Radar System Engineering

36 THE RADAR EQUATION [SEC.2.10
(Vol. 24, Chap. 8)leads tocertain positive statements. First, therapidity
of the fluctuations is determined by the bandwidth @ of the amplifier.
That is to say, in a time short compared with l/(B, it is extremely unlikely
that the output power will change noticeably. 1 On the other hand,
during a time long compared with l/01, fluctuations will almost certainly
occur, and the value of the power P at the end of such a long time will
bear no systematic relation at all to the value it happened to have at the
beginning. This is a roundabout way of saying that values of P determined
at times differing by much more than 1/6 are statistical y independent,
or, in other words, I/@ is a measure of the correlation time of the
fluctuations.
The second statement which can be made concerns the probability
that at some arbitrarily selected instant the output power will be found
to lie between some specified level, P, and P + dP. This probability,
which we shall label WI(P) dP, is given by
W,(P) dP = #o e-$ dP. (22)
PO is the average power, determined over a long time.
A statement which is easily seen to be equivalent to this is: The probability
that the power exceeds some specified level P, at an arbitrarily
P
selected instant, is just e‘F”. Thus there is always a finite chance of
getting a high noise peak. For example, the probability that at a given
time P is greater than 5Po is e-5 or 0.0068. For the discussion that
follows it is convenient to simplify the problem somewhat by dividing
the time base into discrete intervals each l/CB long. In Fig. 2.6a we
would have 50 such intervals. The essential features of the noise background
can then be described by regarding these intervals as independent
and associating some one value of P with each. Again Eq. (22) correctly
expresses the probability that the power, in an arbitrary interval, will lie
between P and P + dP.
The task of detecting a signal amid the noise in this simplified case
amounts to selecting an interval which displays so large a value of P
that one is justified in betting that the peak was due to a combination of
signal and noise, and not to noise alone. This should dispose of any hope
that we shall be able to define once and for all the minimum detectable
1The intermediatefrequencyitselfis assumedto be high comparedwith (B,so that
it is permissibleto speakof the instantaneouspowerwhileactuallymeaningthe power
averagedover one cycle of the intermediatefrequency.
‘ The intervalsso defined are actually not entirelyindependent,for the output
poweris, after all, a continuousfunction of the time. We areheresubstitutingfor a
continuousrandom processa discreterandom process,which is easierto discuss in
elementaryterms,