What exactly is the sample period?

I thought the term sample period referred to the time interval between one measurement and another. But I've come across this article which says

About every tenth of a second, a dataset, $$S(i)$$, of the form:

...

is recorded (of course, the time between taking two datasets must be a multiple of the sample period, i.e., $$\Delta t \mid (t_{i+1}- t_i), \forall i$$).

Now, the way I interpret this is that $$t_{i+1}- t_i$$ is the time interval between a recording of a data set and the successive. So, if that's correct and $$\Delta t$$ (i.e. the sample period) is the time interval between one measurement and another, then $$\Delta t = t_{i+1}- t_i$$, but clearly some definitions must be wrong, otherwise the original author of that article would not have introduced two terms to identify the same thing, and would not have said that $$t_{i+1}- t_i$$ must be a multiple of $$\Delta t$$?

What exactly is the sample period $$\Delta t$$? How is it different from the interval between one measurement and the other (i.e. $$t_{i+1}- t_i$$)? What exactly does it mean that $$t_{i+1}- t_i$$ must be a multiple of $$\Delta t$$ (I guess this will be answered once I know what's exactly the sample period)?

Your definition of the sample period is correct but is misused here. In my opinion, this originates from the notations used in the referred article.

Therefore, let me firstly introduce the following notation $$f[n] \triangleq f(n\Delta t)$$ used to refer to discrete samples from a continuous function $f$ which are regularly distant (in the continuous domain) by an interval $\Delta t$ called the sample period (you already know that). Here, we sample a continuous-time function and obtain a discrete-time function.

Secondly, consider sampling $f[n]$, for example $$f_s[n] \triangleq f[n\ N]$$ where $N$ is also a sample period! But $\Delta t$ and $N$ are different because of having been defined in different contexts. Particularly, notice that $\Delta t$ is a time duration (in seconds) while $N$ is dimensionless. Furthermore, $\Delta t \in \mathbb{R}^+$ while $N \in \mathbb{N}$.

Now, let us translate $f_s[n]$ into a continous-time function $$f_s(t) \begin{cases} \triangleq f_s\Big[\frac{t}{N\Delta t}\Big] & \text{if } \frac{t_n}{N\Delta t} \in \mathbb{Z}\\ \text{undefined} & \text{otherwise} \end{cases}$$ As $f_s$ is defined over $\mathbb{R}$, when we have no corresponding sample for $t$, $f_s(t)$ can not be defined. Actually, you could define it as 0 or something else but this does not matter here because the interesting part is when it is defined: notice that the times at which the function is defined are expressed as $t_n \triangleq n\ N\Delta t$.

Going back to the article, $\Delta t$ is the sample time for the sensors while $S(i)$ is a sample of a set of (already sampled) functions (or a set of samples, depending on the way you look at it) that they call a dataset. When they talk about a sample period for $S$:

About every tenth of a second

they refer to $N\Delta t$, not $\Delta t$! And, when you understand this distinction,

of course, the time between taking two datasets must be a multiple of the sample period, i.e., $\Delta t \mid (t_{i+1}- t_i), \forall i$

as $(t_{i+1}- t_i)$ is $N\Delta t$.

After another quick look at the article, I noticed that they use the symbol $N$ as well. It should be obvious that the meaning of $N$ here is different from what it expresses in the article.

Secondly, as I have less than 50 reputation, I am not allowed to post comments so I will add this to my answer: Another answer states that $\Delta t$ is the total duration (in the article) while my previous definition of it comes from Section 3

Assume that two inertial sensors, one attached to the upper leg and the other attached to the lower leg, measure the accelerations, $a1(t),a2(t) \in \mathbb{R}^3$, and angular rates, $g1(t),g2(t) \in \mathbb{R}^3$, at some sample period, $\Delta t$.

which I assumed to be the definition of $\Delta t$. I would appreciate any clarification of this.

I suggest a grammar problem:

If you read properly the article $\Delta t$ is the total time of the measurement.

So the word 'Period' is here not to describe the wavelength of a repetitive behaviour but the Duration.

This is wrong: $\Delta t = t_{i+1}- t_i$

This is correct $\Delta t = N.(t_{i+1}- t_i)$

($N$ is defined in your article and also 'the time between taking two datasets')

Again there is no repetition in the time period. To make it less confusing, the author should have called it 'time interval' or 'duration'

• Well, I think he just did not knew how to say $t_{i+1}- t_i = \Delta t / N$ with $N$ is an integer and is the number of measurement you plan to do. Because $N$ is not yet really defined and also most likely because in his methods he first chose the time before the number of measurements. And you know when you write (Obviously ...) then most likely you will have something not very mathematical coming. Anyway I can see the question has generated brain twisting answers, so I'll give a + vote and good luck with figuring out. May 24 '17 at 15:25

Collection of the dataset is performed less frequently than sampling - it is performed at certain (unnamed, nor given a symbol) intervals, which are the multiples of the sample period, described by symbol $Δt$, defined as $Δt = t_{i+1}−t_i$.

This is pretty standard - in case of simple sensors, like accelerometers, measurements are performed very frequently (very short sample period), because why not? - their controller can do it, doesn't have anything else to do, and in case step of some analysis needs to be decreased, the higher density measurements are there.

Afterwards, complex analysis of data ignores/discards part (often a major part) of the measurements, and only picks every nth sample for the dataset - for reasons of performance, storage space, lack of accuracy of other inputs, and a range of other reasons - the sample period of the dataset is consciously made a multiple of sample period of the sensor. We have one dataset sample every $n$ samples, and complete dataset comprised of $N$ dataset samples. The whole dataset covers therefore a period equal $N \cdot n \cdot (t_{i+1}−t_i)$

The authors deemed detailing how many sensor samples are dropped unimportant, but the gist of the paragraph (with N >> 1) is "the sensor provides overwhelming amount of data, so we drop some of it regularly, but we still collect a lot."