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EE3TP4_2_CTSignals_Lecture 4

EE3TP4_2_CTSignals_Lecture 4 - Continuous Time...

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1 Continuous Time Signals (Textbook Section 1.1) These overheads were originally developed by Mark Fowler at Binghamton University, State University of New York.
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2 Continuous-Time Signals The mathematical model for a C-T signal is a “function of time”. A C-T signal is defined on the continuum of time values, e.g., f ( t ) for t Real line f ( t ) t Example: Example:
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4 Continuous-Time Signals in Matlab The signal is “sampled” at a sufficiently high level of detail. Matlab interpolates the samples when graphed. (Figure 1.10 in textbook) % Figure 1.10 t=0:0.1:30; x = exp(-.1*t).*sin(2/3*t); plot(t,x) grid axis([0 30 -1 1]); ylabel('x(t)') xlabel('Time (sec)') title('Figure 1.10') x t = e 0.1t sin 2 3 t u t
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5 Unit Step Function u ( t ) u t = { 1, t 0 0, t 0 . . . u ( t ) 1 t Note: A step of height A can be made from Au ( t ) Step and Ramp Functions These are common textbook signals but are also common test signals, especially in control systems. . . . In system analysis, what use are unit step functions?
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6 The unit step signal can model the act of switching on a power source … V s R + C t = 0 R V s u ( t ) C +
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7 Matlab and the Unit Step This is available as the heaviside function in some versions of Matlab. Here is an example: function y = heaviside(x) y = double(x >= 0); This will return a “1” when the argument is “0”. Some implementations will return “0.5”, and some will return “NaN”!
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8 Unit Ramp Function r ( t ) Note: A ramp with slope m can be made from: mr ( t ) . . . r ( t ) 1 t 1 Unit slope r t = { t , t 0 0, t 0 mr t = { mt , t 0 0, t 0
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9 Relationship between u ( t ) & r ( t ) What is ? Depends on t value function of t : f ( t ) What is f ( t )? - Write unit step as a function of λ - Integrate up to λ = t - How does area change as t changes? i.e., Find Area u ( λ ) 1 λ λ = t Area = f ( t ) “Running Integral of step = ramp” −∞ t u λ f t = −∞ t u λ f t = −∞ t u λ = 1 t = t = r t r t = −∞ t u λ
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10 Also note: For we have: Overlooking this, we can roughly say . . .
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