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So the $H_1$ and $H_2$ frequency response estimators for SISO systems are defined according to:

\begin{align} H_1 &= \frac{P_{yx}}{P_{xx}}\\ H_2 &= \frac{P_{yy}}{P_{xy}} \end{align}

Where $\frac{A}{B}$ is an element-wise division of vectors $A$ and $B$. The $H_1$ estimator should be used when the noise is uncorrelated with the input, and $H_2$ should be used when the noise is uncorrelated with the output. I've been trying to experiment with estimating frequency responses but I end up getting the same result regardless of which of the estimators I use. I first define the cross-spectral density according to:

\begin{align} P_{xy} &= X\odot \bar{Y}\\ P_{yx} &= Y\odot \bar{X}\\ P_{yy} &= Y\odot \bar{Y}\\ P_{xx} &= X\odot \bar{X} \end{align}

Where $\bar{A}$ denotes the complex conjugate of $A$, $\odot$ denotes element-wise multiplication. Here $Y$ and $X$ are vectors of the same length and the Fourier transformation of the signals $x$ and $y$.

If the definition of cross-spectral density is inserted into the $H_1$ and $H_2$ estimator formulations as described at the beginning of this post I get:

\begin{align} H_1 &= \frac{Y\odot \bar{X}}{X\odot \bar{X}} = \frac{Y}{X}\\ H_2 &= \frac{Y\odot \bar{Y}}{X\odot \bar{Y}} = \frac{Y}{X} \end{align} And, therefore, $H_1 = H_2 = \frac{Y}{X}$

The $H_1$ and $H_2$ estimators should yield different results as one is supposed to be used when the noise is uncorrelated with the input and the other with the output (as mentioned above).

However, according to my methodology, they are equivalent, at least according to my definitions and calculations. There must, therefore, be something wrong with my approach, but I can't seem to understand what.

EDIT: I realised that if one estimates the PSD or CSD without any averaging, i.e., welch or similar the $H_1$ and $H_2$ estimators are equivalent when I implement them. However, when I use welch averaging they are not equivalent. Although the $H_1$ estimator is better than the $H_2$ estimator regardless if there's noise on the input or output.

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    $\begingroup$ Your distinction between $H_1$ and $H_2$ is based on different properties of "noise" but you have no representation of noise in your models. Add some noise to your input or output and see what happens. $\endgroup$ – Hilmar Dec 5 '20 at 0:50
  • $\begingroup$ @Hilmar Even if I add noise, I still don't see that the equations change. Let's say that there's noise on the output such that $\hat{y} = y + \gamma $ where $\gamma$ is sampled from some random distribution. After I've measured $\hat{y}$ I still get the same result from both estimators. And I don't see how the equations above would change. $\endgroup$ – Pontus S Dec 5 '20 at 14:53
  • $\begingroup$ I think that my starting post proves that the estimators are equivalent for any $X$ and $Y$, regardless of system properties such as noise or similar. However, there must be something wrong as they are obviously not equivalent. I just can't seem to figure out what assumptions I've done wrong. $\endgroup$ – Pontus S Dec 5 '20 at 15:00

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