The autocorrelogram of randomly fading waves

PressLtd., London Journalof Atmospheric andTerre&la1Physics,1955,vol. 6, pp. 50 to 56. Per@xnon

The autocorrelogram of randomly fading waves R. B. BANERJI Ionosphere Research Laboratory, The Pennsylvania

State University,

State College, Pennsylvania

(Received 1 June 1954) ABSTRACT

The power spectrum returned by a completely rough ionosphere having a random motion superposed on a steady drift is deduced. The autocorrelograms of fading patterns corresponding to power spectra returned by the two extreme cases of this spectrum, i.e., pure drift and pure turbulence, are compared and the possibility of statistically distinguishing between them discussed. On the basis of a test of significance of autocorrelograms it is shown that a fading pattern need not contain more than.250 independent points to make a distinction between two possible types of autocorrelograms. The effect of antenna polar diagrams on the autocorrelograms is also discussed qualitatively.

1.

INTRODUCTION

The purpose of the present paper is to investigate the possibility of determining the velocity of drift and turbulence of an irregular ionosphere by studying the autocorrelogram of a fading wave at a single point. The basic principle of t)he method is based on the fact that doppler shifts introduced into the radio waves scattered at different points from an irregular ionosphere endow the wave with a characteristic power spectrum whose form is determined by the magnitude and direction of the velocity as well as the radiation patterns of the transmitting and receiving antennae and the angular spectrum of the returned wave. We deduce, in Section 2, the power spectrum that would be returned from a completely rough ionosphere (one wit’h an angular spectrum determined by Lambert’s Law) having superposed steady and random motion, when the transmit,ting and receiving antennae have omnidirectional characteristics. The power spectrum deduced has been compared to the spectrum obtained for a turbulent ionosphere by RATCLIFFE (1948) and that corresponding to a drifting ionosphere by t’he author in a previous work (BANERJI, 1953). Since the autocorrelograms corresponding to the three power spectra are very similar, we discuss in Section 3 the possibility of stat’istically distinguishing between the autocorrelograms corresponding to t,he two extreme cases of turbulent and drifting motion. It’ is shown, on the basis of a test’ of significance developed by the author, that no more than 250 independent’ points on the fading patt)ern
THE

POWER

SPECTRUM WITH

FROM A STEADILY

SUPERPOSED

RAXDOM

DRIFTING

IONOSPHERE

MOTION

The power spectrum from a completely rough ionosphere. when are undergoing either a purely random or a steadily drifting deduced before. In t’his section we deduce the power spectrum consists of a superposition of random and steady components. We assume, as in the previous work, a plane ionosphere at 50

the irregularit,ies motion has been when the motion a height) h above

The autocorreiopm

of mndomly fading wswes

the observer, and having a uniform distribution of scatterers. The coefficient of back scatter is the same for all scatterers. We also assume, as before, that the antennae are omnidirectional. The drift is ia+ along the negative X direction. The random motion, as in RATCLIFFE’S work, is omnidirectional and has a standard deviation vO. The doppler shift produced at a point (0, 9) by scatterers having a random velocity v along the line of sight, superposed on the drift, is given by

The power scattered depends, as indicated in the previous paper, on the solid angle and, in addition, on the probability that the random part of the velocity lies between v and v + dv. Assuming that the distribution of this velocity is normal, i.e.. -_-

1

- e

-z_ s 2q

(2)

VOX& we

have the expression

W"(f) 133A# dv =

for the power radiated as

---& e 0

on expressing v in equation (2) in terms of frequency and the co-ordinates, and insert,ing the expression for the solid angle. Since the same frequency can be returned from all points with different probability, the total power radiated in the range f to f + df may be obtained as follows: We transform co-ordinates according t‘o the scheme:

(4)

reducing the expression

for the power to

We can now integrate over 8 and #, i.e., over the whole plane, limiting ourselves to the frequency range f to f + df, to obtain the total power scattered in this frequency range. Integration over 8 yields: --_[e-2*

_

~-(z-Kces~)*

where

51

(1. -

2Rx sin2 + co9 (611

(6)

IS.B. BANERJI This int.egral can be shown to reduce to RATCLTFFE'Sand our previous expression, respectively, in the extreme cases where v1 = 0 and v0 = 0. In the general case, however, the integral can only be evaluated approximately. This has been done by dividing the range of integration (# = 0 to + = TT)into six equal parts and approximating the function by the straight lines joining the values at the end of the intervals. The expression for the power then reduces to

