Is there a name for this smoothing formula? Hi: I am reading a book called "lectures on wiener and kalman filtering" by Professor Thomas Kailath. On page 18, it says the following:


  To carry through this approach, let us first note that a smooth random process
  $ \{ Y(\tau), a \le \tau \le b \}$ can be approximately represented as
  
  $$Y(\tau)  = \sum_{i=0}^{n-1} Y(\tau_{i}) \times \sqrt \triangle \times p(\tau - i \triangle) = \sum_{i=0}^{n-1} Y_{i} \times p(\tau - i \triangle)$$
  
  where
  
  $ p(\tau) = \begin{cases}\frac{1}{\sqrt \triangle}, &amp; 0 \le \tau \le \triangle \\  0, &amp; \text{otherwise}\end{cases}$


There is a picture included with the text:



$\triangle$ is the length of the space between the discrete $\tau_{i}$. The $\tau_{i}$ are the discrete versions of the time and are equally spaced.

I have some questions about this formula but I can send someone the pages from the book if they need them since, without a picture, it may be difficult to understand. ( note that I don't even follow it with the picture !!!!! ).


I don't have any intuition for this smoothing formula.  This seems like some kind of windowing process that uses weights that are smaller for further away values and larger for closer values ? Is there a name for it ?
The second equality in the smoothing relation implies that $Y(\tau_{i}) \times \sqrt  \triangle = Y_{i} $. I don't follow that ?
It looks like the weights are going to be negative and positive depending on the value of $\tau$. Is this correct ?
Like I said, I can send the 3 pages ( what I wrote is a part of a longer derivation of a functional relation for least squares ) to anyone who is interested.  Dr. Kalaith makes this approximation sound trivial but it's not to me.
1) I don't have any intuition for this smoothing formula.  This seems like some kind of windowing process that uses weights that are smaller for further away values and larger for closer values ? Is there a name for it ?


The first misconception is that this is not a smoothing formula.  It's a way of representing an already smooth continuous-time signal in discrete samples.

All it's doing is as @robert says in the comments: it's assuming that the value of the signal between $\tau_i$ and $\tau_{i+1}$ is just $Y(\tau_i)$ --- it's piecewise constant.

I believe that $p(t)$ is just a "pulse".


  2) The second equality in the smoothing relation implies that $Y(\tau_{i}) \times \sqrt  \triangle = Y_{i} $. I don't follow that ?


This is just allowing us to simplify $Y(\tau_{i}) \times \sqrt  \triangle$ and allow us to talk about $Y_i$ as shorthand notation.

edit from r b-j: it's just an alternative shorthand notation for $Y(\tau)\bigg|_{\tau = \tau_i}$ if $Y_i$ was anything else (like coefficients from some other set), i can't say that it's a ZOH with a DC gain of 0 dB.  and that "$\sqrt{\triangle}$" is chaff.


  3) It looks like the weights are going to be negative and positive depending on the value of $\tau$. Is this correct ?


The "weights" are either $0$ or $\frac{1}{\sqrt{\triangle}}$. For any value of $\tau$ outside $0 \le \tau - i\triangle \le \triangle$ the "weight" (pulse) is zero.


  4) Like I said, I can send the 3 pages ( what I wrote is a part of a longer derivation of a functional relation for least squares ) to anyone who is interested.  Dr. Kalaith makes this approximation sound trivial but it's not to me..


Please give me feedback on the above answers.