Interpolation Sammanfattning

UPTEC F16 007

Examensarbete 30 hp Mars 2016

Long Range Channel Predictions for Broadband Systems

Predictor antenna experiments and interpolation of Kalman predictions

Joachim Björsell

Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Abstract

Long Range Channel Predictions for Broadband Systems

Joachim Björsell

The field of wireless communication is under massive development and the demands on the cellular system, especially, are constantly increasing as the utilizing devices are increasing in number and diversity. A key component of wireless communication is the knowledge of the channel, i.e, how the signal is affected when sent over the wireless medium. Channel prediction is one concept which can improve current techniques or enable new ones in order to increase the performance of the cellular system. Firstly, this report will investigate the concept of a predictor antenna on new, extensive measurements which represent many different environments and scenarios. A predictor antenna is a separate antenna that is placed in front of the main antenna on the roof of a vehicle. The predictor antenna could enable good channel prediction for high velocity vehicles. The measurements show to be too noisy to be used directly in the predictor antenna concept but show potential if the measurements can be noise-filtered without distorting the signal. The use of low-pass filter and Kalman filter to do this, did not give the desired results but the technique to do this should be further investigated.

Secondly, a interpolation technique will be presented which utilizes predictions with different prediction horizon by estimating intermediate channel components using interpolation. This could save channel feedback resources as well as give a better robustness to bad channel predictions by letting fresh, local, channel predictions be used as quality reference of the

interpolated channel estimates. For a linear interpolation between 8-step and 18-step Kalman predictions with Normalized Mean Square Error (NMSE) of -15.02 dB and -10.88 dB, the interpolated estimates had an average NMSE of -13.14 dB, while lowering the required feedback data by about 80 %. The use of a warning algorithm reduced the NMSE by a further 0.2 dB. It mainly eliminated the largest prediction error which otherwise could lead to retransmission, which is not desired.

Examinator: Tomas Nyberg Ämnesgranskare: Mikael Sternad Handledare: Mikael Sternad

Sammanfattning

Tr˚adl¨os kommunikation ¨ar en stor del av dagens tekniksamh¨alle, d¨ar allt fr˚an sociala rela- tioner, informationstillg¨anglighet och teknikl¨osningar ¨ar mer eller mindre beroende av snabba, p˚alitliga och tillg¨angliga kommunikationsm¨ojligheter. Dagens mobiln¨at har utvecklats fr˚an att enbart m¨ojligg¨ora mobil r¨ost-kommunikation till att idag ¨aven f¨orse oss med n¨ast intill obegr¨ansad tillg¨anglighet av information, underh˚allning och sociala n¨atverk, n¨ar och var som helst. Mobiln¨atet har utvecklats i takt med ett ¨okande krav p˚a prestanda men ocks˚a i f¨ors¨ok att skapa nya behov hos konsumenter. Denna utveckling kommer att forts¨atta och kommer beh¨ova tillfredsst¨alla nya krav och behov, som ett ¨okat antal uppkopplade enheter, l˚ag-latens- kommunikation och ¨okade datahastigheter. F¨or att uppfylla detta kr¨avs nya teknikl¨osningar.

En av dessa teknikl¨osningar ¨ar att kunna estimera eller prediktera kanaler (f¨or¨andringen av radiov˚agen fr˚an s¨andare till mottagare). Kanalprediktioner kan till˚ata flera basstationer att samarbeta med kommunikationen till enskilda anv¨andare eller minska m¨angden kanalinforma- tion som beh¨over skickas mellan s¨andare och mottagare och p˚a s˚a s¨att frig¨ora resurser som kan anv¨andas till att skicka anv¨andardata eller till˚ata b¨attre kommunikation n¨ar anv¨andare f¨ardas med h¨oga hastigheter.

I det h¨ar arbetet unders¨oks m¨ojligheten att anv¨anda kanalprediktioner genom att interpolera mellan kanalprediktioner med olika prediktionshorisonter. P˚a det s¨attet beh¨over endast en br˚akdel av kanalinformationen skickas mellan s¨andare och mottagare, vilket frig¨or resurser. En algoritm f¨or att uppt¨acka och motverka ”d˚aliga” kanalprediktioner, fr˚an interpolering, kommer ocks˚a att unders¨okas. Kanalprediktionerna som anv¨ands till detta ¨ar, redan gjorda, Kalman- prediktioner fr˚an h¨ogkvalitativa m¨atningar.

Konceptet att anv¨anda en extra antenn p˚a taket av ett fordon, s˚a kallad prediktionsantenn, f¨or att m¨ata kanalen i f¨orv¨ag f¨or den efterkommande antennen kommer ¨aven att utforskas vidare p˚a ny, omfattande, m¨atdata fr˚an centrala Dresden. M¨atdatat ¨ar av s¨amre kvalitet ¨an tidigare unders¨okningar och f¨or att f˚a ut s˚a mycket information som m¨ojligt beh¨ovs brusredusering av m¨atdatat. I detta arbete kommer detta g¨oras dels med ett l˚agpassfilter och dels med Kalman filter med, l¨att varierande, auto-regressiva kanalmodeller. Som utv¨ardering av brusfiltreringen kommer f¨orb¨attringen av prediktionsf¨orm˚agan med hj¨alp av prediktionsatennen j¨amf¨oras med teoretiska resultat.

Interpolation

N¨ar det g¨aller interpolationen mellan kanalprediktioner anv¨ands m¨atdata som kan anses vara brusfri och kanalprediktioner fr˚an samma m¨atdata, 8 och 18 tidssteg, motsvarande ca 10 och 23 ms, in i framtiden. B˚ade linj¨ar och andra-ordningens-polynom-interpolation unders¨oks. Skill- naden i prediktionsfel f¨or de olika interpolationsmetoderna visar sig vara f¨orsumbar, men m¨ang- den kanalinformation som m˚aste skickas fr˚an anv¨andaren (d¨ar prediktionerna ber¨aknas) till basstationen ¨ar mindre f¨or linj¨ar interpolation och d¨arf¨or mer f¨ordelaktig. Den l¨agre predik- tionsgr¨ansen, 8 tidssteg, ¨ar den tidsf¨ordr¨ojning systemet antas ha fr˚an att kanalinformationen skickas, fr˚an anv¨andaren till basstationen, till att kanalprediktionen anv¨ands f¨or att skicka data. Interpolationen ger en prediktionsf¨orm˚aga ca 2 dB h¨ogre ¨an att enbart anv¨anda 8-stegs prediktioner men minskade m¨angden kanalinformation som beh¨over s¨andas med ca 80%.

