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I have read about Finite Rate of Innovation signal by Martin Vetterli in here. But i do not understand several basic things

  1. The paper said that finite rate of innovation is the number of degree of freedom of a signal. But i cannot imagine a degree of freedom in signal term. If i have a robot that move lateral in X, Y, and Z, hence i can say it has 3 degree of freedom. How can i imagine it in signal term?

  2. A standard sampling is just sample a signal for every time interval, but in FRI sampling, there is a smoothing kernel, what is the purpose of this?

  3. what is annihilation filter?

Thank you

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If a signal can be exactly represented by $N$ real numbers per time interval, then its number of degrees of freedom for that time interval equals $N$. The most well-known example are band-limited signals, which can be represented by $2B$ samples per second, if their bandwidth is not greater than $B$. I.e., such signals have $2B$ degrees of freedom per second. This is equivalent to saying that these signals have a finite rate of innovation, and that rate of innovation equals $2B$.

A signal doesn't need to be band-limited in order to have a finite rate of innovation (i.e., a finite number of degrees of freedom). You can think of any signal that can be parameterized with a finite number of parameters per time unit. A simple example would be a signal of the form

$$x(t)=\sum_kw_kg(t-kT)\tag{1}$$

where $g(t)$ is a rectangular impulse with a value of $1$ in the interval $[0,T]$ and zero otherwise. You need exactly one sample per $T$ seconds to represent $x(t)$, even though $x(t)$ is clearly not band-limited.

A smoothing kernel is just an optional filter that is applied to the continuous-time signal before sampling. For sampling a (quasi-)band-limited signal, the anti-aliasing filter would be an instance of such a smoothing filter.

An annihilating filter for a specific signal is an LTI system that produces an output of zero when that specific signal is applied at its input. E.g., for a sinusoidal signal with frequency $\omega_0$, a filter with a notch at $\omega_0$ is an annihilating filter.

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  • $\begingroup$ Hello, Thank you for the help. But is it just a single real number for each time interval sampling? such as if i have 10 Hz than for each 100 ms time interval i just can have a single value? $\endgroup$ – Zahi Azmi Dec 18 '18 at 10:46
  • $\begingroup$ @ZahiAzmi: It depends on the time interval you're considering. For a band-limited signal, you have $2B$ samples per second, or one sample per $1/2B$ seconds. The rate is usually measured in number of samples per second. So in general you have more than $1$ number per time unit (which is one second). $\endgroup$ – Matt L. Dec 18 '18 at 10:48
  • $\begingroup$ Thank you, so FRI means the rate for one second? And about the the non-band-limited signal, because the frequency can be infinity so the minimum sampling will be infinity, so how can it have a FRI? Thank you $\endgroup$ – Zahi Azmi Dec 18 '18 at 10:52
  • $\begingroup$ @ZahiAzmi: The rate of innovation is the number of values per second needed to exactly represent the signal. As I've mentioned in my answer, there are non-bandlimited signals that only depend on a finite number of parameters per time unit, so they have a finite rate of innovation, even though they are not band-limited. $\endgroup$ – Matt L. Dec 18 '18 at 10:54
  • $\begingroup$ Thank you, i am still confused about the parameters, what is it related to the signal? $\endgroup$ – Zahi Azmi Dec 18 '18 at 10:56

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