_

8e- p+

3

(cash EC -

j$Kx sinh Kz}

-z-

Fig. 1. Thededuced power

sppectmrmcompared with those eo~~apondillg to drift~ing (a) and turbulent (b) ionosphere. W

In Fig. 1 is shown the value of ihzy/; to the ratio

of frequency

shift

aa a function of z (which is proportional

and the random

~amponent)

fur K =

1 (i.e.,

w1 = V?&,), The same quantity is plotted for TJ~= 0, shown dotted, as well as the most similar plot for the case when v,, = 0. It will be noted that the shape of the power spectrum for the intermediate case is very similar to the case of random motion with a larger r.m.s. velocity. For values of K much greater than 1,i.e., for drifts which are much larger than the turbulent velocities, the spectrum approaches that due to drift and, consequently, the difference between the two cases remains small. We will consider, in the next section, whether it is possible, in view of the statistical fl~~ctuatio~ of autocorrelograms derived from finite fading data, to distinguish bet,ween the aut,ocorrelograms corresponding to the extreme cases of this spectrum, i.e., pure drift and purefy turbulent motion.

52

The autocomiogram of randomly fading WBVBS

3.

THE SHAPE OF THE EXTREMEAUTOCORRELOORAMS: POSSIBILITYOF DISTINCTION

The shapes of the autoco~~lograms corresponding to the extreme cases of rctndom and drifting motion can be deduced as follows: (i) Turbulent motion The power spectrum Wi(f) is given by ~~(~) = A,e

-- &I-fD)* *X+4

(8)

where A is a constant, c the velocity of light, fO the midband frequency, and v. the r.m.s. random velocity. The autocorrelogram with lag 7, denoted by P(T), of the amplitude of the wave is then the oosine Fourier transform of this spectrum, i.e.,

m Ale-

By replacing cos 2n(f -f,,)~ axis, we have p,(7)

by e2G(f-fo)r

and

o'(f-.fo)* iif,pv,' df

integrating over the real

sfi)Qe=nz+ =

e

C*

(9)

To facilitate later comparison, we will write equation (9) as a power series, yielding 8.~%oW?a 32j’04~04~4~4 P&) = I @a) + o4 - *‘* C2 (ii)

Drif%ng motion

The power spectrum is given by w2t.f

1 =

4~~being the velocity of drift. Hence the autocorrelogram, in the same manner, is found to be given by

where J, is the Bessel function of the first order. This, again on expansion into TAYLOR’Sseries, yields 2f o%1%2T2 4f,4v,%r4r4 P2(4 = 1 (llal c2 + @ - * *It will be noticed that the two autooorrelograms are similar for small values of 7. A drift velooity v1 = 2v, would give rise to the same autooorrelogram as a random 53

R. B. BANERJI

velocity

v,, for small values of the lag T. This is clearly seen in Fig. 2 where

f 2f 0%X pl(~) and P&T) have been plotted against __I_- T and -3F

7 respectively. c If, therefore, an autoco~elo~am from an experimental fading pattern is to be compared with the above two theoretical models, with a view to deciding

Fig. 2. (a) Autocorrelogram for a drifting ionosphere. (b) Autooorrelogram for a turbulent ionosphere.

whether the motion of the irregularities is random or drifting, the autocorrelograms must be accurately evaluated at the “tails”, i.e., for large values of the lag T. is subject Unfortunately, however, it is at the “tails” that an autocorrelogram as has been pointed out by KENDALL (1946). This to the greatest inaccuracy;

7-

Fig. 3.

Au~co~elo~8m

obtained by

MCNICOL.

fact has led ionospheric physicists to consider experimentally obtained autoFor example, MCNICOL (1949) obtained an autocorrelograms with suspicion. correlogram for the fading of short waves which is shown in Fig. 3. IIe was of the opinion that the oscillations in the autocorrelogram were spurious and were due to the kind of fluctuations obtained by KENDALL. On the other hand, the smooth nature of the oscillations obtained by him, and also its similarity to the theoretical 54

