signals and systems 8.ppt

7/18/2019 Signals and Systems 8.ppt

http://slidepdf.com/reader/full/signals-and-systems-8ppt 1/26

Linear Systems Khosrow Ghadiri - EE Dept. SJSU 1

Signals and SystemsLinear System Theory

EE 112 - Lecture Eight

Signal classification



Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 2

Signal

A signal is a pattern of variation that carryinformation.

Signals are represented mathematically as a functionof one or more independent variable

A picture is brightness as a function of two spatialvariables, x and y.

In this course signals involving a single independentvariable, generally refer to as a time, t areconsidered. Although it may not represent time in

specific application A signal is a real-valued or scalar-valued function of

an independent variable t.




Example of signals

Electrical signals like voltages, current andEM field in circuit

Acoustic signals like audio or speech signals(analog or digital)

Video signals like intensity variation in animage

Biological signal like sequence of bases ingene

Noise which will be treated as unwantedsignal …




Signal classification

Continuous-time and Discrete-time

Energy and Power

Real and Complex Periodic and Non-periodic

Analog and Digital

Even and Odd Deterministic and Random




A continuous-time signal

Continuous-time signal x(t), the independent variable, t isContinuous-time. The signal itself needs not to be continuous.




A continuous-time signal

x=0:0.05:5;

y=sin(x.^2);

plot(x,y);




A discrete-time signal

A discrete signal is defined only at discrete instances. Thus,

the independent variable has discrete values only. [ ] x n




Sampling

A discrete signal can be derived from a continuous-time signalby sampling it at a uniform rate.

If denotes the sampling period and denotes an integer thatmay assume positive and negative values,

Sampling a continuous-time signal x(t) at time yields asample of value

For convenience, a discrete-time signal is represented by asequence of numbers:

We write

Such a sequence of numbers is referred to as a time series.

t n ( ) x n

[ ] ( ) x n x n n

n




A piecewise discrete-time signal

A piecewise discrete-time signal

0; 0 10

1; 10 20

1; 20 25

2; 25 30

n

n x n

n

n




Energy and Power Signals

X(t) is a continuous power signal if:

X[n] is a discrete power signal if:

X(t) is a continuous energy signal if:

X[n] is a discrete energy signal if:

21

0 lim2

T

T T x t dt

T

210 lim

2 1

N

N n N

x n N

2

0 x t dt

2

0n

x n




Power and Energy in a Physical System

The instantaneous power

The total energy

The average power

21 P t v t i t v t

R

2 2 2

1 1 1

21t t t

t t t P t dt v t i t dt v t dt

R

2 2

1 1

2

2 1 2 1

1 1 1t t

t t P t dt v t dt

t t t t R




Power and Energy

By definition, the total energy over the time interval in acontinuous-time signal is:

denote the magnitude of the (possibly complex) number

The time average power

By definition, the total energy over the time interval in adiscrete-time signal is:

The time average power

1 2t t t x t

2

1

2t

t E x t dt

x t x t

2

1

2

2 1

1 t

t P x t dt

t t

2 1n n n

x n

2

1

2n

n n

E x n

2

1

2

2 1

1

1

n

n n

P x nn n




Power and Energy

Example 1:The signal is given below is energy or power signal.

Explain.

This signal is energy signal

12 2

0

11 1 1 9lim lim 3 lim 9 lim 0

02 2 2 2

T

T T T T T P x t dt dt t

T T T T

2 21

0

13 9 9

0

T

T E x t dt dt t

x t

x t

3

0 1 t




Power and Energy

Example 2:The signal is given below is energy or power signal.

Explain.

This signal is energy signal

x n

2

1

32 2

0

1 1 27lim lim 3 lim 0

2 1 2 1 2 1

n

N N N n n

P x n N N N

2

1

32 2

0

3 27n

n n

E x n

x n

3

0 1 n

2




Real and Complex

A value of a complex signal is a complex number

The complex conjugate, of the signal is;

Magnitude or absolute value

Phase or angle

x t

1 2

1 2Where 1 Re Im

x t x t jx t

j x t x t x t x t

x t x t

1 2

1 1Re Im

2 2

x t x t jx t

x t x t x t x t x t x t j

2 2

1 2

1

2 x t x t x t x t x t

1

2 1tan x t x t x t




Periodic and Non-periodic

A signal or is a periodic signal if

Here, and are fundamental period, which is the smallestpositive values when

Example:

x t x n

0

0

; , any integer

; , any integer

x t x t mT t m

x n x n mN n m

oT o N 1m

0( ) cos( ); ;2o x t A t t f

0 0 0 0

0

0

( ) cos( [ ] ) cos( )

2; 0,1,2......

x t T A t mT A t

T m




Analog and Digital

Digital signal is discrete-time signal whose values belong

to a defined set of real numbers

Binary signal is digital signal whose values are 1 or 0

Analog signal is a non-digital signal

1 2{ , ,..., } N a a a

[ ] ( ) 0 1;n x n x t or n

[ ] ( ) ; 1n i x n x t a i N




Even and Odd signals:Facts

Product of 2 even or 2 odd signals is an even signal

Product of an even and an odd signal is an odd signal

Any signal (continuous and discrete) can be expressedas sum of an even and an odd signal:

( ) ( ) ( ); [ ] [ ] [ ]

( ) ( ) ( ) ( )( ) ; ( )

2 2

[ ] [ ] [ ] [ ][ ] ; [ ]2 2

e o e o

e o

e o

x t x t x t x n x n x n

x t x t x t x t x t x t

x n x n x n x n x n x n




Complex-Valued Signal Symmetry

For a complex-valued signal

is said to be conjugate symmetric if it satisfies thecondition

where

is the real part and is the imaginary part;

is the square root of -1

1 2( ) ( ) ( ) x t x t j x t

*( ) ( ) x t x t

1 2*( ) ( ) ( ) x t x t j x t

1( ) x t 2( ) x t

j




Deterministic and Random signal

A signal is deterministic whose future values can bepredicted accurately.

Example:

A signal is random whose future values can NOT be

predicted with complete accuracy

Random signals whose future values can be statisticallydetermined based on the past values are correlatedsignals.

Random signals whose future values can NOT be

statistically determined from past values areuncorrelated signals and are more random thancorrelated signals.

( ) sin x t A t




Deterministic and Random signal(contd…)

Two ways to describe the randomnessof the signal are:

Entropy:

This is the natural meaning andmostly used in system performancemeasurement.

Correlation:This is useful in signal processing bydirectly using correlation functions.




Taylor series and Euler relation

General Taylor series

Expanding sin and cos

Expanding exponential

Euler relation

2

0

...!

nn

n

f a x a f x f a f a x a f a x a

n

3 5

sin ...3! 5!

x x x x

2 4

cos 1 ...2! 4!

x x x

2 3 4 5

1 ...2! 3! 4! 5!

x x x x xe x

2 3 4 5

1 ...2! 3! 4! 5!

jx jx jx jx jx

e jx

2 3 3 5

1 ... ... cos sin2! 3! 3! 5!

jx x x x xe j x x j x




Example

The signal and shown below constitute the real andimaginary parts of a complex-valued signal .

What form of symmetry does have?

Answer

is conjugate symmetric.

1( ) x t 2( ) x t

( ) x t

( ) x t

( ) x t

2

T

2

T 0

1 ( ) x t

signals and systems 8.ppt

Documents