RADIO SCIENCE, VOL. 44, RS6005, doi:10.1029/2008RS004132, 2009
Orthogonal-polarization alternating codes B. Gustavsson1 and T. Grydeland1,2 Received 22 December 2008; revised 5 May 2009; accepted 21 August 2009; published 3 December 2009.
[1] We propose that by using orthogonal polarizations for alternating bauds of an
alternating code, improvements are available both in cycle length, by means of using weak condition alternating codes, and in code performance through a reduction of autocorrelation function estimate variances by a factor of 4. This means an increase in the speed of incoherent scatter radar experiments by the same factor of 4. Citation: Gustavsson, B., and T. Grydeland (2009), Orthogonal-polarization alternating codes, Radio Sci., 44, RS6005, doi:10.1029/2008RS004132.
1. Background and Introduction [2] The incoherent scatter technique has been successfully used to study the Earth’s ionosphere for almost 50 years, and has demonstrated its usefulness under widely varying conditions. As the technique has been applied to more demanding conditions, huge advances in instrumentation and techniques has improved the precision of the measurements and the speed at which they can be made by orders of magnitude. [3] While there are some special cases where measurement of back-scattered power is enough to be of interest, the more usual case is that the experimenter is interested in the physics embodied in the thermal or excited fluctuation spectrum of the ionospheric plasma, and then the spectral properties of the scattered signal must be measured. It is possible to estimate this spectrum directly, e.g., through a filter-bank technique [Farley, 1969], but this requires a prohibitively long pulse to be used, and the more common approach is to admit a band-limited signal into the radar receiver, and then estimate its spectrum by one of a multitude of techniques, depending on the operating parameters of the radar and the properties of the plasma under observation. [4] When the correlation time of the plasma is much longer than the pulse round-trip time to the observed volume, pulse-to-pulse correlation techniques are used. (low radar frequencies; mesopause and below; not explained or discussed further here). [5] Otherwise, the correlation time of the plasma is short enough that no correlations will remain between independently transmitted and received pulses. In this 1 Department of Physics and Technology, University of Tromsø, Tromsø, Norway. 2 Now at Discover Petroleum, Tromsø, Norway.
Copyright 2009 by the American Geophysical Union. 0048-6604/09/2008RS004132
case, the spectrum must be estimated from the scattering from every transmitted pulse or pulse group. While it is possible to use Fourier transform-based techniques on the received signal, it is more common to estimate the spectrum by way of the signal’s autocorrelation function (ACF), whose Fourier transform is an estimate of the scattered power spectral density, or power spectrum [e.g., Peebles, 1993]. For the scattered spectrum to be accurately measured, the pulse must be longer than 3.5 times the correlation time of the plasma [Vallinkoski, 1988]. At the same time, the pulse must be short enough that it does not average over volumes with too different physical parameters (e.g., it must be shorter than a typical ionospheric scale height), and it must also be short enough that it can be transmitted before the first signal of interest 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]. [6] In many interesting situations, however, these requirements cannot be satisfied simultaneously, and some sort of subpulse technique is necessary. The simplest of these, and the easiest to understand, is the double-pulse technique [Farley, 1969]. To estimate the ACF from range h at lag t with a range resolution of dh, two pulses, each of length dt = 2dh/c are transmitted with an interval of t between the start of the two pulses. The receiver is then opened, and (complex baseband) samples v0 and v1 are taken at times 2h/c and 2h/c + t. The experiment is repeated a number of times, and the quantity hv0v*1i, where hi signifies average and v* is the complex conjugate of v, is our estimate of the desired correlation at this lag. The setup is illustrated in Figure 1. In Figure 1, range increases along the vertical axis and time along the horizontal axis. This type of diagram is called a rangetime diagram, or a Farley diagram. The diagonal lines rising with increasing time (to the right) indicate how the bauds of the transmitted pulse propagate outward with time, while the diagonals which descend with increasing
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Figure 1. Range-time, or Farley, diagram showing a double pulse, and a pair of samples giving one estimate of the ACF from range h together with self-clutter contributions from regions above and below the desired altitude.
