# Estimation

June 2, 2021

Back to: Fundamentals of Signal Processing

We can use an inference of measured data x(n) and y(n) to derive an estimate (Ĥ) of the true H. We must first collect a limited number of data samples x(n) and y(n). From the finite set, we can then infer the likely value of H and use the calculated value for Ĥ. Thus, we can only estimate H.

ObserVIEW offers several estimators; all are calculated using estimates of cross-spectral density S_{yx}(f) and power spectral densities S _{xx}(f) and S_{yy}(f).

We do not know the p(n) and q(n) values in a typical system. We only have the x and y value corrupted by measurement noise w and v.

**What is the best estimate Ĥ(f) of the true transfer function H(f)?**

There is an infinite number of options. Some estimates are better than others, and some are more complex. For example, Ĥ=0 is an estimate but not often a good one.

### Directions for Estimation

No matter the estimator, all estimates start the same, and samples must be collected. In a system, an analog-to-digital converter (ADC) often performs this step. After data sampling, there are additional directions one can take for estimation.

#### Time Series Analysis

Time series analysis typically refers to the direct use of time sample data. This type of analysis attempts to find some pattern in the data.

For example, trend analysis may calculate the line or some higher order polynomial that best fits the x-y data (often in least-squares). This technique can be traced back to Gauss in the 18th century. There are many texts available that deal with time series analysis.

#### Spectral Analysis

In this type of analysis, time data is first projected onto a sequence of basis functions. The result may be referenced as the spectrum or transform of the data. One of the most common bases is the set of complex exponential functions {eiωt}. The resulting transform is typically called the Fourier transform.

The estimates used by ObserVIEW are computed using spectral techniques and are computed using PSD and cross-spectral density estimates.

#### Eigen Analysis

Eigen Analysis is nearly the same as spectral analysis, but the basis functions are eigenfunctions and the transform is eigendecomposition. An example of eigendecomposition is singular value-decomposition (SVD).