# Reference


```python
%matplotlib widget
import rad.css as css
import rad.example as ex
import rad.quiz as qz
from rad.const import c, k
from rad.radar import to_db, from_db, deg2rad, rad2deg
from math import sqrt, sin, asin, cos, acos, tan, atan2, pi, log, log10
css.add_custom_css()
```


<style> th { font-size: 16px }</style>



<style> td { font-size: 16px }</style>


To find material quickly, use the *<font color="DarkBlue">Table of Contents</font>* </img> button on the left sidebar.

## Aperture Beamwidth

The beamwidth of an aperture is lower bounded by:

$$
\theta \geq \mathrm{Shape~Factor}~\cdot\frac{\lambda}{D}
$$

where the shape factor is dictated by the aperture shape, $\lambda$ is the transmit wavelength (in $\mathrm{m}$), and $D$ is the extent of the aperture in the dimension under consideration.

### Circular Aperture

The beamwidth of a circular aperture (in degrees) of diameter $D$ is lower bounded by:

$$
\theta \geq 70^\circ\cdot\frac{\lambda}{D}
$$

where $\lambda$ is the transmit wavelength (in $\mathrm{m}$).

### Rectangular Aperture

The vertical beamwidth, $\Delta \theta_v$ , and horizontal beamwidth, $\Delta \theta_h$ , of a rectangular aperture (in degrees) of height $a$ (in $\mathrm{m}$) and width $b$ (in $\mathrm{m}$) is lower bounded by:

$$
\Delta\theta_v \geq 57.3^\circ\cdot\frac{\lambda}{a}
$$

$$
\Delta\theta_h \geq 57.3^\circ\cdot\frac{\lambda}{b}
$$

where $\lambda$ is the transmit wavelength (in $\mathrm{m}$).

## Aperture Gain

The gain of an aperture in direction $\theta$, $G(\theta)$ , is upper bounded by:

$$
G(\theta) \leq \frac{4\pi A}{\lambda^2}
$$

where $A$ is the aperture area (in $\mathrm{m^2}$) and $\lambda$ is the transmit wavelength (in $\mathrm{m}$).

## Constants

| Constant           | Symbol |
|--------------------|--------|
| Boltzmann constant | `k`    |
| $\pi$              | `pi`   |
| Speed of light     | `c`    |

## Decibels

Converting a number $x$ to its decibel equivalent, $x_d$ :
    
$$
x_d = 10\log_{10}(x)
$$

To return to original units:

$$
x = 10^{x_d/10}
$$

The function `to_db()` can be used to convert to decibels, and the function `from_db()` can be used to convert back to original units in notebook calculations.

### Multiplication

Conversion to decibels turns multiplication into addition:

$$
\begin{align}
z &= x\cdot y\\
z_d &= x_d + y_d
\end{align}
$$

Likewise, division becomes subtraction:

$$
\begin{align}
z &= x/ y\\
z_d &= x_d - y_d
\end{align}
$$


### Exponentiation

Conversion to decibel will change exponents into multiplication:

$$
\begin{align}
z &= x^a\\
z_d &= a \cdot x_d
\end{align}
$$

## Doppler

The Doppler frequency shift, $f_D$ , incurred by a target moving at a range rate $\dot{r}$ relative to a radar transmitting at frequency $f$ is:

$$
f_D = -\frac{2\dot{r}f}{c}
$$

## Operators and Functions

### Mathematical Operators

| Operation           | Symbol |
|---------------------|--------|
| Addition            | `+`    |
| Subtraction         | `-`    |
| Multiplication      | `*`    |
| Division            | `/`    |
| Exponent            | `**`   |
| Square root         | `sqrt` |
| Logarithm           | `log`  |
| Logarithm (Base 10) | `log10`|

### Trigonometric Functions

| Function       | Symbol    |
|----------------|-----------|
| Sine           | `sin`     |
| Arcsine        | `asin`    |
| Cosine         | `cos`     |
| Arccosine      | `acos`    |
| Tangent        | `tan`     |
| Arctangent     | `atan2`   |

## Radar Range Equation


### Received Power

Received power, $P_r$ , (in watts) from a radar transmission is:

$$
P_r = \frac{P_t G_t(\theta) G_r(\theta) \lambda^2 \sigma}{(4\pi)^3 r^4}
$$

where

- $P_t$ is the transmitted power $(\mathrm{W})$
- $G_t(\theta)$ is the transmit gain in the transmit direction $\theta$
- $G_r(\theta)$ is the receive gain in the receive direction $\theta$
- $\lambda$ is the transmission wavelength ($\mathrm{m}$)
- $\sigma$ is the radar cross section of the target ($\mathrm{m^2}$)
- $r$ is the range to the target ($\mathrm{m}$)


