## Measurement Accuracy, Bias, and Resolution

### Definition of Terms

Measurement accuracy and resolution are related concepts that are often confused with one another. Accuracy expresses how close a measurement is to its true value. Resolution is the granularity, or fineness, of a measurement. One description of resolution expresses is how close two objects can be before they can no longer be distinguished as different objects. Resolution used in this sense is sometimes referred to as Rayleigh resolution. Resolution need not refer to two different objects. You can use it to describe errors in how large an object is or how far has an object has moved before its motion can be detected. Accuracy and resolution may apply to radar measurements such as range, azimuth and elevation, and range rate.

### Resolution

Resolution is the ability to distinguish between two objects. For example, range resolution is the ability of radar system to distinguish between two or more targets on the same bearing but at different ranges. The range resolution depends on the width of the transmitted pulse. A radar system should be able to distinguish targets separated by one-half the pulse width.

You can think of resolution as an instrumentation limitation such as signal bandwidth or antenna aperture size. Each measurement has a characteristic quantity that determines that limitation. It is further assumed that the measurement error associated with a particular parameter is independent of the errors in any of the other parameters, the accuracy is limited only by receiver noise, and that all bias errors are accounted for separately. This table lists the defining characteristic for each measured quantity.

 Radar Measurement Characteristic Property Resolution (Δ) Range Bandwidth 1/BW Angle Antenna or array aperture λ/D Speed (Doppler) Coherent integration time λ/T

For range measurements, range resolution is inversely dependent on the signal bandwidth BW. Larger bandwidth provides greater range resolution. Range-rate resolution depends is inversely proportional to the coherent integration time of N pulses. Azimuth and elevation angle resolution is inversely proportional to the antenna or array aperture.

• The range resolution ΔR is the minimum range between two targets the can be resolved. Range resolution depends on the signal bandwidth BW.

`$\Delta R=\frac{c}{2\cdot BW}$`

where c is the speed of light and BW is the signal bandwidth. The `RangeResolution` property of the `radarDataGenerator` object lets you specify the range resolution.

• Azimuth resolution Δazim is primarily determined by antenna or array aperture size, signal frequency, and array tapering. The `AzimuthResolution` property of the `radarDataGenerator` object defines the minimum separation in azimuth angle at which the radar can distinguish between two targets. The azimuth resolution is typically the half-power beamwidth of the azimuth angle beamwidth of the radar. The same considerations apply to elevation resolution Δelev. Elevation resolution Δelev is also determined by antenna or array aperture, frequency, and array tapering. The `ElevationResolution` property defines the minimum separation in elevation angle at which a radar can distinguish between two targets. The elevation resolution is typically the half-power beamwidth of the elevation angle beamwidth of the radar.

• Range rate resolution Δv defines the minimum separation in range rate at which the radar can distinguish between two targets.

`$\delta v=\lambda \cdot PRF/\left(2N\right)$`

where Δv is the range rate resolution, PRF is the pulse repetition frequency, and N is the number of pulses. The `RangeRateResolution` property defines the minimum separation in range rate at which the radar can distinguish between two moving targets.

### Accuracy

The accuracy σmeas of a radar measurement states how precisely the measurement can be made. Accuracy is calculated as the root-mean square value (rms) of the difference between an estimated value of a quantity and its true value. Generally, accuracy is inversely proportional to the square-root of the signal-to-noise ratio and therefore improves with SNR. Accuracy is also a function of resolution. A general relation between resolution and accuracy is

`${\sigma }_{meas}=\frac{k\Delta }{\sqrt{2{\chi }_{meas}}}$`

where χmeas is the received signal-to-noise ratio (in linear units) and 𝑘 is a constant with a value close to unity. A measurement may also have a bias which is a constant offset to its measured value.

The figure below shows the normalized accuracy of a detected object's measurement plotted against received signal SNR. The accuracy is calculated using the Cramer-Rao bound.

The following list shows the general formulas for estimation accuracy of radar parameters

• Range estimation accuracy is:

`${\sigma }_{R}=\frac{1}{BW\sqrt{\chi }}\frac{c}{2\pi }\sqrt{\frac{3}{2}}$`

• Range-rate estimation accuracy is

`${\sigma }_{V}=\frac{1}{N\cdot PRI\sqrt{\chi }}\frac{\lambda }{4\pi }\sqrt{6}$`

where N is the number pulses, λ is the wavelength, and PRI is the pulse repetition interval.

• Azimuth and elevation angle estimation accuracies have the same form

`${\sigma }_{\phi }=\frac{1}{\sqrt{M\cdot \chi }}\frac{\phi }{2\pi k}\sqrt{6}$`

The accuracy improvement factor kmeas represents how much the accuracy of the measurements is improved over the resolution. The equation here expresses the accuracy in terms of the kmeas, SNR, and measurement resolution Δmeas.

`${\sigma }_{meas}=\frac{k{\Delta }_{meas}}{\sqrt{2{\chi }_{meas}}}={f}_{meas}{\Delta }_{meas}$`

This table gives typical accuracy improvement factors for range, range rate, and angle measurements:

 Measurement type Improvement factor kmeas range (derived for LFM chirp) sqrt(3)/π ≅ 0.5513 range rate sqrt(3)/π ≅ 0.5513 angle (azimuth or elevation) 0.628

This figure shows the accuracy improvement factor using parameters specified in the table above for range, range rate, and angle measurements as a functions of SNR. Without biases, the accuracy approaches zero as the SNR increases.

### Bias

Additional terms may be added to the accuracies to account for biases in the estimate of radar parameter. Biases place a lower limit on the estimated accuracies. They may be due to, for example, instrumentation or timing errors.

• Range bias, bR, can be specified as a fraction of the range resolution `RangeResolution` using the `RangeBiasFraction` property.

`${\sigma }_{R}=\sqrt{\frac{3{c}^{2}}{8{\pi }^{2}\cdot \chi \cdot B{W}^{2}}+{b}_{R}^{2}}$`
• The range rate bias, bv, can be expressed as a fraction of the range-rate resolution `RangeRateResolution` using the `RangeRateBiasFraction`.

`${\sigma }_{V}=\sqrt{\frac{6{\lambda }^{2}}{{\left(N\cdot PRI\right)}^{2}\cdot \chi \cdot 16{\pi }^{2}}+{b}_{V}^{2}}$`
• The `AzimuthBiasFraction` property is the azimuth error bias is expressed as a fraction of the azimuth resolution `AzimuthResolution` property. This sets a lower bound on the azimuth accuracy of the radar. The `ElevationBiasFraction` property is the elevation error bias expressed as a fraction of the elevation resolution `ElevationResolution` property. This sets a lower bound on the elevation accuracy of the radar.

`${\sigma }_{\phi }=\sqrt{\frac{6{\phi }^{2}}{4{\pi }^{2}\cdot \chi \cdot M{k}^{2}}+{b}_{\phi }^{2}}$`

This figure shows the accuracy improvement factor using parameters specified in the table above for the range, range rate, and angle measurements as a functions of SNR when there are biases present in the measurements.

With biases present, the accuracy never approaches zero as the SNR increases.