F¨or att minska antalet f¨orv˚anadsv¨art d˚aliga prediktioner, f¨ors¨oker vi identifiera dessa genom att j¨amf¨ora med de b¨asta, tillg¨angliga prediktionerna som skulle hinna anv¨andas utan att bli utdaterade p˚a grund av tidsf¨ordr¨ojning. Eftersom Kalmanfiltret uppdateras varje tidssteg kom- mer det hela tiden finnas nya 8-stegs-prediktioner tillg¨angliga att j¨amf¨ora med. Kalmanfil- tret generar ocks˚a en uppskattad varians p˚a prediktionsfelet som anv¨ands f¨or att identifiera stora prediktionsfel fr˚an interpolationen. Om kvadraten av avst˚andet mellan interpolations-

prediktionen och 8-stegs-prediktionen ¨ar st¨orre ¨an dubbla, Kalman-genererade variansen av 8-stegs-prediktionen klassas interpolations-prediktionen som d˚alig och nya 8 och 18-stegs predik- tioner skickas f¨or att interpoleras p˚a nytt. Den h¨ar varningsalgoritmen reduserar antalet faktiskt d˚aliga prediktioner med ca 20% utan att signifikant p˚averka prediktionsf¨orm˚agan i genomsnitt eller m¨angden kanalinformation som m˚aste s¨andas mellan anv¨andaren och basstationen.

Prediktionsantenn

Konceptet prediktionsantenn drar nytta av att ett fordon r¨or sig i en, mer eller mindre, best¨amd riktning framm˚at och att efterkommande antenner kommer uppleva en v¨aldigt liknande kanal som den f¨orsta antennen. Genom att kolla p˚a korrelationen mellan prediktionsantennen och hu- vudantennen, ¨over ett best¨amt tidsintervall d¨ar kanalen kan anses vara tidsoberoende, kan trans- formationen fr˚an prediktionsantennskanalen till huvudantennskanalen vara den komplexv¨arda korrelationskoefficienten. Absolutbeloppet av den normaliserade korrelationen mellan anten- nerna avg¨or hur pass bra det g˚ar att prediktera huvudantennen med hj¨alp av prediktionsanten- nen.

Eftersom m¨atbrus och annat brus minskar korrelationen mellan antennerna beh¨over detta filtreras bort s˚a mycket som det g˚ar, utan att p˚averka den faktiska kanalen m¨arkv¨art, f¨or att kunna uppskatta den fysiska korrelationen mellan antennerna. Till att b¨orja med beh¨ovs de olika antennerna frekvens-synkroniseras d˚a det visat sig att antennerna har haft separata frekven- sklockor vid m¨atningarna som var aningen osynkroniserade. Sedan l˚agpassfiltreras m¨atdatat med ett FIR-filter. Det l˚agpassfiltrerade m¨atdatat anv¨ands dels som en typ av brusfiltrering men ¨aven f¨or att skapa en kanalmodell som sedan implementeras i ett Kalmanfilter f¨or att ge ytterligare en typ av brusfiltrering.

Brusfiltreringen visar sig ¨oka det uppskattade signal-till-brus-f¨orh˚allandet, det vill s¨aga min- skar bruset, men n¨ar prediktionsf¨orm˚agan f¨or prediktionsantennen utv¨arderas s˚a ¨ar det ingen st¨orre f¨orb¨attring, vilket ¨ar mots¨agelsefullt mot teorin. Det skulle kunna bero p˚a att brus- filtreringen p˚a n˚agot s¨att ¨aven f¨orvr¨anger den sanna kanalen eller att estimeringsmetoden f¨or signal-till-brus-f¨orh˚allandet ¨ar missvisande. Det skulle ocks˚a kunna vara s˚a att frekvenssynkro- niseringen mellan antennerna inte ¨ar tillr¨ackligt finjusterad och att det kvarvarande frekvensfelet ger den minskade korrelationen mellan antennerna.

Contents

1 Introduction 1

1.1 Aim of the Thesis . . . 1

1.2 Structure . . . 2

2 Background 3 2.1 Wireless Transmission . . . 3

2.2 Coding . . . 7

2.3 Signal Fading . . . 7

2.4 Cellular Systems . . . 8

2.4.1 Multiple Antennas . . . 9

2.4.2 Resource Sharing . . . 10

2.4.3 Coordinated Multipoint Transmission . . . 11

2.4.4 Channel Predictions . . . 12

3 Kalman Channel Filtering/Prediction with AR-Models 14 3.1 Channel Modeling . . . 14

3.1.1 Model Estimation . . . 14

3.1.2 Model Estimation from Noisy Measurements . . . 15

3.1.3 Subsampling . . . 16

3.1.4 State Space Form . . . 17

3.1.5 Multi-Channel Modeling in State Space Form . . . 17

3.2 Kalman Filter . . . 19

3.2.1 Stationary Kalman Filter . . . 20

3.2.2 Kalman Predictor . . . 21

4 The Use of Predictor Antennas on Vehicles 22 4.1 Previous Work . . . 22

4.1.1 Purpose . . . 23

4.2 Theory . . . 23

4.2.1 Estimation of Prediction Coefficient . . . 23

4.2.2 Effects of Measurement Noise . . . 24

4.3 Measurements . . . 24

4.4 Noise Cleansing . . . 26

4.4.1 Data Synchronization . . . 26

4.4.2 Low-pass Filtering . . . 27

4.4.3 Noise Estimation . . . 27

4.4.4 Channel Modeling and Kalman Filtering . . . 28

4.5 Noise Cleansing Performance Assessment . . . 29

4.5.1 Prediction Antenna Performance . . . 29

4.6 Results . . . 30

4.6.1 Potential Performance Gains . . . 31

4.6.2 Noise Cleansing Performance . . . 32

4.7 Discussion and Conclusions . . . 32

4.7.1 Further Work . . . 33

5 Prediction Interpolation 35

5.1 Purpose . . . 35

5.2 Measurements . . . 35

5.3 Model and Kalman Filter Design . . . 36

5.3.1 Prediction Performance . . . 37

5.4 Polynomial Interpolation . . . 40

5.4.1 Design Parameters for Polynomial Interpolation . . . 42

5.4.2 Implementation of Prediction Interpolation in CoMP Systems . . . 43

5.5 Warning Algorithm . . . 44

5.6 Results . . . 46

5.6.1 Warning Algorithm Performance . . . 47

5.7 Discussion and Conclusions . . . 49

Appendix A Derivation of Prediction Antenna Theory 51 A.1 Estimation of Prediction Coefficient . . . 51

A.2 Estimation of Prediction Coefficient for Noisy Measurements . . . 52

Appendix B Figures 53

Appendix C Design of Interpolation Techinque 57

Acronyms

AR Autoregressive.

AWGN Additive White Gaussian Noise.

BER Bit Error Rate.

CoMP Coordinated Multipoint.