time indicate what regions of range and time could potentially influence a measurement taken at a particular time, i.e., the time when these lines reach the horizontal axis. [7] Every sample contains scattered signal from both elementary pulses, but the signal contributions from the nonoverlapping regions in Figure 1 do not correlate, and will instead contribute to an increased noise floor in the measurement. This noncorrelating signal we call selfclutter. Even in the absence of noise the standard deviation of an autocorrelation functions estimate is greater than the expected value for single measurements. The variance for correlated products is reduced to tolerable levels through incoherent integration, usually over a few hundred to several thousand realizations (radar pulses). In practice, a number of samples are taken to cover all ranges of interest with every pulse, while the spacing t is varied to obtain a sufficiently densely sampled ACF such that the scattered spectrum can be estimated. If 1000 incoherent integrations are required, and the time between transmitted pulse pairs (the interpulse period, or IPP) is 10 ms, it will take 10 s to estimate a single lag of the ACF. Typical time to complete a spectral measurement is then 2 – 5 min. [8] An interesting variation of the double-pulse technique is to transmit the two elementary pulses with orthogonal polarizations, and to sample both polarizations simultaneously in independent receivers. In this case, every polarized sample contains signal from only the transmitted pulse with that polarization [Farley, 1969], and the variance of the lagged product is reduced by a factor of 4, as explained below. [9] By using several short pulses with nonredundant spacing (i.e., the spacing between any pair of short
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pulses in the sequence appears only once) it is possible to measure several lags with every set of pulses [Farley, 1972]. This dramatically reduces the time necessary to obtain the spectral measurement, at the price of an increase in self-clutter. For an multipulse code with N short pulses, the variance of each lagged product is proportional to N2. [10] The double pulse and multipulse measurements share the drawback that the transmitter is used only for a small proportion of the time theoretically available for transmission. One improvement possible, if the transmitter has sufficient fidelity, is to interleave several multiplepulse patterns at different frequencies. The EISCAT radars were designed to have such capability from the outset, and used this approach for many years [Turunen and Sile´n, 1984]. [11] While this approach keeps the transmitter fully occupied, it is only possible to form correlations between the signal from the elementary pulses at a single frequency. A multiple-pulse sequence can only have a single estimate of each lag at every range, and for N > 4 there will always be lags that are missing in the lag sequence. [12] Ideally, one would want to keep the transmitter at the same frequency and transmit the elementary pulses with no spacing between them, as long as the signal from the elementary pulses can be distinguished on the receive side. (When phase coding is used, we will refer to the shortest constant-phase element of a transmitted pulse as a baud.) It is a remarkable fact that this is actually possible: not in a single pulse measurement, but through a phase coding which is varied from one pulse to the next. The working of this alternating code approach [Sulzer, 1986; Lehtinen, 1986; Lehtinen and Ha¨ggstro¨m, 1987] is repeated more fully in the next section. Then in the following section we show the benefits of transmitting consecutive bits of the code in orthogonal polarization.
2. Decoding of Regular Alternating Codes [13] The explanation and Figure 2 in this section are heavily influenced by those of Farley [1996]. In Figure 2, we show how the cancelation property of the alternating codes work in the case of matched sampling, i.e., when the sampling interval is equal to the bauds of the transmitted code. [14] Consider the 4-bit alternating code shown in Figure 2. The four bauds of the transmitted pulse are coded with the signs a0 through a3, corresponding to phase values of either 0° or 180°. The sampling is matched to the baud length. [15] As an example, we will look at the lagged product formed between the two shaded regions in Figure 2. To
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Figure 2. Range-time diagram showing a few lagged products in an alternating code with matched (integer) sampling. select these regions, we will multiply sample v0 with the sign a0 used in transmission, and likewise we multiply sample v1 with sign a1. These signs can (and will) vary from pulse to pulse in a sequence, either randomly or in a fixed pattern, while the same signs are used when transmitting and decoding the signal received from a single pulse. Averages over products of signs are therefore to be taken over a large number of coded pulses in a set. The samples are separated in time by the baud length, and the complex conjugate product of these two samples will give us an estimate of the correlation of the ionospheric target at a lag corresponding to the baud length, which we will simply call ‘‘an estimate of lag 1’’ henceforth. The total contribution to this lagged product is given by
where Sh is the signal from the volume centered at range h, and the superscript * indicates complex conjugate. [16] Since the back-scattered signal from nonoverlapping regions does not correlate, the expected average from ranges separated by two or more do not contribute to the correlated signal, Si S*j ¼ 0 when ji jj 2;
