Filter Problem

A Filter Problem
Problem Freq1P00CInt

This is a guided problem to help you with first order filters. In two previous notes like this you should have encountered the following filter behavior.

In Problem Freq1P00Int, you should have learned about the effects of a first order filter on sinusoidal signals.
In Problem Freq1POOIntB, you should have learned about the effects of a first order filter on noise signals, and you should also have encountered a first order signal that processed a constant signal corrupted by noise.

In this problem you will encounter a first order filter that has a constant input added to a sinusoidal input, and the sum is corrupted by noise. Once again, we will do that using a simulator. Click here for the filter simulator.

Problems

P1 Start the filter simulator. The default values in the filter simulator are:

The DC component in the signal is always 1.0
The default value of the AC component is 0.2. That's considerably smaller than the DC component, so the effect is to have a signal (1.0) with a smaller fluctuating component.
The default input noise amplitude is 0.2. The noise in this simulator will be uniformly distributed from -0.2 to +0.2 when the noise amplitude is set to 0.2.
The filter DC gain is set to 1.0.
The default value of the filter time constant is 1.0 second.

Question

Q1 With these values, will there be noticable phase shift in the AC signal? (Noticable would be anything above, say, ten degrees.)

Actually, what is the phase shift? When you enter your answer below, don't forget the sign.

Run the simulator using the default values. Do the simulated results agree with your calculations? Note, you can set the noise magnitude to zero, and increase the AC signal magnitude (increasing to 0.5 looks good) so that you can see better what is going on.

There is another problem. The filter can change the magnitude of the output for the sinusoidal component. What is the ratio of output to input for the sinusoidal component at the default frequency?

At this point, there are some conclusions that you should have reached from what you have observed to this point.

Just looking at the results obtained when the simulator is run with the default values, it should be clear that it is possible to do a good job of filtering the noise from the signal.
It should also be clear that the DC component of the signal passes through the filter with no reduction in size.
It should also be clear that the AC component of the signal can be shifted in phase and can experience a reduction in amplitude.

With these observations, there is another question that you should consider. Here is the question.

Is is possible to get good filtering without seriously altering the AC component of the signal?

To answer this question, let's try another simulation. Do the following.

Use the default values for the simulator except for the time constant.
Adjust the time constant to reduce the effect on the AC signal.

Question

Q1 Should we increase the time constant or decrease it?

You should - by now - realize that we need to decrease the time constant. So try doing that. Decrease the time constant by a factor of 2 and redo the simulation. (That means that the filter time constant would be 0.5 seconds. Remember to reset the simulator to the default values if you changed any of those values.)

When we did that simulation, we noticed the following.

The output sinusoid looked to be nearly in phase with the input sinusoid. (The zero-crossings should be right at the grid marks when the input is a sine wave, and they were almost there.)
Looking carefully at the output sinusoid, there did not seem to be too much noise in it.
The response to the DC portion of the signal is faster.

The natural question is whether we might get more improvement with a further decrease in time constant. Reduce the time constant to 0.25 seconds, and run the simulation again. Our observations were as follows.

For this simulation, the noise starts to be very noticable in the output sinusoid, but the output sinusoid seems to be in phase with the input sinusoid.
Looking carefully at the output sinusoid, there did not seem to be too much noise in it.
The response to the DC portion of the signal is faster yet again.

The conclusions that you can reach look something like the following.

When you filter noise from a signal, you may remove some or most of the noise, but there is always a price for that.

The response of the system to a DC component of the signal is slower when the filtering is good.
The reponse of the system to an AC component of the signal gets worse when the filtering is good. You get more phase shift and more reduction in amplitude when the filtering is good.

Essentially, there are tradeoffs between getting rid of the noise, and getting good response to the signal components. You pay your money and you take your choice.

That's it for this worksheet. You should have learned more about filters by being able to play a few "What if?" games. We hope that you enjoyed that, and that you found it to be a good learning experience.