CSI Channel State Information.

CSIT Channel State Information at Transmitter.

FDD Frequency Division Duplex.

FIR Finite Implulse Response.

i.i.d. independent and identically distributed.

JT Joint Transmission.

LOS Line of Sight.

MIMO Multiple-Input Multiple-Output.

MSE Mean Square Error.

NLOS Non Line of Sight.

NMSE Normalized Mean Square Error.

OFDM Orthogonal Frequency Division Multiplexing.

PAM Pulse Amplitude Modulation.

PSK Phase Shift Keying.

SISO Single-Input Single-Output.

SNR Signal-to-Noise Ratio.

TDD Time Division Duplex.

WGN White Gaussian Noise.

Notation

A, Φ Boldface captital letters denote matrices.

a Boldface letters denote vectors.

a, α Normal font letters denote scalars.

A^∗ The complex conjugate of A.

A^| The transpose of A.

Aˆ The estimate of A.

E[A] The expectation value of A.

ˆa(t + τ |t) The estimate of a(t + τ ) given a(k) for k = 0, 1, . . . , t.

A  B Element-wise multiplication.

A  B Element-wise division.

1 Introduction

Wireless communication has become a central part of the modern society. Social media, infor- mation exchange and constant accessibility are all now natural parts of the every-day life of citizens of the wealthier part of the world and is constantly spreading. The rapid growth of world economics as well as the evolution in technical areas will allow for more users and devices to utilize better and cheaper services. In 2011 the number of connected devices was expected to have grown to over 50 billion by year 2020 [1], [2]. This projected number has decreased the last couple of years to about 20 billion but it is still a startling number.

As the technique has evolved, our devices have shrunk in size and mobility has improved significantly. To support communication between the expected increasing number of connected devices, many of them mobile, the current network standards needs to be adapted to new demands. The EU project Mobile and wireless communications Enablers for the Twenty-twenty Information Society (METIS) aims to lay the foundation for the next generation of mobile network (5G). The expected demand for a wider variety of services and an improved quality of those are reflected in the formulation of requirements for this new network. The goal is to achieve a system that will support, for example, 1000 times higher mobile data volume per area, 10-100 times more connected devices and 5 times reduced end-to-end latency, compared to the current mobile network (LTE) [3]. There are prototypes of 5G system that are being tested today and, e.g., South Korea have stated that they will have a working 5G system for the Winter Olympics 2018.

These high ambitions will require a massive technical evolution in both existing and new areas of wireless communication. In this work the focus will be on channel estimation/prediction which could enable a system which communicates with multiple base stations simultaneously.

This would have multiple advantages compared to current systems.

1.1 Aim of the Thesis

In this thesis the aim is to evaluate current and develop new ways to predict radio channels for varying requirements. In a first part, this will be accomplished by evaluating the predictor antenna concept on newly acquired measurements which are more extensive and diverse than what have been available so far. The measurement will be investigated for the purpose of finding possible irregularities that can cause misleading results. The measurements will also be processed in order to reduce the amount of noise so that the potential performance of the predictor antenna concept can be properly assessed.

In a second part, the development of new prediction methods will be based on the idea to interpolate between different, already made, channel predictions in order to potentially reduce channel feedback requirements. The interpolation could also open up for new ways to identify bad channel predictions before they are used. An algorithm to exploit this information, a kind of warning algorithm, will be developed and evaluated in order to reduce dangerously bad channel predictions.

The initial thesis goals also included the use of high quality measurements to evaluate a channel estimation technique which enables the estimation of a large amount of channel from different base stations without consuming a considerable amount of the bandwidth. However, the extent of the previous two assignments turned out to be larger than first thought and this third task was therefore decided to be excluded from the thesis.

1.2 Structure

The report will start with a basic introduction to wireless communication in Section 2 and successively describe how current cellular systems work as well as new techniques which we aim to enable with the work done here.

Secondly, Section 3 will derive the theory used for channel modeling and Kalman filter- ing/prediction in such a way that it can be used for improving channel information in different situations. The channel modeling and Kalman theory will be used in both of the following sections.

After that, Section 4 will present the predictor antenna idea which utilizes the exterior of vehicles to use extra antennas as predictions and could possibly accurately predict channel information for users that travel with a high velocity. The theory will be tested on new, extensive measurement data in order to see how the theory works in practice.

Section 5 will lastly propose an idea to reduce both the amount of bad channel prediction from Kalman predictors as well as the required resources for channel information feedback to the base station. The idea is based on estimating channel information by interpolating between prediction of different prediction horizon in time and use fresh, local, channel estimates to warn about possibly bad predictions.

2 Background

2.1 Wireless Transmission

Radio communication is the exchange of information via electromagnetic waves. The informa- tion can be encoded in and extracted from the frequencies, amplitudes and phases of the waves.

The information bearing signal is then modulated onto a carrier wave with a fixed and know frequency when transmitted, often into the air. The modulation is done by multiplying the information bearing signal x(t) with the carrier signal as

y(t) = x(t)A_ccos(2πf_ct + φ₀), (1) where A_c is the amplitude of the carrier wave, f_c is the carrier frequency and φ₀ is the initial phase offset. An example of a carrier modulation is shown in Figure 1 where the information bearing signal is shown to the left and the modulated signal shown to the right. The transmitted signal is demodulated at the receiver and the information bearing signal is extracted.

0 1 2 3 4

Time [ms]

-2 0 2

x(t)

(a) Information bearing signal consisting of 3 si- nusoids with frequencies f1= 0.1 kHz, f2= 0.7 kHz, f3 = 2 kHz and amplitudes A1 = 0.5, A₂= 0.8, A₃= 0.6.

0 1 2 3 4

Time [ms]

-2 0 2

y(t)

(b) Signal in part (a) modulated onto a carrier wave with frequencyfc= 10 kHz and amplitude Ac= 1.

Figure 1: Example of an information bearing, analogue signal modulated onto a carrier fre- quency.

Current cellular systems communicate with digital data, sequences of bits (ones and zeros).

Therefore the information bearing signal has to represent a bit sequence. The transformation from a bit sequence to an analogue signal is also a part of the modulation. This is usually done by adjusting the amplitude, phase, frequency or a combination of these in correspondence to the sequence. A limited bit sequence may be seen as a sequence of M different, equally long bit sequences, M = {m₀, ..., m_{M −1}}. Each bit sequence m_i is called a message and consists of K bits, m_i = {b₀, ..., b_K−1} where K = dlog₂M e. Before the wireless transmission each message is mapped to a unique analogue signal s(t) ∈ S = {s0, ..., sM −1}, called a symbol. The choise of what symbols to be used will not be discussed here. However, the Gram-Schmidt orthogonalization procedure can be used to construct a set of N orthonormal basis functions, N ≤ M , by which all signals ∈ S can be represented by a linear combinations of basis functions [4].

