Radar equation review

1/19/10

Radar eq (Rayleigh scatter)

€

P r =π 3

16⋅64 ⋅ln 2

⎡

⎣ ⎢

⎤

⎦ ⎥⋅

PtG2θφh

λ2

⎡

⎣ ⎢

⎤

⎦ ⎥⋅

K 2

r 2vol

∑ Di6

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= c ⋅K 2

r 2⋅Z

The only variable is h, the pulse lengthMost radars have a range of h values.

€

Z = c3P rr2

Rewrite the radar eq as:

Convert to log form:

€

Z = C3 + P r + 20log10(r)

Radar equation, Mie scatter

€

P r =1

π 2 ⋅16⋅64 ⋅ln 2

⎡

⎣ ⎢ ⎤

⎦ ⎥⋅ PtG2λ2θφh[ ]⋅

1

r 2vol

∑ σ i

⎡

⎣ ⎢

⎤

⎦ ⎥

= c'⋅η

r 2

Uses of the radar equation

• Convert Pr to Z

• Used for specifications, such as minimum detectable signal (minimum detectable reflectivity at some standard range)

• The general form of the radar equation also applies to sodars and lidars

Important radar parameters• Wavelength (cm vs mm)• Peak transmit power• Pulse vs. continuous wave (CW)• Pulse length • Pulse repetition frequency• Beam width• Minimum detectable signal• Duty cycle: PW/PRT or Pavg/Ppeak

• Receiver bandwidth• Antenna size (gain)• Scan rate

Lidar equation

• But, additional terms representing absorption and extinction are important.

Table 1. Specifications for the dual frequency, dual pol radar system. Frequency 9.4-9.6 GHz 34-36 GHz Polarization H and V (STAR) H and V (STAR) Phase stability (I don’t have a firm number for this, but we will use a low-phase noise design)

±2° ±2°

Pulse lengths 0.2 – 2.0 s 0.2 – 1.0 s PRF 250 – 2500 Hz 2000 – 10,000 Hz Minimum detectable Z at 10 km -30 dBZ (6 /deg s; 10:1

pulse compression, 1 s compressed pulse width;

1% duty)

-35 dBZ (6 /deg s; 20:1 pulse compressio ; n 0.4 s compressed pulse;

3% duty) Antenna dish diameter 8 ft Beam widt h (max) 1° 0.25° First side lobe (one way) -17 dB -17 dB Cross-polar isolation -25 dB spec; -30 dB goal

-25 dB spec; -30 dB

goal Bore site poin tangle error ±0.1° ( weneed to check thi )s azim/elev pointing accuracy ±0.1° Max scanning rate 36 / deg sin azimut ,h 6 /deg s in elevation Range gates (minimum) 1024 1024

Table 2. Specifications for a W-band FMCW radar. Frequency 95 GHz Antenna type zenith pointing, transmit and receive antennas Power 1 W CW PRF 1-10 kHz Range Resolution 5-100 m Polarization (receiving antenna) orthogonal Phase stability (see comment above for X-Ka band spec)

±2°

Minimum detectable Z at 5 km -40 dBZ (1.6 s integration tim ;e 30 m range resolution)

Antenna dish diameter 24” Beam width 0.25 degrees First side lobe (one way) -17 dB Cross-polar isolation -25 dB spec

-30 dB goal

Beam width estimate

€

θ1 =1.27λ

DFor D = 8 ft (2.44 m): if = 3.2 cm, then θ = 1.27(0.032)/2.44 * 57.3 = 0.95 deg

if = 8.6 mm, then θ = 1.27(0.0086)/2.44 * 57.3 = 0.26 deg

For D = 2 ft (0.61 m):if = 3.2 mm, then θ = 1.27(.0032)/0.61 * 57.3 = 0.38 deg

3.2 cm 9.4 GHz8.6 mm 35 GHz3.2 mm 94 GHz

These are estimates; need to conduct test on antenna range to get actual value

Radial profile of Zmin for WSR-88D

At 50 km: Z = 58.4 dB + (-107 dBm) +20 log10(50) = -20.6 dBZ.At 10 km: Z = 58.4 +(-107) + 20log10(10) = -28.6 dBZ.

-45.0

-40.0

-35.0

-30.0

-25.0

-20.0

-15.0

-10.0

-5.0

0.0

0 10 20 30 40 50 60 70 80 90 100

Range (km)

r = 2 km

end