Unfortunately indexing the output of convolution product confuses me. Suppose $x[n]$ is of length $15$ (i.e. $0 <= n <= 14$) and $y[n]$ has the length of $9$ ($0 <= n <= 8$). if $z[n]$ be the output of convolution product of $x[n]$ and $y[n]$, that is $$z[n] = (x * y)[n] = \sum_{k=-\infty}^{+\infty} x[k]y[n-k]$$
As a result, $z[n]$ would has the length of $23$, right?
Now the question is what is the range of $n$ for $z[n]$? Is it $0 <= n <= 22$ or $-4 <= n <= 18$?
From formula for convolution I think the latter is true. But the problem arises when I work with seismic signals.
Suppose $acc[t_i]$ is seismic signal of Kobe earthquake where $0 <= t_i <= 41.99 sec$ with sampling period of $T_s = 0.01 sec$ (length of signal $4200$) as depicted below:
Now it is desired to denoise $acc[t_i]$ using moving average filter with impulse response $h[t_i]$:
$$ h[t_i] =
\begin{cases}
\frac{1}{30}, & \text{if $0 <= t_i <= 0.29$} \\
0, & \text{if $0.3 <= t_i <= 41.99$}
\end{cases}$$
where $T_s = 0.01 sec$. After applying $h[t_i]$ to $acc[t_i]$ by convolution, the output is the sequence $ACC[t_i]$ of length 8399, as shown in following figure.
Obviously the results of filtering process is the interval marked by red bullets. However according to convolution formula, I think I should label time axis between approximately $-21sec$ to $63sec$.
Is that right or something is missing?