t). It is defined as u(t)
u(t )={ 1,t ≥0
0,∧t ≤0

Unit Impulse Function:
Impulse function is denoted by δ(t) and it is defined as δ(t)
δ (t)={1,t=0
0,∧t ≠0
Ramp Signal
Ramp signal is denoted by r(t), and it is defined as r(t)
r(t )={ t ,t ≥0
0,∧t ⋖ 0
Parabolic Signal
Parabolic signal can be defined as x(t)
x(t)=
{(t
2
)/2,t ≥0
0,∧t ⋖0

Signum Function
Signum function is denoted as sgn(t). It is defined as sgn(t)
sgn(t)=
{
1,t>0
−1,∧t ⋖0
0,t=0
Exponential Signal
Exponential signal is in the form of x(t) = eαteαt.
The shape of exponential can be defined by α
Case i: if αα = 0 →→ x(t) = e0
= 1
Case ii: if α < 0 i.e. -ve
then x(t) = e−αt
. The shape is called decaying exponential.

Case iii: if α> 0 i.e. +ve
then x(t) = eαt
. The shape is called raising exponential.
Sinusoidal Signal
Sinusoidal signal is in the form of x(t) = A cos(w0 ±ϕ) or A sin(w0 ±ϕ)
To= 2π/w0
CLASSIFICATION OF SIGNALS
Signals are classified into the following categories:
 Continuous Time and Discrete Time Signals
 Deterministic and Non-deterministic Signals
 Even and Odd Signals
 Periodic and Aperiodic Signals
 Energy and Power Signals
 Real and Imaginary Signals

Continuous Time and Discrete Time Signals
A signal is said to be continuous when it is defined for all instants of time.
A signal is said to be discrete when it is defined at only discrete instants of time
Deterministic and Non-deterministic Signals
A signal is said to be deterministic if there is no uncertainty with respect to its value at any instant of
time. Or, signals which can be defined exactly by a mathematical formula are known as deterministic
signals.
A signal is said to be non-deterministic if there is uncertainty with respect to its value at some instant
of time. Non-deterministic signals are random in nature hence they are called random signals.
Random signals cannot be described by a mathematical equation. They are modeled in probabilistic
terms.

Even and Odd Signals
A signal is said to be even when it satisfies the condition x(t) = x(-t)
Example 1: t2
, t4
… cost etc.
Let x(t) = t2
x(-t) = (-t)2
= t2
= x(t)
∴ t2
is an even function
A signal is said to be odd when it satisfies the condition x(t) = -x(-t)
Example: t, t3
... And sin t
Let x(t) = sin t
x(-t) = sin(-t) = -sin t = -x(t)
∴ sin t is an odd function.
Any function x(t) can be expressed as the sum of its even components xe(t) and odd
components xo(t).
x(t) = xe(t) + x0(t) -----(1)
Replace t by –t in equation (1), then
x(-t) = xe(-t) + x0(-t) = xe(t) - x0(t)-----(2)
by adding equation (1) & (2),
x(t) + x(-t) = xe(t) + x0(t) + xe(t) - x0(t) = 2 xe(t)

xe(t ) = ½[x(t ) + x(-t )]
by subtracting equation (1) & (2),
x(t) - x(-t) = xe(t) + x0(t) - xe(t) + x0(t) = 2 x0(t)
x0(t) = ½[x(t ) - x(-t )]
Example: Find the even & odd components of a given signal.
1. x(t) = cost + sint + cost*sint
x(-t) = cos(-t) + sin(-t) + cos(-t)*sin(-t)
= cost – sint –cost*sint
The even component of the given signal is
xe(t ) = ½[x(t ) + x(-t )] = ½ [cost + sint + cost*sint + cost – sint –cost*sint] =cost
The odd component of the given signal is
x0(t) = ½[x(t ) - x(-t )] = ½ [cost + sint + cost*sint - cost + sint + cost*sint]
= ½ [2sint + 2cost*sint] = sint + cost* sint
Periodic and Aperiodic Signals
A signal is said to be periodic if it satisfies the condition
x(t) = x(t + T) or x(n) = x(n + N).
Where
T = fundamental time period,
1/T = f = fundamental frequency.
The above signal will repeat for every time interval T0 hence it is periodic with period T0.
Example: 1. For a continuous time sinusoidal signal x(t) = A sin(w0t+θ). Find the fundamental time
period T.
x(t)=Asin(w0t+θ)
x(t)=Asin(w0t+θ)−−−−(1)
x(t+T )=A sin(w0(t+T )+θ)
x(t+T )=A sin(w0t+w0T +θ)−−−(2)
Eq(1)=Eq(2)
So
w0T=2π ⇒T=
2π
w0

