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Radio Science , Volume ???, Number , Pages 113,

Orthogonal-polarization multi-pulse sequences

T. Grydeland1,3

and B. Gustavsson,2,4

It is well known that using orthogonal polarizations for sub-pulses in multi-pulsesequences can be used to reduce clutter contributions in these modes. In this paper weshow that further improvements are achieved if the orthogonality is taken into accountalready when constructing the codes. Using orthogonal polarizations one can use densertransmission patterns, including elementary pulses without gaps between them, patternsthat have severe range ambiguities when only a single polarization is used. Furthermore,correlations are computed separately for each combination of elementary pulsepolarizations. Consequently, it is possible to have longer multi-pulse sequences withoutgaps in the lag sequence; it is possible to compute the odd lags as well as the even ones;and it is permissible to have some lags multiply obtained without range ambiguity. Thismeans that using orthogonal polarizations when creating the multi-pulse transmissionpattern gives exibility well beyond the single-polarization case. This exibility can beused to design patterns suited to particular experimental requirements. Furthermore, wepoint out that the improvement in clutter might have a more dramatic impact than isgenerally realized, particularly in high SNR situations where the improvement in speed isup to a factor of 4. Examples are given of single- and multiple-frequency patterns that arenot usable if only one polarization is available. Although all IS radars in use today, exceptJicamarca, lack orthogonal polarization capabilities, designers of the next generation of radars might nd the improvements described herein to be of interest.

1. Background and Introduction

The incoherent scatter technique has been suc-cessfully used to study the Earths ionosphere for

almost 50 years, and has demonstrated its useful-ness under widely varying conditions. As the tech-nique has been applied to more demanding condi-tions, huge advances in instrumentation and tech-niques has improved the precision of the measure-ments and the speed at which they can be made byorders of magnitude. This has been achieved by inge-nious manipulation of the radar pulses transmitted,either by using multi-pulse sequences, dened below,[Farley , 1972; Turunen and Silen , 1984], frequencymodulation [ Sulzer , 1986] or phase coding [Lehtinen and H aggstr om , 1987]. In Gustavsson and Gryde-land

[2009], from which parts of this introductionhave been adapted, it was shown that using orthogo-

1 Northern Research Institute, Troms, Norway.2 Dept. of Physics, University of Southampton,

Southampton, United Kingdom3 Previously at Dept. of Physics and Technology,

University of Troms, Troms, Norway4 Previously at Dept. of Communications System,

Lancaster University, Lancaster, UK

Copyright 2010 by the American Geophysical Union.0048-6604/10/$11.00

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2 GRYDELAND AND GUSTAVSSON: ORTHOGONAL-POLARIZATION MULTI-PULSES

nal polarizations in alternating codes allows the useof shorter alternating codesets in addition to a re-duction in self-clutter, which leads to improved mea-surement statistics.

In this paper we show that transmission of orthog-onal sub-pulses can also be utilized in multi-pulsesequences, and that this makes it possible to obtaina larger number of lag estimates; to have multiplyavailable lag estimates; and also to achieve improvedspeed of the modulation through a reduction in thevariance of the auto-correlation function (ACF) esti-mates.

In most cases, the experimenter is interested in thephysics embodied in the thermal or excited uctua-tion spectrum of the ionospheric plasma, and thenthe spectral properties of the scattered signal must

be measured. This is usually done by admitting aband-limited signal into the radar receiver, and thenestimating its spectrum by one of a multitude of tech-niques, depending on the operating parameters of theradar and the properties of the plasma under obser-vation.

