DC part isolation - matlab

Question

I have a signal consisting of fast oscillating AC part and slowly varying DC part. I need to calcuate its DC part (and envelope, but that's not important now).
I could use the STFT, filter and transform it back, but it's a little inefficient cause I am not looking for a whole spectrum. Any other ideas?
I have read the articles on MathWorks and my maths is good enough to design something general and complicated, but I am looking for tips, hints or smart and elegant simple solutions. Thanks in advance!

Answer 1

DC isolation is basically low-pass filtering. You want to remove all the high-frequency components, while preserving the low. There are tons of approaches to this problem, but it looks like you're looking for something quick and simple.
One of the simplest forms of low-pass filter is the moving average. It's messy in the frequency domain, but for many simple applications it is good enough, trivial to implement, and incredibly fast.
For reasons I'll get to later on, it isn't actually a very good low-pass filter when you consider the frequency domain, but it is a very good smoothing filter in the time domain, which is what it sounds like you need.
Update: After posting this, I realized this was asked with the MATLAB tag, not just signal-processing. I've added a bit at the end showing the simplest way to do it in MATLAB.
Moving Average Filter
Parameters:
N: Filter order, number of samples to average
Input:
xt: Input at time t. For negative t assume x is zero.
Output:
yt: Output at time t. This is the average of the previous N samples.
Algorithm
Initialization (t = 0)
y0 = x0 / N
At each time step (t > 0)
yt = *yt-1 + (xt - xt-N ) / N
Filter Frequency Response
The impulse response of the filter is a rectangular pulse the width of the filter order. This gives a straightforward frequency response:
H[f] = | sin(πfN) / (Nsin(πf)) |
This shows how horrible the frequency response is, but for many smoothing applications it doesn't really matter. The key thing to notice is that the first zero happens at f = 1/N. This gives you a good idea of what order filter to use in order to smooth out components above a specific frequency, as described in the next section.
Parameter Selection
Given a sampling rate fs and a cut-off frequency fc, the normalized cutoff frequency is simply f = fc / fs. This, plus the frequency response above, gives a simple way to come up with the filter order, although tweaking it slightly by hand may improve results:
N = 1/f = fs / fc
Another way to read this formula is that it is important that the filter order is smaller than the fastest DC feature that you wish to preserve. Otherwise it will be partially filtered out.
Time-Domain Response
To illustrate the time-domain response, I'll work through some examples. I'll be using a sample rate of fs = 1000.
I generated a 40ms rectangular pulse with noise added to demonstrate the effect of increasing the filter order.
As you can see, increasing the filter order smooths out the noise, but it has other effects as well. Higher orders mean a longer delay, and a slower response to step changes. Once the order gets too high, the results are worhtless. The N = 51 example shows what happens when the filter order is wider than the event that you want to capture.
Although those sloping edges may look bad, the moving average filter actually gives the steepest possible edges for a given amount of noise reduction that is possible for a linear filter of the same order, and due to the incredibly simple algorithm it is also the fastest linear filter to implement. Because it has only one parameter, it is also simple to tweak to find the optimal order for your application.
Other Tweaks and Implementation Tips
Multipass Filter
The simplest way to improve the moving average filter's response in the frequency domain is to run multiple passes. After a few runs it becomes a very good approximation of a Gaussian filter due ot the central limit theorem. This slows down the step response a bit, but it also rounds it off. Higher frequencies will be attenuated further, although it also affects the roll-off in the low-frequency end. The performance hit for adding a few more passes is minimal.
Circular Buffer
If you are processing a stream of data, then you don't always have an easy way to lookup xt-N. By using a circular buffer to keep track of the last N samples you end up with a simple way to look up the sample you need with only a few steps added to the algorithm - it preserves the constant-time per step regardless of filter order.
Roundoff Error
If you're working with fixed-point or integer values the roundoff can be completely eliminated. Simply don't divide by N in the filter algorithm:
at = *at-1 + xt - xt-N
yt = *at / N
However, you need to take care to make sure that this won't cause an overflow in a.
With floating-point values, the tiny rounding errors could add up over time. The simplest solution for a long-running filter is to calculate y by averaging the entire buffer of the last N samples every so often. This will reset you to a point where there is no accumulated error, while letting you use the fast algorithm for the vast majority of time steps.
Doing it in MATLAB
A moving average is a linear FIR filter with each weight set the same. This means the numerators are all the same and sum to one, and the denominator is 1. It's simple to use the filter function to perform a moving average:
y = filter(ones(1,N)/N, 1, x);
However, that isn't going to use the fast algorithm discussed above. If performance is an issue then you'll want to implement the algorithm yourself.