1 Signal Energy & Power MATLAB HANDLE IMAGE USING single function with 3 Dimensional Independent Variable, e.g. Brightness(x,y,colour)

2 Signal Energy & Power Border or Edge

3 Signal Energy & Power

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6 Instantaneous Power Dissipated By Resistor R ohms. Total energy expanded over time interval between t1 and t2 is :-

7 Average Power Similarly for the automobile, instantaneous power dissipated through friction is:-

8 Common Conventions

9 Energy over infinite interval Let us examine energy over the time interval or number of sample that is infinite:-

10 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.

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

12 Finite total energy signal. -5 < n <+5, x[n] = 1 otherwise x[n]=0. ENERGY =11.

13 Finite average power x[n] = 4, for all n. ENERGY =infinite, Power = 16

14 Neither power nor energy are finite X[n]=0.5n, for all n.

15 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.

16 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.

17 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*.

18 Shifting right or lagging signal x(t) X(t) t X(t-t0) 0 0 t t0 is a positive value

19 Shifting left or leading signal x(t) X(t) t X(t+t1) 0 0 t -t1

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

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

22 Time scaling of continuous signal x(t) x(2t) x(t/2) t t t Compression a>1 Linearly stretching a<1

23 Examples x[n] x[n-5]

24 Examples x[n] x[n+5]

25 Examples x[n] x[-n+5]

26 Example 1.1 x(t) x(t+1), x(t) shifted left by 1sec t t

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

28 Example 1.1 x(t+1) is x(t) shifted left by 1 x(-t+1) is x(t+1) flipped about t=0 t t

29 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

30 Example 1.1, Method 2 x(-t), flip about axis t=0 x(-t+1), shift right (because -t) by 1 t t

31 Example 1.1 x(t) x(t+1), x(t) shifted left by 1sec t t

32 Example 1.1 x(3t/2), x(t) compressed by 2/3 t /34/3 x((3/2)*(t+2/3)), x(t) compressed by 2/3 & shifted left by 2/3 t /3-2/3