Transmission code optimization method for incoherent scatter radar
Juha Vierinen, Markku S. Lehtinen, Mikko Orispaa, Ilkka I. Virtanen
MManuscript prepared for Ann. Geophys.with version 1.3 of the L A TEX class copernicus.cls.Date: 27 October 2018
Transmission code optimization method for incoherent scatter radar
Juha Vierinen , Markku S. Lehtinen , Mikko Orisp¨a¨a , and Ilkka I. Virtanen Sodankyl¨a geophysical observatory University of Oulu
Abstract.
When statistical inversion of a lag profile is usedto determine an incoherent scatter target, the posterior vari-ance of the estimated target can be used to determine howwell a certain set of transmission codes perform. In this workwe present an incoherent scatter radar transmission code op-timization search method suitable for different modulationtypes, including binary phase, polyphase and amplitude mod-ulation. We find that the combination of amplitude and phasemodulation provides better performance than traditional bi-nary phase coding, in some cases giving better accuracy thanalternating codes.
Incoherent scatter radar lag profile measurements can be de-convolved using statistical inversion with arbitrary range andtime resolution as shown by Virtanen et al. (2007a). In thiscase, the transmission does not need to be completely freeof ambiguities. The only important factor is the variance ofthe estimated target autocorrelation function. Even thoughalternating codes are transmission sequences that are opti-mal in terms of posterior variance when integrated over thecode transmission cycle (Lehtinen, 1986), shorter and onlyslightly less optimal code groups are beneficial in many caseswhere an alternating code sequence is too long. Also, ashorter code group offers much more flexibility when de-signing radar experiments, e.g., making it easier to combinemultiple different experiments in the same frequency chan-nel and simplifying ground clutter removal. The use of shorttransmission codes is described in more detail in the compan-ion paper by Virtanen et al. (2007b) submitted to this sameissue.
Correspondence to:
Juha Vierinen(Juha.Vierinen@iki.fi) We have previously studied the target estimation varianceof a coherent target where the target backscatter is assumedto stay constant while the transmission travels through thetarget (Vierinen et al., 2006). We found using an optimiza-tion algorithm that a combination of amplitude and arbitraryphase modulation can achieve very close to optimum cod-ing (the order − less than optimal in terms of normalizedvariance). In this study we apply an optimization method,similar to the one used for coherent targets, to find transmis-sion codes that minimize the variance of incoherent targetautocorrelation function estimates. We compare results ofthe optimization algorithm for several different modulationmethods.All formulas in this paper use discrete time, unless oth-erwise stated. All waveforms discussed are complex valuedbaseband signals. The ranges will be defined as round-triptime for the sake of simplicity. A code with length L can be described as an infinite lengthsequence with a finite number of nonzero bauds with phasesand amplitudes defined by parameters φ k and a k . These pa-rameters obtain values φ k ∈ [0 , π ] and a k ∈ [ a min , a max ] ,where k ∈ [1 , . . . , L ] : k ∈ N . The reason why one mightwant to restrict the amplitudes to some range stems frompractical constraints in transmission equipment. Usually, themaximum peak amplitude is restricted in addition to averageduty cycle. Also, many systems only allow a small numberof phases. The commonly used binary phase coding allowsonly two phases: φ k ∈ { , π } .By first defining δ ( t ) with t ∈ Z as δ ( t ) = (cid:26) when t = 00 otherwise , (1) a r X i v : . [ phy s i c s . d a t a - a n ] J a n Vierinen et.al.: Transmission code optimization method for incoherent scatter radarwe can describe an arbitrary baseband radar transmissionenvelope (cid:15) ( t ) as (cid:15) ( t ) = L (cid:88) k =1 a k e iφ k δ ( t − k + 1) . (2)We restrict the total transmission code power to be con-stant for all codes of similar length. Without any loss of gen-erality, we set code power equal to code length L = L (cid:88) t =1 | (cid:15) ( t ) | . (3)This will make it possible to compare estimator variances ofcodes with different lengths. It is also possible to comparecodes of the same length and different transmission powersby replacing L with the relative transmission power. We will only discuss estimates of the target autocorrelationfunction σ τ ( r ) with lags τ that are shorter than the length ofa transmission code (here r is the range in round-trip time,and it is discretized by the baud length). The lags are as-sumed to be non-zero multiples of the baud length of thetransmission code. Autocorrelation function estimation vari-ance is presented more rigorously in the companion paperby Lehtinen et al. (2007) submitted to the same special issue.The variance presented there also includes pulse-to-pulse andfractional lags, taking into account target post-integration aswell.Lag profile inversion is conducted using lagged productsfor the measured receiver