volume 14 g a special edition/ n ik specijalno izdanje b&h … · 2020. 12. 24. · 6 bh electrical...

1Volume 14 January/December 2020

B&H

Elec

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alE n

g in e

e ri n

g Volume 14 Special edition/Specijalno izdanje2020

ISSN 2566-316X

Bosa

nsko

herc

egov

ačka

e le k

trot

e hn i

ka

Blok Blok 7

www.epbih.ba

- Zamjenski kogeneracijski Blok na lignit- Integrisana snaga

270 MW toplotne energije- Godišnja proizvodnja 2.756 GWh- Blok 7 dio Energetske strategije BiH do 2035. godine- Izgradnja po najsavremenijim (BAT) tehnologijama- Ispunjavanje ekoloških standarda EU

Emisije SO2 < 150 mg/Nm

3

NOx < 200 mg/Nm3

Prašina < 10 mg/Nm3

CO2 924 g/kWh

- Sigurnost snadbijevanja i energetska neovisnost

- Kontinuitet proizvodnje uglja i prestruktuiranje RU „Kreka”


PUBLISHER/IZDAVAČCommittee for Bosnia and Herzegovina

International Council on Large Electric Systems - CIGRÉBH K CIGRÉ

Mula Mustafe Bašeskije 7/4, 71000 SarajevoTel: +387 (0)33 227 036Fax: +387 (0)33 227 037

E-mail: [email protected] site: http://www.bhkcigre.ba

ISSN 2566-316X

Editor in Chief/Glavni urednikTatjana Konjić, B&H

Deputy Editor in Chief/Zamjenik glavnog urednikaSabina Dacić-Lepara, B&H

Gest editors/Gostujući uredniciIsmar Volić, USA

Samir Avdaković, B&H

Deputy Editors/Urednički odborSamir Avdaković, B&HMirza Kušljugić, B&H

Adnan Mujezinović, B&HMustafa Musić, B&H

Amir Tokić, B&H

Editorial Board/Uredničko vijećeDrago Bago, B&H

Mladen Banjanin, B&HElvisa Bećirović, B&HAdnan Bosović, B&HVesna Borozan, MKD

Nada Cincar, B&HIzet Džananović, B&H

Šeila Gruhonjić Ferhatbegović, B&HOmer Hadžić, B&H

Jasna Hivziefendić, B&HMarko Ikić, B&H

Mensur Kasumović, B&HVladimir Katić, SRBAnes Kazagić, B&H

Lidija Korunović, SRBSlavko Krajcar, CROEdin Lapandić, B&H

Ajla Merzić, B&HJovica Milanović, GBViktor Milardić, CROKruno Miličević, CRO

Vladimiro Miranda, PORDragan Mlakić, B&H

Aljo Mujčić, B&HSaša Mujović, MNE

Srete Nikolovski, CROSamir Omanović, B&H

Predrag Osmokrović, SRBSenad Osmović, B&H

Jože Pihler, SLOMirza Sarajlić, SLO

Koviljka Stanković, SRBNermin Suljanović, B&H

Sanela Suljović – Fazlić, B&HMirza Šarić, B&H

Gorazd Štumberger, SLOEmir Turajlić, B&HIrfan Turković, B&HTarik Uzunović, B&H

Language Editor/Lektor jezikaDušica Ikić-Cook, B&H

Technical Editor/Tehnički urednik Dejan Marjanović, B&H

Account: Raiffeisen Bank B&H16100000021500-16

CONTENTS/SADRŽAJ

Original scientific paper/Izvorni naučni radMaja Muftić Dedović, Samir Avdaković, Adnan Mujezinović, Nedis DautbašićCOMPARIOSON OF DIFFERENT METHODS FOR IDENTIFICATION OF DOMINANT OSCILLATION MODE

KOMPARACIJA RAZLIČITIH METODA ZAIDENTIFIKACIJU DOMINANONTNIH OSCILATORNIH MODOVA 43

Review scientific paper/Pregledni naučni radAsim VodenčarevićOVERVIEW OF FACTORIZATION METHODS IN KALMAN FILTERING

PREGLED METODA FAKTORIZACIJE U KALMANOVOM FILTRIRANJU 51

ABOUT THE JOURNAL 61O ČASOPISU 61

TYPE OF THE PAPER 62VRSTA RADA 62

DECLARATION OF AUTHORSHIP 63IZJAVA O AUTORSTVU 63

PAPER WRITING GUIDELINES 64UPUTE ZA PISANJE RADA 67

INITIAL JOURNAL CHECK LISTFOR AUTHORS 70INICIJALNI KONTROLNI SPISAK ZA AUTORE 71

CODE OF ETHICS 72ETIČKI KODEKS 75

Original scientific paper/Izvorni naučni rad

Zlatan AkšamijaNUMERICAL STUDY OF THERMAL

DISSIPATION PROCESSES IN SILICON

NUMERIČKA STUDIJA PROCESA DISIPACIJE TOPLINE U SILICIJUMU 5

Review scientific paper/Pregledni naučni rad

Ismar VolićTOPOLOGICAL METHODS IN

SIGNAL PROCESSING

TOPOLOšKE METODE U PROCESIRANJU SIGNALA 14


Mateo Bašić, Matija Bubalo, Dinko Vukadinović, Ivan Grgić

OPTIMAL POWER FLOW CONTROL IN A STAND-ALONE PV SYSTEM WITH A

BATTERY-ASSISTED QUASI-Z-SOURCE INVERTER

OPTIMALNO UPRAVLJANJE TOKOVIMA SNAGA U OTOČNOM PV SUSTAVU S

IZMJENJIVAČEM KVAZI Z-TIPA I INTEGRIRANIM BATERIJAMA 26


Haris Ahmetović , Adnan Bosović , Ajla Merzić, Mustafa Musić

ANALYSIS OF MICROGRID OPERATION IN STAND-ALONE MODE - SUSTAINABLE SMART

TOURISTIC VILLAGE CASE STUDY

ANALIZA RADA MIKROMOREŽE U SAMOSTALNOM REŽIMU RADA-

STUDIJA ZA SAMOODRŽIVO PAMETNO TURISTIČKO NASELJE 35

Blok Blok 7

www.epbih.ba

- Zamjenski kogeneracijski Blok na lignit- Integrisana snaga

270 MW toplotne energije- Godišnja proizvodnja 2.756 GWh- Blok 7 dio Energetske strategije BiH do 2035. godine- Izgradnja po najsavremenijim (BAT) tehnologijama- Ispunjavanje ekoloških standarda EU

Emisije SO2 < 150 mg/Nm

3

NOx < 200 mg/Nm3

Prašina < 10 mg/Nm3

CO2 924 g/kWh

- Sigurnost snadbijevanja i energetska neovisnost

- Kontinuitet proizvodnje uglja i prestruktuiranje RU „Kreka”

4 B&H Electrical Engineering Bosanskohercegovačka elektrotehnika

Preface/PredgovorB&H Electrical Engineering - Special Issue on ‘Computational, Numerical, and Mathematical Methods in Electrical Engineering’ is a product of years of successful cooperation between BHK CIGRÉ and the Bosnian-Herzegovinian American Academy of Arts and Sciences (BHAAAS).

The objective of this special issue is to bring together researchers from academia and industry with the goal to dis-seminate state-of-the-art research and development results on the applications of computational, numerical, and mathematical methods in engineering.

For this edition six papers that cover a wide range of scientific areas have been selected.

We believe that the continuation of cooperation between BHK CIGRÉ and BHAAAS will be even more pronounced in the future and will involve more researchers and authors who are originally from Bosnia and Herzegovina and who have built successful careers around the world.

Editor in Chief Professor Tatjana Konjić, Ph.D. in EEFaculty of Electrical Engineering, University of Tuzla and University of Sarajevo, Bosnia and Herzegovina

Deputy Editor in Chief

Sabina Dacić-Lepara, M.Sc. in EE JP Elektroprivreda d.d. Sarajevo, Bosnia and Herzegovina

Guest Editors

Professor Ismar Volić, Ph.D. in Mathematics Department of Mathematics - Wellesley College, Boston, United State of America

Associate Professor Samir Avdaković, Ph.D. in EE Faculty of Electrical Engineering, University of Sarajevo, Bosnia and Herzegovina


Abstract: Heat dissipation in nanoelectronics has become a major bottleneck to further scaling in next-generation integrated circuits. In order to address this problem and develop more energy-efficient nanoelectronic transistor, sensor, and storage devices, we must under-stand thermal processes at the atomic scale, which requires numerical simulation of the interaction between electrons and heat, carried by quantized lattice vibrations called phonons. Here we examine in detail the phonon emission and absorption spectra in silicon at several elevated values for the electron temperature. The effect of electric field on the electron distribution and equivalent electron temperature is obtained from full-band Monte Carlo simulation for bulk silicon. The electron distributions are used to numerically compute the phonon emission and absorption spectra and discover trends in their behavior at high electron temperatures. The concept of electron temperature is used to understand the relationship between field and heat emission, and it is found that longitudinal acoustic (LA) phonon emission increases at high electron temperatures. It is also found that emission of slower zone-edge phonons increases for all phonon branches athigh electron temperatures. These conclusions at high electric fields can be used to enable heat-conscious design of future silicon devices.