The autocorrelogrem of randomly fading waves model in Fig. 2(a), tempts one to believe in the reality of these oscillations. TO decide on this matter, as well as to gauge the possibility of distinguishing between the two autocorrelograms shown in Fig. 2, we must enquire into the cause of the inaccuracy in the evaluation of the “tails” of autocorrelograms. Autocorrelograms obtained from finite time series, under certain restrictive conditions, have been studied by BARTLETT (1946) in some detail. He has shown that the dispersion of experimentally obtained autocorrelations is not dependent on the individual value of t,he autocorrelations but on the whole autocorrelogram. Thus, spurious oscillations are introduced at the “tail” when the value of the dispersion is great’er than the value of the autocorrelogram itself. This can only be reduced by a suitable increase in the size of the sample, to which the dispersion is inversely proportional. In the light of the above conclusion of BARTLETT'S we can suspect an oscillating autocorrelogram to be spurious only when the dispersion of the individual points about the mean curve is greater than the ordinate of the mean curve. This does not seem to be the case in MCNICOL’S curve, the dispersion appearing to be very low, and does not support the view that the oscillations of the mean curve are spurious. It is also clear from the above discussion that an increase in the length of the sample is capable of reducing the statistical dispersion of the experimentally obtained autocorrelograms sufficiently to make a distinction between the two theoretical curves shown in Fig. 2 possible. The number of points on the fading pattern required for the necessary reduction of the dispersion can be found in terms of a test of significance developed by the author (1954). In the following we shall use the results of this work. Let us initially assume that an autocorrelogram has been obtained having the shape shown in Fig. 2(a), corresponding to drift. It is our purpose to test whether this could have been obtained by random fluctuations from an autocorrelogram like Fig. 2(b), corresponding to random motion. For this we calculate, on the basis of a theoretical autocorrelogram like Fig. 2(b) and a sample of N points on the fading pattern, the probability of obtaining an autocorrelogram having the shape like Fig. 2(a) due to statistical fluctuations. It can be shown, in this case, that if x, (p = 1, 2, . . . 6) be the deviation of the experimentally obtained autocorrelogram from the theoretical one at 6 points then we must have ;P&DZ

+ f*~i7~i~r

XQ> 11

qfr

if the experimental autocorrelogram is to be considered significantly different from the theoretical one. If the experimentally obtained autocorrelogram corresponds to Fig. 2(a), then by choosing six equally spaced points between 0 and 5 we have for the above inequality 0.05N > 11 i.e.,

N > 220

Hence, if from a fading pattern containing 55

more than 220 points we compute

an

R. B. BANERJI: The autocorrelogram of randomly fading waves

autocorrelogram having the shape of Fig. 2(a), we may say that it is highly improbable that it is due to fluctuations from an autocorrelogram like Fig. 2(b). Thus, a sample of this size can distinguish significantly between the two extreme types of autocorrelograms. For distinction from the intermediate cases, however, larger samples would be necessary. 4.

CoNCLUsIoN

It appears from the above discussion that, provided fading data for an adequate interval are available, it is possible to measure the drift velocity of ionospheric irregularities when present by the study of the autocorrelogram of the fading pattern at a single point instead of from several spaced points. It is also possible to recognize purely random velocities, when present, and measure their magnitude. In the case of superposition of random and drift motions, however, fading data for a very long interval of time would be necessary to make the measurement even theoretically possible. In an actual application of this technique it will, usually, not suffice to use the autocorrelograms of Section 3, which are based on the assumption of omnidirectional antennae and a completely rough ionosphere. The power spectra must be deduced in accordance with the radiation pattern of the antennae and the relative positions of the receivers. In our experiments on winds in Pennsylvania State University, the three spaced receivers have loop antennae which are parallel to one another. The plane of two receiver antennae have almost the same disposition with respect to the transmitter-receiver line while the third is oriented differently with respect to this line. It has been observed that, for certain directions of drift in the ionosphere, the autocorrelograms of the first two receivers are significantly different from that at the third, the sense and magnitude of the difference depending on the direction of the wind. The quantitative theory of this phenomenon is at present being developed and will be published elsewhere. Acknowledgments-The theoretical part of this work was carried out in the Institute of Radio Physics and Electronics, Calcutta, under Dr. S. K. MITRA, with financial subsistence from the Ministry of Education, Government of India. The experimental work is being sponsored by the Geophysics Research Division of the Air Force Cambridge Research Center, Air Research and Development (‘ommand, under contract AF19(122)-44. REFERENCES BANEKJI, R. B. BANERJI, R. B.

1953 1954

B~RTJXTT, M. KENDALL, M. (:.

1946 1946

McN~c~or,, R. k$‘. E. RATCLIFFE, .J. A.

1949 1948

Proc. Phys. Sot. B. 66, 105 Scientific Report No. 59, Ionosphere Research Lab., The Pennsylvania State Univ. J. Roy. Stat. Sot. 8, 27 “Contributions to the Study of Oscillating Time Series” (Camb. Univ. Press) Proc. Inst. Elect. Eng. (Part III) 06, 306 Nature 162, 366

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