Example 2.1:

Let K = 2 and thus M = {00, 01, 11, 10}. Given a bit sequence, {011101000111000110} the messages sent will be {m1, m2, m1, m0, m1, m2, m0, m1, m3}. We use Pulse Amplitude Modula- tion (PAM) to modulate the messages. Each message will then map to a square pulse with a

unique amplitude. Let Ψ be a normalized square pulse of length T . Our symbols s_i ∈ S can than be expressed as

si = aiΨ, (2)

where ai ∈ {−3, −1, 1, 3}. This way our symbols can be geometrically illustrated in a one- dimensional vector space. The geometrical illustration and the information bearing signal can be seen in Figure 2.

s0 Ψ 00

s1

01

s2

11

s3

10

(a) Symbols illustrated in vector space

Time

T 2T 3T 4T 5T 6T 7T 8T

x(t)

-3 -1 1 3

(b) The bit sequence modulated to an analogue signal with PAM

Figure 2: Illustration of PAM modulation from Example 2.1.

As the data and signal processing is performed on digital computers, conversion between digital and analouge signals are crucial for enabling the kind of processing used today in radio communication. The conversion is done with digital-to-analouge and analouge-to-digital con- verters. Transmitted signals are often constellations of predefined pulses of discrete amplitudes generated by specificly designed hardware. Received signals however, need to be measured at discrete samples in order to convert the analogue measurements to digital values, which in turn can be logically processed. The conversion is done by letting a set of bits map to specific values, quantization levels, as in Figure 3. The number of quantization levels are limited in order to restrict the amount of data needed during the measurements. This introduces an error in the sampled signal as values are rounded to the closest quantization level, this error is called quan- tization error. A finer quantization grid will result in less quantization error but will require the use of more data.

Time [ms]

0 0.5 1 1.5 2

y(t)

-1.75 -1.25 -0.75 -0.25 0.25 0.75 1.25 1.75

Bits

000 001 010 011 100 101 110 111

Quantized Sample True

Figure 3: Digital sampling of an analogue signal with 8 uniformly distributed quantization levels.

In addition to the quantization error the signal is exposed to some noise and interference.

The noise can be in form of e.g. thermal noise generated within power amplifiers. The in-

terference can come from other messages leaking over in time or frequency, or it could come from other transmitter sending at the same frequency. The quality of the signal channel, the transmission medium, is often measured in terms of Signal-to-Noise Ratio (SNR). The SNR, γ, is often expressed in dB and is defined as

γ_dB = 10 log₁₀(γ) = 10 log₁₀ P_r Pn



, (3)

where Pr and Pn is the power of the received signal and the noise.

The received signal is demodulated at the receiver. From the noisy signal, the receiver tries to detect which symbol was sent by matching the received signal to the different symbols. In vector space, that means that the closest symbol is selected. The bits of the detected messages are then put together sequentially to reproduce the transmitted bit sequence. A useful measure for the quality of the transmission is the Bit Error Rate (BER), the ratio between the number of bit errors and number of received bits. A lower SNR decreases the chances of detecting the correct symbol a thus increases the BER. By illustrating the detection regions in a vector space, it can be easier to get an intuition of how the demodulation and detection works.

Example 2.2:

Assume the same bit sequence as in Example 2.1, {011101000111000110}. Given K = 2, the resulting message sequence is {m₁, m₂, m₁, m₀, m₁, m₂, m₀, m₁, m₃}. In this example we use Phase Shift Keying (PSK) to map the messages to a analogue symbol. PSK directly modulates the messages onto the carrier frequency and each message will map to a unique phase offset for the carrier. Our symbols s_i ∈ S can then be expressed as

s_i = r2E

T sin (2πf_ct + 2πi/M ), (4)

given M different symbols, each of length T and energy E [5]. The phase shift can be expressed by a linear combination of

Ψ1 = r2E

T sin (2πfc), Ψ2 =

r2E

T cos (2πfc),

(5)

which, for large f_care close enough to being orthonormal [6]. The symbols can then be expressed as

si = a1

r2E

T sin (2πfc) + a2

r2E

T cos (2πfc), (6)

where each message will map to a unique set of a₁and a₂. Thus the symbols can be geometrically illustrated in two-dimensional vector space. The vector space and the modulated signal of the message signal is shown in Figure 4.

The signal is affected by noise as it received. In this example, the SNR is -17 dB at the receiver which tries to detect the message sequence transmitted. A matched filter is used to find the correlation between the received signal and the two orthonormal symbols [7]. Visually, each message can be placed in the two-dimensionally vector space, determined from the matched filter. Each message will be detected as the symbol corresponding to the area in which it is placed. The noisy, received signal can be seen in Figure 5a. The placement of the messages from the matched filter that is fitted in the picture shown in Figure 5b. Missing messages are placed outside of the shown area but they have been correctly identified in this example.

Ψ₁ Ψ₂

s₁ 01

s2

11

s3

10

s0

00

(a) Symbols with corresponding detection re- gions illustrated in vector space.

Time

T 2T 3T 4T 5T 6T 7T 8T

x(t)

0

(b) The bit sequence modulated to an analogue signal with PSK and fc = 3/2T .

Figure 4: Illustration of PSK modulation from Example 2.2.

Time

T 2T 3T 4T 5T 6T 7T 8T

y(t) ₀

(a) The bit sequence modulated to an analogue signal with PSK and fc = 3/2T with added White Gaussian Noise (WGN) and an SNR of -17 dB.

Ψ1

Ψ₂ s₁ 01

s₂ 11

s₃ 10

s₀ y[T, 2T ) 00

y[3T, 4T ) y[4T, 5T )

(b) Identified symbols from the noisy signal, after matched filtering. The messages detected as the wrong symbol is marked as and the correctly detected messages are marked by . Figure 5: Illustration of PSK modulation from Example 2.2.

Each message is detected and the bits of each message are then combined sequentially to reconstruct the transmitted bit sequence. Since two of the messages are detected incorrectly we have symbol errors. The received bit sequence will differ from the one transmitted. In Table 1 the transmitted and detected messages and bit sequence are compared. The two symbol errors result, in this case, in two bit errors and thus a BER of 2/18 ≈ 0.11. Note that this is an illustrative example and a SNR of -17 dB usually results in a much higher BER, of about 50 %.

Table 1: Transmitted and detected messages and bits from Example 2.2.