### Received Energy

Received energy, $\mathcal{E}_r$ , (in joules) from a radar transmission is:

$$
\mathcal{E}_r = \frac{\mathcal{E}_t G_t(\theta) G_r(\theta) \lambda^2 \sigma}{(4\pi)^3 r^4}
$$

where

- $\mathcal{E}_t$ is the transmitted energy $(\mathrm{J})$
- $G_t(\theta)$ is the transmit gain in the transmit direction $\theta$
- $G_r(\theta)$ is the receive gain in the receive direction $\theta$
- $\lambda$ is the transmission wavelength ($\mathrm{m}$)
- $\sigma$ is the radar cross section of the target ($\mathrm{m^2}$)
- $r$ is the range to the target ($\mathrm{m}$)

### Average Noise Energy

Average noise energy, $\mathcal{E}_n$ , (in joules) is:

$$
\mathcal{E}_n = k T_s
$$

where

- $k = 1.38064852 \times 10^{-23} \frac{J} {^\circ\mathrm{K}}$ is the Boltzmann constant
- $T_s$ is the noise temperature of the sensor $(^\circ\mathrm{K})$

### Signal-to-Noise Ratio

The received signal-to-noise ratio is then:

$$
\mathrm{SNR} = \frac{\mathcal{E}_r}{\mathcal{E}_n}
$$

## Scale

Here is a list of the scale prefixes, their values, and code snippets to use them in notebook calculations:

| Symbol            | Prefix       | Value               | Code           |
|-------------------|--------------|---------------------|----------------|
| $T$               | tera         | $1 \times 10^{12}$  | `1E12`         |
| $G$               | giga         | $1 \times 10^9$     | `1E9`          |
| $M$               | mega         | $1 \times 10^6$     | `1E6`          |
| $k$               | kilo         | $1 \times 10^3$     | `1E3`          |
| $m$               | milli        | $1 \times 10^{-3}$  | `1E-3`         |
| $\mu$             | micro        | $1 \times 10^{-6}$  | `1E-6`         |
| $n$               | nano         | $1 \times 10^{-9}$  | `1E-9`         |
| $p$               | pico         | $1 \times 10^{-12}$ | `1E-12`        |


## Symbols

Here is a list of the mathematical symbols used throughout the course and their respective meanings:

| Symbol            | Definition                                             |
|-------------------|--------------------------------------------------------|
| $a$               | Amplitude                                              |
| $A$               | Area, $\mathrm{m^2}$                                   |
| $A_t$             | Area of transmit aperture, $\mathrm{m^2}$              |
| $A_r$             | Area of receive aperture, $\mathrm{m^2}$               |
| $B$               | Transmit bandwidth, $\mathrm{Hz}$                      |
| $c$               | Speed of light, $\mathrm{m/s}$                         |
| $\delta_\phi$     | Target angle resolution, $\mathrm{deg}$                |
| $\delta_r$        | Target range resolution, $\mathrm{m}$                  |
| $\delta_\dot{r}$  | Target range rate resolution, $\mathrm{m/s}$           |
| $\mathcal{E}$     | Energy, $\mathrm{J}$                                   |
| $\mathcal{E}_n$   | Noise energy, $\mathrm{J}$                             |
| $\mathcal{E}_r$   | Received energy, $\mathrm{J}$                          |
| $\mathcal{E}_t$   | Transmit energy, $\mathrm{J}$                          |
| $f$               | Transmit frequency, $\mathrm{Hz}$                      |
| $f_D$             | Doppler shift, $\mathrm{Hz}$                           |
| $f_r$             | Received frequency, $\mathrm{Hz}$                      |
| $f_{rep}$         | Pulse repetition frequency, $\mathrm{Hz}$              |
| $\theta$          | Transmit azimuth angle, $\mathrm{deg}$                 |
| $\Delta\theta$    | Transmit azimuth angle beamwidth, $\mathrm{deg}$       |
| $P$               | Power, $\mathrm{W}$                                    |
| $P_i$             | Incident power, $\mathrm{W}$                           |
| $P_n$             | Noise power, $\mathrm{W}$                              |
| $P_r$             | Received power, $\mathrm{W}$                           |
| $P_t$             | Transmit power, $\mathrm{W}$                           |
| $\phi$            | Target azimuth angle, $\mathrm{deg}$                   |
| $\hat{\phi}$      | Target azimuth angle estimate, $\mathrm{deg}$          |
| $r$               | Target range, $\mathrm{m}$                             |
| $\hat{r}$         | Target range estimate, $\mathrm{m}$                    |
| $\sigma$          | Target radar cross section, $\mathrm{m}^2$             |
| $\hat{\sigma}$    | Target radar cross section estimate, $\mathrm{m}^2$    |
| $\tau$            | Propagation delay, $\mathrm{s}$                        |
| $\hat{\tau}$      | Propagation delay estimate, $\mathrm{s}$               |
| $t$               | Time, $\mathrm{s}$                                     |
| $t_p$             | Pulsewidth, $\mathrm{s}$                               |
| $t_{rep}$         | Pulse repetition interval, $\mathrm{s}$                |
| $v_p$             | Propagation velocity, $\mathrm{m/s}$                   |

## Target Parameter Estimates

### Accuracy and SNR

Each of the observable attributes of target can be estimated with a certain level of accuracy<sup>[1](#foot_acc)</sup>. It can be shown<sup>[[1]](#ref_skolnik)</sup> that the accuracy, $\alpha$, of each estimate is approximately:

$$
\alpha \approx \frac{\delta}{\sqrt{\mathrm{SNR}}}
$$

where $\delta$ is the resolution in that dimension and $\mathrm{SNR}$ is the signal-to-noise ratio.