Keywords: phonons, thermal properties, Boltzmann transport, scattering rates, CMOS, MOSFET

Sažetak: Prenos toplote u nanoelektronici je postalo glavno usko grlo za dalji razvoj u narednim generacijama visoko integrisanih kola. Da bismo pristupili ovom problemu i razvili efikasnije tranzistore, senzore, i ostale elektronske elemente na nanoskali, moramo proučavati ter-malne procese na nivou atoma, što zahtijeva numeričku simulaciju interakcije između elektrona i toplote, koju prenose kvantizovane vibracije kristalne rešetke zvane fononi. U ovom radu detaljno ispitujemo spektre emitovanja i absorpcije fonona u silicijumu pri nekoliko povečanih vrijednosti elektronske temperature. Efekat električnog polja na statističku distribuciju elektrona te odgovarajuće temperature elektrona su postignute putem full-band Monte Carlo simulacije rasprostiranja elektrona. Distribucije elektrona se koriste da se numericki izračunaju rate emitovanja i absorpcije fonona te da se sazna o tendencijama njihovih karakteristika pri visokim temperaturama elektrona. Koncept tempera-ture elektrona se koristi da bi se razumio odnos između polja i emitovanje toplote, i došlo se do saznanja da se emitovanje longitudinalnog akustičnog (LA) fonona povečava pri visokim temperaturama elektrona. Također se došlo do zaključka da se prisustvo sporijih fonona sa višim energijama povećava kod svih fononskih polarizacija pri visokim temperaturama elektrona. Ovi zaključci kod visokih električnih polja se mogu koristiti da bi se omogućio razvoj budućih silicijumskih komponenata sa unaprijeđenim termalnim svojstvima.

Ključne riječi: fononi, termalna svojstva, Boltzmannova jednačina, rate rasijavanja, CMOS, MOSFET

NUMERICAL STUDY OF THERMAL DISSIPATION PROCESSES IN SILICON

NUMERIČKA STUDIJA PROCESA DISIPACIJE TOPLINE U SILICIJUMU

Zlatan Akšamija1

1University of Massachusetts Amherst, 01003 Amherst, MA, [email protected] submitted: May 2020 Paper accepted: June 2020

INTRODUCTION

Thermal budget is quickly emerging as the prominent limitation on future trends in scaling of semiconductor devices. As new generations of central processing units (CPUs) rely on denser and denser integration of billions of transistors for processing and storage, the transistors themselves are scaled smaller and smaller. This trend is following Moore’s Law, coined by Intel’s co-founder Gor-don Moore, that the number of transistors on a chip dou-bles every 18 months.


As integrated circuits (ICs) become denser, the heat theydissipate becomes more concentrated and harder to re-move, causing the IC to self-heat and limiting its peak operating clock frequency and ultimately its performance. Heat removal is further constrained by the small physical dimensions of ICs and the technological need to further combine them into portable electronics, wearable health monitoring, and other consumer devices where both en-ergy consumption and heat dissipation are undesirable.

To address the issue of heat dissipation, device engineersand circuit designers have long sought to develop more energy-efficient switches and to analyze the impact of selfheating on the performance of individual transistors viadevice simulation. As device features such as the length of the active channel region inside Complementary Met-al-Oxide-Semicondcutor (CMOS) Field Effect Transistors


(FETs) dip into the single nanometer range, the need for atomistic simulation arises. The widely-used laws gov-erning macroscopic transport of charge and heat begin to falter at scales smaller than the average distance tra-versed by a carrier between scattering events. The carri-ers of charge in semiconductors are electrons and holes; their transport at macroscopic scales is described by drift-diffusion equations, which are first-order differential equations that relate local current densities to gradients of the electrostatic potential (voltage) and carrier concentra-tion. Analogously, heat conduction is described by Fouri-er’s law, which gives rise to the widely-used heat equationṪ = ĸ∇2 T with ĸ being the thermal conductivity, long taken to be a material-specific property. The two equations of current and heat transport can be coupled at the level of total heat dissipation into an electro-thermal description, but this requires prior knowledge of how materials prop-erties vary with temperature.

Both drift-diffusion and the heat equations are continuumrepresentations that fail to capture the particle nature ofcurrent, made up of indistinguishable electrons, and heatflux, made up of quantized lattice vibrations called pho-nons [1]. Being fundamentally local, these continuum laws cannot represent the behavior of electrons and phonons during socalled ballistic (particle-like) motion between randomizing scattering events where they interact. The underlying microscopic picture is that electrons start out in equilibrium undergoing thermal motion; they are ac-celerated by the electric field coming from the externally applied voltage, whereby they gain energy, followed by scattering of electrons with the lattice vibrations (phonons) when that energy is released as heat via phonon emis-sion. The typical distance an electron travels ballistically between being scattered by a phonon can be only a few nanometers; phonons travel greater distances but at far lowers speeds than electrons (several thousand meters per second compared to the ≈ 107 m/s thermal velocity of electrons), averaging a mean-freepath of about 100-300 nm, depending on temperature and material. At these scales, thermal conductivity becomes nonlocal and strongly a function of device dimensions, geometry, and surfaces [2].

Consequently, at scales of typical transistors in pres-ent-day ICs, we must abandon the continuum description of electrothermal transport and turn to an atomistic and statistical description based on the Boltzmann transport equations of electrons and phonons. Further, detailed un-derstanding of electron-phonon coupling, as well as char-acteristics of the phonon heat generated by the electron current, such as phonon mode and spectrum, are crucial to our understanding of the micro-scale heating issues in semiconductor devices [3]. High electric fields present in ultra-scaled devices pose additional challenges to the understanding of electro-thermal properties as they push the electron population far from equilibrium and can even lead to hot electrons, where the proportion of electrons occupying higher energy states far above the edge of the conduction band increases appreciably. Identifying trends in phonon emission spectra at high fields is crucial to

enabling more thermally efficient devices. More recently, interest in coupled electro-thermal simulation has grown [4]-[7] due to inherent thermal limitations to electron trans-port. It is becoming necessary to model accurately the response of phonons to heating due to the release of ther-mal energy through the electron-phonon interaction. This is significant in the case when electrons are subjected to very high electric fields [8], [9], such as those that are present in the channel region of semiconductor devices.

Previous work in phonon emission in silicon relied on Monte Carlo simulation of electron transport [10]. Such simulations capture the electron-phonon interaction for the purpose of accurately describing the electron current, and usually simplify the treatment of phonons by assum-ing a simple analytic dispersion relationship between pho-non momentum and energy ℏω(q) [8], [11]. This is usually sufficient to describe electron transport with great accu-racy. Monte Carlo simulations then keep track of the inter-actions between electrons and phonons, from which the number of phonons generated during the course of the simulation can be extracted, and the relationship between the net phonon generation rate and the phonon energy can be computed. This was done by Pop et al. [9] for bulk and strained silicon based on analytical isotropic phonon dispersion and analytical multi-valley non-parabolic elec-tron bandstructure. Although the main trends could be identified, lack of full electron bandstructure, and even more importantly the use of analytical isotropic phonon dispersion, did not capture the full detail of the phonon emission spectra. In addition, the Monte Carlo method, due to its stochastic nature, requires a large number of particles and iterations for convergence.

In order to avoid these limitations, we use a deterministicmethod due to Gilat and Raubenheimer, and extend it to calculate the rate of change of the phonon distribution function due to electron-phonon interactions. In this work, we treat the electron-phonon coupling within the defor-mation potentials framework, using values given in the lit-erature [11]. We deploy established numerical algorithms for Brillouin zone integration [12] to compute detailed electron-phonon coupling and obtain the rates of pho-non emission and absorption by electrons. The electronic band structure is obtained from the empirical pseudo-po-tential method [13]. In order to capture the band struc-ture more accurately, our implementation was extended to include non-local pseudo-potentials [14], while Weber’s adiabatic bond-charge model is used for the phonon dis-persion [15]. From our numerical model, we obtain the rates at which phonons are emitted and absorbed by electrons and show that the net emission of phonons scales with the electron temperature, while the spectra of emitted phonons increasingly favor small-wavelength longitudinal phonons. We conclude that heat dissipation in the far-from-equilibrium regime leads to further thermal bottlenecks.


