Radar Resolution Detection Considering the Effects of Thermal Noise

What is a radar?
Radar (acronym for RAdio Detection and Ranging) uses modulated waveforms and antennas to direct electromagnetic energy into space. The modulated waveforms are used to search, track, and extract information of a target. Information such as current velocity, angular position in space relative to the radar, size of the target and much more can be gathered based on the way the electromagnetic energy is reflected off a target or multiple targets. The energy that is reflected, also known as “radar returns” or “echoes” are usually much weaker in power relative to the power of the transmitted waveform.



Not only are there different types of modulated sinusoidal waveforms which are transmitted by radars, but there are also coded types of waveforms. For example there are the Barker Codes, Bi Phase Codes, and MLS (maximum Linear Sequences) Code.



Waveforms can be characterized by their wavelength which is the distance between repeating peaks of a wave for a given frequency.

Radars are generally used for the following: b) airport surface / landing control c) air defense and early warning (over the horizon detection) d) battlefield surveillance e) weapon control (missiles/guns) f) collision avoidance (sea, land, air) g) weather mapping and predicting h) imaging (aircraft, surface) i) astronomy (measure planet and stars) j) ground penetration (seismic, oil, etc)
 * a) air traffic control

Types or Radars
Radars are classified by the types of waveforms modulated and sent into space or by the frequency at which the radar is operation in. Waveforms can be classified further into either Continuous Wave (CW) or Pulsed Radars (PR). Continuous Wave radars are exactly as they sound, they are radars which continually emit electromagnetic waves into space. Pulsed Radars on the other hand use a train of pulsed waveforms similar to the image above in Figure 2.

Continuous Wave Radar: The transmitter and receiver are operating simultaneously and continuously.
 * Advantages: low peak power required, high average power and ideal coherent signal processing capable
 * Disadvantages: requires isolation of received signals from signals being transmitted.

Synthetic Aperture Radar: Radar platform moves during coherent integration of pulses. This is done in order to extend the antenna aperture in order to achieve a narrow beam width which provides higher resolution.
 * Advantage: very high angle resolution
 * Disadvantage: motion of platform which radar is on (needs to be air/space born)

Monstatic Pulsed Radar:The transmitter and receiver are operating simultaneously and continuously.
 * Advantages: very flexible waveforms, excellent transmit and receive isolation, strong anti-jam, wide set of applications
 * Disadvantages: high peak power an significant power dissipation required

Simple Block Diagram of Radar System

 * Exciter Controller: Generates the base carrier frequency.
 * Transmitter: Amplifies the waveform by increasing the signal power
 * Data Processor: Prepares targets for user and control system
 * Signal Processor: Detects, filters, sorts, locates, and tracks the echoes
 * Receiver: Amplifies, filters, and synchronizes the received pulses with the generated pulses



Radar Range Equation
$$\left \lbrack \frac{P_{t}G^{2}\lambda^{2}\sigma}{(4\pi)^{3}kT_{o}BF(SNR)_{omin}} \right \rbrack ^{1/4}= R_{max}$$ where,


 * $$P_{t}$$ is the peak transmitter power from the radar
 * $$G^{2}$$ is the gain of the antenna squared
 * $$\lambda^{2}$$ is the wavelength of the transmitted waveform squared
 * $$\sigma$$ is the radar cross section
 * $$k$$ is Boltzman's constant, $$1.38x10^{-23}$$
 * $$T_{o}$$ is the effective temperature of the radar in degress Kelvin
 * $$B$$ is the operating bandwidth of the radar in Hz
 * $$F$$ is the reciever noise figure $$\frac{SNR_{in}}{SNR_{out}}$$
 * $$(SNR)_{omin}$$ is the minimum detectable signal to noise ratio output of the reciever

What is Thermal Noise
Thermal noise is the random fluctuation of voltage within electrical components such as a resistor due to the thermal motion of electrons. The power spectral density of thermal noise is practically flat over a very large band of frequencies. Thermal noise is also known as thermal noise floor since generally all systems are susceptible and thus will see a almost flat power spectral density over all frequency bands due to noise.

Derivation of Gaussian Thermal Noise Across a simple RC circuit
The power spectral density of thermal noise is given by,

$$ S_{n}(\omega) = 2kTR $$

where,
 * $$k$$ is Boltzmann’s constant, $$1.38x10^{-23}$$
 * $$T$$ is the effective temperature in degrees Kelvin
 * $$R$$ is the resistance of the single resistor

For a resistor R at temperature T, the noisy resistor can be modeled as a noiseless resistor in series with a random white-noise voltage source.



From circuit analysis the transfer function is found to be,

$$ H(\omega) = \frac{1}{1+jwRC}$$

From communication theory it is known that,


 * $$ S_{o}(w) = |\frac{1}{1+jwRC}|^{2}2kTR $$


 * $$ S_{o}(w) = \frac{2kTR}{1+\omega^{2}R^{2}C^{2}}$$

It then follows that the mean square value,


 * $$ \bar{v_{o}^{2}} = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{2kTR}{1+\omega^{2}R^{2}C^{2}} d\omega$$

Using a change of variables lettting,


 * $$ z = \omega RC $$


 * $$ dz = RC d\omega $$


 * $$ \bar{v_{o}^{2}} = \frac{2kTR}{2\pi RC} \int_{-\infty}^{\infty} \frac{1}{1+z^{2}} dz$$


 * $$ \bar{v_{o}^{2}} = \frac{2kTR}{2\pi RC} (tan^{-1}(z) |_{-\infty}^{\infty})$$


 * $$ \bar{v_{o}^{2}} = \frac{2kTR}{2\pi RC} (\frac{\pi}{2}-(-\frac{\pi}{2}))$$


 * $$ \bar{v_{o}^{2}} = \frac{kT}{C} $$

Effects of Thermal Noise on Radar Range Equation
It is also known that the minimum signal to noise ratio in the radar range equation is also the visibility factor. This is because the minimum signal to noise ratio provides the minimum detectable signal resulting in factor describing the minimum required signal power in order to be detected by the radar.

This means that if the thermal noise is large and assuming the signal power remains the same, the signal to noise ratio will decrease. A smaller signal to noise ratio will then mean a larger visibility factor which is required of the radar in order to detect targets at the previous range.

Simple Example Dealing with a 2-D Pulsed Radar
Say a radar with the following characteristics are provided the visibility factor, $$V_{o}$$ can then be determined using the Radar Range Equation,

$$\left \lbrack \frac{P_{t}G^{2}\lambda^{2}\sigma}{(4\pi)^{3}kT_{o}BF(R^{4}} \right \rbrack ^{1/4}= V_{o}$$


 * $$\sigma=1m^{2}$$


 * $$P_{t}=500kW$$


 * $$G^{2}=72dB$$


 * $$\lambda=0.1m^{2}$$


 * $$N_{p}=35 pulses$$


 * $$\tau=2/mu s$$


 * $$k=1.28x10^{-23}$$


 * $$T_{o}=290K$$


 * $$F=4.0dB$$


 * $$R=150km$$


 * $$L=11dB$$

Then changing all the values into dB and using the above radar range equation the resulting visibility factor required for a target detection range of 150km is, $$ V_{o}=16.4dB $$