Here we tackle what a Radar needs to accurately measure a target's position, size and motion. (This is nothing new, just a placeholder for some of my notes.)

First of all, everything depends on $SNR$, and the parameter estimation happens in the main computer of a Radar. The theoretical rms error of a measurement is of the form:

$$\delta M = \frac{k M}{\sqrt{SNR}}$$

where $$k$$ is a constant between 0.5 and 1.

h3. Range

First, let's tackle range. They key metric for range is bandwidth, the higher the bandwidth the better the resolution.

Estimation of the range of a target is based upon using A/D sampled measurements of the round trip time to and the target

$$R = \frac{c T_R}{2}$$

For time delay measurements , such as range, the value of the constant depends on the shape of the radar pulse's spectrum and the pulse's rise time and for a rectangular pulse, whose width is, (T):

$$\delta T = \frac{T}{2 \sqrt{SNR}}$$

or equivalently,

$$\delta R = \frac{c T}{2 \sqrt{SNR}}$$

However, pulses always move in trains or if $T_D$ is the dwell time:

$$\delta R = \frac{c , T}{2, \sqrt{SNR\, \text{PRF} \, T_D}}$$

from Barton and Ward.

So Range depends on $$SNR$$, pulse shape and width, effective bandwidth, number of pulses that can be coherently integrated.

This is explained well by a diagram in the lincoln presentation:

h3. Angle

Now, let's tackle angle. Angle accuracy depends on antenna size. Simple detection provides coarse location in angle by isolating a target's location within beamwidth of antenna (and beamwidth is just $$\lambda / D$$, where $$D$$ is the aperture diameter and $$\lambda$$ is the wavelength. But this is very coarse, 1° beam at 100 km extends across 1,745 meters. In order to provide more accurate angle estimation, you have to measure at different beam positions to improve accuracy. You can do this through sequential lobing, conical scan tracking, or monopulse angle estimation.

h4. Conical Scanning

Conical scanning is a technique to keep the beam pointed at the target to improve angle accuracy. It is based on the principle that the radar receiver will get maximum returned signal strength when the target is in the beam centre, so if the beam is pointed directly at the target, when the target moves it will move out of the beam center and the received signal strength will fall. Circuitry designed to monitor any fall off in received signal strength controls a servo motor that steers the aerial to follow the target motion.

Wikipedia has a nice drawing that explains this technique:

h4. Sequential lobing

While conical scan tracking uses the phase of modulation to get the angle error and amplitude modulation to get the beam displacement, sequential lobing uses two slightly separated antenna elements to send the beam slightly to either side of the midline of the antenna, switching between the two to find which one gave the stronger return, thereby indicating which direction the antenna should be moved in order to point directly at the target.

Here, wikipedia provides another good diagram:

h4. Monopulse

Monopulse radars are similar in general construction to conical scanning systems, but add one more feature. Instead of broadcasting the signal out of the antenna "as is", they split the beam into parts and then send the two signals out of the antenna in slightly different directions. When the reflected signals are received they are amplified separately and compared to each other, indicating which direction has a stronger return, and thus the general direction of the target relative to the boresight. While monopulse, requires a separate receiver for each channel, it improves performance over conical scan and sequential lobing whose performance degrade with time varying radar returns.

All angle errors are summed up well by the following:

and

SNR, type of measurement technique, antenna illumination distribution, antenna size, and frequency.

h3. Doppler

Lastly, let's look at Doppler accuracy. Doppler accuracy depends on coherent integration time $$\Delta t$$:

Doppler frequency is calculated as:

$$f_d = \frac{2 v_r}{\lambda}$$

To accurately measure Doppler, use two closely spaced frequency filters offset from the center frequency of the Doppler filter containing the detection

$$\text{Doppler measurement accuracy} \propto \frac{\lambda}{\Delta t} \frac{1}{\sqrt{SNR}}$$

Doppler accuracy depends on SNR, pulse shape, integration time.

h3. Real world limitations

# Poor calibration leads to poor estimation

h4. Amplitude fluctuations

# Small effect on monopulse and array solutions

h4. Angle noise (angle scintillations, or target glint)

# Complex target return biases angle estimate

h4. Multipath (low angle tracking)

# Reflection off earth's surface combines with direct path return

# Can cause biases in angle estimates for all technique

h4. Good References