# radareqpow

Peak power estimate from radar equation

## Description

estimates the peak transmit power, `Pt`

= radareqpow(`lambda`

,`tgtrng`

,`SNR`

,`tau`

)`Pt`

, required for a radar operating
at a wavelength of `lambda`

meters to achieve the specified
signal-to-noise ratio, `SNR`

, in decibels for a target at a range of
`tgtrng`

meters. `tau`

is the pulse width. The
target has a nonfluctuating radar cross section (RCS) of 1 square meter.

## Examples

### Compute Required Transmit Power

Estimate the required peak transmit power required to achieve a minimum SNR of 6 dB for a target at a range of 50 km. The target has a nonfluctuating RCS of 1 m². The radar operating frequency is 1 GHz. The pulse duration is 1 μs.

```
fc = 1.0e9;
lambda = physconst('LightSpeed')/fc;
tgtrng = 50e3;
tau = 1e-6;
SNR = 6;
Pt = radareqpow(lambda,tgtrng,SNR,tau)
```

Pt = 2.1996e+05

### Compute Required Transmit Power at Specified System Temperature

Estimate the required peak transmit power required to achieve a minimum SNR of 10 dB for a target with an RCS of 0.5 m² at a range of 50 km. The radar operating frequency is 10 GHz. The pulse duration is 1 μs. Assume a transmit and receive gain of 30 dB and an overall loss factor of 3 dB. The system temperature is 300 K.

fc = 10.0e9; lambda = physconst('LightSpeed')/fc; Pt = radareqpow(lambda,50e3,10,1e-6,'RCS',0.5, ... 'Gain',30,'Ts',300,'Loss',3)

Pt = 2.2809e+06

### Compute Required Transmit Power for Bistatic Radar

Estimate the required peak transmit power for a bistatic radar to achieve a minimum SNR of 6 dB for a target with an RCS of 1 m². The target is 50 km from the transmitter and 75 km from the receiver. The radar operating frequency is 10 GHz and the pulse duration is 10 μs. The transmitter and receiver gains are 40 dB and 20 dB, respectively.

fc = 10.0e9; lambda = physconst('LightSpeed')/fc; SNR = 6; tau = 10e-6; TxRng = 50e3; RvRng = 75e3; TxRvRng =[TxRng RvRng]; TxGain = 40; RvGain = 20; Gain = [TxGain RvGain]; Pt = radareqpow(lambda,TxRvRng,SNR,tau,'Gain',Gain)

Pt = 4.9492e+04

## Input Arguments

`lambda`

— Wavelength of radar operating frequency

positive scalar

Wavelength of radar operating frequency, specified as a positive scalar. The wavelength is the ratio of the wave propagation speed to frequency. Units are in meters. For electromagnetic waves, the speed of propagation is the speed of light. Denoting the speed of light by *c* and the frequency (in hertz) of the wave by *f*, the equation for wavelength is:

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

**Data Types: **`double`

`tgtrng`

— Target range

positive scalar | two-element row vector of positive values | length-*J* column vector of positive values | *J*-by-2 matrix of positive values

Target ranges for a monostatic or bistatic radar.

Monostatic radar - the transmitter and receiver are co-located.

`tgtrng`

is a real-valued positive scalar or length-*J*real-valued positive column vector.*J*is the number of targets.Bistatic radar - the transmitter and receiver are separated.

`tgtrng`

is a 1-by-2 row vector with real-valued positive elements or a*J*-by-2 matrix with real-valued positive elements.*J*is the number of targets. Each row of`tgtrng`

has the form`[TxRng RxRng]`

, where`TxRng`

is the range from the transmitter to the target and`RxRng`

is the range from the receiver to the target.

Units are in meters.

**Data Types: **`double`

`SNR`

— Input signal-to-noise ratio at receiver

scalar | length-*J* real-valued vector

Input signal-to-noise ratio (SNR) at the receiver, specified as a scalar or
length-*J* real-valued vector. *J* is the number
of targets. Units are in dB.

**Data Types: **`double`

`tau`

— Single pulse duration

positive scalar

Single pulse duration, specified as a positive scalar. Units are in seconds.

**Data Types: **`double`

### Name-Value Arguments

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

**Example:**

`'RCS',3.0`

`RCS`

— Radar cross section

`1`

(default) | positive scalar | length-*J* vector of positive values

Radar cross section specified as a positive scalar or length-*J* vector of
positive values. *J* is the number of targets. The target RCS is
nonfluctuating (Swerling case 0). Units are in square meters.

**Data Types: **`double`

`Ts`

— System noise temperature

`290`

(default) | positive scalar

System noise temperature, specified as a positive scalar. The system noise temperature is the product of the system temperature and the noise figure. Units are in Kelvin.

**Data Types: **`double`

`Gain`

— Transmitter and receiver gains

`20`

(default) | scalar | real-valued 1-by-2 row vector

Transmitter and receiver gains, specified as a scalar or real-valued 1-by-2 row vector. When
the transmitter and receiver are co-located (monostatic radar),
`Gain`

is a real-valued scalar. Then, the transmit and receive
gains are equal. When the transmitter and receiver are not co-located (bistatic radar),
`Gain`

is a 1-by-2 row vector with real-valued elements. If
`Gain`

is a two-element row vector it has the form
`[TxGain RxGain]`

representing the transmit antenna and receive
antenna gains.

**Example: **`[15,10]`

**Data Types: **`double`

`Loss`

— System losses

`0`

(default) | scalar | length-*J* real-valued vector

System losses, specified as a scalar. Units are in dB.

**Example: **`1`

**Data Types: **`double`

`AtmosphericLoss`

— Atmospheric absorption loss

0 (default) | scalar | two-element row vector of real values | length-*J* column vector of real values | *J*-by-2 matrix of real values

Atmospheric absorption losses for the transmit and receive paths.

