Properties of Delta Function

1. Properties of Delta Function • Delta function is a particular class of functions which plays a significant role in signal analysis. • They have simple mathematical form but they are either not finite everywhere or they do not have finite derivatives of all orders everywhere. They are also known as singularity functions. • Unit impulse function or Dirac delta function is a singularity function of great importance. This function has the property • for any f(t) continuous at for finite t0. Communication Fundamentals

2. The impulse function selects or sifts out a particular value of the function f(t), namely, the value at t=t0, in the integration process. • If f(t) = 1, then the above equation becomes • Therefore (t) has unit area. • Also • The symmetry properties of delta function stipulates that • Time scaling property -- Communication Fundamentals

3. Multiplication by a time function • Relationship with the Unit step function, which is given by • For f(t)=1, we have • Thus the derivative of the unit step function yields a delta function Communication Fundamentals

4. Communication Fundamentals

5. Fourier Transform • We consider an aperiodic function f(t) as shown • We wish to represent this function as a sum o fexponential functions over the entire interval . For this purpose, we construct a new periodic function , with period T so that the function f(t) is forced to repeat itself completely every T seconds. The original function can be obtained by letting Communication Fundamentals

6. The new function fT(t) is a periodic function and consequently can be represented by an exponential Fourier Series • we define Using these definitions we obtain Communication Fundamentals

7. The spacing between adjacent lines in the line spectrum of fT(t) is Using this relation for T we obtain the alternate form • Now as T becomes very large, becomes smaller and the spectrum becomes denser. As , the discrete lines in the spectrum merge and the frequency spectrum becomes continuous. Thus, Communication Fundamentals

8. becomes • similarly • Symbolically • The complex Fourier series coefficients can be evaluated in terms of the Fourier Transform Communication Fundamentals

9. Spectral Density Function • The area under the spectral density function F() gas the dimensions voltage. Each point on the F() curve contributes nothing to the representation of f(t). It is the area that contributes. But each point does indicate the relative weighting of each frequency component. The contribution of a given frequency band to the representation of f(t) may be found by integrating to find the desired area. • A periodic waveform has its amplitude components at discrete frequencies. At each of these discrete frequencies there is some definite contribution. To portray the amplitude components of a periodic waveform on a spectral-density graph requires a representation with area Communication Fundamentals

10. equal to the respective amplitude components yet occupying zero frequency width. This can be done by representing each amplitude component of the periodic function by an impulse function. The area of the impulse is equal to the amplitude component and the position of the impulse is determined by the particular discrete frequency. • Summarizing, a signal of finite energy can be described by a continuous spectral density function. This spectral density function is found by taking the Fourier transform of the signal. Communication Fundamentals

11. e.g. Find the Fourier transform of a gate function defined as • We have Communication Fundamentals

12. Communication Fundamentals

13. Fourier Transform Involving Impulse Functions • The Fourier transform of a unit impulse is • The phase spectrum of the time-shifted impulse is linear with a slope that is proportional to the time shift. Communication Fundamentals

14. Complex Exponentials • We would expect that the spectral density of • as shown Communication Fundamentals

15. Sinusoids • The sinusoidal signals can be written in terms of the complex exponentials using Euler’s identities Communication Fundamentals

16. Communication Fundamentals

17. Signum Function and the Unit Step • The Signum function, sgn(t), changes sign when its argument is zero • The signum function has an average value of zero and is piecewise continuous, but not absolutely integrable. To make it absolutely integrable we multiply sgn(t) by • and then take the limit as Communication Fundamentals

18. Interchanging the operations of taking the limit and integrating we have • The unit step function can be expressed as • Thus Communication Fundamentals

19. Periodic Functions • A periodic function, of period T, can be expressed as • Taking the Fourier transform, we find Communication Fundamentals

20. Communication Fundamentals

21. Communication Fundamentals

22. Properties of Fourier TransformLinearity • This follows directly from the integral definition of Fourier transform Complex Conjugate • For any complex signal we have • If f(t) is real, then Communication Fundamentals

23. Symmetry • Any signal can be expressed as a sum of an even function and an odd function Duality • Duality exists between time and frequency domain as shown below Communication Fundamentals