voltage, defined for lag τ as m τ ( t ) ≡ m ( t ) m ( t + τ ) . (4)As more than one code is used to perform the measure-ment, we index the codes with c as (cid:15) c ( t ) . For convenience,we define a lagged product of the code as (cid:15) cτ ( t ) ≡ (cid:15) c ( t ) (cid:15) c ( t + τ ) . (5)With the help of these two definitions, the lagged productmeasurement can be stated as a convolution of the laggedproduct of the transmission with the target autocorrelationfunction m cτ ( t ) = ( (cid:15) cτ ∗ σ τ )( t ) + ξ τ ( t ) , (6)The equation also contains a noise term ξ τ ( t ) , which is rathercomplicated, as it also includes the unknown target σ τ ( r ) .This term is discussed in detail, e.g., by Huuskonen andLehtinen (1996). In the case of low SNR, which is typicalfor incoherent scatter measurements, the thermal noise dom-inates and ξ τ ( t ) can be approximated as a Gaussian whitenoise process defined as E ξ τ ( i ) ξ τ ( j ) = δ ( i − j ) s , (7) where s is the variance of the measurement noise.In this case, the normalized measurement “noise power”of lag τ can then be approximated in frequency domain as P τ ≈ (cid:90) π N c ( L − τ ) (cid:80) N c c =1 | ˆ (cid:15) cτ ( ω ) | dω, (8)where ˆ (cid:15) cτ ( ω ) = F MD { (cid:15) cτ ( t ) } is a zero padded discrete Fouriertransform of the transmission envelope with transform length M (cid:29) L . N c is the number of codes in the transmissiongroup and L is the number of bauds in a code. Each code ina group is assumed to be the same length.will notFor alternating codes of both Lehtinen (1986) and Sulzer(1993) type, P τ = 1 for all possible values of τ . For constantamplitude codes, this is the lower limit. On the other hand,if amplitude modulation is used, this is not the lower limitanymore, because in some cases more radar power can beused on certain lags, even though the average transmissionpower is the same.To give an idea of how phase codes perform in general,Fig. 1 shows the mean lag noise power for random codegroups at several different code and code group lengths. Itis evident that when the code group is short and the codelength is large, the average behaviour is not very good. Onthe other hand, when there is a sufficient number of codes ina group, the performance is fairly good even for randomlychosen code groups. Thus, we only need to worry about per-formance of short code groups that are sufficiently long. Nearly all practical transmission code groups result in sucha vast search space that there is no possibility for an exhaus-tive search. As we cannot yet analytically derive the mostoptimal codes, except in a few selected situations, we mustresort to numerical means. The problem of finding a trans-mission code with minimal estimation variance is an opti-mization problem and there exist a number of algorithms forapproaching this problem numerically.A typical approach is to define an optimization criteria f ( x ) with a parameter vector x . The optimization algorithmthen finds x min that minimizes f ( x ) . In the case of transmis-sion code groups, x will contain the phase φ ck and amplitude a ck parameters of each code in the code group x = { a , ..., a N c L , φ , ..., φ N c L } . (9)There are many different ways to define f ( x ) in the case oftransmission code groups, but a trivial one is a weighted sumof the normalized lag power P τ , with weights w τ selected insuch a way that they reflect the importance of that lag f ( x ) = (cid:88) τ w τ P τ . (10)ierinen et.al.: Transmission code optimization method for incoherent scatter radar 3In this paper, we set w τ = 1 for all lags. This gives eachlag an equal importance. This is a somewhat arbitrary choiceof weights, in reality they should be selected in a way thereflects the importance of the lag in the experiment. As our search method will also have to work with codes thathave a finite number of phases, we needed an algorithm thatcould also work with situations were an analytic or numer-ical derivative of f ( x ) cannot be defined. We developed analgorithm for this specific task, which belongs to the class of random local algorithms (Lewis and Papadimitriou, 1997).Another well known algorithm belonging to the same classis simulated annealing (Kirkpatrick et al., 1983), which hassome similarities to the optimization algorithm that we used.The random local optimization algorithm is fairly efficientat converging to a minima of f ( x ) and it can also to someextent jump out of local minima. In practice, it is faster torestart the optimization search with a different random initialparameter set, in order to efficiently locate the minima of f ( x ) that can then be compared the different optimizationruns.A simplified description of our code search algorithm thatsearches for local minima of f ( x ) is as follows:1. Randomize parameters in x .2. For a sufficient number of steps, randomize a new valuefor one of the elements of x and accept the change if f ( x ) is improved.3. Randomize all parameters x , accept the change if f ( x ) is improved.4. If sufficient convergence to a local minima of f ( x ) hasbeen achieved, save x and goto step 1. Otherwise goto step 2. The location of