# Synthetic Aperture Radar (SAR) Calculator

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This calculator solves for the Signal-to-Noise ratio (SNR_"image") of a Synthetic Aperture Radar.  Two versions of the Radar Equation are presented, along with auxiliary equations to help the calculation of different parameters.  This page heavily draws on Sandia National Laboratories's Performance Limits for Synthetic Aperture Radar– second edition, which only considers the monostatic case (where the same antenna is used for TX and RX operations).  The findings were also checked against Skolnik's Radar Handbook.

# Using the Calculator

If there's a drop-down box next to the input text field when you click on a button, then you may choose any of those units from the list.  The equations will adapt.  If there is no drop-down box, then please enter the field in the units requested.  If no units are mentioned, then the parameter is dimensionless.

## Red Buttons

The red buttons are two versions of the Radar Equation.  They are equivalent in the context of SAR.
Briefly, SNR1 is the ratio between the signal and noise.  Higher SNR means that the radar has a better picture of the target.  Lower, worse picture.

SNR_"radar" = P_r / N, where:

• P_r is the power received by the antenna (W)
• N is the thermal/background noise the antenna is subjected to (W)

The two versions of the Radar Equation in the calculator will look more complicated since they substitute in for P_r, and N.  There are more details about the Radar Equations below.

## Blue buttons

In case the data you have doesn't precisely match either of these equations, the blue buttons are there to try and solve for certain parameters in terms of others.  For example, the A_e button calculates the effective area of the calculator given the antenna's actual area and aperture efficiency.

The "alpha lookup" button provides estimated two-way atmospheric loss factors from a computer model.  As such, the numbers are a little squishy - don't rely on them being perfectly accurate.

Typically, a maximum duty factor(d) needed by a radar is less than 50%, and usually less than 35% or so2.  Consequently, a reasonable duty factor limit of 35% might be imposed on power amplifiers that could otherwise be capable of more.

This calculator has two versions of the Radar Equation.  This first Radar Equation34 is:

• SNR_"image" = G_a * G_r * SNR_"radar" = (P_t * G_A * A_e * sigma * G_r * G_a)/( (4 pi)^2 * R^4 * L_"radar" * L_"atmos" * (k * T * F_N) * B_N)

Where the parameters are:

• P_t - transmitter signal power (W)
• G_A - transmitter antenna gain factor
• A_e - receiver antenna effective area (m^2)
• sigma - target Radar Cross Section (m^2)
• G_r - SNR gain due to range processing/ pulse compression
• G_a - SNR gain due to azimuth processing/ coherent pulse integration
• R - range vector from target to antenna (m)
• L_"radar" - microwave transmission loss factor due to miscellaneous sources, such as hardware
• L_"atmos" - loss factor due to the wave propagation through the atmosphere
• k - Boltzmann's constant = 1.38 times 10^(-23)(J/K)
• T - nominal scene noise temperature (K)
• F_N - system noise factor for the receiver
• B_N- noise bandwidth at the antenna port (Hz)

The second version of the Radar Equation5 has many substitutions that assume a Synthetic Aperture Radar.

• SNR_"image" = (P_"avg" * ( n_"ap"^2 * A_A^2 ) * rho_r * sigma_(0,ref) * (f/f_"ref")^n * f * a_"wa") / ( 8 pi * c * v_x * (k * T * F_N) * (L_"radar" * L_r * L_a) * [R^3 * sqrt(1 - (h/R)^2) 10^((alpha * R)/10)])

Where the new parameters are:

• P_"avg" - average power transmitted during the synthetic aperture data collection period (W)
• A_A - the physical area of the antenna aperture (m^2)
• n_"ap" - the aperture efficiency of the antenna
• rho_r - the slant-range resolution (m)
• sigma_(0,ref)- distributed target reflectivity (m^2/m^2)
• a_"wa" - azimuth impulse response broadening factor
• f - nominal frequency (Hz)
• f_"ref" - reference nominal frequency (Hz)
• n - RCS frequency dependence
• v_x - the platform velocity, horizontal and orthogonal to the target direction (m/s)
• psi_g - the grazing angle (°)
• L_r - the reduction in SNR gain due to non-ideal range filtering
• L_a - the reduction in SNR gain due to non-ideal azimuth filtering
•  alpha - the two-way atmospheric loss rate (dB/m)
• h - the elevation of the radar (relative to the target) (m)

# References

1. ^ Performance Limits for Synthetic Aperture Radar - Second Edition.  Sandia National Laboratories, Albuquerque, NM.  Printed February 2006.
2. ^ Performance Limits for Synthetic Aperture Radar - Second Edition.  Sandia National Laboratories, Albuquerque, NM.  Printed February 2006.
3. ^ Performance Limits for Synthetic Aperture Radar - Second Edition.  Sandia National Laboratories, Albuquerque, NM.  Printed February 2006.
4. ^ Skolnik, M. I. (2008). Radar handbook (3rd ed.). New York: McGraw-Hill.
5. ^ Performance Limits for Synthetic Aperture Radar - Second Edition.  Sandia National Laboratories, Albuquerque, NM.  Printed February 2006.

This calculator, Synthetic Aperture Radar (SAR) Calculator, is listed in 1 Collection.