Controlling Frequency Resolution of Spectral Models

This section supports the estimation procedures Estimate Frequency-Response Models in the App and Estimate Frequency-Response Models at the Command Line.

What Is Frequency Resolution?

Frequency resolution is the size of the smallest frequency for which details in the frequency response and the spectrum can be resolved by the estimate. A resolution of 0.1 rad/s means that the frequency response variations at frequency intervals at or below 0.1 rad/s are not resolved.

Note

Finer resolution results in greater uncertainty in the model estimate.

Specifying the frequency resolution for etfe and spa is different than for spafdr.

Frequency Resolution for etfe and spa

For etfe and spa, the frequency resolution is approximately equal to the following value:

$$\frac{2\pi }{M}\left(\frac{\text{radians}}{\text{sampling interval}}\right)$$

M is a scalar integer that sets the size of the lag window. The value of M controls the trade-off between bias and variance in the spectral estimate.

The default value of M for spa is good for systems without sharp resonances. For etfe, the default value of M gives the maximum resolution.

A large value of M gives good resolution, but results in more uncertain estimates. If a true frequency function has sharp peak, you should specify higher M values.

Frequency Resolution for spafdr

In case of etfe and spa, the frequency response is defined over a uniform frequency range, 0-F_s/2 radians per second, where F_s is the sampling frequency—equal to twice the Nyquist frequency. In contrast, spafdr lets you increase the resolution in a specific frequency range, such as near a resonance frequency. Conversely, you can make the frequency grid coarser in the region where the noise dominates—at higher frequencies, for example. Such customizing of the frequency grid assists in the estimation process by achieving high fidelity in the frequency range of interest.

For spafdr, the frequency resolution around the frequency k is the value R(k). You can enter R(k) in any one of the following ways:

• Scalar value of the constant frequency resolution value in radians per second.

  Note

  The scalar R is inversely related to the M value used for etfe and spa.

• Vector of frequency values the same size as the frequency vector.

• Expression using MATLAB^® workspace variables and evaluates to a resolution vector that is the same size as the frequency vector.

The default value of the resolution for spafdr is twice the difference between neighboring frequencies in the frequency vector.

etfe Frequency Resolution for Periodic Input

If the input data is marked as periodic and contains an integer number of periods (data.Period is an integer), etfe computes the frequency response at frequencies $$\frac{2\pi k}{T}\left(\frac{\text{k}}{\text{Period}}\right)\text{ where }k=1,2,\mathrm{...},\text{Period}$$.

For periodic data, the frequency resolution is ignored.