Physical or Intuitive understanding of Integration/Differentiation of Signals

Hi,

Integration:
= Summing up over time.
Example: you drive a car. The higher the speed the more distance per time. the total distance is the speed integrated up per time.
You drive with 20 mph for half an hour. Then 30 mph for 20 minutes. Then you pause for half an hour.
The distance first rises with a constant rate (linear, straight line on a chart) to 10 miles after half an hour.
Then it rises additional 10 miles (linear line) to a total of 20 miles after a total of 50 minutes.
Then it stays constant at 20 miles (horizontal line) until 1h 20 min..

Electrically: you have a capacitor connected with a (variable) current source.
Integrate the capacitor current over time and you get (a value proportional to) the capacitor voltage.

***********
Differentiating is the other way round.
= differentiating tells the steepness of a graph at a dedicated location (often: time).

Electrically:
Connect a capacitor to an AC voltage and measure the current.
The faster the voltage rises (the steeper the graph), the higher the current.

Klaus

Added:

For me its always fascinating to recognize how powerful the human brain is.
Learning integration in a math class can be a pain. But indeed the brain uses "integration" quite frequently.

Example:
Merging into the gap between two cars when entering a highway:
* your brain needs to control your car speed to adjust it to the other cars.
* but it also needs "to calculate the integral" of the speed to meet the gap between the cars (and not to collide with the cars)
And to make it hard: both calculations need to be processed in parallel and in real time.

The benefit is, that the human brain is intelligent and adaptive.

Klaus

KlausST said:

Added:

For me its always fascinating to recognize how powerful the human brain is.
Learning integration in a math class can be a pain. But indeed the brain uses "integration" quite frequently.

Example:
Merging into the gap between two cars when entering a highway:
* your brain needs to control your car speed to adjust it to the other cars.
* but it also needs "to calculate the integral" of the speed to meet the gap between the cars (and not to collide with the cars)
And to make it hard: both calculations need to be processed in parallel and in real time.

The benefit is, that the human brain is intelligent and adaptive.

Klaus

Click to expand...

My view is that much of our brain activity works on feedback loops rather than forward precomputations. Just like heating thermostat, it is much more practical to control heating through feedback of temperature than to measure many factors otherwise. This is well documented in the case of hormone control...

Basic RC networks can illustrate differentiators and integrators. They're the same circuits generally used for high-pass or low-pass filters.

Simulation showing how a differentiator and integrator each affect different waveforms: