© Urban & Fischer Verlaghttp://www.urbanfischer.de/journals/aeue

Comparison of Amplitude-Matching and Complex MonopulseAlgorithms with Respect to SNR

Werner Kederer and Jürgen Detlefsen

Dedicated to Professor Peter Russer on the occasion of his 60th birthday

Abstract: Direction of arrival (DOA) determination is an import-ant subject for remote sensing. For radar applications monopulseprinciples are widely used. This article compares the amplitudematching algorithm [1] to the complex monopulse algorithm [2, 3]with respect to the influence of noise. While the overall assessmentof the angular estimation process is performed by simulation, themajor features can be confirmed by analytical analysis. The resultsshow that by using phase and magnitude information a better errorcharacteristic performance within the multibeam-illuminated areacan be achieved.

Keywords: Monopulse, DOA

1. Introduction

Due to the recent progress in computational power andthe availability of monolithic integrated microwave com-ponents which are produced cost-efficiently by the semi-conductor industry, radar systems step by step gain mar-ket shares wherever remote sensing is needed. Precisedetermination of the direction of arrival (DOA) for thebackscattered incident wave is necessary for current com-mercial automotive and other radar sensor systems. Themost typical example requiring an efficient DOA deter-mination can be found in ACC (adaptive cruise control)systems. In most of the systems, angular discriminationis done by evaluating the signals received from severalinterlacing antenna beams. Nearly all angular signal pro-cessing algorithms within those systems are based on orhave their roots in monopulse theory. It has been de-veloped during the first half of the last century espe-cially for two antenna beams considering military angulartracking tasks. A comprehensive description of classicalmonopulse techniques is given in [4] and [5]. This pa-per focuses on two derivates of monopulse techniques,Amplitude Matching (AM) [1, 3] and complex monopulse(CMP) [2, 3] algorithms, which are used in ACC-systemsand will be discussed and compared with respect to theirsensitivity to the influence of noise.

Received October 1, 2002. Revised December 31, 2002.

Lehrstuhl für Hochfrequenztechnik, Technische Universität Mün-chen, Arcisstr. 21, 80333 München, Germany.E-mail: detlefsen@ei.tum.de

Correspondence to J. Detlefsen.

2. Algorithms and simulations

2.1 General evaluation procedure

Since most of the systems today are multibeam configura-tions, all the algorithms have been extended to multibeamradar systems. For the specific implementation, the an-tenna diagramsxi(φ) of each beam must be known. Usingthe complex radiation characteristicsxi(φ), the two-wayantenna diagramsXi(φ) can be calculated for ann0-beamsystem, assuming that all antenna beams are simultan-eously used for transmission and illuminate the area ofinterest while angular specific reception is realized by theindividual beams:

Xi(φ) = xi

n0∑n=1

xn(φ) (1)

Figure 1 shows the two-way antenna diagram for a 3-beam-system in magnitude and phase. The magnitudes are nor-malized to the maximum of the central beam.

2.2 Amplitude matching

The amplitude matching algorithm is characterized bya complete neglect of the phase information. This can bejustified by the fact that the flat phase characteristic withinthe mainlobes of the beams carries only minor informa-tion. The CMP algorithm, however, takes care of all theinformation in order to determine DOA. The expected sig-nal amplitudesU1e . . .U3e in a 3-beam system, which arerelated to the two-way antenna characteristic, depend onthe transmitted powerS and can be described by

U1e(φ) = KX1(φ) ·√S (2a)

U2e(φ) = KX2(φ) ·√S (2b)

U3e(φ) = KX3(φ) ·√S (2c)

whereS: transmitted powerXn : two-way diagram (dimensionless)K : constant to account for propagation and scattering(

in√

Ω)

A suitable normalization [2] gives the expected re-sults independent of the propagation and scattering ef-

Int. J. Electron. Commun. (AEU) 57 (2003) No. 3, 168−172 1434-8411/03/57/03-168 $15.00/0

W. Kederer, J. Detlefsen: Comparison of Amplitude-Matching and Complex Monopulse Algorithms with Respect to SNR169

(a) Magnitude

(b) Phase

Fig. 1. Two-way antenna diagram for a three-beam system.

fects and leads to 3 reference functionsfi , which prin-cipally include the two-way responses. They are a func-tion of angular information only. Each object belonging toa specific angle of incidenceproduces values of the refer-ence functions between 0 and 1, which are generally notunique:

fi(φ) = |Uie(φ)||U1e(φ)|+ |U2e(φ)|+ |U3e(φ)|

= |Xi(φ)||X1(φ)|+ |X2(φ)|+ |X3(φ)|

= |xi(φ)||x1(φ)|+ |x2(φ)|+ |x3(φ)| (3)

The observed magnitudesUim of the received signalsstemming from a scattering center at angleφ0 will be cor-

rupted by noise:

U1m(φ0) = KX1(φ0) ·√S ± K√

N1 (4a)

U2m(φ0) = KX2(φ0) ·√S ± K√

N2 (4b)

U3m(φ0) = KX3(φ0) ·√S ± K√

N3 (4c)

If these values are used as an estimateui for the referencefunction fi according to (5), the result will be dependenton noise and possibly also on further systematic errors.

ui(φ0) = |Uim ||U1m |+ |U2m |+ |U3m | (5)

The differencedi between the observed quantityui andthe expected values of the pertaining reference functionaccording to (5) will exhibit azero respectively a mini-mum in the presence of noise. If these difference func-tionsdi are evaluated for all possible anglesφ, a minimumfor each of them is expected ifφ approachesφ0.

di(φ) = |ui(φ0)− fi(φ)| (6)

R(φ) =∑

i

di(φ) (7)

To take into account the possible ambiguities of the indi-vidual difference functions, an overall evaluation is doneby computing the sumR of the difference functions asa function ofφ according to (7) in the expectation that thecorrect angle will be characterized by the global minimumresponse ofR(φ) (8).

R(φ0) = minR(φ) (8)

These difference functionsdi are plotted againstφ for anincident angleφ0 of 3.9 in Fig. 2 (upper diagram). Theindividual difference functions given there show severalminima, while the global responseR in the lower diagramclearly indicates the actual angle.

The final accuracy of the estimate for the DOA givenby the effective deviation∆φi,eff directly depends on thenoise and its influence on each ofthe reference functions.It can be analytically characterized by:

∆φi,eff =∣∣∣∣∂φi

∂ fi

∣∣∣∣ ·∆ fi,eff(φ) (9)

The effective error in determining the value of the indi-vidual normalized value∆ fi,eff of the reference functioncaused by the noiseNj can be formulated as:

∆ fi,eff(φ) =

√√√√√ n0∑j=1

∣∣∣∣∣∂ fi(φ)

∂√

Nj·∆√

Nj

∣∣∣∣∣2

(10)

According to (4) it can be derived that:

∂ fi(φ)

∂√

Nj= ∂ fi(φ)

∂Uj(φ)· ∂Uj(φ)

∂√

Nj= ± ∂ fi(φ)

∂Uj(φ)· K

= ± ∂ fj(φ)

∂X j(φ)· ∂X j(φ)

∂Uj(φ)· K = ± ∂ fj (φ)

∂X j(φ)· 1√

S(11)

170 W. Kederer, J. Detlefsen: Comparison of Amplitude-Matching and Complex Monopulse Algorithms with Respect to SNR

(a) ( )d i φ

(b) ( )R φ

Fig. 2. Difference functionsdi(φ) and global response functionR(φ).

By combining (10) and (11) and assuming equal noisecontributions

∆√

Nj = √Nj = √

N (12)

we obtain

∆ fi,eff(φ) =√√√√ n0∑

j=1

∣∣∣∣ ∂ fi(φ)

∂X j(φ)

∣∣∣∣2

·√

N

S(13)

The effective angular error∆φi,eff due to the evaluation ofeach reference function can be formulated with (9) as

∆φi,eff(φ) =

√√√√√√√√n0∑

j=1

∣∣∣∣ ∂ fi(φ)

∂X j(φ)

∣∣∣∣2

∣∣∣∣∂ fi

∂φ

∣∣∣∣2 · 1√

S

N

(14)

The signal-to-noise ratio, which is used to characterisesimulation results is the one which is obtained by

SNR= maxn,φ

SNRn(φ) , (15a)

whereas the actual SNRn in each channel depends onangle and direction of incidence:

SNRn = 10 log

(Un

2(φ)

Un,noise2

)= 10 log

(KXn(φ)

√S

K√

Nn

)2

= 10 log

(Xn

2(φ) · S

Nn

)(15b)

The result of angular estimation is clearly influenced bythe channel specific reference and difference functions.According to (14) it can be expected that simulation re-sults confirm problematic angular zones which will existwhere any of the reference functions shows a horizontaltangent.

Figure 3b gives a typical simulation result (100 sam-ples) for the case of Gaussiannoise, which has been addedto the true valuesUie according to (4). The noise has beendefined according to (15) [3]. The root-mean-square (solidline) and the maximum (broken line) angular error con-firm that in the central angular region, where always anyreference functionfi has horizontal tangents, the error isenlarged. At the boundaries of the evaluation area the in-creasing errors of the algorithm are caused by less SNRand are influenced by the horizontal tangents off1 and f3.

2.3 Complex Monopulse Algorithm (CMP)

The CMP algorithm [2, 3] can be determined in a similarway. For this algorithm, different reference functions aredefined, which incorporate the phase information evaluat-ing the complex signal amplitudes. Further, the evaluationis restricted to the two neighboring beams of higher am-plitude, while the AM-algorithm takes into account simul-taneous information from all beams. This approach omitsthe contribution from noise of the channel without any tar-get in its illuminated area.

All functions involved are now complex. The CMP al-gorithm’s reference functions [1, 3] are given by

fi(φ) = xi+1(φ)− xi(φ)

xi+1(φ)+ xi(φ). (16)

The estimate for the reference functionui and the re-sponse functionR are given accordingly by

ui = U(i+1)m −Uim

U(i+1)m −Uim(17)

R(φ) = di(φ) , (18)

wheredi is again given by (6). The CMP algorithm pro-duces onlyn0 −1 reference functions. Only one of them

W. Kederer, J. Detlefsen: Comparison of Amplitude-Matching and Complex Monopulse Algorithms with Respect to SNR171

(a)

(b)

Fig. 3. Reference functions and simulated angular error for a three-beam configuration according to Fig. 1, SNR= 26 dB (according to(15a)).

is used for the final angular estimation. The selection offiwhich will be used, is made by a suitable amplitude cri-terion. Typically, the adjacent channels with the highestamplitudes will be chosen for angular evaluation [2, 3].

The actual angular position of the target is estimatedby the angleφ0 obtained from finding the minimum of theresponse functionR.

R(φ0) = minR(φ) (19)

The reference functions, which are plotted in Fig. 4, canbe plotted in the complex plane, and are shown for thethree-beam system characterized in Fig. 1.

The reference functionsfi again are analytical func-tions and can be evaluated with respect to error using(14). Since the term∂ fi/∂φ never will become zero (seeFig. 4) for the reference functions used in CMP evalu-ation, no error singularities will occur. Figure 5 shows

(a)

(b)

Fig. 4. Complex reference functions for the CMP algorithm fora three-beam system (see Fig. 1).

Fig. 5. RMS and maximum error characteristic of the CMP algo-rithm using the multibeam system of Fig. 1, SNR= 26 dB.

172 W. Kederer, J. Detlefsen: Comparison of Amplitude-Matching and Complex Monopulse Algorithms with Respect to SNR

the simulated error characteristic of a three beam sys-tem with CMP algorithm evaluation. As expected, thereare no significant singularity influences within the eval-uation region. The transition of the RMS error value tothe unusable area is quite smooth. The peaks in the max-imum error characteristic are caused by assigning thenormalized valuesui to the wrong part of the referencecurve. This effect can be minimized by a suitable choiceof the evaluation region for each reference curve. Es-pecially for multibeam systems this effect will not dis-turb the results in the main evaluation region, since theproblematic behavior of the reference curves near theirboundaries is not used. By a conformal transformationof the reference curve such boundary effects can alsobe minimized [3].

3. Conclusion

A comparison between AM and CMP angular determin-ation algorithms has been performed by simulation show-ing better performance in terms of accuracy and the widthof the angular field for the CMP algorithm.

The major role of the reference functions is identi-fied and explained by an analytical analysis. The resultshows that it is of great benefit in spite of the necessaryadditional hardware efforts to use phase and amplitude in-formation for estimating the directions of arrival. This isalso true for cases in which,at a first glance, phase infor-mation does not seem to carry much information. This is,e.g., typical for any multibeam lens system using feeds inthe focal plane.

The AM algorithm suffers from the singularities whichare unavoidably present in the real evaluation domain.They can be avoided by carrying out the angular esti-mation through comparing information in the complexplane.

References

[1] Wagner, K. P.: Winkelauflösende Radarverfahren für Kraft-fahrzeuganwendungen. Dissertation. München: TechnischeUniversität München, 1997.

[2] Kederer, W.; Detlefsen, J.: Direction of Arrival Determination(DOA) based on Monopulse Concepts. Asia Pacific MicrowaveConference Proceedings, Sydney, 2000. 120–123.

[3] Kederer, W.: Mehrkeulen-Monopuls-Verfahren für Radarsen-soren. Dissertation. München: Technische UniversitätMünchen, 2002.

[4] Rhodes, D. R.: Introduction to Monopulse. McGraw-Hill, NewYork 1959.

[5] Sherman, S. M.: Monopulse Principles and Techniques. Ded-ham: Artech House, 1984.

Werner Kederer was born 1968 inVilseck, Germany. After his studies inelectronics, he joined the Institute for highfrequency engineering at Technische Uni-versität München. Where he reached hisPhD. His main interests were directed tothe needs of commercial adaptive cruisecontrol radar systems, where he developedeffective algorithms for DOA determin-ation based on monpulse concepts.

Jürgen Detlefsen graduated in 1967from the Technische Universität München,where he also received his Ph.D. in 1971.Since 1980 he has been a professor at thesame university, since 1988 responsiblefor the field of “Electromagnetic Fieldsand Circuits”. His main research activitiescomprise microwave and millimeter-waveradar-sensor systems and also to the spe-cific antennas, microwave componentsand the signal processing necessary forthese applications.