When the absorption is a scalar or length-

*J*column vector, the loss specifies the atmospheric absorption loss for a one-way path.When the absorption is a 1-by-2 row vector or

*J*-by-2 column vector, the first column specifies the atmospheric absorption loss for the transmit path and the second column of contains the atmospheric absorption loss for the receive path

**Example: **`[10,20]`

**Data Types: **`double`

`PropagationFactor`

— Propagation factor

0 (default) | scalar | two-element row vector of real values | length-*J* column vector of real values | *J*-by-2 matrix of real values

Propagation factor for the transmit and receive paths.

When the propagation factor is a scalar or length-

*J*column vector, the propagation factor is specified for a one-way path.When the propagation factor is a 1-by-2 row vector or

*J*-by-2 column vector, the first column specifies the propagation factor for the transmit path and the second column of contains the propagation factor for the receive path

Units are in dB.

**Example: **`[10,20]`

**Data Types: **`double`

`CustomFactor`

— Custom factor

0 (default) | scalar | length-*J* column vector of real values

Custom loss factors specified as a scalar or length-*J* column vector of real
values. *J* is the number of targets. These factors contribute to the
reduction of the received signal energy and can include range-dependent STC, eclipsing,
and beam-dwell factors. Units are in dB.

**Example: **`[10,20]`

**Data Types: **`double`

## Output Arguments

`Pt`

— Transmitter peak power

positive scalar

Transmitter peak power, returned as positive scalar. Units are in watts.

## More About

### Point Target Radar Range Equation

The point target radar range equation estimates the power at the input to the receiver for a target of a given radar cross section at a specified range. The model is deterministic and assumes isotropic radiators. The equation for the power at the input to the receiver is

$${P}_{r}=\frac{{P}_{t}{G}_{t}{G}_{r}{\lambda}^{2}\sigma}{{(4\pi )}^{3}{R}_{t}^{2}{R}_{r}^{2}L}$$

where the terms in the equation are:

*P*— Peak transmit power in watts_{t}*G*— Transmit antenna gain_{t}*G*— Receive antenna gain. If the radar is monostatic, the transmit and receive antenna gains are identical._{r}*λ*— Radar wavelength in meters*σ*— Target's nonfluctuating radar cross section in square meters*L*— General loss factor in decibels that accounts for both system and propagation loss*R*— Range from the transmitter to the target_{t}*R*— Range from the receiver to the target. If the radar is monostatic, the transmitter and receiver ranges are identical._{r}

Terms expressed in decibels, such as the loss and gain factors, enter the equation in the form 10^{x/10} where *x* denotes the variable. For example, the default loss factor of 0 dB results in a loss term of 10^{0/10}=1.

### Receiver Output Noise Power

The equation for the power at the input to the receiver represents the
*signal* term in the
signal-to-noise ratio. To model the noise term, assume the
thermal noise in the receiver has a white noise power spectral
density (PSD) given by:

$$P(f)=kT$$

where *k* is the Boltzmann
constant and *T* is the effective noise
temperature. The receiver acts as a filter to shape the white
noise PSD. Assume that the magnitude squared receiver frequency
response approximates a rectangular filter with bandwidth equal
to the reciprocal of the pulse duration, *1/τ*.
The total noise power at the output of the receiver is:

$$N=\frac{kT{F}_{n}}{\tau}$$

where *F _{n}
* is the receiver

*noise factor*.

The product of the effective noise temperature and the receiver noise factor is referred to as the *system temperature*. This value is denoted by *T _{s}*, so that T

_{s}=

*TF*.

_{n}### Receiver Output SNR

Define the output SNR. The receiver output SNR is:

$$\frac{{P}_{r}}{N}=\frac{{P}_{t}\tau \text{}\text{\hspace{0.05em}}{G}_{t}{G}_{r}{\lambda}^{2}\sigma}{{(4\pi )}^{3}k{T}_{s}{R}_{t}^{2}{R}_{r}^{2}L}$$

You can derive this expression using the following equations:

Received signal power in Point Target Radar Range Equation

Output noise power in Receiver Output Noise Power

### Theoretical Maximum Detectable Range

Compute the maximum detectable range of a target.

For monostatic radars, the range from the target to the transmitter and receiver is identical. Denoting this range by *R*, you can express this relationship as $${R}^{4}={R}_{t}^{2}{R}_{r}^{2}$$.

Solving for *R*

$$R={(\frac{N{P}_{t}\tau {G}_{t}{G}_{r}{\lambda}^{2}\sigma}{{P}_{r}{(4\pi )}^{3}k{T}_{s}L})}^{1/4}$$

For bistatic radars, the theoretical maximum detectable range is the geometric mean of the ranges from the target to the transmitter and receiver:

$$\sqrt{{R}_{t}{R}_{r}}={(\frac{N{P}_{t}\tau {G}_{t}{G}_{r}{\lambda}^{2}\sigma}{{P}_{r}{(4\pi )}^{3}k{T}_{s}L})}^{1/4}$$

## References

[1] Richards, M. A.
*Fundamentals of Radar Signal Processing*. New York: McGraw-Hill,
2005.

[2] Skolnik, M.
*Introduction to Radar Systems*. New York: McGraw-Hill,
1980.

[3] Willis, N. J. *Bistatic
Radar*. Raleigh, NC: SciTech Publishing, 2005.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Does not support variable-size inputs.

## See Also

`phased.Transmitter`

| `phased.ReceiverPreamp`

| `noisepow`

| `radareqrng`

| `radareqsnr`

| `systemp`

**Introduced in R2021a**

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