which leads to * i ha0 v0 a1 v*1 i ¼ hSh Sh*i þ ha0 a2 ihSh1 Sh1 * þ ha0 a1 a2 a3 ihSh2 Sh2i * i þ hSh1 Sh*i þ ha0 a1 i½hSh Shþ1 * i þ hSh3 Sh2 * i þ hSh2 Sh1 * i þ ha0 a3 ihSh1 Sh2 * þ ha1 a2 ihSh Sh1i:
ða0 v0 Þða1 v1 Þ* ¼ a0 ða0 Sh þ a1 Sh1 þ a2 Sh2 þ a3 Sh3 Þ a1 ða0 Shþ1 þ a1 Sh þ a2 Sh1 þ a3 Sh2 Þ*;
ð1Þ
ð2Þ
ð3Þ
The term hShS*h i is the one we are interested in, the correlation between the signals from range h at the two
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different times. The codes must therefore be chosen such that the averaged sign sequences for the undesired terms all disappear. For this particular lagged product, the sign averages that must cancel are ha0 a1 i ¼ ha0 a2 i ¼ ha0 a3 i ¼ ha1 a2 i ¼ ha0 a1 a2 a3 i ¼ 0;
ð4Þ
but there are (N 1)(N 2)/2 possible lagged products in an N-bit code, all of which can be obtained through an appropriate choice of decoding bits on the receive side. Each of these is accompanied by a different set of undesired terms, and all the undesired terms must cancel in every case. This means that all products of signs that arise from overlapping volumes must be required to cancel after averaging over the code sequence. [17] A correlated product can be indicated by bits ai and ai0 used to code the transmitted bauds, while bits aj and aj0 are used to decode the received samples. If the symbol Aiji0j0 is used to indicate the sum over the entire code sequence of the product aiajai0aj0, the condition for necessary cancelation can be written [Lehtinen and Ha¨ggstro¨m, 1987] Aiji0 j0 ¼ 0;
if i j ¼ i0 j0 or i j ¼ i0 j0 1ðand i 6¼ i0 Þ
ð5Þ
[18] If the contributions from neighboring bauds do not overlap in range for a given sample, e.g., if the impulse response function of the receiver can somehow be made to have infinitesimal width, the relaxation (2) due to nonoverlapping volumes is relaxed further, to Si Sj* ¼ 0
when i 6¼ j:
ð6Þ
In this case, the condition for necessary cancelation is written Aiji0 j0 6¼ 0; i 6¼ j and i j ¼ i0 j0 ; Aiji0 j0 ¼ 0; otherwise
ð7Þ
[19] Following the terminology of Lehtinen and Ha¨ggstro¨m [1987], condition (5) is called the strong condition, while the more relaxed condition (7) is called the weak condition. [20] Obviously, ideal zero-width impulse response functions cannot be achieved in practice. Barker codes [Ioannidis and Farley, 1972] decode to give an impulse response function which is very narrow and with low sidelobes. Weak-condition codes can be used when each baud of the alternating code has been subcoded with a Barker code [Lehtinen and Ha¨ggstro¨m, 1987]. However, the imperfections in the decoded Barker code impulse
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response leads to undesirable long-range ambiguities, in exactly the same way as for Barker-coded multipulse codes [Huuskonen et al., 1988]. Of course, if strong condition codes are used with Barker codes, these ambiguities disappear. To our knowledge, experiments with Barker-coded alternating codes have used strong condition codes [e.g., Wannberg, 1993]. [21] Leaving the weak condition for the moment one way of achieving the necessary cancelation (5) is to choose all signs ai in all pulses to be +1 or 1 at random. Every product of pair of code signs aiaj is then ±1, randomly, and Aiji0j0 ! 0 whenever i 6¼ j, unless i j = i0 j0. This random coded long pulse approach was introduced by Sulzer [1986]. [22] Another way of achieving the cancelation (5) is to construct a set of codes such that the necessary cancelation occurs exactly for a complete set of codes. It is not immediately obvious that such code sets can be constructed. However, through clever use of orthogonality properties of binary sequences achieved through the so-called Walsh matrix, Lehtinen [1986] was able to create sets which had these cancelation properties. These code sets are commonly called alternating codes and form the basis for a great portion of the experiments used on incoherent scatter radars today. [23] The alternating codes use code sets containing a number of codes which is a power of two. When satisfying the strong condition, (5), the number of codes is twice the power of two greater than or equal to the number of bits in the code. When satisfying the weak condition, (7), the number of codes can be halved, to the power of two equal to or greater than the number of bits in the code. [24] It is worth noting that from a signal processing point of view, the only difference between random codes and alternating codes is the way the code sets have been created. The signal processing and decoding is performed identically in the two cases [Grydeland et al., 2005]. [25] Using alternating codes, it is possible to increase resolution in the lag and range directions almost arbitrarily by shortening the bauds and increasing the code length. This carries with it a price in increased bandwidth in the receiver, and in self-clutter, both of which reduce the SNR, and also a price in an increased cycle length. With an IPP of 10 ms and a 64-bit code, the scattering targets must remain stationary (in a statistical sense) for 1.28 s for the cancelation property of the alternating codes to work. If 2-pulse ground clutter removal is used [Turunen et al., 2000], the time required is doubled. [26] The coding, when stationarity requirements are met, eliminates all the correlated signal from the lag estimates (i.e., all the hSiS*j i terms not selected by the code in expressions like equation (3) above). Uncorre-
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Figure 3. Timing diagram showing a few lagged products in an orthogonal polarization alternating code with matched (integer) sampling. lated signal still remains, and in a high SNR case is the dominant contribution to the noise floor.
