But the functional is the ratio of two voltages, Vin and Vout , so it should be dimensionless. The system functional is dimensional nonsense, and cannot be valid. Signals and Systems Fall 41 A third method, and the one that we use here because it illustrates the integration operator A, is to represent each tank as a power series in A, and then feed the output of the first tank into the series for the second tank. The leftward factor computes a weighted average of those samples. Appendix Data for problem 5 also available as a.
We have practiced using it to find the impulse response. Write a program to determine how much tracer is in the sherry bottled in year n for a B-barrel solera system, where B is a parameter to your program. Signals and Systems Fall 15 2. Each new tank is three times slower than its corresponding tank in the original cascade. However, the bare A in the denominator represents a differentiator. Here we want the the step response.
The sum is 1. We can use the forward-Euler method to generate the following approximation: Assume that one unit of tracer is added in year 0 and no tracer is added in any other year. The capacitor integrates current into stored charge, which generates a voltage across the capac- itor. Therefore what can you conclude about the poles of the system? Homework on the plotted data points, what solutions can you make?
Determine the unit sample response of this system. Make sure the pole loca- tions are consistent with your answers to parts a and c!
Notice that we can just run the difference equation backwards using the desired output: The leftward factor computes a weighted average of those samples. R Now replace I in the functional expression for the output voltage: Use the forward-Euler method yomework develop a set of difference equations that approximate this system of differential equations.
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Signals and Systems Fall 19 Draw a block diagram, write the system functional, and give a pole—zero diagram. Both poles are inside the unit circle but off the positive real axis, so both are decaying and oscillatory.
What is the maximum usable sensor delay i. That problem is equivalent to the impulse response uomework a cascade of 1 A and 2 the ODE system. There are many numerical methods.
Signals and Systems Fall 42 0. The block diagrams have similar parts, but the topologies are completely different.
Signals and Systems Fall 20 e. Here are thumbnails of the images: Use difference equations to express the relation between deposits and balances for each bank.
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The leaky tank has the same equation as the RC circuit 66.003 Recitation 1. Which if any of the plots A, B, C, or D correspond to the following block diagram?
Now add one more sensor delay. Feel free to use a computer, graphing calculator, or paper and pencil.
Appendix Data for problem 5 also available as a. Which bank has the larger balance in the year ? You can use a computer or, if you are skilled with series, you can do it analytically. One way to check stability is to write a simulation.
So this functional cannot be right. Due in recitation on Wednesday, 17 October Find the impulse response of the two systems analytically and sketch the results. Impulse responses Find the impulse responses of the following systems: The resulting pictures can be unblurred by passing each row through a system with a zero at 0.
An alternative, which we use here for variety, is to ask maxima to find the poles. Then check the behavior in between the extremes.