SKCT. 5] ANALYSIS AND STATISTICS 585 



total energy in E{a) is estimated. The standard method of evaluating E{g) is 

 in fact to operate on ^(/)(0 ^t ^T) by a process whose result has expectation 



Too 



8((7) = f{a'-a)E{a')da', 



where /(a>) is a filtering function of total integral unity and is large only within 

 a fairly narrow band centred at a> = 0. The C.V. for £(ct) is 



1^ p{<y'-a)E'\a')da'lTZ'^{a), (31) 



as in (30), and can again be expressed as 27TlojeT, where cog is the equivalent 

 width of the filter/(cu). Now as a rule we can alter cue arbitrarily within certain 

 limits, and so for a given duration T it is possible to reduce the C.V. by choosing 

 a large filter width cDg. But if oje is too large, £(ct) gives but a blurred picture of 

 E{a), thereby losing detail. On the other hand, if T is increased beyond a certain 

 limit; E{a) itself becomes blurred because of non-stationarity of ^(^). In practice 

 then we have to accept a reasonable limit for T and choose a compromise for 

 oje so that £(ct) gives a fairly detailed picture of E{a) while C.V. is not too great. 

 It is usually difficult to reduce C.V. much below 0.05 (standard error 22% of 

 expectation) without blurring the spectrum. If "confidence limits" are re- 

 quired, the estimates of £(o-) may be regarded as samples from the distribution 

 of ix^lf), where /= 2/C.V. is the number of degrees of freedom associated with 

 the well-known x^ distribution (Tukey, 1949). 



B. Practical Methods of Filtering 



The choice of /(a>) depends on the method of analysis, analogue or digital. 

 In analogue methods, l,{t) is converted to an electrical voltage (greatly speeded 

 up) which is fed through a tuned circuit. The characteristic curve of the power 

 output of the circuit is thenf{a>), and is varied by altering the components of 

 the circuit. Chang (1954) discusses the criteria for the choice of various types of 

 tuned circuits. To cover the whole spectrum, ll,{t) may be fed through a number 

 of circuits in succession, each tuned to a different frequency, or as in the case 

 of a photo-electric analyser used at the National Institute of Oceanography 

 (Barber et al., 1946; Tucker, 1956) a single filter is used and the time scale of 

 t,{t) varied. The latter method actually gives the Fourier series amplitudes of 

 ^(^)(0 ^t^T) spaced in frequency by o- = "Ztt/T. The square of any one of these 

 amplitudes has C.V.= 1, but the sum of squares of n consecutive amplitudes 

 has C.V.= 1/n and gives a fairly good approximation to a square-topped filter 

 with coe = 27TnlT (Tucker, 1957). 



The Fourier series method (often called periodogram method) can also be 

 applied to digital records of (,{t), but if the series is long the computing time 

 becomes prohibitive. It is more economical to compute first the autocovariance 

 0(t), for m equal steps of t, and then the cosine periodogram of these m+1 



