Radsr signels have reeently been reflected from the planet Venus. Suppose that in sucb an experiment s pulse of electromsguetic radistion of durstion $\tau$ is sent from the earth toward Venus. A time $t$ later (which corresponds to the tixne necessary for light to go from the earth to Venus and beck again) the receiving antenns on the earth is turned on for a time $T .$ The returning echo ought then to register on the recording meter, placed at the cutput of the electronic equipment following the receiving antenns, as s very faint signal of defnite amplitude $a_{4}$ But a fluctusting randon signal (due to the inevitahle fuctustions in the radiation fiek in outer space and due to enrrent fluctuations alwaye existing in the sensitive receiving equipment itself) slso registers as a signsl of smplitude $a_{n}$ on the reeording meter. This meter thus registers a total amplitucle $a=a_{x}+a_{n}$.
Although $a_{n}=0$ on the average, since $a_{n}$ is as likely to be positive as negstive, there is considerable probsbility that $a_{n}$ attains values considerably in excess of $a_{a}$; i.e, the root-mean-squsre smplitude $\left(\overrightarrow{a_{m}^{2}}\right)^{\text {tan be considersbly }}$ greater than the signsl $a_{4}$ of interest. Suppose that $\left(\overline{\left.a_{n}^{3}\right)^{3}}-1000 a_{4}\right.$ Tben the fluctuating signsl $a_{n}$ eosstitutes a background of "noise" which makes observation of the desired echo signal essentially impossible. On the other hsnd, suppose that $N$ such radsr pulses are sent out in succession and that the total amplitudes $a$ picked up at the recording equipment after esch pulse are sll added together before being displaved on the recording meter. The resulting amplitude must then have the form $A=A_{n}+A_{n}$, where $A_{n}$, represents the resultant noise smplitude (with $A_{n}=0$ ) and $A=A$, represents the resultant echo-signal kmplitude. How many pulses must be sent out before $\left(A_{m}^{3}\right)^{1}=A$, so that the echo signal becomes detectsble?