Signals and Systems

Prof. Satheesh Monikandan.BHOD-ECE

INDIAN NAVAL ACADEMY, EZHIMALA

sathy24@gmail.com

92 INAC-L-AT15

• Course Code : ECL427 • Course title : SIGNALAS AND SYSTEMS• Credit Hours : 3

• Semester : AT2015

• Refernece Book : A.V.Oppenheim, A.V.Willsky and

S.Hamid Nawab, “Signals and Systems,” PHI, 2nd Edition, 2013.

Syllabus - I• Introduction to Signals• Spectral Analysis

– Fourier Series– Fourier Transform– Frequency Domain Representation of Finite Energy

Signals and Periodic Signals– Signal Energy and Energy Spectral Density – Signal Power and Power Spectral Density

• Signal Transmission through a Linear System– Convolution Integral and Transfer Function

Outline• Signals and Systems

– Signals and Systems– What is a signal?– Signal Basics– Analog / Digital Signals– Real vs Complex– Periodic vs. Aperiodic– Bounded vs. Unbounded– Causal vs. Noncausal– Even vs. Odd– Power vs. Energy

CAUSAL AND NON-CAUSAL SIGNALS

CASUAL AND NON-CAUSAL SYSTEM

The Bands

VLF LF MF HF VHF UHF SHF EHF

Su

bm

illime

ter

Ra

ng

eELF

3MHz 30MHz300MHz 3GHz 30GHz 300GHz

FarInfra-Red

300KHz30KHz 3THz

300m

Radio Optical

3KHz

NearInfra-Red

700nm

1PetaHz

Red

Orange

Yellow

Green

Blue

Indigo

Violet

600nm 400nm500nm

Ultraviolet

1ExaHz

X-Ray

1500nm

Introduction to 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.

Elementary Signals

Sinusoidal & Exponential Signals• Sinusoids and exponentials are important in signal

and system analysis because they arise naturally in the solutions of the differential equations.

• Sinusoidal Signals can expressed in either of two ways :

cyclic frequency form- A sin 2Пfot = A sin(2П/To)t

radian frequency form- A sin ωot

ωo = 2Пfo = 2П/To

To = Time Period of the Sinusoidal Wave

Sinusoidal & Exponential Signals Contd.

x(t) = A sin (2Пfot+ θ)= A sin (ωot+ θ)

x(t) = Aeat Real Exponential

= Aejω̥t = A[cos (ωot) +j sin (ωot)] Complex Exponential

θ = Phase of sinusoidal wave A = amplitude of a sinusoidal or exponential signal fo = fundamental cyclic frequency of sinusoidal signal ωo = radian frequency

Sinusoidal signal

Signal Examples• Electrical signals --- voltages and currents in a

circuit• Acoustic signals --- audio or speech signals

(analog or digital)• Video signals --- intensity variations in an image

(e.g. a CT scan)• Biological signals --- sequence of bases in a

gene• Noise: unwanted signal

:

Measuring Signals

2468101214161820222426283032343638404244464850525456586062646668707274767880828486889092949698100102104106108110112114116118120122124126128130132134136138140142144146148150152154156158160162164166168170172174176178180182184186188190192194196198200202204206208210212214216218220222224226228230232234236238240242244246248250252254256258260262264266268270272274276278280282284286288290292294296298300302304306308310312314316318320322324326328330332334336338340342344346348350352354356358360362364366368370372374376378380382384386388390392394396398400402404406408410412414416418420422424426428430432434436438440442444446448450452454456458460462464466468470472474476478480482484486488490492494496498500502504506508510512514516518520522524526528530532534536538540542544546548550552554556558560562564566568570572574576578580582584586588590592594596598600602604606608610612614616618620622624626628630632634636638640642644646648650652654656658660662664666668670672674676678680682684686688690692694696698700702704706708710712714716718720

-1

-0.8

-0.6

-0.4

-0.2

5.55111512312578E-017

0.2

0.4

0.6

0.8

1

Period

Am

plitude

Definitions

• Voltage – the force which moves an electrical current against resistance

• Waveform – the shape of the signal (previous slide is a sine wave) derived from its amplitude and frequency over a fixed time (other waveform is the square wave)

• Amplitude – the maximum value of a signal, measured from its average state

• Frequency (pitch) – the number of cycles produced in a second – Hertz (Hz). Relate this to the speed of a processor eg 1.4GigaHertz or 1.4 billion cycles per second

Signal Basics Continuous time (CT) and discrete time (DT) signals

CT signals take on real or complex values as a function of an independent variable that ranges over the real numbers and are denoted as x(t).