Message sequence Bit sequence

Transmitted m1, m2, m1, m0, m1, m2, m0, m1, m3 0 1 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 0 Detected m₁, m₂, m₁,m₁,m₀, m₂, m₀, m₁, m₃ 0 1 1 1 0 1 01 00 1 1 0 0 0 1 1 0

2.2 Coding

To make sure that the correct message is sent, bit errors need to be detected and corrected. In wireless systems the detection, and sometimes the correction, is performed by a decoder in the receiver [6]. This is done by coding the digital message at the transmitter, before modulation, in such a way that the decoder can detect if one or a few bits are misinterpreted in the received signal. Depending on the coding scheme used, the decoder could either correct the error itself or ask for a retransmission of the signal. A better coding scheme, e.g. the possiblity to detect larger errors, often comes at the cost of transmitting more bits for the same message, a larger coding overhead.

As the SNR affects the quality of the received signal, different coding schemes could be used at different SNRs to compensate for the increased amount of symbol errors associated with decreasing SNR. Such adaptive coding schemes can be used to reduce the variations of the remaining BER after decoding, for channels with varying SNR. This comes at the cost of varied performance, or throughput, data bits per time unit.

2.3 Signal Fading

As a wave travels it will spread in space and interact with matter, e.g., by reflection, refraction, diffraction or scattering. The attenuation of the signal at the receiver will be affected by the three main phenomena [8], path loss, shadowing and multipath fading. As the wave travels and spreads, the energy of the wave will also be divided over and increasing area and thus the strength of the signal will decrease with distance. This is referred to as path loss. Thus, a shorter distance between transmitter and receiver generally increases the signal strength. However, shadowing, the decrease of signal strength due to objects partly or completely absorbing the signal, can prevent the signal strength to increase, or even make it decrease, as the distance to the transmitter decreases.

A large object does not only absorb the signal but also reflects parts of it. The reflection of objects allow the signal to take multiple, different paths and still reach the receiver. The different paths will have different phases, amplitudes, and angles of arrival. Due to superposition, the signals might add constructively, increasing signal strength, or destructively, decreased signal strength, depending on the phases of the different signals. Changes due to constructive and destructive addition of multipath components are called multipath fading. Changes can occur by moving either the transmitter or the receiver or by movement of the objects contributing to any signal path. Moving the transmitter, the receiver or reflectors/scatterers will also bring Doppler shifts to different multipath components. Since the angle of arrival is most likely different for the different signal components the Doppler shifts will vary with the angle of arrival. The Doppler shift ∆fdfor a given velocity v and transmitted wavelength λ is given by

∆f_d= v

λcos θ (7)

where θ is the angle the incoming signal with respect to the direction of motion of the receiver. A Doppler spectrum can be made, showing the energy of the different frequency shifts depending on the energy and angle of arrival of all incoming signals. In Figure 6 two different scenarios of incoming signals are shown as well as the corresponding Doppler spectrum.

Rayleigh fading is a model, frequently used in urban environments, where there is assumed to be no Line of Sight (LOS) components and equal incoming energy from all angle of arrivals, due to existence of lots of reflective areas, such as buildings. The Rayleigh fading model assumes that the phases of the multipath components are evenly distributed between −π and π.

v

E θ

(a) Incoming energy E from different an- gles of arrival, θ, with respect to the di- rection of motion with the velocity of v for two different cases and .

-15 -10 -5 0 5 10 15 Frequency shift [Hz]

0

Normalizedenergy

(b) The Doppler spectrum for the cases of incoming energy from different angles of arrival in Figure 6a.

Figure 6: Illustration of the Doppler spectrum from two different cases of incoming signals, and . The velocity of the receiver v is 7 km/h and the frequency of the transmitted signal is 2.66 GHz.

As electromagnetic waves can behave differently for different frequencies when interacting with matter, the fading of a signal can vary over the entire spectrum of the signal. This phenomenon is known as frequency selective fading or a wideband system and can occur if the bandwidth of the signal is relatively large. If the signal instead has a narrow frequency band, the fading will be the same or very similar over the entire spectrum, referred to as flat fading or narrowband system.

2.4 Cellular Systems

A transmitter antenna can be constructed to have directivity. This means that it generates waves that travel mainly in a set of directions. However, due to the nature of wave propagation, a signal will not only be transmitted to the intended position but also in a cone-shaped area originating from the transmitter. If multiple receivers are located close to each other, on the same side of a transmitter, any signals transmitted to each of those receiver will also be picked up by the other receivers, see Figure 7a. Any signals picked up to some extent by a receiver which is not meant for it is called interference and will decrease the ability to read any signal meant to be received by the receiver.

The effective distance for which communication is possible between a receiver and a trans- mitter is limited because waves attenuate as the distance increases. To provide large areas with telecom coverage, multiple transmitters need to be placed in different positions. If all trans- mitters would send with the same frequency/ies, a receiver located at the same distances of two transmitters can experience large interference from one of the transmitters, see Figure 7b.

To limit the interference between transmitters, geographical areas are divided into cells. Each cell is an area associated with a set of receive and transmit antennas located at a base station.

The cells are often geometrically illustrated as hexagons, the most circle-like shape that is not overlapping when covering a larger area [6]. The base stations handles all transmission inside the cell and tries to minimize the interference inside the cell, intra-cell interference. To avoid interference from close by base stations, inter-cell interference, different frequencies are often used in different cells. As the strength of the signals decreases with the distance traveled, fre- quencies can be reused in cells far enough away. In Figure 8 cells with frequency reuse in every third cell, can be seen.

In the earlier stages of the cellular systems the base stations were expensive and placed

sparsely but designed to provide coverage over a large area. As the demand on the cellular systems has increased and will continue to do so, improvements to the old system needs to be done. As the price of base stations has gone down, it is now possible to place more cells in more crowded areas, such as cities, malls and arenas. The increased amount of base stations increases the amount of possible served users or/and possibly the performance for each user in the area. The new cells, called micro cells or pico cells, are smaller and designed to only cover minor areas compared to the original cells, called macro cells, which still provides coverage over less crowded areas. The increase amount of base stations requires fast hand-over between the different base stations. The pico cells and micro cells will be located inside the macro cells. To limit the inter-cell interference, different frequencies could be used for different types of cells.

1

N1

N2

(a) Illustration of how a second receiver, N², can be affected by the signals meant for another receiver, N¹, sent from a transmitter close by, 1.

1 2

N1

N2

(b) Illustration of how two transmitters, 1and 2, sending on the same frequency, each interfere a user located close to their target receivers,N¹respectivelyN².

Figure 7: Illustration of signal propagation and interference.

f1

f2

f₃

f2

f3

f₁

f1

f2

f₃

f2

f3

f₁

f1

f2

f₃

f2

f3

f₁

Figure 8: Illustration of cells and frequency reuse in every third one.

