In the above p(fm) ^^ ^^® transform of C(t) by equation (11) and 



Z Qi^[f - -T— J is the transform of V ,^ (T;At) (Blackman and Tukey, 

 1958). The function QgCf) is (Blackman and Tukey, with T^, = NAt/2) : 



Qo(f3 = ^^"^;f '^^ . (26) 



rience. 



p(fj ^ I \ Qo(fm - h- yJp(y)dy . (27) 



= _oo J -00 



This shows that p(f^), to the extent that Cj^ behaves like C(kAt), is 

 the result of smoothing with the Qq Cf ) function and then summing or 

 aliasing as indicated by the summation. 



The function QgCf) has an effective width of (1/NAt). Hence, p(fn,) 

 represents a smoothing approximately over a frequency interval of (1/NAt). 

 This is the spacing between the f^ in equation (6). It follows that 

 sach FFT spectral estimate represents approximately a smoothing of p(f) 

 3ver the interval fjj, ± (l/2NAt). 



It should be noted that problems related to the side lobes of QQ(f), 

 the aliasing, and the sampling variability have been ignored in the above 

 discussion. These effects will now be discussed briefly. The distortion 

 due to aliasing can be mitigated by choosing the Nyquist frequency suf- 

 ficiently large. The sampling variability shows up in the statistical 

 fluctuations of the estimates and can be examined from that perspective. 

 If the true spectra are relatively smooth and linearly changing, the side 

 lobes of Qo(f) will compensate for each other somewhat. However, if the 

 true spectra are not smooth, there will be unavoidable leakage of large 

 spectral departures or spikes into neighboring spectral estimates. 



IV. FFT AVERAGING NEEDED TO GIVE SPECTRA EQUIVALENT TO THOSE 

 FROM THE COVARIANCE METHOD 



From the previous section, the covariance spectral estimates (after 

 hamming) represent a smoothing over the frequency interval of width, 

 (1/kj^ At), while the FFT spectral estimates involve a smoothing over a 

 frequency interval of width, (1/N At). Hence, the number of FFT spectral 

 lines that should be averaged together to yield an estimate with approxi- 

 mately the same smoothing as the covariance spectral estimates is: 



(1/km At) 



number = = N/km . (28) 



(1/N At) 



15 



