TR No. 22 
Since the e; are assumed random, statistically independent variables , 
the u' and the e, are uncorrelated, as are the wt and the e,. 
j +k ja bs J 
Therefore 
Nek 
N-k 
milk, ‘ | a a | = 
NK & ay SMe 7 hes e WS ON 
Uiail is 
| (27) 
In addition, the e. are uncorrelated with the e. nes unless k = 0. 
Then we have J ¥ | N 
\ 2) 
ee a Totes ' es an Cy an 
Rey Nel a Ci+k 7 NL {3 =e 
o, otherw ‘se 
IN 
(28) 
Nek 
\ 
oe beni Ly ‘ 
KX (kos ) +Rey, = Mi . ™\ ‘e C Sie ai 
J 
where 
eu ©, otherwise 
This demonstrates that the presence of random error in the digitized 
velocity data has an effect on only the value of the autocovariance 
series at k = 0 (the variance). The expected form of the autocovariance 
function for small values of 5 is (Batchelor, 1) 
te 
Comparison of this with the autocovariance series given in figures 18 
through 22 indicates that the sample variances are larger than expected 
by around 3 cm2-sec”2. The Fourier transform of equation (29) is 
ae IRkes)+ moe di ee Cos KK Os 
Sana 4s N 
AS Fie: separ 
See eh Ss j 
e us N J=I (31) 
1416 om 
= @(k) + sues [2 omr-see” ) 
i 
2a 
