signals and systems 8

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 carry

information. Signals are represented mathematically as a function

of one or more independent variable A picture is brightness as a function of two spatial

variables, x and y. In this course signals involving a single independent

variable, generally refer to as a time, t are considered. Although it may not represent time in specific application

A signal is a real-valued or scalar-valued function of an independent variable t.

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Example of signals Electrical signals like voltages, current and

EM field in circuit Acoustic signals like audio or speech signals

(analog or digital) Video signals like intensity variation in an

image Biological signal like sequence of bases in

gene Noise which will be treated as unwanted

signal …

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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

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A continuous-time signal Continuous-time signal x(t), the independent variable, t is

Continuous-time. The signal itself needs not to be continuous.

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A continuous-time signal

x=0:0.05:5; y=sin(x.^2); plot(x,y);

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A piecewise continuous-time signal

A piecewise continuous-time signal

0; 0 10

1; 10 20( )

1; 20 25

2; 25 30

t

tx t

t

t

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A discrete-time signalA discrete signal is defined only at discrete instances. Thus,

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

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Mathlab file

x = 0:0.1:4; y = sin(x.^2).*exp(-x); stem(x,y)

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Sampling A discrete signal can be derived from a continuous-time signal

by sampling it at a uniform rate. If denotes the sampling period and denotes an integer that

may assume positive and negative values, Sampling a continuous-time signal x(t) at time yields a

sample of value For convenience, a discrete-time signal is represented by a

sequence 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

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A piecewise discrete-time signal

A piecewise discrete-time signal

0; 0 10

1; 10 20

1; 20 25

2; 25 30

n

nx n

n

n

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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

TTx t dt

T

2

10 lim

2 1

N

Nn N

x nN

2

0 x t dt

2

0n

x n

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Power and Energy in a Physical System The instantaneous power

The total energy

The average power

21P t v t i t v t

R

2 2 2

1 1 1

21t t t

t t tP 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 tP t dt v t dt

t t t t R

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Power and Energy By definition, the total energy over the time interval in a

continuous-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 a discrete-time signal is:

The time average power

1 2t t t x t

2

1

2t

tE x t dt

x t x t

2

1

2

2 1

1 t

tP 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

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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

TT T T TP x t dt dt t

T T T T

2 21

0

13 9 9

0

T

TE x t dt dt t

x t x t

3

0 1t

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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 Nn n

P x nN N N

2

1

32 2

0

3 27n

n n

E x n

x n

3

0 1n

2

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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 tj

2 21 2

1

2x t x t x t x t x t

12 1tanx t x t x t

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Periodic and Non-periodic A signal or is a periodic signal if

Here, and are fundamental period, which is the smallest positive 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 oN

1m

0( ) cos( ); ; 2ox t A t t f

0 0 0 0

00

( ) cos( [ ] ) cos( )

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

x t T A t mT A t

T m

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Analog and DigitalDigital 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{ , ,..., }Na a a

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

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

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Even and Odd

Even Signals The continuous-time signal /discrete-time signal is

an even signal if it satisfies the condition

Even signals are symmetric about the vertical axis Odd Signals The signal is said to be an odd signal if it satisfies the

condition

Odd signals are anti-symmetric (asymmetric) about the time origin

( ) ( );x t x t t

( ) ( ); [ ] [ ];x t x t t x n x n n

( )x t x n

[ ] [ ];x n x n n

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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 expressed

as 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 tx t x t

x n x n x n x nx n x n

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Complex-Valued Signal Symmetry For a complex-valued signal

is said to be conjugate symmetric if it satisfies the condition

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

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Deterministic and Random signal A signal is deterministic whose future values can be

predicted accurately. Example: A signal is random whose future values can NOT be

predicted with complete accuracy Random signals whose future values can be statistically

determined based on the past values are correlated signals.

Random signals whose future values can NOT be statistically determined from past values are uncorrelated signals and are more random than correlated signals.

( ) sinx t A t

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Deterministic and Random signal(contd…)

Two ways to describe the randomness of the signal are:

Entropy: This is the natural meaning and mostly

used in system performance measurement.

Correlation: This is useful in signal processing by

directly using correlation functions.

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Taylor series and Euler relation General Taylor series

Expanding sin and cos

Expanding exponential

Euler relation

2

0

...!

nn

n

f a x af x f a f a x a f a x a

n

3 5

sin ...3! 5!

x xx x

2 4

cos 1 ...2! 4!

x xx

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 jxe jx

2 3 3 5

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

jx x x x xe j x x j x

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Example The signal and shown below constitute the real and

imaginary 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

T0

1( )x t

signals and systems 8

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