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Image Processing Eng. Ahmed H. Abo absa E-mail: a.absa@up.edu.ps. Outline the lecture. Signal Operations Time Shifting Time Scaling Time Inversion. Important Functions Mean value, Mean square value, variance, standard deviation. Signal and Vector Correlation. Signal Operations.

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outline the lecture
Outline the lecture
  • Signal Operations
        • Time Shifting
        • Time Scaling
        • Time Inversion.
  • Important Functions
  • Mean value, Mean square value, variance, standard deviation.
  • Signal and Vector
  • Correlation
signal operations
Signal Operations
  • Time Shifting: Consider a

signal g(t) and the same

signal delayed by T seconds

which we shall denote by ¢(t).

slide4

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).

slide5

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).
slide6

x(t)

x

T

time, t

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.
mean square value variance standard deviation

x

x(t)

x

T

time, t

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
unit impulse
Unit Impulse
  • 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 cont
Unit Impulse (cont.)
  • 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 εand height 1/ εso that its area is 1.

The unit impulse is the quintessential tall and narrow pulse!

unit step
Unit Step
  • Definition

Integration of the unit impulse yields the unit step function

which is defined as

unit impulse vs unit step
Unit Impulse vs. Unit Step
  • As an example of the method for dealing with generalized functions consider the generalized function
  • Since u(t) is discontinuous, its derivative does not exist as an ordinary function, but it does as a generalized function. To see what x(t) means, put it in an integral with a smooth testing function

and apply the usual integration-by-parts theorem

unit impulse vs unit step cont
Unit Impulse vs. Unit Step (cont.)
  • The result is that

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.

plotting the signal
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
signal and vector
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)
correlation
Correlation
  • indicates the strength and direction of a linear relationship between two random variables