March 30, 2004 1Pulse Doppler Radar Assume a target at a distance R and has a radial velocity component of Vr. The round-trip distance to target is 2R. This is equivalent to 2R/λ wavelengths or (2R/λ)2π = 4πR/λ radians. If the phase of the transmitted signal is φo, then the phase of the received signal is λπφφR40+= The change in phase between pulses is rVdtdRdtdλπλπφ44== . The left hand side of the equation is equal to the frequency 2πfd, so that λλππrdrdVfVf242 ==>= (1) Alternatively, let the transmitted frequency be ft. The received signal can be represented as ))(2sin(rec RtTtfKA −=π. The round-trip time TR is equal to 2R/c . With a radial velocity of Vr the round-trip time is changing as R = R0 – Vrt. Thus, the received signal is −+=cRftcVfKAtrt0rec4212sinππ Thus, the received frequency changes by a factor 2ftVr/c = 2Vr/λ., which is the same as before. The Nyquist criterion says that fmax = PRF/2, combining this with Equation (1), results in 4maxλPRFV = (2) This is the maximum unambiguous velocity. Higher velocities cause velocity folding or velocity aliases. Maximum unambiguous range is PRFcR2max= . This causes range folding or range aliases. Combining this with Equation (1) results in This summarizes the Doppler dilemma: a large Rmax implies a small Vmax and vice versa. 8maxmaxλcRV =March 30, 2004 2 Recognizing and dealing with range an velocity aliases. • Look outside. • Examine horizontal and vertical shapes of object. For example range-aliased storms become skinny close-by objects. Storm heights are suspicious – convective storms are 10–15 km tall. Aliased they would be say 2 km tall, which is unrealistic. • Examine reflectivities in conjunction with other factors, • Change PRF–real echoes will not change position, but aliases will. This is not alwas an option. • Velocity folding causes a change in sign, which is relatively easy to spot if the folding is within a larger region. • Watch out, one can get multiple velocity foldings. Example 1. A 3-cm radar with PRF of 1000 Hz is pointed towards a storm located 200 km from the radar. The radar display will show an alias at 200 km - c/(2PRF) = 50 km. The storm is moving away from the radar with a radial velocity of 25 miles per hour, or 11.1 m/s. The radar will display the storm velocity as (PRF×λ)/4 – 11.11 = 7.5-11.1 = 3.6 m/s towards the radar. Example 2. A scanning radar with a PRF of 1000 Hz observes two identical distributed targets located at R1 = 80 km and another at R2 = 210 km (see below). Sketch and dimension a PPI up to 200 km. Answer. The unambiguous range for the radar is c/(2PRF) = 150 km. The first target will not alias, the radar measures P1 and R1 and its processor outputs a reflectivity factorMarch 30, 2004 321131RPcz = . The radars measures a power P2 =P1(R1/R2)2 for the second target. However, this target aliases to Ra = 210-150 = 60 km, so the processor outputs 22122113221132232)/(====RRzRRRPcRRRPcRPczaaaa Substituting the numerical values we get z2 = 0.082z1. Thus, z2 is about –10 dBZ less that z1. Example 3 Reflectivity (dBZ) Velocity