EAS487: Radar Remote Sensing Spring 2014

HOMEWORK 6: Detection

Reference: Lectures 31-34 Due: 2014 April 11

Problem 1. In class, we discussed (in the contextof target detection, false alarms, and missed detec-tions) the fact that, if the probability density func-tions (PDFs) for raw receiver voltages V = x + iyare Gaussian, the PDF for the magnitude of thosevoltages |V | is something different.Consider the output of a quadrature receiver re-

ceiving noise only. Let the PDFs of the in-phase andquadrature components of the output voltages haveGaussian distributions so that

P (x)dx =12pi

ex2

22 dx

P (y)dy =12pi

ey2

22 dy

Since the two components are statistically indepen-dent, P (x, y) = P (x)P (y), and so the probabilityof producing a voltage in a given dx, dy interval isP (x, y)dxdy = P (x)P (y)dxdy.

Now, we can change variables from x and y to |V |and where |V |2 = x2+y2 and = atan(y/x). Con-verting everything to polar coordinates and recallingthat dxdy = |V |d|V |d, we can rewrite the PDF forthe receiver output as

P (|V |, )d|V |d = 12pi2

e|V |2

22 |V |d|V |d

Integrating over all phase angles then, we can writethe PDF for V as

P (|V |) = |V |2

e|V |2

22

which is a Rayleigh distribution.

Now the question: suppose that, in addition tonoise, a deterministic signal is also present thatmakes a constant added contribution to the outputvoltage of the form V = a + ib, with V

2

= a2 + b2

and = atan(b/a). Find the resulting PDF P (|V |).Sketch it for the cases where the deterministic signalis much weaker than and much stronger than thenoise.

Hint:

2pi

0exp(c cos()) = 2piI(c) where I is

the modified Bessel function of the first kind of orderzero, I(0) = 1, and I(x) exp(x)/

2pix for large

x. The result you find is called a Rician distribution.

Problem 2. A 300 MHz radar transmits 20 spulses, giving a range resolution of 3 km, towards atarget 75 km away. The IPP is 2 ms, and we as-sume that range aliasing is not occurring. The radarcross section of the target is such that the S/N ra-tio for each individual pulse is -20 dB. The target isapproaching the radar at 40 m/s, and the Dopplerspread caused by the target tumbling is about 2 Hz.Design a data processing scheme, consisting of coher-ent FFT processing (discrete Fourier transform of 2n

samples, giving 2n frequency bins) followed by inco-herent averaging of the resulting power spectra, thatwill give the best output S/N ratio. In other words,what is the largest useful value of n, and how manyspectra can you average incoherently? What is theresulting output S/N ratio, roughly? The idea is tomake the FFT such that all the signal is more or lessin one frequency bin, giving the best S/N ratio inthat bin, and then to incoherently average as manyof these spectra as possible. How many can you av-erage, given that you want the target to remain inone range gate during the averaging?

Problem 3. Range aliasing. A radar searchingfor aircraft transmits a long series of pulses withan IPP of 500 s. An echo is observed at delay of300 s. The radar operator suspects that the targetmay actually be range aliased, however, and checksthis by changing the IPP to 400 s. The echo de-lay changes to 200 s. What is the smallest possibletarget range?

Problem 4. Frequency aliasing. Suppose that,in the above example, the observed Doppler shift ofthe target also changed from +400 Hz for the orig-inal IPP to +900 Hz for the second IPP. What isthe smallest possible (in absolute value) true Dopplershift consistent with these observations?

CORNELL UNIVERSITY c DAVE HYSELL (14/03/22)HW 61