the minima can be furtherfine tuned using gradient-based methods, if a gradient isdefined for f ( x ) .In practice, our algorithm also included several tunablevariables that were used in determining the convergence of f ( x ) to a local minima. Also, the number of local minima tosearch for depends a lot on the number of parameters in theproblem. In many cases we are sure that the global minimawas not even found as the number of local minima was sovast. In order to demonstrate the usefulness of the optimizationmethod, we searched for code groups that use three differenttypes of modulation: binary phase modulation , polyphasemodulation , and the combination of amplitude and polyphase modulation, which we shall refer to as general modulation .In this example, we used a k = 1 for the constant amplitudemodulations and allowed amplitudes in the range a k ∈ [0 , for general modulation codes, while still constraining the to-tal transmission code power in both cases to be the same.The results are shown in Fig. 1. In this case the re-sults are shown in terms of mean lag noise power P =( L − − (cid:80) τ P τ . It is evident that significant improvementcan be achieved when the code group length is short. Forlonger code groups, the optimized groups do not differentthat much from random code groups. Also, one can see thatoptimized polyphase codes are somewhat better than binaryphase codes; ultimately general phase codes are better thanpolyphase codes – in some cases the mean lag noise power isactually better than unity. The reason for this is that ampli-tude modulation allows the use of more power for measuringsome lags, in addition to allowing more freedom in remov-ing range ambiguities. It should also be noted, that whenthe code or code group length is increased, the difference be-tween modulation methods also becomes less significant. We have introduced an optimization method suitable forsearching optimal transmission codes when performing lagprofile inversion. General radar tranmission coding, i.e.,modulation that allows amplitude and arbitrary phase shifts,is shown to perform better than plain binary phase modula-tion. Amplitude modulation is shown to be even more ef-fective than alternating codes, as the amplitude modulationallows the use of more radar power in a subset of the lags.For sake of simplicity, we have only dealt with estima-tion variances for lags that are non-zero multiples of thebaud length, with the additional condition that the lags areshorter than the transmission pulse length. It is fairly easy toextend this same methodology for more complex situationsthat, e.g., take into account target post-integration, fractionalor pulse-to-pulse lags. This is done by modifying the opti-mization criterion f ( x ) .In the cases that we investigated, the role of the modulationmethod is important when the code length is short. Whenusing longer codes or code groups, the modulation becomesless important. Also, the need for optimizing codes becomessmaller when the code group length is increased.Further investigation of the high SNR case would be ben-eficial and the derivation of variance in this case would beinteresting, albeit maybe not as relevant in the case of inco-herent scatter radar. Acknowledgements.
This work has been supported by the Academyof Finland (application number 213476, Finnish Programme forCentres of Excellence in Research 2006-2011). The EISCAT mea-surements were made with special programme time granted for Fin-land. EISCAT is an international assosiation supported by China
Vierinen et.al.: Transmission code optimization method for incoherent scatter radar C ode g r oup l eng t h l og10 ( C ode l eng t h ) M ean no i s e po w e r Random C ode g r oup l eng t h l og10 ( C ode l eng t h ) M ean no i s e po w e r Binary−phase C ode g r oup l eng t h l og10 ( C ode l eng t h ) M ean no i s e po w e r Polyphase C ode g r oup l eng t h l og10 ( C ode l eng t h ) M ean no i s e po w e r General
Fig. 1.
The mean lag noise power for random codes, optimized binary phase codes, optimized polyphase codes and optimized amplitude a k ∈ [0 , and arbitrary phase modulated (general modulation) codes. The largest improvements are achieved for short code groups. Also,it is clear that the combination of amplitude and phase modulation provides the best lag variance.(CRIRP), Finland (SA), Germany (DFG), Japan (STEL and NIPR),Norway (NFR), Sweden (VR) and United Kingdom (PPARC). References
Huuskonen, A. and Lehtinen, M. S.: The accuracy of incoherentscatter measurements: error estimates valid for high signal lev- els, Journal of Atmospheric and Terrestrial Physics, 58, 453–463,1996.Kirkpatrick, S., Gelatt, C. D., and Vecchi, M. P.: Optimization bySimulated Annealing, Science, 220, 671–680, 1983.Lehtinen, M.: Statistical theory of incoherent scatter measurements,EISCAT Tech. Note 86/45, 1986.Lehtinen, M. S., Virtanen, I. I., and Vierinen, J.: Fast Comparisonof IS radar code sequences for lag profile inversion, Submitted to ierinen et.al.: Transmission code optimization method for incoherent scatter radar 5ierinen et.al.: Transmission code optimization method for incoherent scatter radar 5