2.4.1 Multiple Antennas

The most basic form of radio transmission consists of one transmit antenna and one receive antenna, Single-Input Single-Output (SISO). The received baseband signal y could then, for narrowband systems, be express as

y = hx + n, (8)

where h, x and n are complex scalars, representing the channel, the transmitted baseband signal and the noise (and possible interference). The noise is often modeled as Additive White Gaussian Noise (AWGN). The theoretical limit to the channel capacity, C bits/s (bps), with a fixed bandwidth B and a time-invariant narrowband channel, is given by the Shannon theorem

C = B log₂(1 + γ), (9)

where γ is the SNR of the channel (3). It is possible to achieve near-capacity transmission by using so called turbo codes but only if the Channel State Information (CSI), the value of h in (8), is known at the receiver, and the SNR is known at the transmitter.

The idea of using multiple antennas at the transmitter as well as the receiver has proven beneficial when it comes to e.g. fading robustness and increasing the SNR at the receiver [9].

The idea is to divide the data stream into parts and transmit the parts simultaneously over different subchannels. The subchannels are a combination of the physical channels between the transmit and receive antennas. If the CSI is known at both the transmitter and the receiver the number of subchannels can at best be the least number of antennas at either the transmitter or the receiver.

A Multiple-Input Multiple-Output (MIMO) system with n receiver antennas and m trans- mitter antennas can represented by





 y1

... y_n





=







h11 · · · h1n

... . .. ... h_m1 · · · h_mn











 x1

... x_m





+





 n1

... n_m





 (10)

or y = Hx + N where y and x is the received and transmitted signal, H is the channel matrix and N is a vector of noises, often assumed to be WGN. The corresponding orthogonal subchannels [˜h₁, . . . , ˜h_r] may be obtained by using a precoding matrix in the transmitter and a combining matrix at the receiver. These matrices are derived from the channel matrix H.

Both the original channels and the decomposed orthogonal (non-interfering) subchannels are illustrated in Figure 9. The capacity of a decomposed MIMO system can thus be derived from the sum of the capacities of each subchannel as

C = B

r

X

j=1

log₂(1 + γ_i), (11)

where γi is the SNR for subchannel i and an equal bandwidth B is assumed to be used in all subchannels. The SNR of each subchannel is affected by the noise, the subchannel gain ˜hi

and the transmission power, of which the last can be controlled. The transmission power is limited due to both legislation and economic reasons. The waterfilling algorithm can optimize the transmission power for a known noise floor, the amount of noise in the channel, and a fixed transmission power restriction. For high SNR the power will be the same for all subchannels and the capacity increases linearly with an increased amount of antennas. For multiple users, the transmission resources have to be shared among the users and the capacity of such a system will be even more complex to derive but will still benefit from multiple antennas. The performance gains come at a cost of larger complexity of the system. Future cellular systems are proposing massive MIMO with up to 256x256 antenna elements, possibly improving throughput by 20 times alone [10].

2.4.2 Resource Sharing

It is essential for a cellular system to handle both downlink (base station → user) and uplink (user → base station) transmission as effective as possible. Because simultaneous communication over the same frequency, full duplex, is not yet a working technique, the downlink and uplink communication needs to be divided either in frequency, Frequency Division Duplex (FDD), or time, Time Division Duplex (TDD), or both [11]. For multiple users inside the same cell, the intra-cell interference needs to be limited or eliminated both in the downlink and the uplink.

This can be done by using orthogonal or semi-orthogonal division of frequencies, time or coding.

x1

x2

x_n

y1

y₂

y_m h_nm

h₁₁

(a) A MIMO system with n transmit antennas and m receiving antennas.

˜ x1

˜ x2

˜ x_r

˜ y1

˜ y₂

˜ y_r

˜h_r

˜h1

˜h2

(b) A decomposed MIMO system with r paral- lel subchannels.

Figure 9: Illustration of a MIMO channel and the derived subchannels.

Time

Frequency

Figure 10: Illustration of time-and-frequency slots for and OFDM system with 8 subcarriers over 8 time slots. The second time slots for all subcarriers are marked by yellow squares .

For code-division, time and frequencies are used simultaneously for all users and the different signals are instead separated by the structure of the code used at the transmitters and the receiver. Frequency and time division instead sends the signal at parts of the frequency band or at specific time slots. One of the most basic forms of frequency division is Orthogonal Frequency Division Multiplexing (OFDM). The idea of OFDM is to instead of sending all data on one carrier frequency fc with a large bandwidth B, experiencing frequency selective fading, the bandwidth is divided into K subcarriers evenly spaced over the original spectrum, each with a bandwidth of BK = B/K and centered and fk = fc+ k(2BK), k = 0, . . . , K − 1. The bandwidth BK of each subcarrier will thus be narrow enough to experience relatively flat fading.

The OFDM resources can be divided into time-and-frequency slots, where each slot stretches over B_K Hz in frequency domain and the symbol time in time domain. The time-and-frequency slots can be illustrated as a grid where each square corresponds to a resource where one symbol can be transmitted.

2.4.3 Coordinated Multipoint Transmission

Since frequency reuse allows each cell to use only parts of the entire spectrum, the spectral efficiency could be improved by letting all cells communicate on all frequencies. To allow the use of all frequencies everywhere, a new approach has to be taken to reduce the inter-cell interference, especially with the expected growth of micro and pico cells. Coordinated Multipoint (CoMP) aims to reduce, or even take advantage of the inter-cell interference, by letting base stations co-operate. By sharing information among a group of base stations surrounding a user, the transmission can be adapted in different ways to control the inter-cell interference. There are

two categories of CoMP, Joint Transmission (JT) and Joint Scheduling/Beamforming [12]. Joint Scheduling/Beamforming shares the Channel State Information at Transmitter (CSIT) and scheduling information, e.g., via a backhaul, a network connecting the different base stations, and adapts the transmission on these information to limit any inter-cell interference.

JT takes another approach and instead sends the same data from multiple base stations to one user in order to improve coverage of high-speed transmission, increasing performance for cell-edge users [13]. In addition to CSIT and scheduling information, data also needs to be shared over the backhaul, increasing the overhead in the backhaul further compared to Joint Scheduling/Beamforming. The gain from JT is hugely dependent on precise CSIT. A common way of estimating channels is by transmitting predefined pilots at specified time-and-frequency slots. The sharing of CSIT via the backhaul, the decision making calculations and distribution of transmission information all introduce delays with respect to when the channel first was estimated. The channel estimates that the transmission calculations are made on are thus outdated at the time of the decision making and even more so at the time of transmission.

The majority of the delay originates from the transmission over the backhaul and therefore the delay will be affected by backhaul capacity and the amount of data that needs to be shared. An upcoming research area is the ability to predict the CSI. Precise CSIT prediction would allow for utilizing the potential of JT CoMP, a larger prediction horizon would reduce the capacity requirements of the backhaul which is limited in many systems today.