3. Orthogonal-Polarization Alternating Codes [27] The only reason for the difference between the strong and weak cancelation requirements, (5) and (7), respectively, is that each sample contains signal from overlapping volumes in the former case and not in the latter. [28] Taking inspiration from the orthogonal-polarization double-pulse experiment, we suggest that a novel way of avoiding such overlap is to transmit every other baud in the alternating code set with orthogonal polarizations, and then receive both polarizations simultaneously and independently. This reduces unwanted self-clutter by roughly a factor of 2 as mentioned by D. T. Farley and T. Hagfors (unpublished manuscript, 1999). At each instant of sampling the received signal, these two samples now contain signal from exactly half of the transmitted code. One sample contains only the signal from the even bauds, the other one the signal from the odd bauds, as shown in Figure 3. [29] If we label the two polarizations L and R, and the samples taken for each of these li and ri, respectively, the contributions to the first pair of samples are then l0 ¼ a0 Sh þ a2 Sh2 r0 ¼ a1 Sh1 þ a3 Sh3 :
[30] To calculate the lagged product for altitude h from the first two sample pairs we see that the samples containing signal from h separated by one baud are l0 and r1. When replacing the expressions for ordinary samples in equation (1) with the expressions for those two samples, the lagged product becomes: ða0 l0 Þða1 r1 Þ* ¼ a0 ða0 Sh þ a2 Sh2 Þ a1 ða1 Sh þ a3 Sh2 Þ*;
ð8Þ
This expands nicely to ha0 l0 a1 r1 *i ¼ hSh Sh *i þ ha0 a1 a2 a3 ihSh2 Sh2 *i: ð9Þ
where all terms with contributions from disjoint altitude ranges have been discarded, since they do not contribute to the correlation. Decoding for any given lagged product proceeds exactly like for ordinary alternating codes, except that care must now be taken to correlate li with li+k for k even, and with ri+k for k odd, and similarly for correlations with ri. [31] The benefit from this approach is twofold: Firstly, since the signal in each sample arises from bauds that do not overlap, only the weak condition on overlapping volumes (7) must be fulfilled. Hence, weak condition alternating codes can be used, halving the code cycle time. Secondly, the self-clutter is reduced roughly by half. This self-clutter reduction leads to a variance reduction for the ACF estimates.
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[32] The variance of an unnormalized ACF estimate, ^ ij is R D E ^ ij Þ ¼ jR ^ ij j2 j R ^ ij j2 ; varðR ð10Þ ^ ij is where the expected value of R * + K X 1 ðkÞ* ðkÞ ^ ij ¼ R A B ¼ Sr; K k¼1 i j
ð11Þ
(k)* (k)* (k)* where A(k) are l(k) or r(k) if i j is odd i Bj i rj i lj (k) (k)* (k) (k)* and li lj or ri rj if i j is even. Here S is the backscattered signal power, r is the correlation and k denotes the summation over independent estimates, within the code group or across different pulses. In the absence of external noise (or equivalently, in the case of high SNR) it can be shown that for this unnormalized estimator D E ^ ij j2 ¼ S 2 r2 þ Na Nb S 2 ; jR ð12Þ K
where Na is the number of bits with polarization A, i.e., the same polarization as bit i in each pulse, and Nb is the number of bits with polarization B. With alternating bits in orthogonal polarization, Na = Nb = N/2, this leads to ^ ij Þ ¼ varðR
Na Nb 2 N 2 2 S ¼ S : K 4K
N2 2 S : K
[34] The technique requires a transmitter which is capable of switching between orthogonal polarizations as well as changing the phase of the transmission within a microsecond or so. Linear polarizations are probably less useful than oppositely rotating circular polarizations, since the former will undergo Faraday rotation. The technique also requires a receiver capable of receiving both polarizations simultaneously and independently, or two independent receivers that are synchronized to a very high degree, one for each polarization.
