Reducing the computational requirements of time-reversal device optimizations

INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS

Int. J. Numer. Model. 2010; 23:458–469Published online 8 December 2009 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/jnm.746

Reducing the computational requirements of time-reversaldevice optimizations

Ian Scott�,y, Ana Vukovic and Phillip Sewell

George Green Institute for Electromagnetics Research, University Park, University of Nottingham, Nottingham,NG7 2RD, U.K.

SUMMARY

Time-reversal simulations using conventional numerical algorithms provide the basis for a simplecomponent optimization procedure. However, the computational requirements of the approach canbecome excessive, requiring the recording of the complete field time histories on a surface surrounding theproblem space. In this work temporal, spatial and modal filtering methods are employed to significantlyreduce the computational resources demanded by the time-reversal process. The design of a number ofWR90 (X-band) waveguide components is used to quantify the balance between the computationalefficiency gains and deterioration in the final design quality. Copyright r 2009 John Wiley & Sons, Ltd.

Received 13 July 2009; Revised 9 October 2009; Accepted 24 October 2009

KEY WORDS: design optimization; time-reversal; transmission-line modeling method

1. INTRODUCTION

Automated optimization techniques are often sought for the design of electromagnetic devicesdue to the increasing complexity and demanding performance requirements of modern systems[1]. Time-reversal simulations are a physically based approach to this problem, alternatingbetween forward-time simulations of trial designs and reverse-time simulations which re-inject amore desirable response of the system as an excitation in order to identify the componentgeometry that would produce it. This scheme can employ any standard time domain numericalalgorithm for the forward and reverse simulations and has been successfully demonstrated withboth the transmission-line modeling (TLM) and the finite difference time domain (FDTD)methods [2, 3]. In the case of lossless materials Maxwell’s equations are time reversible and so

*Correspondence to: Ian Scott, George Green Institute for Electromagnetics Research, University Park, University ofNottingham, Nottingham, NG7 2RD, U.K.yE-mail: [email protected]

Contract/grant sponsor: U.K. Engineering and Physical Sciences Research Council (EPSRC)

Copyright r 2009 John Wiley & Sons, Ltd.

the time-reversed simulations actually employ the same time stepping algorithm as the forwardsimulations.

Although simple to implement, the disadvantage of the approach is its computationalintensity. Besides the inherent requirements of the simulations, both in memory and runtime,it is necessary to record the full-time history of the fields scattered to a surface bounding theproblem space in the forward simulations as a perturbed version of these are then used toexcite the reverse simulations. The bounding surface on which this information is collected isoften referred to as a time-reversal-mirror (TRM) [4]. The objective of this work is to reducethe quantity of the data that is recorded by the TRMs without compromising the quality ofthe designs produced and this is achieved by a variety of filtering techniques.

We initially propose and investigate the performance of the time-reversal design techniquewhen more than one forward/reverse iteration is performed, as the concept was originallypresented employing just a single forward and reverse iteration [2].

2. TLM TIME-REVERSAL OPTIMIZATION

In this work, the TLM numerical time stepping method [5] has been used to perform thesimulations although the conclusions are equally applicable to other algorithms such as FDTD.TLM is an explicit time-domain differential modeling method based on the equivalence betweenthe evolution of voltage impulses on a mesh of interconnected transmission-lines andelectromagnetic waves. The method is fully described in [5, 6] and its use for time-reversaldesign was originally presented in [2].

In particular, we focus the discussion on the class of components comprising metal scattererssuch as waveguide septa. A target performance is specified in terms of the field scattered to theTRMs when the device is excited in a prescribed manner. Usually the TRMs coincide with theinput and output ports of, for example, a waveguide structure.

Given the full-time history of the fields incident on the TRMs from the forward simulation,that part radiated by the sources induced on the surface of the scatterer is identified bysubtracting the field directly radiated by the primary sources. This radiated field is then adjustedtowards the desired distribution and is used to define equivalent sources on the TRMs thatsubsequently excite the reverse simulation. The boundary of the correspondingly perturbedmetal scatterer is then identified by the nulls in a map of the maximum observed Poynting vectormagnitude during the reverse simulation [2, 7].

As an arbitrary scattered field distribution may not be physically realistic, it is necessary thatthe changes to the TRM equivalent sources are perturbational. Moreover, even smallperturbations will not in general lead to a physically distinct scatterer, rather its surface willbe blurred and thus can be extracted in maximal likelihood manner.

For example, if the performance of a waveguide based component is specified in terms of thefundamental mode return loss over a finite operating bandwidth Df then the perturbations arecalculated as follows: the performance error after the kth forward simulation is evaluated in thefrequency domain

Gðf Þ ¼SD11ðf Þ � Sk

11ðf Þ f 2 Df

0 f =2Df

(ð1Þ

TIME-REVERSAL DEVICE OPTIMIZATIONS 459

Copyright r 2009 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2010; 23:458–469

DOI: 10.1002/jnm

where the superscript D denotes the desired response. Let

k

Hðx; tÞ

Eðx; tÞ

!;

H0ðx; tÞ

E0ðx; tÞ

!and

hðx; tÞ

eðx; tÞ

!ð2Þ

denote the TRM transverse field values recorded during the kth forward simulation, during asimulation with the scatterer removed and of the fundamental mode profile, respectively. TheTRM equivalent sources for the kth reverse simulation are defined by

k

~Jðx; tÞ

~Mðx; tÞ

!1

n̂�H0ðx; tÞ

n̂� E0ðx; tÞ

!¼ ð1� aÞk

n̂�Hðx; tÞ

n̂� Eðx; tÞ

!

� a k

n̂�Hðx; tÞ

n̂� Eðx; tÞ

!� jgðtÞj �

n̂� hðx; tÞ

n̂� eðx; tÞ

!" #ð3Þ

where ~J and ~M are equivalent electric and magnetic sources. g(t) is the inverse Fourier transform ofG(f), and a is a damping factor that controls the convergence of the optimization algorithm: If a5 0,no perturbation is made and the reverse simulation will reveal the original scatterer geometry [2]. Ingeneral the damping value begins with a51, and the iterative time-reversal simulation is performed.The damping remains constant throughout a particular optimization run and a suitable valueyielding stable and accurate convergence was determined by examining the behavior of the routinewith different values. In general, a damping factor in the range 0:05oao0:2 has been found toprevent the algorithm becoming unstable for all the examples in this paper.

3. TEMPORAL, SPATIAL AND MODAL FILTERING

During every forward simulation of the waveguide example a total of NY �Nt sampled values arerecorded on each of two TRMs, one at the input and one at the output of the device, and this rapidlyconsumes memory. NY is the number of spatial samples in the waveguide cross-section and Nt is thenumber of forward time steps. If the data is not recorded in its entirety then a reverse simulation willnot exactly reconstruct the scatterer. However as discussed above, the recorded data is actuallyperturbed towards the desired scattered field before the reverse simulation and that this causes a lossof resolution of the scattering object anyway. Therefore, the key issue is whether the quantity of databeing recorded can be significantly reduced in order to save memory without causing further loss ofresolution in practice. Here we investigate, temporal, spatial, and modal filtering.

In the cases of temporal and spatial filtering only every mth value is stored in time and space,respectively, during the forward simulation and simple linear interpolation is used to generatethe missing values needed to excite the reverse simulations.

In many cases, a more appropriate representation of the scattered fields for the purposes ofcomplexity reduction is a local modal decomposition. This may be in terms of the modes of anexplicit waveguide structure that exists or for example, in terms of the spherical harmonicsaround a relatively small object. The key feature being exploited is the appearance of a cut-offphenomenon that effectively distinguishes the near and far fields of the scattered field.Therefore, with modal filtering only a simple time varying scalar amplitude for each physicallysignificant mode present in the far field need be stored rather than a full time varying vector ofspatial samples across the waveguide.

I. SCOTT, A. VUKOVIC AND P. SEWELL460


DOI: 10.1002/jnm

4. CASE STUDIES

In the following illustrations the extent to which the TRM data can be filtered is quantified bycomparing the success of the overall time-reversal synthesis algorithm in minimizing thedifference between the achieved and desired responses.

