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Signal Energy & Power

TotalEnergyt1

MATLAB HANDLE IMAGE USING single function with 3 DimensionalIndependent Variable, e.g. Brightness(x,y,colour)

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Signal Energy & Power

TotalEnergyt1

Border or Edge

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Signal Energy & Power

TotalEnergyt1

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Signal Energy & Power

TotalEnergyt1

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Signal Energy & Power

TotalEnergyt1

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Signal Energy & Power

TotalEnergyt1

).(*1

)(*)()( 2 tvR

tvtitp

Instantaneous Power Dissipated By Resistor R ohms.

Total energy expanded over time interval betweent1 and t2 is :-

2

1.)(

t

tdttpenergy

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Average Power

•

2

1

22

1.)(

1

12

1)(

12

1 t

t

t

tdttv

Rttdttp

tt

Similarly for the automobile , instantaneous powerdissipated through friction is:-

2

1

2

1

2

12

1 t

t

t

t

p(t)dttt

poweraverage

p(t)dt.energytotal

(t)bvp(t)

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Common Conventions

2

1

2

2

1

2

|][|

:][

|)(|

:)(

.)(&

n

nn

t

t

nxenergyTotal

nxsignalcomplexdiscreteFor

dttxenergyTotal

txsignalcomplexContinuousFor

unityntcoeifficiefrictionb

RusetocommonGenerally

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Energy over infinite interval

Let us examine energy over the time intervalor number of sample that is infinite:-

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Infinite energy

• If x(t) or x[n] equals a nonzero constant value for all time or sample number, the integral or summation will not converged, therefore the energy is infinite.

• Otherwise it will converge and the energy will be finite if x(t) or x[n] tends to have zero values outside a finite interval.

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Time-average power over infinite interval

N

NnN

T

TT

nxN

P

dttxT

P

2

2

|][|12

1lim

|)(|2

1lim

With this 3 classes of signals can be identified:-1) Finite total energy,2) Finite average power,3) Neither power nor energy are finite,

02

lim,

TP

T ,0 P

,P

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Finite total energy signal.

-5 < n <+5, x[n] = 1otherwise x[n]=0. ENERGY =11.

02

lim,

TP

T

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Finite average power

x[n] = 4, for all n.ENERGY =infinite, Power = 16

,0 P

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Neither power nor energy are finite

X[n]=0.5n, for all n.

,P

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Transformation of Independent Variable.

• Central concept in signal & system is the transformation of a signal.

• Aircraft control system:-– Input correspond to pilot action– these action are transformed by electrical &

mechanical system of the aircraft to changes to aircraft trust or position control surfaces such as the rudder & ailerons.

– finally these changes affect the dynamics & kinematics such as the aircraft velocity and heading.

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High fidelity audio system

• Input signal representing music recorded on cassette or compact disc.

• This signal is modified or transformed to enhanced the desirable characteristics.

• Such as, remove recording noise and to balance the several components of the signal e.g. treble and bass.

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Modification of independent variable (time axes)

• Introducing several basic properties of signals & systems through elementary transformations.

• Examples of elementary transformation:-– time shift, x(t-t0), x[n-n0]– time reversal, x(-t), x[-n].– time scaling, x(0.5t), x[2n].– and combinations of these. x(at+b), x[an-b], where

a & b are signed constants*.

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Shifting right or lagging signal x(t)

X(t)

t

X(t-t0)

0

0 t

t0is a positive value

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Shifting left or leading signal x(t)

X(t)

tX(t+t1)

0

0 t

-t1

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Folded or Flipped x(t) =x(-t), time reversal

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Signal Flip about y- axes X[-n], time reversal

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Time scaling of continuous signal

x(t)

x(2t)

x(t/2)

t

t

t

Compression a>1

Linearly stretching a<1

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Examples

x[n]

x[n-5]

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Examples

x[n]

x[n+5]

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Examples

x[n]

x[-n+5]

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Example 1.1

x(t)

x(t+1), x(t) shifted left by 1sect

t

1

1

0

0

1 2

1 2-1

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Tables of x(t) & x(t+1) & x(-t+1)

t x(t) x(t+1) x(-t+1)-2 0 0 0-1 0 1 00 1 1 11 1 0 12 0 0 03 0 0 0

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Example 1.1

x(t+1) is x(t) shifted left by 1

x(-t+1) is x(t+1) flipped about t=0t

t

1

1

0

0

1 2

1 2-1

-1

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Example 1.1 Alternative 1

x(t-1) is x(t) shifted right by 1sec

x(-t+1)=x(-1(t-1))Flip about axis t=1

t

t

1

1

0

0

1 2

1 2-1

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Example 1.1, Method 2

x(-t), flip about axis t=0

x(-t+1), shift right (because -t) by 1t

t

1

1

0

0

1 2

1 2-1

-1

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Example 1.1

x(t)

x(t+1), x(t) shifted left by 1sect

t

1

1

0

0

1 2

1 2-1

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Example 1.1

x(3t/2), x(t) compressed by 2/3

t

1

0 1 2-1 2/3 4/3

x((3/2)*(t+2/3)), x(t) compressed by 2/3& shifted left by 2/3

t

1

0 1-1 2/3-2/3