When the correlation time of the plasma is shortenough that no correlations will remain between in-dependently transmitted and received pulses, thespectrum must be estimated from the scattering fromevery transmitted pulse or pulse group. This is com-monly done by estimating the spectrum by way of the signals auto-correlation function whose Fouriertransform is an estimate of the scattered power spec-tral density, or power spectrum [e.g. Peebles , 1993].For the scattered spectrum to be accurately mea-sured, the pulse must be longer than 1.5 to 3.5 timesthe correlation time of the plasma, depending on con-ditions, [ Vallinkoski , 1988]. At the same time, thepulse must be short enough that it does not averageover volumes with too different physical parameters(e.g. it must be shorter than a typical ionosphericscale height), and it must also be short enough thatit can be transmitted before the rst signal of in-terest begins to arrive at the receiver. When these

requirements can all be met simultaneously, a long,uncoded pulse can be used [e.g. Farley , 1969].In many interesting situations, however, these re-

quirements cannot be satised simultaneously, andsome sort of sub-pulse technique is necessary. Thesimplest of these, and the easiest to understand, isthe double-pulse technique [Farley , 1969]. To esti-mate the ACF from range h at lag with a range res-olution of h, two pulses, each of length t = 2h/care transmitted with an interval of between thestart of the two pulses. The receiver is then opened,and (complex baseband) samples v0 and v1 are takenat times 2 h/c and 2h/c + . The experiment is re-

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GRYDELAND AND GUSTAVSSON: ORTHOGONAL-POLARIZATION MULTI-PULSES 3

peated a number of times, and the quantity v0 v1 ,where signies average and v is the complex con- jugate of v, is our estimate of the desired correla-tion at this lag. The setup is illustrated in gure 1.In this gure, range increases along the vertical axisand time along the horizontal axis. This type of dia-gram is called a range-time diagram , or a Farley dia-gram . The diagonal lines rising with increasing time(to the right) indicate how the bauds of the trans-mitted pulse propagate outward with time, while thediagonals which descend with increasing time indi-cate what regions of range and time could potentiallyinuence a measurement taken at a particular time,i.e., the time when these lines reach the horizontalaxis.

Every sample contains scattered signal from both

elementary pulses, but the signal contributions fromthe non-overlapping regions in the gure do not cor-relate, and will instead contribute to an increasednoise oor in the measurement. This non-correlatingsignal we call self-clutter . Even in the absence of noise the variance of an auto-correlation functionsestimate is greater than the expected value for sin-gle measurements. The variance for correlated prod-ucts is reduced to tolerable levels through incoherentintegration, usually over a few hundred to severalthousand realizations (radar pulses). In practice, anumber of samples are taken to cover all ranges of in-terest with every pulse, while the spacing is variedto obtain a sufficiently densely sampled ACF suchthat the scattered spectrum can be estimated.

A variation of the double-pulse technique of partic-ular interest to the present study, currently possibleto use with the Jicamarca IS radar, is to transmitthe two elementary pulses with orthogonal polariza-tions, and to sample both polarizations simultane-ously in independent receivers. In this case, everypolarized sample contains signal from only the trans-mitted pulse with that polarization [ Farley , 1969],and the variance of the lagged product is reduced byas much as a factor of four, as explained below. A

requirement for this technique to work is that the or-thogonality is preserved during the entire round-tripfrom transmitter to scatterer to receiver. For this tobe full-lled for propagation through a magnetizedplasma the magneto-ionic eigenmodes, i.e. circularpolarization, can be used.

By using several short pulses with nonredundantspacing (i.e., the spacing between any pair of shortpulses in the sequence appears only once, that is thepulse sequence is equivalent to a Golomb ruler [e.g.Sidon , 1932; Babcock , 1953] ) it is possible to measureseveral lags with every set of pulses [ Farley , 1972; Tu-runen and Silen , 1984]. This dramatically reduces

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4 GRYDELAND AND GUSTAVSSON: ORTHOGONAL-POLARIZATION MULTI-PULSES

the time necessary to obtain the spectral measure-ment, at the price of an increase in self-clutter. Fora multi-pulse sequence with M short pulses one ob-tains M (M 1)/ 2 lagged products with a varianceproportional to M 2 .

