Hello,
how can I get:
$\displaystyle y[k](\frac{T}{T_v}+1) = Kx[k] + \frac{T}{T_v}y[k-1]$
to this:
$\displaystyle y[k]= \frac{T_v K}{T+T_v}x[k] + \frac{T}{T_v + T}y[k-1]$
? I would need detail derivation.
Thanks.
I can believe you are serious. You seriously need to learn algebra! "Dividing by $\displaystyle \frac{T}{T_v}+ 1$" is not "multiplying by $\displaystyle \frac{T_v}{T}+ 1$"! $\displaystyle \frac{T_v}{T}+ 1= \frac{T_v}{T}+ \frac{T}{T}= \frac{T_v+ T}{T}$. Dividing by that is the same as multiplying by $\displaystyle \frac{T}{T_v+ T}$.
Thanks, sorry if I am an idiot but I don't use math in my daily life.
I have other one which is a bit difficult:
How do we get this:
$\displaystyle y[k] = K x[k] + K \sum_{i=0}^{k-1}x[i]$
to:
$\displaystyle Kx[k] + y[k-1]$
This last one is a recursive equation.
This last one isn't an equation at all! Are you trying to solve for Kx[k] + y[k - 1]? If then just write out what y[k - 1] will be and then you can write out an expression for x[k] and y[k - 1]. This isn't the hard part. The hard part is to try to simplify the answer (I'm assuming you need to do that) and for that you'll have to find a way to handle the summation.
Hint: The summation just adds up a series of terms x[0] + x[1] + ... + x[k - 1].
-Dan