Csdrhrt

I was trying to demonstrate the discrete-time expression of a lowpass filter:

$ y_i = alpha x_i + (1-alpha) y_{i-1} $

However my result is completely off the target so I am wondering which of my assumptions is/are wrong...

So we have the input signal:

$ f(t) <=> F(s) $

The lowpass filter (static gain 1 and, Fc = alpha):

$ g(t) <=> G(s) $

with

$ G(s) = alpha / (s + alpha) $

Thus

$ g(t) = alpha * e^{-alpha t} * u(t) $

Our filtered signal is given by:

$ H(s) = F(s) G(s) $

When converting that to temporal, this boils down to a convolution:

$ h(t) = (f * g)(t) $

Which can be turned into a discreet convolution (I'm not exactly sure of this step):

$ h_i = sum_{k = -infty}^{+infty} f(k)g(i-k) $

For $ forall k > i, g(i-k) = u(i-k) = 0 $, and, assuming our signal is null before 0, $ forall k < 0, f(k) = 0 $ so our previous expression becomes:

$ h_i = sum_{k = 0}^{i} f(k) * alpha * e^{-alpha (i-k)} $

It's pretty clear the result is wrong already, but if one tries to express $h_i$ from $h_{i-1}$ here's what one would obtain:

$ h_i = alpha f(i) + alpha sum_{k = 0}^{i-1} f(k) * e^{-alpha (i-k)} $

$ h_i = alpha f(i) + alpha e^{-alpha} sum_{k = 0}^{i-1} f(k) * e^{-alpha (i-1 -k)} $

$ h_i = alpha f(i) + e^{-alpha} h_{i-1} $

Thanks!