A multi-pulse sequence can only have a single es-timate of each lag at every range, and for M > 4there will always be lags that are missing in thelag sequence. Thus the longest multi-pulse sequencegiving even/regular spacing between lag estimates isobtained e.g. with a four-pulse sequence with sub-pulse transmitted at 0,2,8,12 giving lag estimatesfor 2,4,6,8,10 and 12 . One possible improvement,if the transmitter has sufficient delity, is to makea multi-pulse code by interleaving several multi-pulse patterns at different frequencies. The EISCAT

radars were designed to have such capability from theoutset, and used this approach for many years [ Tu-runen and Silen , 1984]. While this approach keepsthe transmitter fully occupied, it is only possible toform correlations between the signal from the ele-mentary pulses at a single frequency.

Provided the requirement of nonredundant spac-ing is met the estimates of the auto-correlation func-tion are easily obtained from products between sam-ples with corresponding lag time. As an example, welook at the lagged product formed between samplesv0 and v2 shown in left panel of gure 2:

(v0 )(v2 ) = ( S h + S h 2 + S h 8 ) (S h +2 + S

h + S

h 6 ) . (1)

where S h is the signal from the volume centered atrange h, S is the back-scatter at the appropriatelydelayed time, and denotes complex conjugate asexplained previously. When the range indices in alagged product differs by two or more, the volumesare disjoint, which means that the signals are uncor-related,

S i S j = 0 when | i j | 2, (2)

so the product (1) averages to

v0 v2 = S h S h . (3)

When a multi-pulse sequence with redundant spac-ing is transmitted, the complex conjugate productswill have correlated contributions from a range of altitudes Lehtinen and Huuskonen [1996]. The ex-pected value of the complex conjugate product be-tween the samples v0 and v2 from the multi-pulsesequence shown in the right panel of gure 2 becomes

v0 v2 = S h S h + S h S

h 1 +

S h 2 S h 1 + S h 7 S

h 6 (4)

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GRYDELAND AND GUSTAVSSON: ORTHOGONAL-POLARIZATION MULTI-PULSES 5

Here, the lagged product v0 v2 has contributionsfrom three separate altitude regions, and otherlagged products will be affected similarly. This isundesirable.

After K pulses of a multi-pulse sequence with non-redundant spacing the desired estimate, R, of theauto-correlation function for a time lag of 2 is

R(2 ) = 1K

K

i

v0 v2 = S. (5)

where S is the back-scattered power and is thecorrelation. The variance of R(2 ) is, by denition:Var( R) = E (| R|

2 ) | E ( R)|2

= 1K 2

K

i

vi v

i

K

j

vj vj S

2 2 (6)

where E (X ) denotes the expected value of the ran-dom variable X . The rst term can be rewritten as

1K 2

K

i

vi v

i

K

j

vj vj =

1K

K

i

vi v

i vi vi +

K 1K

K

i,j

vi v

i vj vj i = j

(7)

These two terms can be expanded with help of thefourth moment expansion [ Reed , 1962] to

vi v

i vi vi =

S

vi v i S

vi vi +P S

vi vi P S

v i vi +0 vi vi

0

v i vi (8)and

vi v

i vj vj =

S

vi v i S

vj vj +0

vi vj0

v i vj +0 vi vj

0

v i vj (9)where P is the number of sub-pulses transmitted in apulse, with P = 4 in this example. Thus the varianceof the ACF estimate becomes:

Var( R) = P 2 S 2

K (10)

Here and in the following we have assumed that back-scattered power is equal over the range of the pulse-length, this simplies the algebra but does not effectthe general validity of the results.

2. Multi-Pulse with Orthogonal Polarization= MPOP

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6 GRYDELAND AND GUSTAVSSON: ORTHOGONAL-POLARIZATION MULTI-PULSES

When using orthogonal polarizations for a multi-pulse sequence, the sub-pulses are transmitted onone or the other polarization, and both polarizationsare received simultaneously and independently, as forthe polarized double-pulse experiment. We proceedas if circular polarizations are used, and label oneL, the other R. The requirement that the pulse se-quence intervals be nonreduntant now applies sepa-rately to each combination of polarizations, e.g. forthe four cases L-R (rst sample L, second sampleR), L-L, R-L and R-R. This leads to improvementsover single-polarization multi-pulse sequences in vedifferent ways:

1. Sequences can be denser, i.e. a higher portion of the total transmission time can be spent transmittingon one frequency.