4.1. Microwave band-pass filter

A simple microwave band pass filter formed by two inductive septa separated by a quarter-wavelength section of 2D parallel metal plate waveguide at the center of the pass band is shownin the inset of Figure 1. The waveguide has widthW5 22.86mm and hence a fundamental modecut-off of 6.56GHz. The degrees of freedom for the design are the widths of the septa, WS1 andWS2. The TLM simulations are performed using a uniform mesh of size D with NX and NY cellsalong the length and width of the waveguide, respectively. In order to examine convergence, wespecify a target performance that ought to be almost exactly realizable. This is determined froma simple transmission-line circuit model of the filter with the two septa replaced by ideal lumpedinductors. The only physical reason that this target performance cannot be exactly realized inthe waveguide is due to the small frequency dependences of the septa inductances and the almostnegligible higher mode interactions between them.

4.2. Waveguide bend

Metal tuning posts are often placed within 901 waveguide bends to improve the return lossas shown in the inset of Figure 3. Using the same parallel plate waveguide as above, the degreeof freedom for the design is the displacement D of the post from the inside corner of the bend. Thetarget performance is to achieve zero return loss and unit transmission at a given frequency.

2 4 6 8 10 12

0.15

0.20

0.25

0.30

0.35

0.40

TR Iterations

FoM

short circuit

short circuit

TR

M

TR

M

NYΔ=

W

WS2

NX = L

WS1

/4

Δ

Figure 1. Convergence of the TR design on case study of microwave band pass filter. Optimization iscompleted in 8 TR iterations.



DOI: 10.1002/jnm

5. RESULTS

5.1. Time-reversal design convergence

This section will initially demonstrate the quality of design optimization achieved with theunfiltered time-reversal approach and then proceed to show the effect of introducing spatial,temporal and modal filtering.

For the band pass filter case study, the target specification is defined as the response of thelumped equivalent circuit model over the band 10–14GHz with inductances L1 5 2.7 nH andL2 5 2.85 nH; values which ought to be achievable with the waveguide septa [7]. Theoptimization starts with initial septa widths of WS1 ¼WS2 ¼ 3D ¼ 0:89 mm which, as can beseen in Figure 2, produce a poor approximation to the target performance. The TLMsimulations employ D ¼ 0:297 mm, Dt ¼ 0:7 ps, NY ¼ 77, NX ¼ 67 and Nt ¼ 16384 time steps.The fundamental waveguide mode excites the structure at one end with standard TLM matchedboundaries terminating the problem space [5]. The scalar figure of merit

FoM ¼1

2Df

Z Df

0

jS11 � SD11j1jS21 � SD

21j� �

df ð4Þ

(a) (b)

(c) (d)

11 12 13 140

0.2

0.4

0.6

0.8

1

|S|

Frequency (GHz)

|S21|

|S11|

11 12 13 140

0.2

0.4

0.6

0.8

1

|S|

Frequency (GHz)

|S21|

|S11|

11 12 13 140

0.2

0.4

0.6

0.8

1

|S|

Frequency (GHz)

|S21|

|S11|

11 12 13 140

0.2

0.4

0.6

0.8

1

|S|

Frequency (GHz)

|S21|

|S11|

Figure 2. Comparison of the scattering parameters during time-reversal optimization of the waveguide bandpass filter, heavy are optimized, dotted are the target parameters. (a) Initial, (b) step 2, (c) step 6, (d) optimized.



DOI: 10.1002/jnm

is defined to assess convergence where the superscript D denotes the desired response. Figure 1shows the convergence of the FoM with respect to the number of time-reversal iterations and amonotonic improvement is observed for the first 8 iterations after which no furtherimprovement is seen. The flat sections that occur during convergence are due to the roundingof the septa widths to the nearest spatial sample. The optimized septum widths are WS1 ¼2:08 mm andWS2 ¼ 2:67 mm and an exhaustive sweep of the degrees of freedom reveals that theoptimized design found is indeed the best possible.

