Th e probability dist ribu Lion of t he am pli tude and phase o f the s um o f a large num ber of random two-di m ensional vecto rs is derived under t he fo llowing general condit io ns : Both t he ampli tudes a nd th e phases of the component vectors a re random, t he di sLribu tions b eing a rbitrary within the vali dit.\7 of t he Central Lim it Th eorem ; in part icular, the d ist ributions of t he individual veetors need not be identical, the amplitude and phase of each co mponent v ec tor n eed not be indep end e nt a nd the di tribu tion s Ileed not b e symmel ri cal. T he di sLributions form erly d erived by R ayleigh , nice, H oy t, a nd Heckmanll are shown lo be special cases of this di sLribu t ion .

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Department of Ele ctric al Enginee ring, University of C olor ado, Boulde r, C olo.(R eceived D eccmber 7, ] 963) Th e Rayleigh d istribu tion is the di stribution of thc sum of a large num be r of coplanar (or t im c) vectors \\'ith r andom. amplitudes alld Ull ifO rlllly distributed phases. As s uch, it is the lim iti ng case of di stributions a~soci at('d II ith 111 0re gl'IlPral v pctor surn s t hat arise i n practical problems. S uch cases arC' til(' folloll' ing: (a) 'I'll(' p hasC' distribu tions of the vec tor tC'rms are not uni fo rm , e.g., in the case of scatt l' ring fro m rough surface'S; (b) One or more vector tcrms predominaLP, their nwan sq uare va lue' not I>l'illg neglig;ble co m pared to th e m ean square valup of Uw SU Ill , c.g., in tJle ea~e of sigl als propagat ed in eil i (,'. ll1<'teor-scatter, and atmosplwrie noi se; (0) Th e numiJer of \'('ctO I' t('r n ' j~ small, ('.g., in rada r ret ur ns from several clo",(' targets; (d) '1'110 Jlumi)('r of vector le l'm o is ilsl'H randolll , e.g., in at mosp her ic t u rbuh'nce, m ekor-scatter and atm ~pl'(' .. ic noise'. The ('('suI ting distributio ns for t hese cases and their deviatio ns from the R ayleig h distr ibution will be co nsidered.

Th e sta ti s ti cal di s tributi ons of th e a ngle of arrival, th e s pot locati on, the cross section, th e amp li . tude , th e carrier phase and th e modulation phase of a la se r bea m tra ve rs in g an anisotropicaliy turbule nt at mos ph ere are de rive d in ter ms of th e s pace co rre la tio n fun c ti o n of th e at mos ph eri c ind ex of refractio n a nd the wind spee d. Th e Limit a ti o ns imposed by th e turbul e nt atm osp he re and th e lo ss ,of c ohe re nce o n th e de pth and bandwidth of th e modulation , o n th e le ng th of th e path a nd o n th e a pe rture of th e rece iv· in g appara tu s are a na lyzed . Experim e nt s to ob tai n num e ri ca l param e ters a nd tn c hf'c k fun c tional depe nde ncies are propose d.

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