1. NUMERICAL APPROACH

As noted before, phonon transport is described by the Boltzmann transport equation (BTE), which describes the evolution in time of the phonon distribution function. This distribution function, denoted by N(r,q,t), represents the average number of phonons which occupy a state q in po-sition r and at time t. The rate of change of the phonon distribution has several contributions. One of the dominant contributions is due to the anharmonic phonon process-es, which arise due to third and higher order components of the bond forces. This is a phonon-phonon interaction which allows the phonon population to return to equilibrium when perturbed by a temperature gradient or a genera-tion source [16]. A second contribution which is strongly present in semiconductor devices is due to the generation of phonons by the electron-phonon interaction. This inter-action occurs when strong electric fields drive the electron population out of equilibrium and impart energy on the elec-trons. The electronphonon interaction is the primary means for electrons to release the energy accumulated from the accelerating effect of the applied electric field and to re-turn to equilibrium. This is especially important in nanoscale transistor devices, where electrons are accelerated by the strong fields in the channel region of the transistor, and then the electrons have to release energy in the form of phonons in order to return to quasiequilibrium in the drain region be-fore they leave the transistor through the contacts.

Generation as well as absorption of phonons couple to-gether the electron and phonon populations and allow energy exchange between them, leading to a net flow of energy from the electron gas to the phonon bath. The pho-non population under the heating effect of electrons can be described by the phonon BTE with the added term due to the electronphonon interaction [17]. The phonon distri-bution function, notated by N(r,q,t), increases every time an electron-phonon scattering event takes place, and de-creases for every anharmonic phonon decay. Therefore we will have two terms on the right-hand-side, one owing to the interactions of phonons with electrons, and the other due to interactions of phonons with other phonons through anharmonic decay, boundary scattering, isotope scatter-ing, and other interaction mechanisms which are beyond the scope of this work:

In considering this challenging seven-dimensional trans-port equation, having, in general, 3 dimensions for po-sition, 3 for crystal momentum, often called reciprocal space, and 1 for time, we will restrict our attention to homogenous materials in the steady state and drop the position variable r. In steady-state, all variation with time has vanished, so that the first term containing a partial

derivative with respect to time can be removed. The ho-mogenous restriction means we are considering the case of bulk material, so that all gradients with respect to the spatial coordinate are zero. This removes from consider-ation the second term of the phonon Boltzmann Trans-port equation containing spatial gradients of the phonon distribution function. Finally, the term due to anharmonic decay is usually treated in the relaxation time approxi-mation, and the relaxation rates have been presented by several researchers [18], [19] based on analytical models of the phonon dispersion, as well as detailed numerical computations of decay rates of optical phonons based on an empirical model of phonon dispersion [20] and Den-sity Functional Theory (DFT) [21]. On the other hand, the generation term due to electron-phonon interaction has not been given the same amount of attention, as it does not play a significant role in purely thermal transport, and contributes little when there are no electric fields present because the electron population is in equilibrium with phonons.

The rate of change of the phonon distribution function dueto electron-phonon scattering can can be computed by combining the contributions from the electron-phonon scattering over all electron momenta. We notice that for every transition an electron makes between an initial state k to a final state k’, a phonon with a momentum q = ±(k−k’) is either absorbed or emitted, depending on the energies of the two electron states. Every emis-sion increases the phonon occupancy and adds one to N(k), while every phonon absorption does the opposite, and decreases N(q) by unity [22]. The electron states are weighted by their probability of occupancy, given by the electron distribution function fe(k). The total rate of change of the phonon distribution function is obtained by sum-ming over all the electron states [23], as shown in (2):

In this work, we are interested in bulk silicon crystal, which is taken to be infinite in extent. This implies that momentum states are also infinite in number, so they form a countable continuum. In this limit, the summation over all momentum states q has been shown to approach an integral over the entire first Brillouin zone for the purpose of calculation [24]:

Energy conservation is expressed by the δ-function, whichdefines a complex energy-conserving surface in the first Brilluoin zone. Using properties of the δ-function, the in-tegral over k is converted into a surface integral over the entire energy-conserving surface (4):

( ) ( ) ( ) ( ), , , ,, ,el ph

N t dN tN t

t dt−

∂ + ∇ = ∂

r

r q r qv q r q

( ), ,ph ph

dN tdt

−

−

r q

( ) ( ) ( )( ) ( )

2 2 1 12 2

el ph

D I qdNN

dt ρω−

= + ±

∑k

qqq

q

( ) ( ) ( ) ( )( )ef E Eδ ω− ± ±k k k q q

( ) ( ) ( )( ) ( )

2 22 1 12 2

el ph

D I qdNd N

dtπ

ρω−

= + ± Ω ∫∫∫

qqk q

q

( ) ( ) ( ) ( )( )ef E Eδ ω− ± ±k k k q q

(1)

(2)

(3)


where the closed surface of integration is defined by en-ergy conservation to include all points k such that the equality E(k) −E(k) = ±q is satisified. At the same time, this surface contains contributions from all the final mo-mentum states k’ which are also able to conserve crystal momentum with a given inital state k. The contribution to the integral from each small cube in momentum space is given by the area of the intersection between this ener-gy-conserving surface and that particular cube, weighted in our case by the contribution from the matrix element. Such an integral over the entire first Brillouin zone can be computed numerically by dividing the Brillouin zone into a regular mesh of small cubes or tetrahedra, and computing the area of intersection of the energy isosurface with each particular cube or tetrahedron. There are several algorithms that are well-suited for the purpose of such a calculation. We choose the algorithm due to Gilat and Raubenheimer, which uses a division of the crystal momentum into reg-ular small cubes, then expanding all the terms in the in-tegrand to first order for the purpose of obtaning a very accurate result and taking into account the strong gra-dients in electron energy and distribution functions. Due to the inclusion of up to linear terms, this approach was also termed Linear Analytic [12]. This algorithm was first proposed for the purpose of calculating spectral prop-erties, such as the density of states. The Gilat-Rauben-heimer (GR) approach has also been applied to a very accurate calculation of the electron-phonon scattering rates in silicon and other semiconductors by Fischetti and Laux [11].

In the GR algorithm, the portion of the energy-conservingsurface in each small discretization cube is assumed to bea plane so that every point on the plane is perpendicularto the gradient at the center of the cube. The shape of theintersection of a plane and a cube then becomes either atriangle, quadrangle, or pentangle, depending on the an-gle of the gradients of the electron energy bandstructure in each particular cube, which are precomputed on the same mesh as the energies, as described earlier. The contribution Si arising from each small cube centered at ki and indexed by i in the entire Brillouin zone can then be computed based on geometric considerations, and summed together to obtain the total rate of change of the phonon distribution function N(q), as given in (5). Further details of the Gilat-Raubenheimer Linear Analytic algorithm, and the expressions for the areas of intersections of equal

energy surfaces with cubes in momentum space are given in [25]. This algorithm is very accurate since it uses extrap-olation and well-suited to using tabulated data for band-structures and dispersions [12]. Unlike the work of Fischetti and Laux, who used an analytical expression for the dis-persion relationship of phonons, we use the full dispersion computed from Weber’s Adiabatic Bond Charge model in order to accurately obtain the dependance on momentum of the phonon energy ℎ̄ω(q) in the integrand of equation (3). This model allows a very accurate representation of all phonon branches, shown in Figure 1, even along directions other than the main symmetry directions [26].

The deformation potential for acoustic phonon interaction, due to the crystal symmetry, reduces to two independent parameters: the dilatation potential Ξd and the uniaxial shear

Figure 1: Phonon dispersion relationship (left) and phonon den-sity-of-states (DOS) for each of the 6 phonon branches and total (right). The Adiabatic Bond Charge model is used to compute the phonon frequencies on a regular grid in momentum space, thereby allowing the DOS to be computed by the method of Gi-lat and Raubenheimer. The phonon DOS is found to be dominat-ed by optical and longitudinal acoustic (LA) modes due to their flat dispersion curves and low group velocities, especially near the Brillouin zone edge. The DOS also shows that the optical modes are found above 48meV, transverse acoustic (TA) modes

are below 25meV, and LA modes are between.