Figure 2(a–d) compares the target and achieved performances of the band pass filter duringthe design optimization process. It is seen that the first few time-reversal optimization stepsproduce the most improvement in performance that is practically valuable for large problemsthat may preclude more than a few iterations. It is also seen that the convergence is generallyuniform with frequency, although clearly a weighting factor could be introduced to influencethis if appropriate. The remaining difference between the target and achieved performances isattributable to the weak frequency dependence of the septa inductances not taken account in thelumped circuit model of the filter.

As discussed above and in reference [2] the boundary of the perturbed septum is defined bythe locus of nulls of the maximum Poynting vector observed during the reverse simulation.However as true nulls are never actually generated a threshold value must be used for thispurpose and it has been found that the most appropriate choice depends upon the dampingfactor, a used. The precise choice of threshold value is not discussed further here although it isnoted that more sophisticated techniques for edge extraction from 2D data can also be used forthis purpose, e.g. the Lagrange operator often used in digital photography [8].

Figure 3 shows the convergence of the time-reversal design process of the placement ofa bend tuning post to optimize the return loss over the operating range Df ¼ 6:7� 8:8 GHz.

5 10 15

-25

-20

-15

-10

-5

TR Iterations

Ret

urn

loss

(dB

)

N ΔY = W

N ΔX = L

D

NΔ

X=

L

NY

= W 45˚

Δ

Figure 3. Convergence of the TR design of case study of microwave bend. Optimization is completed in13 TR iterations. The initial post displacement was D5 15.5mm, and optimized was D5 20.43mm.The TLM simulation used D5 0.18mm, Dt 5 0.431 ps, NY 5 125, NX 5 220, W5 22.86mm, L5 40.23mm,

and NF 5 16384.



DOI: 10.1002/jnm

A fine spatial sampling was used in the TLM simulations to allow more accurate placement ofthe post and the post size was chosen as 5:5� 5:5 mm. This second example exhibits the samegeneral features as the previous one, identifying an optimal design after 13 iterations whichpossesses a minimum return loss of �28.07 dB. In Figure 4 selected examples of the return lossduring the optimization process are shown.

5.2. Application of temporal and spatial filtering

The two examples considered briefly illustrate the capability of time-reversal optimization. Thissection will focus on the central theme of this work, reducing the memory requirements byfiltering the data collected by the TRMs.

Figure 5 presents the performance of the approach when temporal and spatial filtering areused. For each particular type and extent of filtering, the number of iterations required toachieve convergence and the final design quality (4) are given. The extent of filtering is expressedas the percentage of the full data set recorded.

It is seen that temporal filtering is not acceptable. Unless almost all the data is recorded theFoM is notably poor. However, spatial filtering is more successful and is seen to only slightlydeteriorate (increase) the FoM when at least 33% of the TRM is stored. Furthermore, when50% of the data is stored this is achieved without deteriorating the convergence rate above. Thisis seen in more detail in Figure 6 when 80% spatial linear interpolation is used.

Figure 7 shows the case of both the temporal and spatial filtering when applied to thedesign optimization of the tuning post in the waveguide bend. Interpolation of 50%, ispossible, below this the method is largely erroneous. A direct correlation between theoptimized solution and design iterations is seen, where in general, fewer design steps results ina poor design.

7.0 7.5 8.0 8.5-30

-25

-20

-15

-10

-5

0

Ret

urn

loss

(dB

)

Frequency (GHz)

Initial

Step 7

Step 9

Optimized(Step 13)

Figure 4. Return loss during time-reversal optimization of the tuning post placement in 901 bend. Initial,step 7, step 9 and optimized are shown.



DOI: 10.1002/jnm

(a) 20 40 60 80 100

0

2

46

8

1012

1416

Iter

atio

ns N

TR20 40 60 80 100

0

0.1

0.2

0.3

0.4

FoM

% Temporal data stored

FoM

NTR

(b)

20 40 60 80 1000

2

4

68

10

12

1416

Iter

atio

ns N

TR

20 40 60 80 1000

0.1

0.2

0.3

0.4

FoM

% Spatial data stored

FoM

NTR

Figure 5. Total time-reversal iterations (NTR) and the convergence error of the final solution as a functionof percentage of TRM stored, with missing values interpolated using (a) temporal and (b) spatial

interpolation. Microwave band pass filter example.

