1. Digital Media Eng. Ahmed H. Abo absa E-mail: a.absa@up.edu.ps a.absa@up.edu.ps 1

2. Outline the lecture Outline the lecture ◦ Signal Operations 1.Time Shifting 2.Time Scaling 3.Time Inversion. ◦ Important Functions ◦ Mean value, Mean square value, variance, standard deviation. ◦ Signal and Vector ◦ Correlation

3. Signal Operations Time Shifting: Consider a signal g(t) and the same signal delayed by T seconds which we shall denote by ¢(t).

4. 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).

5. 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).

6. Mean 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x T

7. Mean square value, variance, standard deviation Mean square value Variance: (average of the square of the deviation of x(t) from the mean value  x) Standard deviation,  x, is the square root of the variance time, t x(t) xx T xx

8. 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 Impulse

9. 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.)

10. Definition Integration of the unit impulse yields the unit step function which is defined as Unit Step

11. 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 Step

12. Combination of Building Block Signals

13. Plotting the signal Plot t<-2  f(t)=0 -2<t<-1  f(t)=3[t+2] -1<t<1  f(t)=-3t 1<t<3  f(t)=-3 3<t<  f(t)=0

14. Signal 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)

15. Correlation indicates the strength and direction of a linear relationship between two random variablesrandom variables

16. Questions?