# Question about linear combination of non-stationary signals

Suppose $$f_1,\ f_2,\ ...\ f_{n-1},\ f_n$$ are $$n$$ non-stationary discrete signals of same length and linearly independent. Each $$f_i$$ is an intrinsic mode function (IMF) of a seismic signal evaluated by empirical mode decomposition (EMD). That is, they are amplitude-modulated, frequency-modulated signals. Linear combination of these signals can be expressed as follow:

$$F(t) = \sum_{j=1}^{j=n} \alpha _j f_j(t)$$

where, $$\alpha_js$$ are real numbers and they are to be selected such that $$\sum_{j=0}^{j=n} \alpha_j\ne0$$. Is it true to claim that $$F(t)$$ is also a non-stationary signal?

No, try with $$\alpha_j = 0$$, $$\forall j$$.

I suspect this to be a tricky question, to deceive fast thinking. For instance, having the "same length" does not mean a lot, for a continuous variable $$t$$. Misleading fake clues to make the question look serious.

Without more hypotheses on the notion of stationarity (in law, wide-sense) and the statistical interrelations between the so-called signals (it would be wiser to talk about processes). See Marcus' answer for more examples.

• Let me to clarify the question. Suppose $f_1, ..., f_n$ be $n$ non-stationary discrete signals of same length and linearly independent. Also, $\alpha_js$ are to be selected such that $\sum_{j=1}^{j=n}\alpha_j\ne0$. With these conditions, what is the answer of my question? May 18 '20 at 6:12
• Do you have more hypotheses about the type of non-stationary in $f_i$? May 18 '20 at 6:40
• Each $f_i$ is an intrinsic mode function (IMF) of a seismic signal evaluated by empirical mode decomposition technique (EMD, developed by Prof. Huang). That is, they are amplitude-modulated, frequency-modulated signals. May 18 '20 at 12:10

And to extend Laurent's excellent, since on-point, answer:

non-stationarity doesn't say anything about correlation.

For example, a totally valid scenario would be:

$$f_{j} = -f_{j-1}\,\text{for } j>1$$

and then, for any even $$n$$ and a lot of potential sequences $$(\alpha_i)$$, the result is a constant, too.

Whenever you see combinations of multiple stochastic entities, you should intuitively start wondering about correlations.

• Great addition, I answered a bit too fast before lunch May 17 '20 at 13:23
• no, don't think so, the question was "is A?" and you proved by counter example that NOT A is true. May 17 '20 at 13:24
• While you were answering, I added some bits about unstated hypotheses May 17 '20 at 13:24