2 4 6 8 10 12

0.15

0.20

0.25

0.30

0.35

0.40

TR Iterations

FoM

Classical (Full T RM)

80% Spatial TRM Stored

short circuit

short circuit

80%

80%WS2WS1

/4

Figure 6. Convergence of the TR design of the case study of microwave band pass filter using 80% linearspatial interpolation in comparison to full TRM stored.



DOI: 10.1002/jnm

This technique is simple and reduces the memory requirements of the time-reversal designprocess significantly as the interpolation increases. However, the method is seen to result inincreased error when less than 75% of the TRM is stored and erratic convergence when morethan 75% is stored, and is hence only suited to design optimization when the modal content ofthe device is unknown.

5.3. Application of modal filtering

As the set of the local waveguide modes for the example case studies of this paper is known, thiscan also be exploited to reduce data storage. The data is projected onto these modes and onlythose whose amplitude is expected to be non-trivial at the TRMs are stored.

In Figure 8 the effect of such modal filtering on the design performance is investigated. Thenumber of design iterations required for convergence and the FoM of the optimized solution forthe band pass filter example is shown. It is seen the number of time-reversal iterations untilconvergence is reached changes with the number of stored modes, and stabilized when morethan 80% of the modes were stored. Figure 8 also shows that at 80% stored modes the FoM ofthe optimization has reduced to 0.1 implying that the optimal solution has been found. At thispoint a reduction in storage has been achieved which is a significant reduction in the resourcerequirement.

In Figure 9 the modal filtering is applied to the second case study of the 901 bend. In this caseit is seen that 70% of the modes yields good performance and a reduction in the storage of 30%,respectively.

(a) 20 40 60 80 100

2

4

6

8

10

12

14

Iter

atio

ns N

TR

20 40 60 80 100-35

-30

-25

-20

-15

-10

-5

0

Ret

urn

loss

(dB

)

% Temporal data stored

Return loss

NTR

(b) 20 40 60 80 100

2

4

6

8

10

12

14

Iter

atio

ns N

TR

20 40 60 80 100-35

-30

-25

-20

-15

-10

-5

0

Ret

urn

loss

(dB

)

% Spatial data stored

Return loss

NTR

Figure 7. Total time-reversal iterations (NTR) and return loss of the final solution as a function ofpercentage of TRM stored (a) temporal and (b) spatial samples. 901 waveguide bend example.



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6. CONCLUSION

Time-reversal as a technique for microwave component design has notable computationalrequirements. In this paper a number of approaches have been introduced to ameliorate thisproblem without compromising the performance of the iterative optimization process.

1000

5

10

15

20

Iter

atio

ns N

TR

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

FoM

NTR

FoM

% modal amplitudes stored20 40 60 80 100

Figure 8. Total time-reversal iterations (NTR) w.r.t. the percentage of modes stored at the TRM, andconvergence error in the final solution. Figure shows only 80% of the excited modes require storage incomparison to all spatial values using classical time-reversal technique. Microwave band pass filter example.

20 40 60 80 100

10

12

14

Iter

atio

ns N

TR

20 40 60 80 100-35

-30

-25

-20

-15

-10

-5

0

Ret

urn

loss

(dB

)

NTR

Return loss

% modal amplitudes stored

Figure 9. Total time-reversal iterations (NTR) w.r.t. the percentage of modes stored at the TRM, andreturn loss in the final solution. Figure shows only 70% of the excited modes require storage in comparison

to all spatial values using classical time-reversal technique. 901 waveguide bend example.



DOI: 10.1002/jnm

The implementation of the iterative time-reversal has been discussed and for the first time theuse of repeated forward and backward propagation steps has been demonstrated to yieldsmooth convergence.

Linear interpolation, both in the spatial and temporal domains has been used to reduce thememory required, with the former offering valuable gains. However, temporal filtering has provedless successful. In scenarios amenable to a local modal decomposition it was demonstrated thatthis provided an alternative filtering technique providing up to 20–30% reduction in the memoryrequirements without compromising the convergence rate or the final design performance.