### Range

The range estimate of a target, $\hat{r}$ , is given by
$$
\hat{r} = \frac{c\cdot \hat{\tau}}{2}
$$

where $\hat{\tau}$ is the estimated propagation delay from the detection. Range resolution, $\delta r$ , is

$$
\delta r = \frac{c}{2B}
$$

where $B$ is the bandwidth of the transmitted waveform. Thus, the accuracy of a range estimate from a detection, $\alpha_r$ , is

$$
\alpha_r = \frac{c}{2B\sqrt{\mathrm{SNR}}}
$$

### Angle

Coarse angle estimation can be done using *mainlobe angle estimation* which places the observed target within the main beam of the radar system. In this case, the target angle estimate, $\hat{\phi}$, is simply the transmit-receive steering direction, $\theta$, or

$$
\hat{\phi} = \theta
$$

The accuracy is then roughly the beamwidth of the radar aperture:

$$
\alpha_\phi = \Delta \theta
$$

Fine angle measurement can be performed using multiple different techniques, e.g., *beamcenter angle estimation*, *monopulse amplitude comparison*, or *monopulse phase comparison*. With these, the target angle estimate accuracy is increased approximately to

$$
\alpha_\phi = \frac{\Delta \theta}{\sqrt{\mathrm{SNR}}}
$$

### Range Rate

Range rate measurements are derived from measuring the frequency shift caused by the Doppler effect, $f_D$. The range rate estimate, $\hat{\dot{r}}$ , can be calculated as

$$
\hat{\dot{r}} = -\frac{c f_D}{2f}
$$

Similar to the range and angle measurements dicussed above, the range rate accuracy of a radar system, $\alpha_{\dot{r}}$, can be approximated by:

$$
\alpha_{\dot{r}} = \frac{c}{2f\Delta t\sqrt{\mathrm{SNR}}}
$$

where $\Delta t$ is the observation time. The highest measureable Doppler shift is given by the rate that the multiple pulses are transmitted. For a set of pulses with pulse repetition frequency, $f_{rep}$ , the magnitude of Doppler shifts that can be unambiguously measured is bounded by:

$$
|f_D| < \frac{f_{rep}}{2}
$$

This can be translated to the span of range rates that can be unambiguously observed:

$$
|\dot{r}| < \frac{c f_{rep}}{4f}
$$

### Radar Cross Section

Radar cross section can be calculated from the estimated received energy, $\hat{\mathcal{E}}_r$ , and range, $\hat{r}$ , by:

$$
\hat{\sigma} = \frac{(4\pi)^3}{\mathcal{E}_t G_t(\theta) G_r(\theta) \lambda^2} \cdot \hat{\mathcal{E}}_r \cdot \hat{r}^4
$$

where

- $\mathcal{E}_t$ is the transmitted energy $(\mathrm{J})$
- $G_t(\theta)$ is the transmit gain in the transmit direction $\theta$
- $G_r(\theta)$ is the receive gain in the receive direction $\theta$
- $\lambda$ is the transmission wavelength ($\mathrm{m}$)
- $r$ is the range to the target ($\mathrm{m}$)

Likewise, it can be calculated efficiently using a reference SNR by:

$$
\hat{\sigma} = \frac{\mathrm{SNR}}{\mathrm{SNR}_0} \cdot \left(\frac{\hat{r}}{r_0}\right)^4 \cdot {\sigma_0}
$$

## Units

| Symbol              | Unit           | Definition                                             |
|---------------------|----------------|--------------------------------------------------------|
| $\mathrm{Hz}$       | hertz          | Unit of frequency                                      |
| $\mathrm{J}$        | joule          | Unit of energy                                         |
| $^\circ \mathrm{K}$ | degree kelvin  | Unit of temperature                                    |
| $\mathrm{m}$        | meter          | Unit of length                                         |
| $\mathrm{s}$        | second         | Unit of duration                                       |
| $\mathrm{W}$        | watt           | Unit of power                                          |


## Wavelength

The frequency and wavelength of a transmitted wave are related by the following:

$$ f = \frac{v_p}{\lambda} $$

and

$$ \lambda = \frac{v_p}{f} $$

where $v_p$ is the propagation velocity.

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