DT signals take on real or complex values as a function of an independent variable that ranges over the integers and are denoted as x[n].

Note the subtle use of parentheses and square brackets to distinguish between CT and DT signals.

Analog Signals

• Human Voice – best example

• Ear recognises sounds 20KHz or less

• AM Radio – 535KHz to 1605KHz

• FM Radio – 88MHz to 108MHz

Digital signals

• Represented by Square Wave• All data represented by binary values

• Single Binary Digit – Bit• Transmission of contiguous group of bits is a bit

stream• Not all decimal values can be represented by

binary1 0 1 0 1 0 1 0

Analogue vs. Digital

Analogue Advantages• Best suited for audio and video• Consume less bandwidth• Available world wide

Digital Advantages• Best for computer data• Can be easily compressed• Can be encrypted• Equipment is more common and less expensive• Can provide better clarity

Analog or Digital

• Analog Message: continuous in amplitude and over time– AM, FM for voice sound– Traditional TV for analog video– First generation cellular phone (analog mode)– Record player

• Digital message: 0 or 1, or discrete value– VCD, DVD– 2G/3G cellular phone– Data on your disk

A/D and D/A

• Analog to Digital conversion; Digital to Analog conversion– Gateway from the communication device to the

channel

• Nyquist Sampling theorem– From time domain: If the highest frequency in the

signal is B Hz, the signal can be reconstructed from its samples, taken at a rate not less than 2B samples per second

Real vs. ComplexQ. Why do we deal with complex signals? A. They are often analytically simpler to deal with than real

signals, especially in digital communications.

Periodic vs. Aperiodic Signals Periodic signals have the property that x(t + T) = x(t) for all t.

The smallest value of T that satisfies the definition is called the period.

Shown below are an aperiodic signal (left) and a periodic signal (right).

A causal signal is zero for t < 0 and an non-causal signal is zero for t > 0

Right- and left-sided signals

A right-sided signal is zero for t < T and a left-sided signal is zero for t > T where T can be positive or negative.

Causal vs. Non-causal

Bounded vs. Unbounded Every system is bounded, but meaningful signal is always

bounded

Even vs. Odd Even signals xe(t) and odd signals xo(t) are defined as

xe(t) = xe(−t) and xo(t) = −xo(−t). Any signal is a sum of unique odd and even signals. Using

x(t) = xe(t)+xo(t) and x(−t) = xe(t) − xo(t)

Another Classification of Signals (Waveforms)

• Deterministic Signals: Can be modeled as a completely specified function of time

• Random or Stochastic Signals: Cannot be completely specified as a function of time; must be modeled probabilistically

Unit Step Function

( )1 , 0

u 1/ 2 , 0

0 , 0

t

t t

t

>= = <

Precise Graph Commonly-Used Graph

Signum Function

( ) ( ) 1 , 0

sgn 0 , 0 2u 1

1 , 0

t

t t t

t

> = = = − − <

Precise Graph Commonly-Used Graph

The signum function, is closely related to the unit-step function.

Unit Ramp Function

( ) ( ) ( ), 0ramp u u

0 , 0

tt tt d t t

tλ λ

−∞

> = = = ≤ ∫

•The unit ramp function is the integral of the unit step function.•It is called the unit ramp function because for positive t, its slope is one amplitude unit per time.

Rectangular Pulse or Gate Function

Rectangular pulse, ( )1/ , / 2

0 , / 2a

a t at

t aδ

<= >

Representation of Impulse Function

The area under an impulse is called its strength or weight. It is represented graphically by a vertical arrow. An impulse with a strength of one is called a unit impulse.

