Channelidenticationmethodsforcomplex-valuedtransmissions PhD.Thesis

Department of Measurement and Information Systems

PhD. Thesis

Channel identication methods for complex-valued transmissions

Author:

Barna Csuka

Electrical Engineer M.Sc.

Supervisor:

Dr. Zsolt Kollár

Associate Professor

Budapest

2021

All rights reserved. No part of this thesis or thesis booklet may be reproduced or used in any manner without written permission of the copyright owner.

Budapest University of Technology and Economics Department of Measurement and Information Systems H-1117 Budapest XI., Magyar Tudósok körútja 2., Building I

E-mail: csuka{at}mit.bme.hu

In the last decades, one of the most growing communication technology is the wireless data transmission.

These developments provide facilities that the devices can be always online, can be always connected to the network(s) to share data, voice, position, images, or videos, wherever the device and its user is: at home, on a vehicle, at the top of a mountain, or in the middle of an ocean. To serve these demands, the transmission methods are permanently improved in many aspects: the data rate of transmission and the amount of users in a network cell. While a voice transmission during a phone call or a short text message require kbit/s as transmission speed, on the other hand, a live HD stream does not work if at least 5-10 MBit/s is not available as data rate. The second aspect is that thousands of users can be connected to a cell for example in a stadium; therefore, the aggregated requested data rate can be even in the range of 10-100 Gbps/s.

These scenarios call for high quality communication systems, where the unfavourable transmission param- eters have to be measured and compensated. One of the main parameter is the transfer characteristic of the transmission channel, which is inuenced by the given stationary objects of the environment (like buildings, mountains, bridges, etc.), and the eect of the temporary, moving objects is also signicant. If the network would be wired, or at least, the environment would be stationary, then it would be enough to measure the characteristic once. Having changing environment, this measurement, the channel estimation have to be periodically repeated to retrieve a current description about the transmission conditions. To perform a channel estimation, either the transmitted signal has to be measured, or the receiver has to know the transmitted signal. Considering that the measuring of the transmitter is not possible, it has to be send such patterns that are known for the receiver as well.

The patterns that are applied in the transmissions are special data sequences, which can be dened either by given generation rules or by certain values. These assumptions are known by the receiver as well: it knows not only the received signal but the original one as well; thus, the channel estimation and its compensation can be performed. Depending on the demands of the transmission network, many types of these sequences have been developed and described in the past 50-70 years. Some of them provides large number of the connections, an other part of them ensure very good correlation properties to perform high precision channel estimation. In this thesis, the Golay sequences are investigated: they have very favourable auto-correlation function for channel estimation purposes.

One of the main aspect of these channel estimation calculations is the complexity (or runtime). Considering that the high data rate networks operate with short frame time, the estimation algorithms have to be enough fast to t into the frame time. To full these requirements, the complexity of the calculations has to be reduced, or the calculation steps has to be distributed over the frames. Both of these approaches are applied in this thesis.

For the estimation, a commonly used algorithm is the Wiener ler whose application is require to perform a matrix inversion. To accelerate this operation in sliding window manner, Kraker presented a new evaluation method. Using his algorithm, the Wiener-Hopf equation can be continuously calculated and updated making possible to use it for high data rate applications as well.

A new challenge of the last 20 years is the IQ data transmission, where the baseband signal is complex- valued to provide application with modulations having such symbols that are located on the complex plane. For processing these signals and for performing the channel estimation, the conventional methods and algorithms have to be improved to handle complex-valued data as well. As a new contribution of this thesis, all of the presented estimation methods are extended for complex-valued inputs. Their complexity is also derived, how it increases by using complex-valued arithmetical units. To illustrate and investigate the behaviour of the presented methods and algorithms, a simulation program package is also developed as part of the research.