( )( ) ( )( ) ( ) ( )

2

E Eel ph

dN ddt E Eω

π

− ± =±−

= Ω ∇ −∇ ±

∫∫k kk k q q

q Sk k q

( ) ( )( ) ( ) ( )

2 2 1 12 2 e

D I qN f

ρω + ±

qq k

q

( ) ( ) ( )( ) ( ) ( ) ( )

2 2 1 12 2 i i e iiel ph

D I qdNN S f

dt ρω−

= + ±

∑qq

q k kq

(4)

potential Ξd [27], [22]. The relationship between the defor-mation potential and these two parameters is described by the Herring-Vogt relation, which expresses the acous-tic deformation potential in terms of the dilatation and shear potentials, along with an angular dependence. The angular dependence is on Θ, the angle between the phonon momentum q and the longitudinal axis of the conduction-band valley [28]. The acoustic deformation potential also depends on the phonon polarization, which can be either longitudinal or transverse, and is expressed in (5):

( ) ( )( ) ( )

2d uLA q

uTA q

D cos q

D sin cos q

= Ξ +Ξ Θ

= Ξ Θ Θ(5)

For optical phonons, the interaction is of the zero-th order and the deformation potential, expressed in (6), is taken tobe a constant with values taken from [11]. The overlap in-tegral between the cell portions of the Bloch states [28] is computed in the Nordheim, sometimes called the spher-


The electron and phonon populations are assumed to be initially in equilibrium, with a Fermi distribution (8) for electrons, and a Bose-Einstein distribution (9) for phonons [30]. The electric field is introduced through a fullband Monte Carlo ensemble electron simulation, described in earlier work [31], [32]. The term full-band here means that the simulation uses a numerically computed electronic structure to accurately describe electronic states an the relationship between their crystal momentum, veloci-ty, and energy without resorting to approximations such as effective mass. The simulation tracks an ensemble of electrons, typically 100000 in size, as they are accel-erated by the electric field and undergo a sequence of electron-phonon scattering events until it converges to a non-equilibrium electron distribution which includes the effect of the applied electric field. This provides us with an accurate distribution for electrons [33]. The effect of the applied electric field can then be understood through the concept of equivalent electron temperature Te [24], which captures the heating of the electron population from the energy imparted on them by the electric field. Increasing the applied field imparts more energy on the electron pop-ulation and pushes the electron temperature further up above the lattice (phonon) temperature Tph. The imbalance means electrons and phonons are out of equilibrium and results in a net transfer of energy from electrons to pho-nons. This transfer is our microscopic description of heat dissipation.

Fitting the semi-logarithmic plot of the Monte Carlo results for the electron distributions at various applied electric fields, as shown in Figure 2, allows us to extract the elec-tron temperature from each distribution and relate it to the applied electric field. In order to account for the effect of electric field on the electron, and ultimately the pho-non population, we use a quasi-equilibrium distribution for electrons based on the Fermi-Dirac distribution with the appropriate electron temperature determined from the fit to Monte Carlo results at each given electric field.

The gradient of the electron energy can be large and the numerical accuracy of the Brillouin zone algorithm in [25] can be improved by expanding the electron distribution function up to first order and including the linear terms, as given in [34]. The gradient of the quasi-equilibrium electron distribution function, (8), can be determined analytically as in (10), while the gradients of the electron energies can be computed numerically and tabulated along with the results for the electron energies. This improvement means we can incorporate the gradient into the summation in (5) in order to obtain a more accurate method, thereby improving the quality of final results for phonon emission and absorption rates calculated from the electron distributions.

Because the integral in (5) is over all electron states k, terms depending only on the phonon momentum q have been re-moved from the integral. In addition, due to its dependence on the phonon momentum q only, the phonon distribution function N(q) can be factored out of the integral in (5) and divided by it to obtain a rate Γ(q) = 1/N(q) (dN(q)/dt). The de-pendence of this rate on the phonon energy ω, rather than momentum q, is obtained by averaging over all modes ear a given energy ω using the same Gilat-Raubenheimer ap-proach (11). The phonon emission spectrum Γ is then scaled by the phonon density-of-states (DOS), shown in Figure 1:

This converts the distribution from momentum space into an energy spectrum Γ(ω), and enables us to examine the results. Each of the phonon types can be identified with a

( ), opLO TO qD DK=

( )( )

( ) ( )33

s s ss

I q sin qR qR cos qRqR

= −

( ) ( )

( ) ( )

1

1

1

F

B e

B ph

E Ef exp

k T

N expk Tω

−

−

− =

= −

kk

qq

( ) ( ) ( )f Ef EE

∂∇ = ∇

∂kk k

( ) ( )( ) ( )1 1B

f f Ek T

= − − ∇k k k

( )( ) ( )( )

( )( )d

d

ω ωω

ω ω

Γ ∂ −Γ =

∂ −∫∫

q q q

q q

(7)

(8)

(9)

(10)

(11)

(6)

Figure 2: Logarithmic plot of the electron distribution histogram. This figure demonstrates that the electron distributions remain linear over the range of energies of interest, even at high applied electric fields. This allows the definition of an equivalent electron temperature Te which reflects the heating of the electron popula-

tion by the applied electric field.

ical cell, approximation [29]. This provides an analytical expression for the integral (7), which depends only on the magnitude of momentum of the phonon being exchanged q = |k − k’|. The Rs = aSi[3/(16π)]

1/3 is the radius of the Wigner-Seitz cell.


Figure 3: Plot of phonon velocities. Group velocity is given by the gradient of the phonon dispersion relation. Phonon velocities dictate how fast each mode can propagate heat through the sil-icon lattice. Longitudinal modes (dashed line), especially the lon-gitudinal acoustic mode, have the highest velocity. Transverse acoustic (TA) modes are slower, while the optical modes are very

slow due to their flat dispersion.

Figure 4: Plot of phonon velocity spectra. For each energy value on the xaxis, the velocity is computed by numerically averaging over the energy isosurface. This plot shows the average pho-non velocity at each particular energy rather than momentum or direction. Longitudinal acoustic mode (dashed line) has the highest velocity. Transverse acoustic (dash-dot) and the optical modes (dotted line) are slower due to their flat dispersion, es-pecially at higher energies corresponding to phonons near the

Brillouin zone boundary.

The transverse acoustic (TA) modes have larger veloci-ties, and approach the speed of sound in the long-wave-length limit. The highest velocities are present for the longitudinal acoustic mode (LA) due to the fact that lon-gitudinal waves propagate along the directions of the bonds in the crystal lattice. Phonon velocities can be averaged over all modes near a particular energy to ob-tain a velocity spectrum v(ω) which depends on phonon energy ω rather than the three-dimensional momentum vector q, and is therefore more suitable for plotting and comparisons. This is done by integrating phonon veloc-ities over the energy isosurface, and then scaling by the density-of-states, as in (12). The velocity spectrum v(ω), shown in Figure 4, demonstrates that phonon velocities for each mode are highest at low energy, for phonons

( )( ) ( )( )

( )( )d

vd

ω ω ωω

ω ω

∇ ∂ −=

∂ −∫

∫q q q

q q

near the Brillouin zone center, and lower for higher ener-gy zone-edge phonons.

(12)

particular energy range. The DOS in Figure 1 shows that the optical modes are found above 48 meV, transverse acoustic (TA) modes are below 25 meV, and LA modes are in between. We can therefore examine trends in these three phonon types by looking at their respective energy ranges, low for TA, high for optical, and middle for LA phonons.

2. RESULTS AND DISCUSSION

Due to the large differences in the directions and veloci-ties of phonon propagation for different phonon modes, examining how much of each phonon polarization is gen-erated is important for understanding heating in silicon at the nanoscale. Transverse optical modes (TO) have very flat dispersion curves, and consequently have the lowest velocities, as shown in Figure 3. The longitudinal optical mode (LO) also tends to zero in the long-wavelength limit, but its velocity increases for short wavelength phonons.

In Figure 5 and 6, we find that phonon absorption is strong-ly dominated by transverse acoustic (TA) phonons. Curves are presented for electron temperatures ranging from 300 K (bottom curve) up to 10000 K (top curve), corresponding to electric fields from 1 kV/cm up to around 300 kV/cm.

Both emission and absorption are in units of inverse sec-onds. Increasing the applied electric field has the effect of increasing the difference between the electron tem-perature Te and the lattice temperature Tph. This differ-ence acts to increase both the emission and absorption spectra, and favors higher energy acoustic phonons. There is an overall flattening of the emission spectrum, where the largest increases can be noted in zone-edge (short wavelength) acoustic (TA and LA) modes, while, to a lesser extent, the trend reverses for optical modes. The trend is especially pronounced for modes near the Brillouin zone edge, at energies around 48 meV for LA phonons, and 55 meV for optical phonons. TA phonon emission is strongest at low electron temperatures, where both emission and absorption are dominated by small-en-ergy (long-wavelength) phonons, while the opposite is true at high electron temperatures, where the largest increase is also in the zone-edge phonons, at energies around 25 meV. LA phonons have the highest propagation velocities of all phonon polarizations, Figure 3, so they are expected to carry the emitted thermal energy fastest. Further, opti-cal phonons which have very short lifetimes, and quickly


Figure 7: Logarithmic plot of the net rate of phonon generation (per second) for electron temperatures ranging from 300 K (bot-tom curve) up to 10,000 K (top curve), corresponding to elec-tric fields from 1 kV/cm up to around 300 kV/cm. This rate is computed from the difference between the total emission and absorption rates, and represents the net increase of the phonondistribution function due to coupling between electrons and pho-nons. The logarithmic plot is linear in the optical region indicating an exponential energy dependence of emission in the optical branches. This plot also shows that the most dramatic increase with increasing electic field and electron equivalent temperature is in the longitudinal acoustic branch, especially near the end of

the Brillouin zone at energies around 48 meV.

decay into combinations of acoustic phonons, while LA phonons have longer mean-freepaths of around 100 nm [5] and longer lifetimes; even after anharmonic decay, they do not necessarily fully equilibrate [35].