In summary, time-reversal device optimization with large problems will become intractable dueto the need to store the full-time history of the forward phase of the algorithm. With the techniquesproposed in this paper, the computational requirements of the algorithm are significantly reduced,making the technique more suitable for complex microwave component design.

ACKNOWLEDGEMENTS

This work was supported by the U.K. Engineering and Physical Sciences Research Council (EPSRC).

REFERENCES

1. Liu D, Vasudevan S, Krolik J, Bal G, Carin L. Electromagnetic time-reversal source localization in changing media:Experiment and analysis. IEEE Transactions on Antennas and Propagation 2007; 55:344–354.

2. Forest M, Hoefer WJR. A novel synthesis technique for conducting scatterers using TLM time reversal. IEEETransactions on Microwave Theory and Techniques 1995; 43:1371–1378.

3. Lerosey G, de Rosny J, Tourin A, Derode A, Montaldo G, Fink M. Time reversal of electromagnetic waves. PhysicalReview Letters 2004; 92:193904-1–193904-3.

4. Cassereau D, Fink M. Time reversal of ultrasonic fields—part III: Theory of the closed time-reversed cavity. IEEETransactions on Ultrasonics, Ferroelectrics, and Frequency Control 1992; 39:579–592.

5. Christopoulos C. The Transmission-line Modeling Method: TLM. John Wiley & Sons/IEEE Publications: New York,NY, 1995.

6. Hoefer WJR. The transmission-line matrix method—theory and applications. IEEE Transactions on MicrowaveTheory and Techniques 1985; 33:882–893.

7. Forest M, Hoefer WJR. TLM synthesis of microwave structures using time reversal. IEEE Microwave Theory andTechniques Symposium Digest, Albuquerque, NM, U.S.A., 1–5 June 1992; 779–782.

8. Sonka M, Hlavac V, Boyle R. Image Processing, Analysis and Machine Vision (2nd edn). Thomson Learning:Toronto, Ontario, Canada, 1998.

AUTHORS’ BIOGRAPHIES

Ian Scott was born in Norwich, U.K., in 1985. He received BSc in Computer Sciencewith Electronics with starred first class honours from the University of East Anglia(UEA), Norwich, U.K., in 2006 and he is currently working towards the PhD inElectrical and Electronic Engineering at the University of Nottingham, U.K. He iscurrently with the George Green Institute for Electromagnetics Research, Universityof Nottingham. His research interests are in the area of inverse numerical modeling,with applications to electromagnetics.



DOI: 10.1002/jnm

Ana Vukovic (M’97) was born in Nis, Yugoslavia, in 1968. She received the Diplomain Engineering in electronics and telecommunication from the University of Nis, Nis,Yugoslavia, in 1992 and the PhD degree from the University of Nottingham,Nottingham, U.K., in 2000. From 1999 to 2001 she was working as a ResearchAssociate at the University of Nottingham. She joined the School of Electrical andElectronic Engineering at the University of Nottingham as a Lecturer in 2001. In2008 she was promoted to Associate Professor at the same University. Her researchinterests are in the area of electromagnetics, with particular emphasis on applicationsin optoelectronics, microwaves and electromagnetic compatibility.

Phillip Sewell (M’89, SM’04) was born in London, U.K., in 1965. He received theBSc Degree in electrical and electronic engineering with first class honours and thePhD degree from the University of Bath, Bath, U.K., in 1988 and 1991, respectively.From 1991 to 1993, he was a Post-Doctoral Fellow at the University of Ancona,Ancona, Italy. In 1993, he was appointed as a Lecturer in the School of Electricaland Electronic Engineering, University of Nottingham, Nottingham, U.K. In 2001and 2005 he was promoted to Reader and Professor of Electromagnetics at the sameUniversity. His research interests involve analytical and numerical modeling ofelectromagnetic problems, with application to optoelectronics, microwaves andelectrical machines.



DOI: 10.1002/jnm

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Reducing the computational requirements of time-reversal device optimizations