2. All lags can be included in the sequence, not just even lags.3. Longer sequences can be created without gaps

in the lag sequence.4. Lags can appear in several polarization combi-

nations.5. Variances are reduced by up to a factor of four

over the single-polarization case in the high SNRlimit.

(The last of these improvements is well knownfor the double-pulse case [ Farley , 1972], but it wasexplicitly ruled out for the multiple-pulse case. Itwas, however, hinted at in an unpublished textbookmanuscript by Hagfors and Farley titled Berndraft,August 1999, in Chapter 6, Measurement Tech-niques). We will address each of these improvementsin turn, using a polarized version of the ambiguoussequence shown in the right panel of gure 2. This5-pulse sequence is shown in gure3. Using a pe-riod character to indicate an interval where neitherpolarization is transmitted, the complete transmit-ted sequence is R.RL...LR for a complete length of 9 units of time. We label the samples li and r i , formeasurements in polarizations L and R, respectively,and the lagged product between samples r0 and r2

now becomes(r 0 )( r 2 ) = ( S h + S h 2 + S h 8 )

(S h +2 + S h + S h 6 ) (11)

with the desired expected value:

r 0 r 2 = S h S h . (12)

In addition, the orthogonality makes it possible toobtain ACF estimates for a lag of 1 . In our examplepulse scheme, the lagged product between samples r8

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GRYDELAND AND GUSTAVSSON: ORTHOGONAL-POLARIZATION MULTI-PULSES 7

and l7 is

(l7 )( r 8 ) = ( S h + S h +4 ) (S h + S

h +6 + S

h +8 ) , (13)

with an expected value

l7 r 8 = (S h + S h +4 )(S h + S h +6 + S

h +8 )

= S h S h , (14)

A second estimate of the ACF for lag one is obtainedfrom samples r2 and l3 with clutter contributionsfrom h 4 (L-mode) and h + 2 and h 6 (R-mode).The L-L lagged product (between the two samples l3and l7 ) gives an ACF estimate of lag 4, the L-R lagproducts give ACF estimates for lags 1, 3, 5, and 7,the R-L lag products give estimates for lags 1 and 5,and the R-R lag products give ACF estimates for lags2, 6 and 8. The sequence is demonstrably denser thanany non-redundant multi-pulse sequence of the samelength, since the only 5-pulse sequence with singlepolarization (say, R) of length 9 is R.R.R.R.R , whichis highly redundant, while the shortest nonredun-dant single-polarization 4-pulse sequence is 13 bitslong (this proves point 1 above). All lags from 1 to8 are covered in the MPOP sequence, while everysingle-polarization 5-pulse sequence has gaps in itslag sequence, and only gives even lags (this demon-strates points 2 and 3 above). Lags 1 and 5 appearsboth for L-R and R-L, without ambiguity. If parts

of the ACF are of particular importance, MPOP se-quences can be selected with multiply-dened lags atdesirable points (this illustrates point 4 above). TheMPOP sequence giving the longest lled sequence of lag products we have found (for sequences no longerthan 30 times the length of a unit pulse) is the 8-pulse sequence of length 30 shown in gure 4 thatgives samples of all lags up to 19 , and in additionlags for 24-29 .

Finally, using orthogonal polarizations, the self-clutter is kept to a level corresponding to a multi-pulse sequence of only half as many pulses. Thisagain reduces the variance of the ACF estimates, asexplained below.

For the polarized case the calculation of ACF esti-mator variance is slightly more complicated than forordinary multi-pulse sequences. Let A i be samples of process i with polarization a, likewise for B i . a and bcan be the same or orthogonal polarizations. In thesequence there are P a elementary pulses with polar-ization a, and P b with polarization b. For balancedcodes P a P b P/ 2, but this is not necessary forthe following.