4. Summary [35] In this paper we have presented an improved principle of coding long pulses for incoherent scatter use, employing alternating orthogonal polarization and phase coding. This leads to complete suppression of sidelobes and elimination of unwanted correlated signal using only weak alternating codes. This reduces the cycle length of a complete code set with a factor of 2. In addition to this the variance of ACF estimates is reduced by a factor of 4 in the high SNR case. This means that statistics can be collected at up to four times the rate possible using only a single polarization. The alternating orthogonal polarization phase coding can be combined with other coding schemes, e.g., subcoding of bauds with Barker codes.
ð13Þ
This should be compared to the ACF estimate variance for an ordinary alternating code experiment: ^ ij Þ ¼ varðR
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ð14Þ
In the special case of the double-pulse experiment, we see that in the unpolarized experiment, each lagged product has a variance of 4S2/K, while for the polarized experiment, the variance is S2/K; a reduction by a factor of 4, as mentioned above. Using normalized estimators eliminates the factors of S2 in the results, but makes the calculations more involved without adding to the present discussion. Including noise adds a term not proportional to Na and Nb, which means that the improvement will be smaller for lower SNRs. [33] For clarity we have ignored the effect of the bit signs in these calculations. To include the effect of the signs, we refer to equation (3) of Huuskonen and Lehtinen [1996], and observe that to calculate the covariances of all lag profile matrix elements all of the sc sc Kt0,t 0 terms needed are to be computed from products Kt,t of samples which contain signals from half of the transmitted bauds, and therefore have their variance reduced by up to a factor of 4 in exactly the same way as indicated for Rij above.
[36] Acknowledgments. The authors want to thank the referees for their helpful and encouraging comments that improved the quality of the paper.
References Farley, D. T. (1969), Incoherent scatter correlation function measurements, Radio Sci., 4, 935 – 953. Farley, D. T. (1972), Multiple-pulse incoherent-scatter correlation function measurements, Radio Sci., 7, 661 – 666. Farley, D. T. (1996), Incoherent scatter radar probing, in Modern Ionospheric Science, edited by H. Kohl, R. Ru¨ster, and K. Schlegel, pp. 415 – 439, Eur. Geophys. Soc., KatlenburgLindau, Germany. Grydeland, T., F. D. Lind, P. J. Erickson, and J. M. Holt (2005), Software radar signal processing, Ann. Geophys., 23, 109 – 121. Huuskonen, A., and M. S. Lehtinen (1996), The accuracy of incoherent scatter measurements: Error estimates valid for high signal levels, J. Atmos. Terr. Phys., 58, 453 – 463. Huuskonen, A., P. Pollari, T. Nygre´n, and M. S. Lehtinen (1988), Range ambiguity effects in Barker-coded multipulse experiments with incoherent scatter radars, J. Atmos. Terr. Phys., 50, 265 – 276. Ioannidis, G., and D. T. Farley (1972), Incoherent scatter observations at Arecibo using compressed pulses, Radio Sci., 7, 763 – 766.
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Lehtinen, M. S. (1986), Statistical theory of incoherent scatter measurements, Ph.D. thesis, Univ. of Helsinki, Helsinki, Finland. Lehtinen, M. S., and I. Ha¨ggstro¨m (1987), A new modulation principle for incoherent scatter measurements, Radio Sci., 22, 625 – 634. Peebles, P. Z., Jr. (1993), Probability, Random Variables, and Random Signal Principles, Electr. Eng. Ser., 3rd ed., McGraw-Hill, New York. Sulzer, M. P. (1986), A radar technique for high range resolution incoherent scatter autocorrelation function measurements utilizing the full average power of klystron radars, Radio Sci., 21, 1033 – 1040. Turunen, T., and J. Sile´n (1984), Modulation patterns for the EISCAT incoherent scatter radar, J. Atmos. Terr. Phys., 46, 593 – 599.
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Turunen, T., J. Markkanen, and A. P. van Eyken (2000), Ground clutter cancellation in incoherent radars: Solutions for EISCAT Svalbard radar, Ann. Geophys., 18, 1242 – 1247. Vallinkoski, M. (1988), Statistics of incoherent scatter multiparameter fits, J. Atmos. Terr. Phys., 50, 839 – 851. Wannberg, G. (1993), The G2-System and general purpose alternating code experiments for EISCAT, J. Atmos. Terr. Phys., 55, 543 – 557.
T. Grydeland, Discover Petroleum, PO Box 690, N-9257 Tromsø, Norway. (
[email protected]) B. Gustavsson, Department of Physics and Technology, University of Tromsø, N-9037 Tromsø, Norway. (bjorn.
[email protected])
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