To start, I have absolutely no background in electrical engineering, I am a computer scientist by trade.
I'm currently working on a project where I am attempting to code a function that uses the power spectrum density of a particular signal at a given frequency.
Specifically, I want to calculate the following function
$$F(x,f) = \int [P(f) * X(x,f)] df$$
where $F$ is the function I want to calculate, $f$ is the frequency, $X$ is an function that doesn't pertain to my question, $x$ is a variable that also doesn't pertain to the question, and $P$ is the power spectral density of a signal.
I don't have a function describing the signal, instead I have a vector of samples of the signal. What I'm interested in calculating is $P$ using my sample vector where $P$ has been normalized such that
$$ \int [P(f)] df = 1$$
How might I go about this in MATLAB? I have looked at the
pwelchfunction (Welch's power spectrum density estimate) and I think it does what I want. However, the values seem to be scaled to $[0,\pi]$ which makes me a bit wary. In addition, just calling the function will not scale the values so that $ \int [P(f)] df = 1$. Is using
pwelchan acceptable approach for what I'm trying to do and how can I get things scaled correctly?
In addition to Computer Science, I also have a background in math. However, I'm a bit thrown off by the integrals that I have been provided since the result has been "evaluated" despite no upper and lower limits ($F$ should evaluate to an actual value). Is it common notation to integrate from negative to positive infinity in signal processing using this notation? I understand that this question is a bit vague since I haven't really explained much of what I am trying to do. This question is meant more as a general notation question regarding convention in the signal processing / electrical engineering community. Specifically, is there a common convention as to what limits to integrate over when the limits have been omitted?