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Step, Ramp and Impulse Functions These two signals are commonly used in signal processing and control systems. A- Unit Step Function ( ) { u(t) 1 0 t If the signal has an amplitude other than unity, then the function is scaled as follows: If the amplitude is K, then Ku(t) (show example) Unit Step function acts as a switch in circuits Show examples. A- Unit Ramp Function ( ) { r(t) slope =1 t 0 If the function has a slope other than unity, say m, then the function is expressed as: ( ) { Observation: - the derivative of the ramp function step function - the integration of the step function ramp function Time Shift Properties Time delay occurs in signals due to propagation delays. In signals and systems, sometimes it is required to switch different signals at different times. Mathematical expression of time-delay helps in the analysis of such situations. - You have seen some of these properties when dealing with trigonometric functions. Suppose you have the time-domain function f(t), then what is the meaning of f(t-t0), and what does f(t+t0) mean ? example: suppose f(t) = 5sin(t), and suppose t0 = 0.5. Then, f(t-0.5) = 5*sin(t-0.5) As can be seen, when t = 0 in the original function, the shifted function will produce the same result at +0.5 seconds later. See Matlab® and figures below >> t=0:0.01:4*pi; >> y1=5*sin(t); >> y2=5*sin((t-.5)); >> plot(t,y1,t,y2);grid 5 4 y1 3 y2 2 1 0 -1 -2 -3 -4 -5 0 2 y2 is delayed by 0.5 seconds. - In class examples. 4 6 8 10 12 14 Thus: +t0 yields a left shift – time advance - t0 yields a right shift, time delay C- Impulse Function Discuss practical and theoretical scinarios Discuss examples.