math/Proof of Output of Low pass filter with Pulses converges into the average
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Definitions
The system focused on here is also called 1st order system. It is formulated with the following equation.
\frac{\od Q}{ \od \ot} = \frac{1}{\tau}\br{- Q + q}
where \(Q\) is a capacity of the system, \(\tau\) is a time constant, \(q\) is an input. The followings are assumed in this discussion. For any \(t\) on the time axis \(\ot\)
\frac{1}{T}\int_{0}^T q(t + \ot )\od \ot \approx \bar{q} \in \sR,
where \(T\) is a interval that average of \(q\) in any interval with the length \(T\) is approximately \(\bar{q}\). More specifically, for all \(a\in \sR\) there is a \(C \in \sR\) such that
\abr{\frac{1}{a}\int_{0}^a q(t + \ot )\od \ot - \bar{q}} \leq C\ce^{-\frac{a}{T}} ,
that is, error will decrease exponentially with respect to \(a\). The following \(h\) is introduced along with the constraint,
T \ll \tau \ll h,
and,
\abr{Q(t)}\leq 1,\abr{q(t)} \leq 1, Q(0) = 0
note that \(\abr{Q(t)}\leq 1\) can be assured from the other 2 conditions above and the equation .
Proof
With the above mentioned definitions, is investigated. It can be,
\frac{\od Q}{ \od \ot}\ce^{\frac{\ot}{\tau}} + \frac{1}{\tau}\ce^{\frac{\ot}{\tau}} Q = \frac{1}{\tau}q\ce^{\frac{\ot}{\tau}}
so,
Q(t + h) \ce^{\frac{t + h}{\tau}} = Q(t) \ce^{\frac{t}{\tau}} + \frac{1}{\tau}\int_{t}^{t+h}\ce^{\frac{t}{\tau}} q(\ot)\od \ot,
multiplying \(\ce^{\frac{t + h}{\tau}} \) on both side of the equation, we have,
Q(t + h) = Q(t) \ce^{-\frac{h}{\tau}} + \frac{1}{\tau}\ce^{-\frac{h}{\tau}}\int_{0}^{h}\ce^{\frac{x}{\tau}} q(t + x)\od x.
From the assumption, \(Q(t) \ce^{-\frac{h}{\tau}} \approx 0\). Let,
N = \max\cbr{i; i \leq h/T, i \in \sN}, L = h/N,
Then,
Q(t + h) &\approx \frac{1}{\tau}\ce^{-\frac{h}{\tau}}\sum_{i=0}^{N-1}\int_{0}^L\ce^{\frac{iL + y}{\tau}} q(t + iL+ y)\od y\\
&= \frac{1}{\tau}\ce^{-\frac{h}{\tau}}\sum_{i=0}^{N-1}\ce^{\frac{iL}{\tau}}\int_{0}^L\ce^{\frac{ y}{\tau}} q(t + iL+ y)\od y\\
&= \frac{1}{\tau}\ce^{-\frac{h}{\tau}}\sum_{i=0}^{N-1}\ce^{\frac{iL}{\tau}}\int_{0}^Lq(t + iL+ y)\od y\ \because \ce^{\frac{ y}{\tau}} \approx 1\\
&= \frac{1}{\tau}\ce^{-\frac{h}{\tau}}L\bar{q}\sum_{i=0}^{N-1}\ce^{\frac{iL}{\tau}}\ \because \\
&= \frac{1}{\tau}\ce^{-\frac{h}{\tau}}L\bar{q}\frac{\ce^{\frac{NL}{\tau}}-1}{\ce^{\frac{L}{\tau}}-1}\\
&= \frac{1}{\tau}\ce^{-\frac{h}{\tau}}L\bar{q}\frac{\ce^{\frac{h}{\tau}}-1}{\frac{L}{\tau}}\ \because \frac{L}{\tau}\ll 1, L = h/N\\
&= \bar{q}\br{1-\ce^{-\frac{h}{\tau}}}
Therefore, considering \(\ce^{-\frac{h}{\tau}} \approx 0\) because \(\tau \ll h \),
Q(t+h)\approx \bar{q}
Note and limitation
Error is not discussed rigorously. The proof uses \(\approx\) notation, that is not precise. This result can be observed in a system uses Pulse Width Modulation and Pulse Density Modulation.