A digital filter is given by the following difference equationy[n] = x[n] − x[n − 2] −1/4y[n − 2].(a) Find the transfer function
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Answer:
Explanation:
According to the first thermodynamic law, the energy must be conserved so:
Where Q is the heat transmitted to the system, U is the internal energy and W is the work done by the system.
This equation can be solved by integration between an initial and a final state:
(1)
As per work definition:
For pressure the force F equials the pressure multiplied by the area of the piston, and considering dx as the displacement:
Here A*dx equals the differential volume of the piston, and considering that any increment in volume is a work done by the system, the sign is negative, so:
So the third integral in equation (1) is:
Considering the gas as ideal, the pressure can be calculated as , so:
In this particular case as the systems is closed and the temperature constant, n, R and T are constants:
Replacion this and solving equation (1) between state 1 and 2:
The internal energy depends only on the temperature of the gas, so there is no internal energy change , so the heat exchanged to the system equals the work done by the system: