0
$\begingroup$

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

$$ \int [P(f)] df = 1$$

Questions:

  1. How might I go about this in MATLAB? I have looked at the pwelch function (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 pwelch an acceptable approach for what I'm trying to do and how can I get things scaled correctly?

  2. 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?

$\endgroup$

1 Answer 1

0
$\begingroup$

In Matlab, you are dealing with sampled signals, so your frequency variable goes from $0$ to Nyquist. This is why the x-axis is labeled in normalized frequency that goes from $0$ to $\pi$. If you like you can use [P, f] = pwelch(___, fs) where fs is your sampling rate, and then the function will return the power spectral density (PSD) estimate P as a function of analog frequencies in the vector f. See example here.

pwelch is not the only method for estimating PSD. The algorithm choice will depend on your application. A nice summary of other competing algorithms can be found in the answer to this question.

As for these integrals, the limits should go from $-\infty$ to $\infty$ for continuous signals. But if you know that the signal is bandlimited to some range of frequencies $(f_1, f_2)$ then you can get away with integrating from $f_1$ to $f_2$. Recall the definition of PSD is inspired by continuous time signals $x(t)$ with finite energy: $$ E = \int_{-\infty}^{\infty} |x(t)|^2 dt < \infty $$ and denoting $X(f)$ to be the Fourier transform of $x(t)$, by Parseval's theorem we have: $$ \int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df $$ This leads to the intuition that $|X(f)|^2$ is the "density" of "spectral energy" around the frequency $f.$

Requiring that the PSD integrate to 1 implies that your signals of interest are unit energy. For a non-unit-energy finite energy signal, you can simply divide the signal $x(t)$ by $\sqrt{E}$ to transform it into a unit energy signal.

For implementation purposes, all your signals and transforms are discrete. So you will have to approximate these integrals with sums.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.