GANDHINAGAR INSTITUTE OF TECHNOLOGY

SIGNALS & SYSTEMS PPTGUIDED BY:PROF.HARDIK PATEL

PREPARED BY: DINESH SIRVANI(130120109054)VIPUL SOLANKI(130120109061)KULDEEP THAKKAR(130120109058)

SIGNAL• A signal is defined as as a function of one or more

independent variables• Basically it is a physical quantity• So it can be defined as A physical quantity which

contains some information and which is function of one or more independent variables.

Classification of Signals

• Deterministic & Non Deterministic Signals• Periodic & A periodic Signals• Even & Odd Signals• Energy & Power Signals• Continuous and discrete time signal

Deterministic & Non Deterministic Signals

• Deterministic signals • These signals can be expressed mathematically. look up

tables or some or some well defined rules. For example x(t) = sin(3t) is deterministic signal.

Deterministic & Non Deterministic Signals

Non Deterministic or Random signals

• These signals can’t be expressed mathematical rules.• For example Thermal Noise generated is non

deterministic signal.

Continuous and discrete time signals

• A Signal is the function of one or more independent variables that carries some information to represent a physical phenomenon.

• A continuous-time signal, also called an analog signal, is defined along a continuum of time.

A discrete-time signal is defined at discrete times.

Periodic and Non-periodic Signals

• Given x(t) is a continuous-time signal • x (t) is periodic iff x(t) = x(t+Tₒ) for any T and any integer n• Example– x(t) = A cos(t)– x(t+Tₒ) = A cos[t+Tₒ)] = A cos(t+Tₒ)= A cos(t+2) =

A cos(t)– Note: Tₒ =1/fₒ ; fₒ

Periodic and Non-periodic Signals Contd.

• For non-periodic signals x(t) ≠ x(t+Tₒ)• Example of non periodic signal is an

exponential signal

Even and Odd SignalsEven Functions Odd Functions

g t g t

g t g t

Various Combinations of even and odd functions

Function type Sum Difference Product Quotient

Both even Even Even Even Even

Both odd Odd Odd Even Even

Even and odd Neither Neither Odd Odd

Energy and Power Signals Energy Signal• A signal with finite energy and zero power is called

Energy Signal i.e.for energy signal 0<E<∞ and P =0• Signal energy of a signal is defined as the area

under the square of the magnitude of the signal.

2

x xE t dt

Energy and Power Signals Contd.Power Signal• Some signals have infinite signal energy. In that

caseit is more convenient to deal with average signal power.

• For power signals 0<P<∞ and E = ∞• Average power of the signal is given by

/ 2

2

x

/ 2

1lim x

T

TT

P t dtT

Energy and Power Signals Contd.

• For a periodic signal x(t) the average signal power is

• T is any period of the signal.• Periodic signals are generally power signals.

2

x

1x

TP t dt

T

What is System?

• Systems process input signals to produce output signals

• A system is combination of elements that manipulates one or more signals to accomplish a function and produces some output.

system output signal

input signal

Types of Systems• Causal & Anticausal• Linear & Non Linear• Time Variant &Time-invariant• Stable & Unstable • Static & Dynamic• Invertible & Inverse Systems

Causal & Anticausal Systems

• Causal system : A system is said to be causal if the present value of the output signal depends only on the present and/or past values of the input signal.

• Example: y[n]=x[n]+1/2x[n-1]

Causal & Anticausal Systems Contd.

• Anticausal system : A system is said to be anticausal if the present value of the output signal depends only on the future values of the input signal.

• Example: y[n]=x[n+1]

Linear & Non Linear SystemsA system is said to be linear if it satisfies the

principle of superpositionFor checking the linearity of the given system,

firstly we check the response due to linear combination of inputs

Then we combine the two outputs linearly in the same manner as the inputs are combined and again total response is checked

If response in step 2 and 3 are the same,the system is linear othewise it is non linear.

Time Invariant and Time Variant Systems

• A system is said to be time invariant if a time delay or time advance of the input signal leads to a identical time shift in the output signal.

Eample:(1)y(t)=t x(t)(2)y(t)=t x(t-T)

Stable & Unstable Systems

• A system is said to be bounded-input bounded-output stable (BIBO stable) iff every bounded input results in a bounded output.

i.e.

| ( ) | | ( ) |x yt x t M t y t M

Stable & Unstable Systems Contd.

Example: The system represented by y(t) = A x(t) is unstable ; A˃1 Reason: let us assume x(t) = u(t), then at

every instant u(t) will keep on multiplying with A and hence it will not be bonded.

Static & Dynamic Systems

• A static system is memoryless system• It has no storage devices• its output signal depends on present values of the

input signal• For example

Static & Dynamic Systems Contd.• A dynamic system possesses memory• It has the storage devices• A system is said to possess memory if its output

signal depends on past values and future values of the input signal

Invertible & Inverse Systems• If a system is invertible it has an Inverse System

• Example: y(t)=2x(t)– System is invertible must have inverse, that is: – For any x(t) we get a distinct output y(t)– Thus, the system must have an Inverse

• x(t)=1/2 y(t)=z(t)

y(t)System

Inverse System

x(t) x(t)

y(t)=2x(t)System(multiplier)

Inverse System

(divider)

x(t) x(t)

Application Areas

• Control• Communications• Signal Processing

Control Applications

• Industrial control and automation (Control the velocity or position of an object)

• Examples: Controlling the position of a valve or shaft of a motor

• Important Tools: – Time-domain solution of differential equations– Transfer function (Laplace Transform)– Stability

Communication Applications

• Transmission of information (signal) over a channel

• The channel may be free space, coaxial cable, fiber optic cable

• A key component of transmission: Modulation (Analog and Digital Communication)

Digital Modulation

• Used in CDs, digital cellular service, digital phone lines and computer modems.

• Advantages: – Can be encrypted– Electronic routing of data is easier– Digital storage faster– Multimedia capability

Signal Processing Applications

• Signal processing=Application of algorithms to modify signals in a way to make them more useful.

• Goals:– Efficient and reliable transmission, storage and display of

information– Information extraction and enhancement

• Examples: – Speech and audio processing– Multimedia processing (image and video)– Underwater acoustic– Biological signal analysis

Multimedia Applications

• Compression: Fast, efficient, reliable transmission and storage of data

• Applied on audio, image and video data for transmission over the Internet, storage

• Examples: CDs, DVDs, MP3, MPEG4• Mathematical Tools: Fourier Transform,

Quantization, Modulation

Biological Signal Analysis

• Examples:– Brain signals (EEG)– Cardiac signals (ECG)– Medical images (x-ray, PET, MRI)

• Goals: – Detect abnormal activity (heart attack, seizure)– Help physicians with diagnosis

• Tools: Filtering, Fourier Transform

Example

• Brain waves are usually contaminated by noise and hard to interpret