Properties of the Impulse Function

( ) ( ) ( )0 0g gt t t dt tδ∞

−∞

− =∫

The Sampling Property

( )( ) ( )0 0

1a t t t t

aδ δ− = −

The Scaling Property

The Replication Property

g(t) ⊗ δ(t) = g (t)

Unit Impulse TrainThe unit impulse train is a sum of infinitely uniformly-spaced impulses and is given by

( ) ( ) , an integerTn

t t nT nδ δ∞

=−∞

= −∑

The Unit Rectangle Function

The unit rectangle or gate signal can be represented as combination of two shifted unit step signals as shown

The Unit Triangle Function

A triangular pulse whose height and area are both one but its base width is not, is called unit triangle function. The unit triangle is related to the unit rectangle through an operation called convolution.

Sinc Function

( ) ( )sinsinc

tt

t

ππ

=

Signal Properties: Terminology

• Waveform• Time-average operator• Periodicity• DC value

• Power

• RMS Value

• Normalized Power• Normalized Energy

Power and Energy Signals

• Power Signal– Infinite duration– Normalized power

is finite and non-zero

– Normalized energy averaged over infinite time is infinite

– Mathematically tractable

• Energy Signal– Finite duration– Normalized energy

is finite and non-zero

– Normalized power averaged over infinite time is zero

– Physically realizable

The Decibel (dB)

• Measure of power transfer

• 1 dB = 10 log10 (Pout / Pin)

• 1 dBm = 10 log10 (P / 10-3) where P is in Watts

• 1 dBmV = 20 log10 (V / 10-3) where V is in Volts

What is a communications system?

• Communications Systems: Systems designed to transmit and receive information

Info Source

Info Source

Info Sink

Info Sink

CommSystem

Block Diagram

ReceiverRx

receivedmessage

tosink

m̃( t )

TransmitterTx s(t)

transmittedsignal

Channelr(t)

receivedsignal

m(t)message

from source

Info Source

Info Source

Info Sink

Info Sink

n(t)noise

Telecommunication

• Telegraph

• Fixed line telephone

• Cable

• Wired networks

• Internet

• Fiber communications

• Communication bus inside computers to communicate between CPU and memory

Wireless Communications

• Satellite

• TV

• Cordless phone

• Cellular phone

• Wireless LAN, WIFI

• Wireless MAN, WIMAX

• Bluetooth

• Ultra Wide Band

• Wireless Laser

• Microwave

• GPS

• Ad hoc/Sensor Networks

Comm. Sys. Bock Diagram

m̃( t )Txs(t)

Channelr(t)

m(t)

Noise

RxBaseband

SignalBaseband

SignalBandpassSignal• “Low” Frequencies

• <20 kHz• Original data rate

• “High” Frequencies• >300 kHz• Transmission data rate

Modulation

Demodulationor

Detection

Discrete-Time Signals

• Sampling is the acquisition of the values of a continuous-time signal at discrete points in time

• x(t) is a continuous-time signal, x[n] is a discrete-time signal

[ ] ( )x x where is the time between sampless sn nT T=

Discrete Time Exponential and Sinusoidal Signals

• DT signals can be defined in a manner analogous to their continuous-time counter partx[n] = A sin (2Пn/No+θ)

= A sin (2ПFon+ θ)

x[n] = an n = the discrete time

A = amplitude θ = phase shifting radians, No = Discrete Period of the wave

1/N0 = Fo = Ωo/2 П = Discrete Frequency

Discrete Time Sinusoidal Signal

Discrete Time Exponential Signal

Discrete Time Sinusoidal Signals

Discrete Time Unit Step Function or Unit Sequence Function

[ ] 1 , 0u

0 , 0

nn

n

≥= <

Discrete Time Unit Ramp Function

[ ] [ ], 0ramp u 1

0 , 0

n

m

n nn m

n =−∞

≥ = = − <

∑

Discrete Time Unit Impulse Function or Unit Pulse Sequence

[ ] 1 , 0

0 , 0

nn

nδ

== ≠

[ ] [ ] for any non-zero, finite integer .n an aδ δ=

Operations of Signals

• Sometime a given mathematical function may completely describe a signal .

• Different operations are required for different purposes of arbitrary signals.

• The operations on signals can be Time Shifting Time Scaling Time Inversion or Time Folding

Time Shifting• The original signal x(t) is shifted by an

amount tₒ.

• X(t)X(t-to) Signal Delayed Shift to the right

Time Shifting Contd.

• X(t)X(t+to) Signal Advanced Shift to the left