## 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:

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

*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*is the sampling frequency—equal to twice the Nyquist frequency. In contrast,

_{s}`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 $${\scriptscriptstyle \frac{2\pi k}{T}}\left({\scriptscriptstyle \frac{\text{k}}{\text{Period}}}\right)\text{where}k=1,2,\mathrm{...},\text{Period}$$.

For periodic data, the frequency resolution is ignored.