# what is an alpha filter?

Currently, I'm working on adaptive beamforming using LMS approach, so they change the value of the step factor adaptively in which one of the steps is to pass the weight vector through an alpha filter. SO I'm unable to get any appropriate texts on what it is. Can anyone help me out? Thanks in advance.

my guess is $$y[t]=\alpha y[t-1] + (1-\alpha) x[t]$$ This is a very common form in array processing. The $y$ and $x$ can be scalers, vectors, or matrices.

It is sometimes called a leaky integrator, a forgetting average. I haven’t seen it called an alpha filter but there or only a few things that can be specified with a single parameter.

The other possibility, is an alpha trimmed mean. This uses a threshold mechanism to reject out laying extreme samples. Actual arrays do experience bumps and klunks that can corrupt a covariance estimate.

edit: why guess?

looking at the paper, Equation 3 is their "alpha" filter (can't be anything else) $$\mu_{n+1}= \begin{cases} \alpha \mu_n + \sigma \epsilon_n & \text{if}\; 0< \mu_{n+1} < \mu_{max} \\ \mu_{max} & \text{otherwise} \end{cases}$$ so, some smoothing and some trimming. there is no discussion on how you pick alpha and epsilon. The paper is cited in a patent, so there is probably some special sacred ritual involved.

My guess is that it is an alpha filter, as defined in the context of alpha/beta filtering.

• if beta=0, it doesn’t filter. – user28715 Jun 4 '18 at 16:17
• If (4) in Carlos's link is re-written, it becomes simple exponential smoothing with parameter $\alpha$, which is what Stanley wrote. So, there are many names but simple exponential smoothing is probably the most common. – mark leeds Jun 4 '18 at 18:26
• Actually, for future readers, the link that carlos pointed out is confusing. In equation (4), the LHS should be written as $\hat{x}_{k+1}$ rather than $\hat{x}_{k}$. In fact, I would not use that link if trying to understand ES. There are better explanations all over the internet. – mark leeds Jun 4 '18 at 20:08
• Also, just to try to unconfuse the link as much as possible, in standard simple ES, there wouldn't be an equation (5). Equation (4) would be all that was needed. ( but the $x_k$ on the LHS would be $x_{k+1}$ ). – mark leeds Jun 4 '18 at 20:14
• @markleeds Just to be clear, I never claimed in my answer that the alpha filter described in the link is an exponential smoother, although as you point out, there are similarities. – Robert L. Jun 4 '18 at 20:26