2.4.4 Channel Predictions

By using channel predictions the decision calculations would be done on estimates of the future channel and at the time when the calculated signals are transmitted the channel should be as similar to the prediction as possible. Predictions however, are often a complicated problem, especially for something as complex as a Rayleigh fading channels. A typical cellphone user in an urban area receives signals from all possible angles, reflected and diffracted from the surrounding buildings and environment. The signals could be said to form a standing wave pattern, as the base station transmitters and a lot of the surrounding environment are non- moving. Smaller objects that may move, such as cars or people usually affects the channel less than the static objects. This means that the downlink channel will be more or less static for a limited time as long as the receiver, the cellphone, does not move.

Because of the high frequencies used in current cellular systems [11], up to 3.6 GHz, and the even higher frequencies expected to be taken into use in future systems, the signal changes over very small distances. The wavelength λ, given by

λ = v_c

f , (12)

where v_cis the speed of the wave, for electromagnetic waves this is the speed of light, 3·10⁸m/s, noted by c and f is the frequency of the wave. The wavelength of signals with a frequency of 3.6 GHz is 8.3 cm. The prediction horizon, the length of the prediction, can be measured in two relevant quantities, time and distance. The prediction horizon in time is often the relevant measure in a running system as the system delay due to information exchange can be modelled as limited by a maximum allowed delay. However, the prediction horizon in time can be misleading when evaluating prediction performance as a majority of the changes in the channel originate from changes of the position of the receiver. Thus the amount of changes in the channel is hugely affected by the velocity of the receiver. A faster moving receiver would generally be harder to predict over a specific time interval than a slow moving, or static receiver.

The prediction horizon in time d_T can be expressed as a linear relation to the prediction horizon as a distance dλ, expressed in wavelengths, as

d_T = d_λc

vf_c, (13)

where c is the speed of light, v is the speed of the receiver and f_c is the carrier frequency of the signal.

Different methods has been proposed and tested for prediction channels. One idea is to use previous time samples to form a time-invariant model of the Doppler spectrum and predict the future channel from it [14]. Since the model is only an approximation of the reality and because the channel is not time-invariant, new measurements have to be taken into account for every predicted time step. In [14] the model and measurements are weighted together by a Kalman filter to give the optimal linear prediction given white noise. As the model is assumed to be time-invariant the model will only hold for short times, on the order of a second for pedestrian velocities, < 10 km/h, of the receiver. During this time the model has to be estimated and then used in the filter. Thus, new models has to be estimated and used in re-adjusted Kalman filters regularly.

Another idea is making use of extra antennas to predict channels for fast moving vehicles.

Since velocities above 10 km/h often occur when people are traveling by e.g. car, bus or train it would allow for the use of extra instruments to construct a local, moving access point for the travelers inside the vehicle. The use of such extra antennas, so called predictor antennas, placed in front of the actual receiving antenna can measure the channel in a position where the receiving antenna will be moments later. By redirecting all incoming and outgoing communication via the local access point, the channel estimation of the predictor antenna will be used for all user in the vehicle.

Both of these ideas are still in the development phase and even though there are still problems to solve they seems promising and necessary to to allow the gains of JT CoMP.

3 Kalman Channel Filtering/Prediction with AR-Models

Kalman filtering/prediction is the optimal linear filter/predictor given WGN. The Kalman fil- ter/predictor combines a given state space model with measurements in order to estimate the current state or predict future states. In this report the Kalman filter will be used both for noise cleansing of noisy data as well as channel predictions from, what can be considered, noise free data. The equations used to model and filter/predict the channels are mostly the same for the two purposes mentioned above.

3.1 Channel Modeling

In this work the channel will be modelled by Autoregressive (AR) models, a parametrized class of models where the current state is derived from a limited number of previous states, as explained e.g. in [15]. This decision is based on previous work in [16] and [14].

Assume a narrowband transmission channel with scalar, complex-valued channel gain h.

Assume also that a sequence of estimates h_t, h_t−1, h_t−2 of a time-varying scalar channel com- ponent htare available at a discrete time index t. The time between two time indexes typically corresponds to the duration between two available known transmitted symbols (pilot symbols), that are used to obtain an estimate of h_t. An AR model of order n_afor h at time step t can be written as an autoregressive difference equation

h_t= −a₁h_t−1− a₂h_t−2− . . . − a_nah_t−na + v_t, (14) or in vector form

ha − v˜ t= −ht (15)

where ˜h = [ht, ht−1, . . . , ht−na] and where a = [a1, . . . , ana]^|are the complex-valued parameters in the model which determine its nature and vt is complex-valued, zero mean, white noise at time step t. By applying z-transformation on (14), the model can be expressed in the z-domain as

h(z) = z^na

z^na+ a₁z^na−1+ . . . + a_nav(z), (16) and rewritten as

h(z) = z^na

(z − p1)(z − p2) . . . (z − pna)v(z), (17) where p₁, . . . , p_na are the the poles of the transfer function of the model.

3.1.1 Model Estimation

There exist multiple methods for obtaining the model parameters. Here, the autocorrelation method, also known as the Yule Walker method and the Covariance method will be used. Given N_yw, noise free, channel measurements in time, {h₀, h₁, . . . , h_Nyw−1}, called the training segment which is used to model the channel, we can, by using (15) for t = 0, 1, . . . , Nyw− 1 with v_t= 0, form an equation system

Aywa = byw (18)

where a = [a₁, . . . , a_na]^|is the vector of stationary model parameters. For example, for a system with N_yw = 50 and n_a= 3 we have

Ayw=



































0 0 0

h0 0 0

h1 h0 0 h2 h1 h0

h₃ h₂ h₁ ... h₄₈ h₄₇ h₄₆ h₄₉ h₄₈ h₄₇ 0 h₄₉ h₄₈ 0 0 h49



































, byw= −

































 h₀ h1

h2

h3

h₄ ... h₄₉

0 0 0



































, (19)

and thus (18) is a overdetermined system, with more equations than unknowns, and can be solved with a least square method. This can be done by multiplying both sides in (18) by A^∗_yw which gives us the parameter estimates

ˆ

a = (A^∗_ywA_yw)⁻¹A^∗_ywb_yw. (20) In the example (19), if all the values in (19) are used,

A^∗_ywAyw =



 ˆ

r_h(0) rˆ_h(−1) rˆ_h(−2) ˆ

rh(1) ˆrh(0) rˆh(−1) ˆ

r_h(2) ˆr_h(1) ˆr_h(0)



, A^∗_ywbyw=



 ˆ r_h(1) ˆ rh(2) ˆ r_h(3)



, (21)

and

ˆ r_h(τ ) =







XNyw−1−τ

i=0 h^∗_ih_i+τ, τ ≥ 0 ˆ

r^∗(−τ ), τ < 0

, (22)

Thus, A^∗_ywA_yw is an estimate of the autocorrelation matrix of h_t, hence it is called the auto- correlation method, or the Yule-Walker method. If only the values inside the dashed lines ( ) in (19) are used, then the method is called the Covariance method.

