# How to index output of convolution product

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?

If $$x[n]$$ and $$y[n]$$ are both causal and starting at index $$0$$, then the result of convolution will also be causal and it will start at index $$0$$. Just plug in $$n=-4$$ in the expression for $$z[n]$$, you will find that it will be $$0$$. $$z[n] = \sum^{\infty}_{k=-\infty}x[k]y[n-k]$$ First non-zero term in above expression is at $$n=0$$. Because for $$k<0$$, $$x[k]$$ will be $$0$$, you can re-write the above expression as: $$z[n] = \sum^{\infty}_{k=0}x[k]y[n-k]$$ In the above expression, put $$n<0$$, you will see that $$y[n-k]$$ will always be 0.
There will be a transient delay of $$N-1$$ samples, when you are using Moving-Average FIR Filter of length $$N$$, not an advance. The delay is due to the fact that you need $$N-1$$ previous samples of the input before you can produce an average of $$N$$ samples. You cannot have an output before you feed input to a causal FIR filter. Expecting an output at $$-21sec$$, is expecting to look into future which is not correct here.
The figures you have posted is showing that you have 8399 samples of output starting at $$n=0$$. As you can see, approximately at $$5sec$$, the seismic activity starts, Moving average just smoothens-out the original input signal with a transience of $$0.3sec$$.
• @Pirooz In that wiki example, both signals are starting at $t=-0.5sec$, and that is why when one is time-reversed and being sled along timeline, the overlap begins at $t=-1sec$. Look closely at the red function which is moving on time-axis. The dotted line in its middle is the value of $t$ in convolution integral. Check closely that when the overlap starts, at that moment dotted line is at $t=-1sec$. Whereas in your question both signals start at $n=0$, hence the overlap also cannot start before that. Non-causal Filters can use before-hand stored input samples in advance to produce output. Jun 23, 2020 at 6:48