Example: 2. For a continuous time complex exponential signal . Find the
fundamental time period T.
Energy and Power Signals
A signal is said to be energy signal when it has finite energy.
Energy E=∫
−∞
∞
x
2
(t)dt
A signal is said to be power signal when it has finite power.
Power P=
lim
T → ∞
1
T
∫
−∞
∞
x2
(t)dt
NOTE: A signal cannot be both, energy and power simultaneously. Also, a signal may be
neither energy nor power signal.
Power of energy signal = 0
Energy of power signal = ∞
Real and Imaginary Signals
A signal is said to be real when it satisfies the condition x(t) = x*(t)
A signal is said to be odd when it satisfies the condition x(t) = -x*(t)
Example:
If x(t)= 3 then x*(t)=3*=3 here x(t) is a real signal.
If x(t)=cost then X(-t)=cos(-t)=cost then x(t) is even
If x(t)= 3j then x*(t)=3j* = -3j = -x(t) hence x(t) is a odd signal.
Note: For a real signal, imaginary part should be zero. Similarly for an imaginary signal,
real part should be zero.
x(t )=e
( jw0 t+θ)
x(t)=e
( jw0
t+θ)
−−−(1)
x(t+T )=e
(jw0
t + jw 0
T +θ )
=e
(jw0
t+θ)
e
(jw0
T )
−−−(2)
Eq.(1)=eq(2)
so,
e
( jw0 T)
=e
j2π
(cos2π+ jsin2π=1)
⇒w0T=2 π
⇒T=
2π
w0

There are two variable parameters in general:
1. Amplitude
2. Time
The following operation can be performed with amplitude:
Amplitude Scaling
C x(t) is a amplitude scaled version of x(t) whose amplitude is scaled by a factor C.
Addition
Addition of two signals is nothing but addition of their corresponding amplitudes. This can
be best explained by using the following example:
As seen from the diagram above,
-10 < t < -3 amplitude of z(t) = x1(t) + x2(t) = 0 + 2 = 2
-3 < t < 3 amplitude of z(t) = x1(t) + x2(t) = 1 + 2 = 3
3 < t < 10 amplitude of z(t) = x1(t) + x2(t) = 0 + 2 = 2
Subtraction

Subtraction of two signals is nothing but subtraction of their corresponding amplitudes. This
can be best explained by the following example:
As seen from the diagram above,
-10 < t < -3 amplitude of z (t) = x1(t) ×x2(t) = 0 ×2 = 0
-3 < t < 3 amplitude of z (t) = x1(t) ×x2(t) = 1 ×2 = 2
3 < t < 10 amplitude of z (t) = x1(t) × x2(t) = 0 × 2 = 0
The following operations can be performed with time:
Time Shifting
x(t ± t0) is time shifted version of the signal x(t).
x (t + t0) →→ negative shift
x (t - t0) →→ positive shift
Example:

Time Scaling
x(At) is time scaled version of the signal x(t).
where A is always positive.
|A| > 1 →→ Compression of the signal
|A| < 1 →→ Expansion of the signal
Note: u(at) = u(t) time scaling is not applicable for unit step function.
Examples
Time Reversal
x(-t) is the time reversal of the signal x(t).

Example

CLASSIFICATION OF SYSTEMS
Systems are classified into the following categories:
 Linear and Non-linear Systems
 Time Variant and Time Invariant Systems
 Linear Time Variant and Linear Time Invariant systems
 Static and Dynamic Systems
 Causal and Non-causal Systems
 Stable and Unstable Systems
 Continuous and Discrete time systems
 Lumped Parameter and distributed parameter systems
 Analog and Digital Systems
 Invertible and Non-Invertible Systems
LINEAR AND NON-LINEAR SYSTEMS
A system is said to be linear when it satisfies superposition and homogenate principles.
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,
T [a1 x1(t) + a2 x2(t)] = a1 T[x1(t)] + a2 T[x2(t)]
∴, T [a1 x1(t) + a2 x2(t)] = a1 y1(t) + a2 y2(t)
From the above expression, is clear that response of overall system is equal to response of
individual system.
Example:
(t) = x2
(t)
Solution:
y1 (t) = T[x1(t)] = x1
2
(t)
y2 (t) = T[x2(t)] = x2
2
(t)
T [a1 x1(t) + a2 x2(t)] = [ a1 x1(t) + a2 x2(t)]2
Which is not equal to a1 y1(t) + a2 y2(t). Hence the system is said to be non linear.
TIME VARIANT AND TIME INVARIANT SYSTEMS
A system is said to be time variant if its input and output characteristics vary with time.
Otherwise, the system is considered as time invariant.
The condition for time invariant system is:
y (n , t) = y(n-t)
The condition for time variant system is:
y (n , t) ≠ y(n-t)
Where y (n , t) = T[x(n-t)] = input change

