Signal Energy & Power
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Transcript of Signal Energy & Power
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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