As shown in Figure 4, small energy TA phonons are faster than their zone-edge counterparts, so we can expect an increasing proportion of the heat to be car-

ried by LA phonons at high electron temperatures, cor-responding to high electric fields. Similar conclusions can be drawn from the logarithmic plot of the net pho-non generation rate, shown in Figure 7. The logarithmic plot shows a nearly linear dependence of the optical phonon generation on phonon energy, indicating an ex-ponential relationship between phonon frequency and generation rate. The most drastic dependence on the electric field strength and electron temperature is again in the LA phonon branch, which increases from very small contributions at low fields to nearly even levels with the optical phonon generation at very high electric field strengths. This is especially true for higher energy LA and LO phonons with energies near 48 meV corre-sponding to short wavelength phonons near the Brilloin zone edge. These results and trends agree with Mon-te Carlo calculations of emission spectra in silicon [9] and similar results for a silicon device [5]. Both of the prior Monte Carlo calculations showed that the larg-est increase in the phonon emission at high electric fields is in the LA modes around 40 meV, in qualitative agreement with our results. More recently, Fischetti et al. [35] showed that the total energy loss from electrons is strongest to the TA phonon branch, followed by the optical (TA and TO) branches, with LA phonons taking up the least dissipation; while we did not evaluate the total energy by phonon branch, our results do support the view that acoustic phonons dominate the net emis-sion spectra.

Figure 5: Plot of the rate of phonon absorption (per second) for electron temperatures ranging from 300 K (bottom curve) up to 10000 K (top curve), corresponding to electric fields from 1 kV/cm up to around 300 kV/cm. Increase in the applied field causes a difference between Te and lattice temperature Tph and acts to increase the rate of absorption, especially in the LA and optical

modes.

Figure 6: Plot of the rate of phonon emission (per second) for electron temperatures ranging from 300 K (bottom curve) up to 10000 K (top curve),corresponding to electric fields from 1 kV/cm up to around 300 kV/cm. Rate of phonon absorption in-creases with the temperature of electron distribution, and favors higher energy and zone-boundary phonons. At the highest elec-tron temperature of 10000 K, we can see sharp increases in the emission of phonons near the highest energy for each mode. This is most prominent in LA phonons, around 48 meV, and TA

modes near 25 meV.


Figure 8: Plot of the logarithm of the ratio of phonon emission to absorption for electon temperatures ranging from 1000 K (top curve) up to 10000 K (bottom curve), corresponding to electric fields from 50 kV/cm up to around 300 kV/cm. The slope of each curve corresponds to the electron temperature of the dis-tribution from which it was calculated, indicating that the ratio of emission and absorption is given by a Boltzmann factor at the

electron temperature Te.

Lastly, we examine the relationship between emission andabsorption spectra. Figure 8 shows that the ratio of emis-sion and absorption follows a Boltzmann factor exp(ℏω/kBTe) at the electron temperature Te, rather than the lattice, or phonon, temperature. This ratio is smaller for high elec-tron temperatures, and for optical phonons. When elec-tron and phonon temperatures are equal, we have equilib-rium, and detailed balance holds, producing a ratio equal to the Boltzmann factor. When the electron and phonon populations are out of equilibrium, the electron tempera-ture is higher, producing a net excess of emission over ab-sorption so that the net heat dissipation is ultimately pro-portional to the difference between electron and phonon temperatures. Nonetheless, the shape of the spectrum of the emitted phonons is not static; rather, it evolves with the electron temperature and depends on the difference between the electron and the lattice temperatures.

3. CONCLUSION

We examined the nature of phonon generation in silicon at several strengths of applied electric field. We identified sev-eral trends, in particular that phonon emission at low and intermediate fields is dominated by low-energy transverse acoustic phonons. As the temperature of the electron pop-ulation increases under the acceleration of the applied elec-tric field, the rate of net emission of longitudinal acoustic and optical phonons approaches that of TA modes. Over-all, high electron temperatures push the phonon spectra towards higher energies. This favors LA modes the most, and may shift the subsequent transport of heat more from TA to LA modes. Due to longer lifetimes, lower velocity, and focused propagation of LA modes, design optimizations

may be possible and necessary in future devices. These may include changing the orientation of the active channel region of a nanowire MOSFET device to coincide with the propagation directions of the LA phonons to avoid trapping phonons inside the critical active region of MOS transistors.

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BIOGRAPHy

Zlatan Akšamija was born on 6 July, 1980 in Sarajevo. He received his Bachelor of Science in Computer Engineering (Summa Cum Laude) in 2003, and his M.S. and doctorate in Electrical Engineering in 2005 and 2009, respective-ly, all from the University of Illinois at Urbana-Champaign, USA. His doctoral dissertation on “Thermal effects in semiconductor materials and devices” was supported by a Department of Energy Computational Science Graduate Fellowship. From 2009 to 2013, Zlatan was a Computing Innovation Postdoctoral Fellow and an NSF CI TraCS Fel-low in the ECE department at the University of Wiscon-sin-Madison. In 2013, he became an Assistant Professor in the Electrical and Computer Engineering Department at the University of Massachusetts-Amherst and founded the NanoEnergy & Thermophysics lab, where he studies thermoelectric energy conversion and heat transfer in 2-dimensional materials and nanostructures. He was pro-moted to Associate Professor with tenure in 2019.


Abstract: This article gives an overview of the applications of algebraic topology methods in signal processing. We explain how the no-tions and invariants such as (co)chain complexes and (co)homology of simplicial complexes can be used to gain insight into higher-order interactions of signals. The discussion begins with some basic ideas in classical circuits, continues with signals over graphs and simplicial complexes, and culminates with an overview of sheaf theory and the connections between sheaf cohomology and signal processing.

Keywords: signal processing, graph, simplicial complex, chain complex, homology, cohomology, sheaf, Laplacian, spectral analysis

Sažetak: Ovaj članak daje pregled primjena metoda algebarske topologije u procesiranju signala. Objašnjavamo kako se pojmovi i invarijante kao sto su (ko)lančani kompleksi i (ko)homologija simplicijalnih kompleksa mogu koristiti za razumijevanje interakcija signala višeg reda. Diskusija počinje sa nekim osnovnim idejama klasičnih strujnih kola, nastavlja se sa signalima defisanim na grafovima i simplicijalnih kom-pleksa, a kulminira pregledom teorije snopova i veze između kohomologije snopa i obrade signala.

Ključne riječi: procesiranje signala, graf, simplicijalni kompleks, lančani kompleks, homologija, kohomologija, snop, Laplaceova matrica, spektralna analiza

TOPOLOGICAL METHODS IN SIGNAL PROCESSING

TOPOLOšKE METODE U PROCESIRANJU SIGNALAIsmar Volić1

1Department of Mathematics, Wellesley College, Wellesley, MA, [email protected] Paper submitted: May 2020 Paper accepted: June 2020

INTRODUCTION

Signal processing is concerned with incorporating and in-terpreting a set of measurements into coherent and use-ful information about a system. These measurements are typically taken over time or space, but, in recent years, it has become clear that it is desirable to understand signals over more complicated structures. Neural, social, and sen-sor networks are just some of the examples where signals interact and depend on each other in more complicated ways than the standard theory could accommodate.

To deal with this, one direction in which the theory de-veloped was graph signal processing where the signal is now measured over the vertices of a graph and the edges are meant to encode the interactions between those signals. Despite its success in fields such as sen-sor networks, biological networks, image processing, and machine learning (a comprehensive review of de-velopments and challenges in graph signal processing can be found in [18]), this theory had a shortcoming, and that was that the edges of a graph could only en-code pairwise relations between the signals. Possible triple or higher oder interactions could not be account-ed for in a system with such a simple underlying struc-ture.


The natural next step, and a subject or much recent investi-gation, is to study signals over simplicial complexes [3], [4], [12]. These are generalizations of graphs where, rather than using just vertices and edges, one also puts together trian-gles, tetrahedra, and their high-dimensional generalizations (simplices) to create an object called a simplicial complex. Such an object is both geometric and combinatorial, and this dual nature endows it with a rich structure that can be exploited from many points of view. Simplicial complexes are an excellent framework for studying signals with multi-ple interactions. For example, if signals over, say, three ver-tices interact in some way, that can be represented with atriangle as the underlying structure. For four vertices, the representation is a tetrahedron, etc.

Bringing simplicial complexes into the picture is also where it becomes useful to bring in emloy topology. Topology has a lot to say about simplicial complexes (and more general spaces) since it can extracts those features that are un-changed under and independent of deformations. The no-tion of homology is particularly useful in this context as it captures the existence of essential ”holes” in the space, and its dual, cohomology, carries even more structure. As we will demonstrate, these concepts turn out to have re-markable connections to some standard tools from signal processing such as the Laplacian, which is crucial in spec- ral signal analysis.

It might seem counterintuitive that topology, which is known for its lack of geometric rigidity, would be useful in signal processing which is often constrained by geometry.