Once the model parameters are found (16) and (17) can be used to find the poles. If the matrices in (20) are badly conditioned the least squares solution could yield unstable poles, which should be reflected in the unit circle, while preserving the spectral density, in order to stabilize the model.

3.1.2 Model Estimation from Noisy Measurements

In order to utilize the above methods for real data, which often contains measurement noise, some additional measures have to be taken. Assume that N_yw noisy channel measurements in time, {y0, y1, . . . , yNyw−1}, are given and that the noise e_t is uncorrelated with the channel ht. The measurements are then given by

yt= ht+ et. (23)

If the autocorrelation r_h(τ ) can be estimated for τ = {0, 1, . . . , n_a} from the measurements, (18) can solved the same way as in (20). The autocorrelation of ytis given by

ry(τ ) = E [y^∗_tyt+τ] = E [(ht+ et)^∗(ht+τ+ et+τ)]

= E [h^∗_tht+τ] + E [e^∗_tet+τ] = rh(τ ) + re(τ ), (24)

as the noise is uncorrelated with the channel. This means that if the autocorrelation of the noisy measurements is estimated, as in (22), and we have an estimation of the autocorrelation of the noise, we also have an estimate of the autocorrelation of the channel, from (24). For white noise

r_e(τ ) =

(σ²_e, τ = 0,

0, τ 6= 0, (25)

where σ_e² is the variance of the noise. In this case the model estimation in the above example will be given by

ˆ a =



 ˆ

r_y(0) − σ²_e rˆ_y(−1) rˆ_y(−2) ˆ

r_y(1) rˆ_y(0) − σ_e² rˆ_y(−1) ˆ

ry(2) rˆy(1) rˆy(0) − σ_e²





−1

 ˆ r_y(1) ˆ r_y(2) ˆ ry(3)



. (26)

One needs to be careful when subtracting from an invertible matrix as it can cause the matrix to no longer be positive (semi)definite and hence no longer invertible. If the true values of ry(τ ) and σe are known this should not be the case. However, estimation errors can generate such problems and if so, the subtracted amount could be decreased slightly to have the matrix remain positive (semi)definite.

For the case of multiple subchannels that are assumed to experience the same statistics, the estimation can be improved by incorporating all subcarriers when estimating the autocorrelation of y_t. This can be done by averaging ˆr_y(τ ) for all measured subchannels and use that in (26).

3.1.3 Subsampling

These model estimation methods minimizes the one step prediction error, meaning that the model tries to predict the output ht+1 at the next time step, based on the model structure (14).

If the time is shifted one step (t → t + 1), (14) is given by

ht+1= −a1ht− a₂ht−1− . . . − a_naht−na−1+ vt+1, (27) where the parameters {a1, . . . , ana} are adjusted so that the variance of the residual sequence v_t+1 is minimized.

When performing channel prediction, the desired prediction horizon will often be longer than the basic time-step in (14), which corresponds to the time duration between two utilized channel measurements. An AR model that is adjusted to minimize the one-step prediction error might work sub-optimally when it is utilized to generate longer-range predictions. It is therefore desirable to let the model prediction horizon be of the same order of magnitude as the prediction horizon of any Kalman predictor. To retain the optimality of the model estimator for longer prediction horizons, every τ :th time step can be used instead of every single time step. Implementing this, a subsampled model with the structure (14) can be rewritten as

h_t= −b₁h_t−τ − b₂h_t−2τ − . . . − b_nah_t−naτ + v_t, (28) on which a z-transform would yield a transfer function with the poles p_sub,1, . . . , p_sub,na, like in (17) when a z-transform is applied to (14). The model estimation uses the same equation as before (18) but the time step between each channel measurement on the same row in Ayw will be τ instead of 1. An example can be seen in Appendix 8.A in [16]. The model parameters {b_i} and corresponding poles {p_sub,i} derived from the subsampled data can be modified to update the model at each time step as in (17), instead of every τ :th time step, but still optimized for minimizing the model error for τ :th time steps ahead. This is necessary for the model to work

with the Kalman filter as it is updated for every time step. The modified poles are then given by

pi = p^1/τ_sub,i, (29)

for i = 1, . . . , na, according to [14]. The subsampling can produce aliasing effects if τ is too large relative to the fastest time-variations of the channel [17]. The aliasing can be avoided by reducing the subsampling, τ in (28) and (29).

3.1.4 State Space Form

The channel model in (14) needs to be expressed in state-space form, in order to fit the Kalman equations. This is done by forming three state space matrices with the help of the poles of the model {p₁, . . . , p_na} in (17) or in (29) if subsampling is used. Here, the model is realized on state space form using a diagonal realization,

xa(t + 1) = Axa(t) + Bva(t),

h(t) = Cxa(t), (30)

where

A =







p₁ 0

. ..

0 p_na





, B =







[(p₁− p₂)(p₁− p₃) · · · (p₁− p_na)]⁻¹ ...

[(p_na− p₁)(p_na− p₂) · · · (p_na− p_na−1)]⁻¹





, C =





 pⁿ₁^a−1

... pⁿ_n^a_a⁻¹





, (31) and where xa(t) and h(t) is the state vector and the channel component at time step t. The dimensions of the matrices are given by

xa∈ C^na×1, A ∈ C^na×na, B ∈ C^na×1, C ∈ C^1×na.

The process noise of the model, va is assigned with a constant covariance matrix Q that is adjusted to measured data. Thus, the statistical properties are assumed not to change over time. The noise is also assumed white, zero mean, independent and identically distributed (i.i.d.) and independent of the state vector xa. The covariance matrix is given by

Q = E [va(t)va(t)^∗] . (32)

3.1.5 Multi-Channel Modeling in State Space Form

Under the assumption that multiple subchannels experience the same statistics and that the subchannels of an OFDM system are some what correlated, the Kalman filtering/prediction can be improved by including multiple subchannels in one model. The extra information will come from the correlation between the process noises of each subchannel, va(t) in (30). These assumptions will be assumed to be true for all subchannels from one base station to one receiver.

The joint modeling of multiple subchannels is done by increasing the dimension of the model.

Due to the increased computational complexity of the Kalman filter related to increased model order, the number of included subchannels in one model needs to be limited. However, by modeling a limited number of adjacent subchannels in each model, we get most of the gain at the same time as we limit the computational complexity. This is due to the relatively