# Test for equality of means for vector-valued random process with different variances

I am studying linearity range of an RF amplifier (henceforth DUT). For that I am stimulating the DUT with a periodic deterministic probe signal (which is known only roughly) and measuring the DUT's output. The power of the probe signal is controlled with the help of attenuators.

For a given level of stimulus power, I am able to obtain discretized snapshots at the DUT's output. The snapshots are perfectly aligned in time with the probe signal, therefore they repeat quite precisely, with only a slight variation due to additive thermal noise. Thus, a series of snapshots can be modeled as a vector-valued i.i.d. Gaussian process with some mean and covariance $$\sigma I$$. The series of snapshots itself is treated as a finite-length sample of the vector-valued random process.

So here is how I am trying to check for device linearity. Given two stimulus power levels $$p_x$$ and $$p_y$$, I measure two sets of samples: $$\bar{x}_1,\dots,\bar{x}_N \sim \mathcal{N}(\bar{\mu}_x,\sigma_x^2I)$$ and $$\bar{y}_1,\dots,\bar{y}_N \sim \mathcal{N}(\bar{\mu}_y,\sigma_y^2I)$$.

I am trying to check the linearity condition. If the DUT is under both power levels, then $$\frac{\bar \mu_x}{\sqrt{p_x}} = \frac{\bar \mu_y}{\sqrt{p_y}}$$

What test do I need to use to check for equality of samples' means? If my random processes were scalar, it would probably be some sort of t-test. But what's about different levels of variance?