But in many real-life situations, relationships between near-by signals are what matters most, and topology does pay attention to local information. Furthermore, topology is also able to put together local information into a global picture via its invariants, and such approach is precisely the novelty that topology is introducing into signal processing.

This expository paper aims to provide just the highlights of the role of topology in signal processing. We barely scratch the surface but provide ample references for further reading. We progress both historically and mathematically, skipping the details and rigorous proofs for brevity and readability.

In Section 2, we begin by introducing the basic notions from topology - spaces, maps, homeomorphisms, sim-plicial complexes, chain complexes, and (co)homology. Section 3 then treats some classical results about circuits in the language of (co)homology. This is meant to indicate that using algebraic topology in electrical engineering, and signal processing in particular, is not a new notion and that it has existed, albeit in somewhat of a background role, for some time.

The following three sections are natural gradual generaliza-tions: We move from particular kinds of signals on circuits to arbitrary signals defined over a graph in Section 4. This is where we review some of the main features of the spectral study of the graph Laplacian. We then move to signals over simplicial complexes in Section 5, where topology really starts to come into play. In particular, the relationship be-tween spectral analysis and cohomology of the underlying simplicial complex is elucidated.

The final generalization is that to the language of sheaves. Sheaf theory allows for the possibility that signals are not single, but multiple measurements at each vertex. They are also designed to facilitate the passage from local to global information. Sheaf-theoretic language allows for a vast uni-cation of many of the concepts found in signal processing. The main reference for this part of the paper is [23].

1. A BRIEF INTRODUCTION TO TOPOLOGy

In this section, we give an extremely brief introduction to the basic notions of (algebraic) topology. Some standard texts on the subject are [1], [11], [17].

1.1. Topological Spaces and Maps

A topological space X is a set endowed with a structure of open subsets, namely a collection of subsets of X that contains the empty set and X itself, and is closed under arbitrary unions and finite intersections.

If a set X has a metric d on it, then the metric can be used to induce a topology on X by first defining an open ball of radius ϵ of a point x ∈ X as the set consisting of all points

y ∈ X such that d(x; y) < ϵ and then using all open balls as basis for a topology on X (so open sets of X are all unions of all possible balls).

Example 1.1. The n-dimensional Euclidean space ℝn has the standard topology induced by the usual Euclidean distance function.

A subset A of a topological space X is a subspace if it is endowed with the topology where open sets are of the form A ∩ U where U is open in X. This is the induced topology on A.

Example 1.2. The n-dimensional sphere Sn and the n-di-mensional torus S 1× ...×S 1 (where “×” stands for the usual cartesian product of sets) are thought of as subsets of ℝn+1 and are topologized as such.

A function f : X → Y between topological spaces X and Y is continuous if the preimage of an open set in Y is an open set in X (i.e. f pulls back open sets to open sets). For a metric space and the induced topology, this coincides with the usual definition of continuity from analysis.

A continuous function between topological spaces is called a map.

Topology is the study of those properties of spaces that are invariant under continuous deformations. Depending on the context, there are several ways to define a defor-mation, but the most basic one is a homeomorphism; a map f : X → Y that is a bijection and has a continuous inverse. If there is a homeomorphism between X and Y, then those spaces are said to be homeomorphic and we write X ≅ Y.

A weaker notion than a homeomorphism is homotopy equivalence; instead of requiring that f has an inverse, we only ask that there be a map g : X → Y such that f ◦ g and g ◦ f are homotopic to the identity map (rather than equal to the identity map). This means that there exist a one-param-eter family of maps starting with f ◦ g (resp. g ◦ f) and ending with the identity map on Y (resp. X). We write X Y if X and Y are homotopy equivalent.

Spaces that are homeomorphic or homotopy equivalent are thought of as the same from the topological point of view. Here are some basic examples.

Example 1.3.

- A circle and a square are homeomorphic.- A sphere S n with a point removed is homeomorphic to ℝn.- If m ≠ n, then ℝm and ℝn are not homeomorphic.- An open interval in (a, b) ⊂ ℝ and ℝ are homeomorphic.- Euclidean spaces ℝn are homotopy equivalent for all n ≥ 0

(ℝ0 is a one-point space).- The standard torus S1× S1 with a point removed is ho-

motopy equivalent to two circles touching at a point (figure-8).


1.2. Simplicial Complexes

One issue with the general definitions above is that mak-ing concrete computations with them is dificult. One way to get around this is to restrict attention to spaces that look like (i.e. are homeomorphic to) spaces that are put together from triangles, tetrahedra, and their generaliza-tions. This turns out to cover most spaces that are ordi-narily encountered in topology (and certainly in any prac-tical applications of topology). At the same, this point of view affords a combinatorial approach to their study.

To make this more precise, we make the following definition.

Definition 1.4. For n ≥ 0, a (geometric) n-simplex ∆n is the convex set spanned by n + 1 affinely independent points {v0, v1, ..., vn} in ℝ for some N ≥ n. Independence means that the points are not contained in any hyperplane, i.e. the vectors vi - vn, 1 ≤ i ≤ n are linearly independent.

A 0-simplex ∆0 is thus a point, 1-simplex ∆1 is a line seg-ment, 2-simplex ∆2 is a (filled-in) triangle, 3-simplex is a (solid) tetrahedron, etc. An n-simplex is said to have dimension n.

The definition does not specify how the n + 1 points are embedded in ℝN but this does not matter since all n-sim-plices are homeomorphic.

A simplex is topologized as a subspace of ℝN.

A face of an n-simplex is the convex set spanned by some subset of its vertices. A tetrahedron thus has 4 triangle faces, 6 line segment faces, and 4 vertex faces.

Definition 1.5. A (geometric) simplicial complex X con-sists of a collection of simplices such that - If a simplex is in X, so is its every face; - Simplices in X intersect only in common faces.

Figure 1 gives examples of simplicial complexes. Note that the simplices in a simplicial complex need not be of the same dimension. Even though the top figure depicts a disconnected space, a simplicial complex will for our purposes always be connected. Thus a simplicial com-plex can be thought of as some number of tetrahedra of various dimensions glued together along common faces.

The dimension of a simplicial complex X is the highest di-mension of a simplex appearing in X. A connected simpli-cial complex of dimension 1 is thus a collection of edges that meet along some vertices, i.e. it is precisely a graph, as illustrated in the bottom picture of Figure 1.

For a fixed n, the collection of n-simplices of X together with their faces is called the n-skeleton of X. This is a sub-complex of X.

Many spaces are homeomorphic to simplicial complexes.For example, Figure 2 exhibits the circle S1 as the simpli-cial complex consisting of three 1-simplices (any number

of 1-simplices could have been used, as long as they form a polygon) and S2 as the simplicial complex consisting of four 2-simplices (triangles) put together into a (hollow) tet-rahedron.

All the information, up to homeomorphism, about a simplicial complex, is contained in the following com-binatorial information: If the vertices of X are labeled V = {v0, v1, ..., vk}, then each n-simplex in X can be labeled by some subset of those vertices, denoted by [vi0, ..., vin]. Conversely, specifying subsets of V tells us precisely how to build the simplicial complex X. For example, if we are given the subset [v2, v4, v7], then we can say that [v2, v4, v7] is a 2-simplex in X.

Figure 1: Examples of a simplicial complex of dimension 3 (top) and of dimension 1 (bottom). A connected 1- dimensional

complex is a graph.

Figure 2: Simplicial complexes homeomorphic to thecircle S1 and the 2-sphere S2.

If [vi0, ..., vin] is a simplex, then all subsets of {vi0, ..., vin} are also simplices because of the requirement that, if a simplex is in X, so is its every face. Intersections of subsets that represent simplices also have to represent simplices because of the requirement that simplices must meet along common faces. We can thus make the following definition, in parallel to Definition 1.5.


Definition 1.6. An (abstract) simplicial complex X is a set of points (vertices) V = {v0, v1,..., vn} and a collection of sub-sets of V called simplices such that - The singletons {vi}, 0 ≤ i ≤ k, are in the collection;- If a subset is in the collection, so are all its subsets; - If two subsets are in the collection, so is their intersection.

It should be clear that an abstract complex determines a geometric complex up to homeomorphism; it suffices to select k affinely independent points in ℝN and pro-ceed to fill in the simplices according to the recipe of the subsets that are supplied by the abstract complex. One can of course go the other way as well, and simply just retain the sets describing the vertex structure of the simplices in a geometric complex while forgetting its geometry. The two notions of geometric and abstract complexes are thus interchangeable.

Built into the simplex notation [vi0, ..., vin] is the order in which vertices are listed. A choice of such an order is called an orientiation. If we list the elements differently, then we will say that the orientation is the same if the new list differs from the original list by an even permutation of vertices. Otherwise the orientation is different. There are thus always two possible orientations of a simplex.

