Digital MediaDigital Media

Eng. Ahmed H. Abo absaEng. Ahmed H. Abo absa

E-mail E-mail : :a.absa@up.edu.ps

1

Outline the lectureOutline the lecture◦ Signal Operations

1. Time Shifting2. Time Scaling3. Time Inversion.

◦ Important Functions◦ Mean value, Mean square value,

variance, standard deviation.◦ Signal and Vector◦ Correlation

Signal OperationsSignal OperationsTime Shifting: Consider a signal g(t) and the same signal delayed by T secondswhich we shall denote by ¢(t).

Time Scaling: The compression or expansion of a signal in time

Example: in below Figures a and b shows the signals g(t) and z(t), respectively. Sketch: (a) g(3t); (b) z(t /2).

Time Inversion: Time inversion may be considered a special case of time scaling with a = -1.

To invert g(t), we rotate this frame 180 deg about the vertical axis.

Example: For the signal g(t) shown in the figure, sketch g(-t).

MeanMean

• The mean value,x , is the height of the rectangular area having the same area as that under the function x(t)

• Can also be defined as the first moment of the p.d.f.

time, t

x(t)

x

T

T

0Tx(t)dt

T

1Limx

Mean square value, variance, Mean square value, variance, standard deviationstandard deviation

Mean square value Variance:

(average of the square of the deviation of x(t) from the mean valuex)

Standard deviation, x, is the square root of the variance

time, t

x(t)

x

T

x

T

0

2

T

2 (t)dtxT

1Limx

22x xx(t)σ

T

0

2

Tdtx-x(t)

T

1Lim

2

Definition: The unit impulse δ(t) is not a function in the ordinary sense. It is defined by the integral relation

and is called a generalized function. The unit impulse is not defined in terms of its values, but is defined by how it acts inside an integral when multiplied by a smooth function f(t). To see that the area of the unit impulse is 1, choose f(t) = 1 in the definition. We represent the unit impulse schematically as shown below; the number next to the impulse is its area.

Unit ImpulseUnit Impulse

Unit impulse — narrow pulse approximation

To obtain an intuitive feeling for the unit impulse, it is often helpful to imagine a set of rectangular pulses where each puls has width ε and height 1/ ε so that its area is 1.

The unit impulse is the quintessential tall and narrow pulse!

Unit Impulse (cont.)Unit Impulse (cont.)

Definition

Integration of the unit impulse yields the unit step function

which is defined as

Unit StepUnit Step

which, from the definition of the unit impulse, implies that

That is, the unit impulse is the derivative of the unit step in a generalized function sense.

Unit Impulse vs. Unit Unit Impulse vs. Unit StepStep

Combination of Building Combination of Building Block SignalsBlock Signals

Plotting the signalPlotting the signal

Plot

t<-2 f(t)=0-2<t<-1

f(t)=3[t+2]-1<t<1 f(t)=-3t1<t<3 f(t)=-33<t< f(t)=0

)3(3)1(]1[3)1(]1[6)2(]2[3)( tututtuttuttf

Signal and VectorSignal and Vector

A vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are defined and satisfy certain natural axioms.

Signal represented by weighted sum of vectors Concept of orthogonality

Sin, cos, FFT Exp(-jz), DFT X, Taylor series DCT (JPEG, MPEG, MP3) Subspace Wavelet (not quite orthogonal)

CorrelationCorrelation

indicates the strength and direction of a linear relationship between two random variables

QuestionsQuestions??