The estimator for unnormalized ACF between

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samples of polarization a and b:

R ab =

1K

K

i

Ai B

i (15)

with expected value E ( R ab ) = S, as before. Wealso need E (Ai Ai ) = P a S (16)and

E (| R ab |2 ) =

1K 2

K

i,j

E (Ai B

i Aj Bj )

= 1K

K

i

E (Ai B

i Ai Bi ) +

K 1K

K

i,j E (Ai B

i Aj Bj )i = j

= 1K

[

S

E (Ai B i )S

E (Ai B i ) +P a S E (Ai Ai )

P b S

E (B i B i ) +0 E (Ai B i )

0

E (B i Ai ) ]+

K 1K

S 2 ||2 (17)

which gives us the varianceVar( Rab ) = E (| R ab |

2 ) | E ( Rab )|2 =

P a P bS 2

K . (18)

Here we see that when only one polarization is used(a = b, and P a = P b = P ) equation 18 is identicalto equation 10. When P a = P b = P / 2 the varianceis reduced to 1/4 of the single polarization variance.For the double-pulse experiment the variance for sin-gle polarization is ( a = b, P = 2)

Var( R2 ) = 22 S 2 /K, (19)

while for the orthogonally polarized double pulse ex-

periment ( a orthogonal to b; P a = 1 , P b = 1)Var( R2 ) = 1

2 S 2 /K, (20)

which is a factor of 4 smaller than the single polar-ized code. This relation holds for all pulse lengths,provided that the number of elementary pulses ineither polarization is equal. When there is noise(N ) in the measurements (that is uncorrelated tothe back-scattered signal) the expression for the vari-ance, equation 18, is modied to

Var(

R ab ) = E (|

Rab |2 ) | E (

Rab )|2 =

(P a P bS + N )2

K

(21)

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GRYDELAND AND GUSTAVSSON: ORTHOGONAL-POLARIZATION MULTI-PULSES 9

and the ratio of improvement is reduced. As forsingle-polarization multi-pulse codes the correlationbetween different lag-estimates should be low [ Hu-uskonen and Lehtinen , 1996], most likely even lowersince the number of pulses in each polarization willbe fewer.

3. Figure of Merit

To properly compare the efficiency of different cod-ing schemes both the number of ACF estimates andtheir variances from one transmitted pulse should betaken into account. This we do by applying a gureof merit number for IS-radar codes:

F OM =

f

K f (S + C f + N )2

(22)

where K f is the number of ACF samples obtainablefrom a given modulation, that is K f = M f (M f 1)/ 2, where M f is the number of sub-pulses trans-mitted at frequency f . For an alternating code [e.g.Lehtinen and H aggstr om , 1987] where all bauds aretransmitted at the same frequency, M f is the sameas the number of bauds; while for codes comprisingmulti-pulse sequences interlaced at several frequen-cies the summation is to be taken over the number of pulses transmitted at each frequency. For a single po-larization code the clutter term, C f is approximately

S (M f 1), and for a code using two orthogonal po-larizations C f S (M f / 2 1) - if there is an equalnumber of sub-pulses transmitted in each polariza-tion. Thus, the efficiency of different coding schemescan be compared.

The number of lag estimates that can be obtainedfrom a code with four interlaced four-pulse sequencesis equal to 4 4 (4 1)/ 2 = 24 ACF samples,and a hypothetical code with two interlaced 8-pulsesequences would give 2 8 (8 1)/ 2 = 56 ACFsamples.

The number of MPOP sequences for a given num-ber of sub-pulses (or length, l(MPOP)), is biggerthan the corresponding number of ordinary multi-pulse sequences, as shown in table 1. In table 1 thereare three facts that warrant mentioning: l(MPOP)is far shorter than l(MP) with the l(MPOP) /l (MP)decreasing with increasing number of sub-pulses atleast for N < 10. For N > 4, l(MP) > N (N 1) withl(MP) N (N 1) growing, i.e. an increasing num-ber of lags are missing for increasing length of ordi-nary MP sequences. On the other hand l(MPOP)