A simplicial complex is completely determined by how its faces t together. More formally, generalizing the no-tion of the adjacency matrix for graphs, we have the following construction: Given a simplicial complex X of dimension d, the incidence or connection matrices Ai, 0 ≤ i ≤ d, associated to X are the matrices whose entries are

We can then consider the Laplacians of X:

The Laplacians completely capture the structure of X. Their eigenvalues are very informative and the spectral analysis of the Laplacians is hence very useful. We will say more about this in the following sections.

1.3. Chain Complexes and Homology

At the broadest level, algebraic topology attempts to de-velop invariants, namely algebraic objects that can be associated to spaces in such a way that, if two spaces are homeomorphic or homotopy equivalent, these invari-ants are the same. The most basic topological invariants are homotopy and homology groups. The former are easy to define: The nth homotopy group of a space X, πn(X), n ≥ 0, is the set of homotopy equivalence classes of maps Sn→ X. However, these groups are notoriously difficult to compute.

On the other hand, homology groups Hn(X), n ≥ 0, are harder to define but easier to compute. These are more useful for our purposes. The intuition behind them is that they measure the structure of ”holes” in a space. For example, H0(X) counts the number of components of X, H1(X) counts the number of ”circular holes” of X, and H2(X) gives the number of ”2-spherical holes” of X. In general, Hn(X) keeps track of ”n-dimensional holes” of X.

We will give a brief description of the definition of homol-ogy for simplicial complexes and will make a comment about how to define it for general spaces, although we will not need the more general definition. For details, see, for example, [11, Chapter 2].

Definition 1.7. Suppose n ≥ 0. Let Cn(X;ℝ), called the group of n-chains of X, be the free group generated by the open n-simplices of X (which we will by abuse of notation also denote by ∆n). An n-chain, i.e. an element of Cn(X;ℝ), is thus a formal finite sum

0 1 1

1 1

;1 1;

.

T

T Tn n n n n

Td d d

, n d+ +

=

= + ≤ ≤ −

=

L A AL A A A AL A A

( )

1

1

1

0,, 1,

-1,

α β

α β

α β

α β

−

−

−

∆ ∆= ∆ ∆ ∆ ∆

n n

n nn

n n

Aif isnota faceof

if isa faceof andorientationsmatch

if isa faceof andorientationsdonotmatch

naαα

α

∆∑

where Naα ∈

There is a map

called the boundary operator defined as follows: For a simplex [v0, ..., vn],

( ) ( ) [ ]0 0 1 10

1n

in n i i n

iv ,...,v v ,...,v ,v ,...v .− +

=

∂ = −∑

Thus, for example, for a 2-simplex [v0, v1, v2],

[ ]( ) [ ] [ ] [ ]2 0 1 2 1 2 0 2 1 2v ,v ,v v ,v v ,v v ,v∂ = − +The boundary operator can be extended linearly to Cn(X:ℝ), and this is how the map (4) is finally defined.

It is a simple combinatorial exercise to show that the boundary of a boundary is zero, i.e. we have

Proposition 1.8.

A shortened notation for this result is ∂2 = 0. This makes sense intuitively since, for example, the boundary of a 2-simplex (triangle) is the three edges of the triangle, but that closed path of three edges does not itself have a boundary.

It is also not hard to see that, in terms of the incidence matrices (1), equation (5) can be restated as

An-1An = 0 (6)

1 0n n−∂ ∂ =

(1)

(2)

(5)

(3)

(4)( ) ( )1: C ; C ;n n nX X−∂ →

.


Because ∂2 = 0, the collection of n-chains and boundary maps forms a chain complex

Note that the condition ∂2 = 0 also means that the image of each boundary map is necessarily contained in the kernel of the next one, i.e.

where ker stand for the kernel and im for the image. Elements of the image of ∂n+1 are called n-boundaries and those in the kernel of ∂n are called n-cycles. We are now finally ready to define homology.

Definition 1.9. For X a simplicial complex, the nth homol-ogy group of X is the quotient group

( ) ( )1im kern n ,+∂ ⊂ ∂

The interpretation of this is that Hn(X;ℝ) records n-dimen-sional holes in the sense that those n-chains that look like they bound something, but they do not, appear in the homology, i.e. they are not ”killed” in the quotient.

For example, the closed path of three edges that bounds a 2-simplex would not get recorded in H1(X;ℝ) since that path has an ”inside”. But if that path appears just as a path of 1-simplices without the presence of a 2-simplex in its interior, that becomes a non-trivial element of H1(X;ℝ). This is because there is an essential ”1-dimensional hole” that is bounded by those 1-simplices.

Each homology group Hn(X;ℝ) is isomorphic to the vec-tor space ℝr for some r ≥ 0. This number r is called the rank of the homology group, or the nth Betti number Bn. The rank records the number of holes of the appropriate dimension. The rank of the zeroth homology group is the number of connected components of X.

It should be clear from the definitions that, if X is a d-di-mensional complex, Hd(X;ℝ) = 0 since such a complex does not have nonntrivial chains of dimension higher than d.

Example 1.10. Looking back at Figure 2, which up to homeomorphism depicts the circle S1 and the sphere S2, we have

Homology also has an interpretation in terms of the incidence matrices (1). Namely, it turns out that the Laplacians (2) contain all the homology information. Recall that the nth Betti number Bn is the rank of Hn(X;ℝ).

( ) ( )( ) ( ) ( )

1 10 1

2 2 20 1 2

H ; H ;

H ; H ; H ;

S , S

S , S S

= =

= = =

Theorem 1.11. We have

where ker stands for the usual kernel (or nullspace) of the Laplacian matrix Ln and dim is the dimension of this subspace.

To define the homology groups for an arbitrary space that is not a simplicial complex, we take all possible maps

( )( )dim kern nB = L

n:n Xσ ∆ →

as the generating set for the n-chains (i.e. replace by in the definition of Cn(X:ℝ)).

Example 1.12. Figure 3 shows a (hollow) torus T 2 whose homology groups are

nα∆

nασ

( ) ( )( ) ( )

2 2 20 1

2 22 3

H ; , H ;

H ; , H ; 0

T T

T T≥

= =

= =

The two circles that can be taken as generators of H1(T 2;ℝ)

are also pictured. Each of them bounds a 1-dimensional hole that is essential to the topology of T 2.

There is a dual notion of cohomology of a space X which associates to it a graded ring. It is often more computable than homology and carries more structure. First define the group of n-cochains of X as

Figure 3: A torus T 2 showing the two generators of the first homology.

namely the group of linear functions from Cn(X:ℝ) to ℝ. In short, this is the dual group of the n-chains. Since these are linear maps, it suffces to define an f ∈ Cn(X:ℝ) on the simplices of a chain. The boundary ∂n is replaced by the natural dual coboundary map

( ) ( )1: ; ;n n nC X C X+∂ →

(7)(8)

(9)

( ) ( )( ) ( )

1

2 1

1

1 0

... C ; C ; ...

... C ; C ; .

n nn nX X

X X

+∂ ∂−

∂ ∂

→ → →

→ →

( ) ( )( )1kercyclesH :

boundariesn

nn

nXn im +

∂−= =

− ∂

( ) ( )( )C : Hom C : ,n nX X=


Definition 1.13. For X a simplicial complex, the nth coho-mology group of X is the quotient group

Since homology and cohomology of a simplicial complex are finitely generated, the ranks of these groups are the same for each n. In other words, the Betti numbers are the same for homology and cohomology, and we will interchangeably use this notation and terminology. In short,

( )( ) ( )( )rank H ; rank H ;nn nB X X= =

As is always the case with duality, there is a natural bilinear pairing between homology and cohomology given by

( ) ( )H ; H ;n nX X× →

( ), ,nf c a f c f cα αα

= ∆ → =

∑

( )na fα αα

= ∆∑

This will be useful in the next section.

2. CLASSICAL CIRCUITS AND HOMOLOGy

The idea that the language of (co)homology is useful in electrical engineering is not new. In fact, some basic ideas about classical circuits can be expressed in this way. This notion appears as early as 1923 [28] and is by now well-established and covered in a number of standard sources, such as [2], [8], [14], [26].

For example, if one thinks of a circuit as a 1-dimensional complex, i.e. a graph, Kirchhoff’s voltage and current laws can be restated as saying that voltage is a 1-coboundary and current is a 1-cycle.

To explain a little, regard an elementary circuit as a con-nected, oriented (or directed) graph X. In other words, X is an oriented connected 1-dimensional simplicial complex consisting of the set of vertices (or nodes) V = {v1,..., vn} (the 0-simplices), and edges (or branches, representing conducting wires) E = {e1,..., em} (the 1-simplices).

A 1-chain I ∈ C1(X:ℝ) is called the current and a 1-cochain V ∈ C1(X:ℝ) is called the voltage. Unravelling what this means, I is really the sum

1

n

I a eα αα=

=∑

where aα is the current on the edge eα (with appropriate signs reflecting the orientation of the edges).

On the other hand, V is the sum

1

n

V b eα αα=

′= ∑

where bα is the voltage on the edge eα and e΄α is the cochain which, by duality, equals 1 when evaluated on eα and it equals zero on all other edges (see (11)).

