A Slight Extesio of Coheret Itegratio Loss Due to White Gaussia Phase oise Mark A. Richards March 3, Goal I [], the itegratio loss L i computig the coheret sum of samples x with weights a is cosidered. The coheret sum w is w= ax () = Whe the samples are cotamiated by statioary white Gaussia phase oise of variace σ, L is give by the formula L = σ + = = m= m = m a e aa a () Referece [] is icluded i its etirety i Sectio 5 of this ote, ad i the remaider we refer to the derivatios ad equatios there to avoid repeatig them. As oted i [], the Gaussia probability desity fuctio (PDF) assumed for the phase oise is ot physically possible, sice it is ot cofied to the iterval [,). While the Gaussia PDF is still a very good approximatio for small σ, i this ote we add a small extesio to that aalysis by re-derivig the result for a more realistic phase oise PDF. The Phase oise PDF The PDF of iterest is the Tikhoov distributio give by Va Trees []. I the otatio of [], this is αcosφ pφ ( φ; α) = e I (3) ( α) This PDF evidetly describes the statistics of the steady state phase estimate of a first order phase locked loop trackig a siusoid i additive white Gaussia oise. For our A Slight Extesio to Coheret Itegratio Page of March 3,
purposes, it provides a family of phase PDFs that vary from uiform (α = ) to, i the limit, a oradom phase of zero as α, but is always cofied to [,) ad thus is always a strictly valid PDF for phase. Thus, we ca cosider the effect of varyig degrees of phase oise, previously achieved by varyig the variace i the Gaussia PDF model of [], by istead varyig α i the Tikhoov PDF. Figure illustrates the Tikhoov PDF for several differet values of the parameter α. Relative Probability 3.5.5.5 α = 5 α = 5 α = α = -3 - - 3 θ (radias) Figure. The Tikhoov PDF for phase. 3 Coheret Itegratio Loss Followig the process i [], we eed to compute the quatity Φ, which is φ jφ αcosφ α j { } I ( ) Φ= E e = e e dφ (4) The itegral portio of Eq. (4) ca be put ito a more coveiet form as follows: A Slight Extesio to Coheret Itegratio Page of March 3,
jφ αcosφ jφ αcosφ jφ αcosφ φ = φ + φ e e d e e d e e d jφ αcos ( φ ) jφ αcosφ = e e dφ + e e dφ jφ αcosφ jφ αcosφ αcosφ e e dφ e e dφ cos( φ) e dφ = + = = I ( α) The last step follows from oe itegral form of the defiitio of the modified Bessel fuctio of the first kid [3]. Thus, ( α ) ( α ) I Φ= (6) I It the follows from the argumet i [] that the coheret itegratio loss is (5) L = I ( α ) a + ( ) = I α = m= m = a m aa (7) ad, for the case where all the a, ( α) ( α) ( ) + I I L = (8) Fially, whe is large, ( α) ( α) L I I, (9) Figure illustrates the loss for the case where all a =, i.e., Eq. (8), as a fuctio of the umber of samples itegrated ad the PDF spread parameter α. For large α (phase early costat), the loss approaches L = ( db), as would be expected. As the phase PDF becomes more spread (α approachig zero), idicatig a greater degree of phase oise, the loss icreases, becomig relatively severe for α < 5. A Slight Extesio to Coheret Itegratio Page 3 of 3 March 3,
L.9.8.7.6.5.4 = 5 L (db) - -4-6 -8 = 5.3 -.. - 5 5 α -4 α Figure. Loss as a fuctio of α ad. Left: liear scale. Right: decibel scale. The radom phase case correspods to α =. For this case, Eq. (8) shows that the loss becomes, for a =, L = /. This example shows that whe the phase of the summed samples are all radom, they add o a power basis istead of a voltage basis. That is, the power of the sum is times the power of a sigle sample, istead of, a reductio i gai by the factor. This, of course, is exactly the way radom oise samples behave. 4 Refereces [] M. A. Richards, Coheret Itegratio, IEEE Sigal Processig Letters, vol., o. 7, pp. 8-, July 3. [] H. L. Va Trees, Detectio, Estimatio, ad Modulatio Theory, Part, Wiley, 968, p. 338. [3] M. Abramowitz ad I. A. Stegu, editors, Hadbook of Mathematical Fuctios. U.S. atioal Bureau of Stadards, Applied Mathematics Series, th pritig, Dec. 97. See formula 9.6.9, p. 376. 5 Origial Paper A copy of referece [] is icluded i the ext three pages. A trivial correctio cocers Eq. (5) i []. That expressio for L is ot i decibels, so the db at the ed of the equatio should be deleted. A Slight Extesio to Coheret Itegratio Page 4 of 4 March 3,
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