The coherence function may be constructed from experimental data to check the influence of uncorrelated system noise on test results. The coherence function involves averaging the power spectra from different segments (or blocks) of the data record. Each block data must contain an integral number of periods for periodic signals. The coherence function is defined as (4)

(2.10.1)

where

and N is the number of blocks of the data record, Fn + еи the Fourier trans­form of [input signal plus input noise] for segment i, F0i + e0i the Fourier transform of [output signal plus output noise] for segment i, and * denotes complex conjugate.

From these definitions, we obtain

P, o = (1/ЛГ) І FnF*0i + FIi(*0i + (iiF*0i + e, i(*oi

І= 1

p„ = №) £ F„F,*; + F, tft + ‘„FT, + Cjtf,

І= 1

Poo = (1/W) Y FoiFoi + Роі€оі + €oiFoi + eo;eoi

i= 1

Each of these terms is recognized as a power-spectrum estimate. As the number of blocks in the average increases, the contributions of the cross­power spectra of uncorrelated signals diminish toward zero. Thus:

P’I0 = lim PI0 = lim (1/ЛГ) £ FnF*0i

N-oo N-oo (tTj

P’„ = lim P„ = lim (1/ЛГ) Y FnF*i + (n(*i

N-*oo N -»oo

_ N

Poo = lim P00 = lim (1 /N) Y FoiFoi + tofoi

N~* oo N -* oo j _ j

Then the coherence function based on these limiting values is

„_________ ИГ-, ___________

II,"., F, n + <n«SI Ig., Vl‘ + <<„<11

If the process is stationary (Fu and F0i have the same values for any block of the data record), then the coherence function goes to its maximum value of unity as the noise contamination goes to zero. The departure from unity of (y2)’ is taken as a measure of the influence of noise on the test results (assum­ing a stationary signal).

Several words of caution are appropriate in connection with the use of coherence functions. First, the interpretation presented above depends on the disappearance of uncorrelated terms as a result of a long data analysis, but actual tests use finite data records. Furthermore, it can be shown that the coherence function has a value of unity for a record of zero length and decreases to its limiting value as the quantity of data analyzed increases. This indicates that errors due to using insufficient data records will cause the estimated coherence function to be too high (nonconservative). The second
point is that the interpretation of the coherence function depends on the existence of exact Fourier coefficients. If the data analysis is imperfect (i. e., if incorrect harmonic frequencies are used in the analysis because of inaccu­rate timing), then the coherence function may be meaningless.