Recall that we have (co)boundary operators

We then have

Theorem 2.1.

- Kirchhoff’s First Law: Voltage V is a 1- coboundary, namely there is a 0-cochain P such that

0V P= ∂

- Kirchhoff’s Second Law: Current I is a 1-cycle, namely

1 0I∂ =

The first law thus says that there exists a functional P on the vertices of X (linearly extended to formal sums of vertices) whose coboundary is the voltage. This functional is called the potential. The second law says that the currents along the edges at each vertex add up to zero.

One can also regard resistance as a function

:r E +→

that defines an inner product on C1(X:ℝ), i.e. a function C1(X:ℝ) × C1(X:ℝ) → ℝ by linearly extending the assignment

( ) ;,,.0,

ii j

i jr ee e

i j≠

= =

But an inner product like that defines a map

( ) ( )( ) ( )11 1: ; Hom ; , ;c

R C X C X C X

c R

→ =

→

where Rc is the function

( )1: ;,

cR C X

c c c

→

′ ′→

This is of course the usual bilinear pairing of a vector space with its dual.

In particular, we can see where the current I ∈ C1(X:ℝ) is sent under this map. Recall that the voltage V is an ele-ment of C1(X:ℝ). The situation where

( )V R I= or V RI=precisely corresponds to Ohm’s Law.

(10)

(11)

( ) ( )( ) ( )

1 1 0

0 0 1

: C ; C ;

: C ; C ; .

X X

X X

∂ →

∂ →

( ) ( )( )1kercocyclesH :

coboundaries

nn

n

nXn im −

∂−= =

− ∂


With this line of investigation, various other results about classical circuits can easily be reframed and investigated.

3. SIGNALS OVER GRAPHS

The story of classical circuits and homology can be natu-rally generalized in a number of directions. As mentioned in the Introduction, one can work in the more general framework of signals defined over a graph, i.e. a 1-di-mensional complex, rather than restricting the attention to voltage and current. A rich field of graph signal process-ing (GSP) arose from this more general framework where a graph is the structure over which data is defined and signals measurements are taken at each vertex. Vehicle movement, internet traffic, cell phone measurements, IoT sensor measurement, and social media relations are just some of the examples. In fact, a special case of GSP is the standard discrete time signal processing (DPS) where a signal is adjacent to exactly two neighbors. This can be modeled by taking the graph that is a cycle (a closed loop of vertices and edges). Other familiar situations fall under the umbrella of GSP; for example, a graph that is a rect-angular lattice grid models digital imaging, etc.

We now very briefly review some of the most salient fea-tures of GSP to motivate the generalization to simplicial complexes. More details can be found in [18], [20], [27].

Given a graph (V,E) where V is the set of vertices and E the set of edges as before, a signal is simply a function

(the reason for the superscript in s0 will become clear in the next section). If the signal at vertex i is si, then one can represent s as a vector or as a formal sum, i.e.

But, according to (3) and (9), the second representation is simply a 0-cochain on X. Signals on X thus precisely correspond to 0-cochains on X, and GSP can thus in a sense be regarded as the study of C0(X:ℝ).

Now recall the notion of the incidence matrices of a sim-plicial complex from (1). In case of a graph, we only have A1, which is of couse the usual adjacency matrix. Another matrix that is relevant is the degree matrix D1, a diagonal matrix whose ith diagonal entry is the degree, or valence, of vi (taken with signs according to the orientation of edges at vi). It is not hard to see that the Laplacian L0 = A1 from (2) is also equal to

0 1 1= −L A D

1TA

One common modification in signal processing is that edges might be assigned weights. In that case, the entries of the adjacency matrix A1 carry the positive and negative weight values instead of the usual ±1. The weight might reflect the actual distance between signal sources, or the

expected similarity between sources, or some other infor-mation. For the sake of brevity, we will ignore the weighted case, although not much would change in the rest of the discussion if we did not.

The spectral study of L0 is one of the cornerstones of GSP. Namely, consider the eigenvalue decomposition

( )00

,...,n

n i ii

a a a v=

↔∑

where U0 is the orthonormal matrix of eigenvectors and Λ0 is the diagonal matrix of eigenvalues. The eigenvalues are important in the study of the clustering of the graph. Clustering is essentially a phenomenon that has to do with connected components. From the point of view of (co)ho-mology, clustering is thus precisely detected by H0(X;ℝ), as the rank of the zeroth (co)homology gives the number of connected components of a space.

From this, one can define the graph Fourier transform (GFT) of the signal s as

The vector X0 represents the projections of the signal s onto the eigenvectors of the Laplacian. I.e. GFT is a de-composition of the diagonal onto the eigenvectors of Λ0. This is analogous to the classical Fourier transform that decomposes the signal onto an orthogonal basis of ei-genfunctions (for more details, see, for example [25]).

Solving for s in the above (and using ) gives

This can be thought of as an expansion of the signal in terms of the eigenvalues of the Laplacian.

It is interesting to observe that, even though a Fourier transform is a tool from analysis, in this context it also depends on the topological structure of the underlying graph X.

4. SIGNALS OVER SIMPLICIAL COMPLEXES

Graph signal processing models many situations, but it is limited in the sense that it only allows for at most two sig-nals to interact, with the interaction represented by an edge between two vertices. Many signal systems interact in more complex ways and cannot be represented by simple flow models that graphs provide. Instead, there might be three or more signals interacting, in which case a natural con-struct to model such interactions are simplicial complexes. In this way, an n-simplex represents a signal defined over an n-tuple of points. Some examples are neural networks [7], [13], collaboration networks [16], [19], and discourse networks [9]. Many more examples are given in [3].

Much of the theory of signals over simplicial complexes extends the one over graphs [3], [4], [12]. Namely, let X(n) be the set of n-simplices of a complex X (i.e. the set of el-

0 0 0To= ΛL U U

0 .To s=X U

10 0

TU U− =(12)

(13)

(15)

(14)

( )0 :s V → ors = U0X0


ements of X of cardinality n + 1; this is not to be confused with the n-skeleton X n which contains the n-simplices but also their faces).

Definition 4.1. A signal over X(n) is a function

( ):ns X n →The collection {s0,s1,...,sd}, where d is as usual the dimension of X, is a signal over the complex X.

Note that s0 is precisely the signal from (12), with X(0) = X.

In analogy with (14), we can consider the eigendecompo-sition of the higher Laplacians (2):

1 1T T T

n n n n n n n n+ += + = ΛL A A A A U U

Then, for each 0 ≤ n ≤ d, define the (complex) Fourier trans-form of order n to be

This of course generalizes the graph Fourier transform (15) and represents the projection of the signal sn onto the eigen-vectors of the nth Laplacian Ln. A signal over the n-simplices can then be decomposed in terms of those vectors as

.T nn n s=X U

The interaction of the Laplacians of various orders is of inter-est in the spectral analysis of the signal; more can be found in [3, III.A].

Now recall the vector space of cochains Cn(X:ℝ) from (9). This is a real vector space of dimension equal to the cardinality of the set X(n) (as it is generated over ℝ by the n-simplices).

Just like a graph signal s0 can be identified with a 0-cochain (13), a signal over the n-simplices can be identified with an n-cochain, i.e.

Since An is a linear transformation of this space, by stan-dard linear algebra we can use it to decompose Cn(X:ℝ) as

Furthermore, we also have a well-known isomorphism

Now recall from (6) that AnAn+1 = 0. In other words, we have that im(An+1) is a subspace of ker(An). This in turn means that a vector in ker(An) can be decomposed into its pro-jection onto im(An+1) and onto its orthogonal complement im(An+1)

┴. Putting this together ultimately implies

Proposition 4.2. The space of n-cochains decomposes as

.n n ns = U X

( )C ;nns X∈

( ) ( ) ( )C ; ker kern n nX⊥

≅ ⊕A A

( ) ( )ker ker Tn n⊥ ≅A A

( ) ( ) ( ) ( )1C ; im ker im .n Tn n nX +≅ ⊕ ⊕A L A

One way to see this is the following: For an a × k matrix A and an k × b matrix B satisfying AB = 0, we in general have

For details, see for example [15]. In our case, where A = An and B = An+1, A

TA+BBT is precisely Ln by (2).

Equation (19) is called the Hodge decomposition of the real vector space Cn(X:ℝ). (The original Hodge decomposition concerns the deRham cohomology of differential forms on a Riemannian manifold, and what we have here can be re-garded as an adaptation to simplicial complexes.)

But basic linear algebra then shows that the middle term in (20) is precisely the nth cohomology group of X (see (10)). We thus have

( ) ( ) ( )im ker imk T T T= ⊕ + ⊕A A A BB B

This is consistent with the familiar and useful notion that Ln has something to do with clustering, which in terms of zeroth cohomology has to do with the number of con-nected components of a space. The above can be under-stood as as generalization of that observation, providing insight into the

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