y (n-t) = output change
Example:
y(n) = x(-n)
y(n, t) = T[x(n-t)] = x(-n-t)
y(n-t) = x(-(n-t)) = x(-n + t)
∴y(n, t) ≠ y(n-t). Hence, the system is time variant.
LINEAR TIME VARIANT (LTV) AND LINEAR TIME INVARIANT (LTI)
SYSTEMS
If a system is both linear and time variant, then it is called linear time variant (LTV) system.
If a system is both linear and time Invariant then that system is called linear time invariant
(LTI) system.
STATIC AND DYNAMIC SYSTEMS
Static system is memory-less whereas dynamic system is a memory system.
Example 1: y(t) = 2 x(t)
For present value t=0, the system output is y(0) = 2x(0). Here, the output is only dependent
upon present input. Hence the system is memory less or static.
Example 2: y(t) = 2 x(t) + 3 x(t-3)
For present value t=0, the system output is y(0) = 2x(0) + 3x(-3).
Here x(-3) is past value for the present input for which the system requires memory to get
this output. Hence, the system is a dynamic system.
CAUSAL AND NON-CAUSAL SYSTEMS
A system is said to be causal if its output depends upon present and past inputs, and does not
depend upon future input.
For non causal system, the output depends upon future inputs also.
Example 1: y(n) = 2 x(t) + 3 x(t-3)
For 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.
Example 2: y(n) = 2 x(t) + 3 x(t-3) + 6x(t + 3)
For present value t=1, the system output is y(1) = 2x(1) + 3x(-2) + 6x(4) Here, the system
output depends upon future input. Hence the system is non-causal system.
INVERTIBLE AND NON-INVERTIBLE SYSTEMS

A system is said to invertible if the input of the system appears at the output.
Y(S) = X(S) H1(S) H2(S)
= X(S) H1(S) ·
1
H 2(s)
Since H2(S) =
1
H 1(s)
∴,Y(S) = X(S)
→ y(t) = x(t)
Hence, the system is invertible.
If y(t) ≠x(t), then the system is said to be non-invertible.
STABLE AND UNSTABLE SYSTEMS
The system is said to be stable only when the output is bounded for bounded input. For a
bounded input, if the output is unbounded in the system then it is said to be unstable.
Note: For a bounded signal, amplitude is finite.
Example 1: y (t) = x2
(t)
Let the input is u(t) (unit step bounded input) then the output y(t) = u2
(t) = u(t) = bounded
output.
Hence, the system is stable.
Example 2: y (t) = ∫x(t)dt
Let the input is u (t) (unit step bounded input) then the output y(t) = ∫u(t)dt
ramp signal (unbounded because amplitude of ramp is not finite it goes to infinite when
t →→ infinite).
Hence, the system is unstable.
CONTINUOUS AND DISCRETE TIME SYSTEMS
• Physically, a system is an interconnection of components, devices, etc., such as a
computer or an aircraft or a power plant.
• Conceptually, a system can be viewed as a black box which takes in an input signal
x(t) (or x[n]) and as a result generates an output signal y(t) (or (y[n]). A short-hand
notation: x(t)  y(t) or x[n] y[n] .

• A continuous-time system is one which operates on a continuous-time input signal
and produces the continuous-time output signal.
• A discrete-time system is one which operates on a discrete-time input signal and
produces the discrete-time output signal.
 A continuous-time system: Example – Filter
 A discrete-time system: Example-Investment
LUMPED PARAMETER AND DISTRIBUTED PARAMETER SYSTEMS
• A lumped-parameter systems is one in which the dependent variables of interest are a
function of time alone. In general, this will mean solving a set of ordinary differential
equations (ODEs)
• A distributed- parameter system is one in which all dependent variables are functions
of time and one or more spatial variables. In this case, we will be solving partial
differential equations (PDEs)
• For example, consider the following two systems:

 The first system is a distributed system, consisting of an infinitely thin string,
supported at both ends; the dependent variable, the vertical position of the string
y(x, t) is indexed continuously in both space and time.
 The second system, a series of “beads” connected by massless string segments,
constrained to move vertically, can be thought of as a lumped system, perhaps an
approximation to the continuous string.
 For electrical systems, consider the difference between a lumped RLC network and a
transmission line
 The importance of lumped approximations to distributed systems will become
obvious later, especially for waveguide-based physical modelling, because it enables
one to cut computational costs by solving ODEs at a few points, rather than a full
PDE (generally much more costly)
DIFFERENCES BETWEEN ANALOG AND DIGITAL SYSTEMS
S.
No.
Key Digital System Analog System
1
Signal Type Digital System uses discrete signals
as on/off representing binary
format. Off is 0, On is 1.
Analog System uses continuous
signals with varying magnitude.
2 Wave Type Digital System uses square waves. Analog system uses sine waves.
3 Technology Digital system first transform the
analog waves to limited set of
An analog system records the
physical waveforms as they are

S.
No.
Key Digital System Analog System
numbers and then record them as
digital square waves.
originally generated.
4
Transmission Digital transmission is easy and can
be made noise proof with no loss at
all.
Analog systems are affected badly
by noise during transmission.
5
Flexibility Digital system hardware can be
easily modulated as per the
requirements.
Analog systems hardware are not
flexible.
6
Bandwidth Digital transmission needs more
bandwidth to carry same
information.
Analog transmission requires less
bandwidth.
7
Memory Digital data is stored in form of
bits.
Analog data is stored in form of
waveform signals.
8
Power
requirement
Digital system needs low power as
compare to its analog counterpart.
Analog systems consume more
power than digital systems.
9
Best suited
for
Digital system is good for
computing and digital electronics.
Analog systems are good for
audio/video recordings.
10 Cost Digital system is costly. Analog systems are cheap.
11
Example Digital systems are: Computer, CD,
DVD.
Analog systems are: Analog
electronics, voice radio using AM
frequency.

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