24. The proof can be done by interchanging t and  in the Fourier transform integral. Communication Fundamentals

25. e.g. It is given that find • Let Communication Fundamentals

26. Coordinate Scaling • The expansion or compression of a time waveform affects the spectral density of the waveform. For a real-valued scaling constant  and any pulse signal f(t), Communication Fundamentals

27. If  is positive and greater than unity, f(t) is compressed, and its spectral density is expanded in frequency by 1/  . The magnitude of the spectral density also changes -- an effect necessary to maintain energy balance between the two domains. If  > 0 but less than unity, f(t) is an expanded version of f(t) and its spectral density is compressed. When  < 0, f(t) is reversed in time compared to f(t) and is expanded or compressed depending whether | | is greater than or less than unity. Communication Fundamentals

28. Time Shifting Frequency Shifting Communication Fundamentals

29. Differentiation and Integration • If df/dt is absolutely integrable, then • The corresponding integration property is Communication Fundamentals

30. Consider the function g(t) defined as • Let g(t) have Fourier transform G(). Now • However, for g(t) to have a transform G() must exist. One condition is that This means • which is equivalent to F(0) = 0. If then g(t) is no longer an energy function and the transform will include an impulse function Communication Fundamentals

31. Time Convolution • There are two ways of characterizing a system -- frequency response and impulse response. The two can be related using the principle of convolution. • For the test signal , the system impulse response is defined as where  is the delay or age variable. If the system is time-invariant, h(t,) takes the special form . The input signal f(t) may be expressed in terms of impulse functions by • If we define Communication Fundamentals

32. From integration theory, we can rewrite this as • Using the principle of superposition, we move the system operator inside the summation. Also, the f(n) are the weights (areas) of the impulse functions and are constants for each impulse. Therefore we have • Therefore we have • This is a key result in signal analysis for it links the input to the output by means of an integral operation. Communication Fundamentals

33. The equation reduces to • This is known as the convolution integral. • An important property of the Fourier transform is that it reduces the convolution integral to an algebraic product. • Proof Communication Fundamentals

34. Changing the order of integration and integrating with respect to t first yields Communication Fundamentals

35. Frequency Convolution • A dual to the preceding property can be established Communication Fundamentals

36. Some Convolution Relationships • The convolution integral holds as long as the system is linear, time-invariant, and causal. Thus h(t) = 0 for all t < 0 and there is no contribution to the integration for (t-) < 0. • Often the input, f(t), also satisfies f(t) = 0 for t < 0. • Properties of Convolution • Commutative Law -- • Distributive Law -- • Associative Law -- Communication Fundamentals

37. Convolution involving Singularity Functions • The unit step response is the indefinite integral of the unit impulse response as shown • This provides a technique for determining the impulse response of a system in the laboratory. • Convolution with the unit impulse function gives Communication Fundamentals

38. Example: Find as shown: Communication Fundamentals

39. Graphical Interpretation of Convolution • The graphical interpretation of convolution permits to understand visually the results of the more abstract mathematical operations. For instance • The required operations are as listed below: • Replace t by  in f1(t) giving f1() • Replace t by (- ) in f2(). This folds the function f2() about the vertical axis passing through the origin of the  axis. • Translate the entire frame of reference of f2(- )by an amount t. Thus the amount of translation, t, is the difference Communication Fundamentals

40. between the moving frame of reference and the fixed frame of reference and the fixed frame of reference. The origin in the moving is at  = t, the origin in the fixed frame is at  = 0. The function in the moving frame represents f2(t- ).The function in the fixed frame represents f1(t). • At any given relative shift between the frames of reference, e.g. t0, we must find the area under the product of the two functions Communication Fundamentals

41. This procedure is to be repeated for different values of t=t0 by successively progressing the movable frame and finding the values of the convolution integral at those values of t. • If the amount of shift of the movable frame is along the negative  axis, t is negative. If the shift is along the positive  axis, t is positive. Communication Fundamentals

42. Example: • Find the convolution of a rectangular pulse and a triangular pulse Communication Fundamentals

43. Communication Fundamentals