List of gures 6

List of tables 7

List of symbols 8

1 Introduction 13

1.1 Motivation . . . 13

1.2 Overview of special sequences . . . 14

1.3 Overview of IQ transmission model . . . 16

1.4 List of publications . . . 18

2 Complementary Golay sequences 19 2.1 General description of the Golay sequences . . . 19

2.2 Generation and correlation of Golay sequences . . . 23

2.3 Channel estimation based on Golay sequences . . . 25

2.4 Complexity of channel estimation . . . 34

2.5 Simulation results . . . 35

2.6 Summary and new scientic results . . . 39

3 Investigation and optimisation of the RSC4BI-algorithm 40 3.1 Principle of adaptive ltering . . . 41

3.2 Recursive calculation of the Wiener-Hopf equation . . . 42

3.3 Complexity analysis for updating the correlation matrices . . . 51

3.4 Complexity analysis for the RSC4BI-algorithm . . . 56

3.5 Simulation results . . . 60

3.6 Summary and new scientic results . . . 68

4 Complex-valued extension of RSC4BI-algorithm 70 4.1 Principle of adaptive ltering for complex-valued signals . . . 70

4.2 Recursive calculation of the complex Wiener-Hopf equation . . . 71

4.3 Complexity analysis for complex-valued recursive Wiener-Hopf equation . . . 74

4.4 Simulation results . . . 75

4.5 Summary and new scientic results . . . 82

5 Conclusion and future work 84 A Denition of Golay sequences using polynomials 86 B Addition and multiplication of complex numbers 87 B.1 Addition of two complex numbers . . . 87

B.2 Multiplication of two complex numbers . . . 87

C Inverse of a hypermatrix with 4 submatrices 89

E.1 Hermitian auto-correlation matrix . . . 93 E.2 Inverse of a Hermitian matrix . . . 94

Bibliography 95

1 Structure of real-valued data modulation and transmission . . . 16

2 Structure of IQ data modulation and transmission . . . 16

3 Structure of Golay lter given by Budi²in . . . 23

4 Structure of ecient Golay lter given by Popovi¢ . . . 23

5 Application of Golay sequences in SISO and MIMO transmissions . . . 24

6 Structure of optimised Golay lter given by Donato . . . 25

7 Normalized correlations and their sum over time with ideal transfer channel . . . 26

8 Normalized correlations and their sum over time with multipath propagation . . . 26

9 Structure of the applied preamble in IEEE 802.11ad standard . . . 27

10 Structure of Golay lter used in IEEE 802.11ad standard . . . 27

11 Example for pair of correlations to the time domain channel estimation using Golay lter 28 12 Parts ofX_A andX_B used for channel estimation . . . 29

13 Parts ofXA andXB for extended channel estimation . . . 29

14 Normalized correlations over time with complex-valued multipath propagation . . . 31

15 Normalized sum of correlations over time with complex-valued multipath propagation . . 31

16 Structure of the Golay simulation program . . . 35

17 Method of single and continuous transmission . . . 35

18 BER of real channel estimations using Golay sequences with length of 32 . . . 36

19 BER of complex channel estimations using Golay sequences with length of 32 . . . 37

20 BER of complex channel estimations using Golay sequences with length of 64 . . . 37

21 BER of complex channel estimations using Golay sequences with length of 128 . . . 38

22 BER of complex channel estimations using Golay sequences with length of 256 . . . 38

23 Block diagram of an adaptive lter . . . 41

24 Eect of the permutation matrix . . . 44

25 Column-wise update of the observation matrix . . . 44

26 Row-wise update of the observation matrix . . . 45

27 Update of the auto-correlation matrix using permutation . . . 45

28 Structure of the correlation matrices and their inverses . . . 48

29 Update of of the auto-correlation matrix using modied permutation . . . 50

30 Number of additions to update the auto-correlation matrix . . . 54

31 Number of multiplications to update the auto-correlation matrix . . . 54

32 Number of additions to update the auto-correlation matrix . . . 55

33 Number of multiplications to update the auto-correlation matrix . . . 55

34 Number of additions to update the inverse of the auto-correlation matrix . . . 58

35 Number of multiplications to update the inverse of the auto-correlation matrix . . . 58

36 Average runtime of RSC4BI on computer . . . 61

37 Avergae runtime of RSC4BI on microcontroller . . . 61

38 RMSD of RSC4BI-algorithm for multisine signal over time . . . 63

39 RMSD of RSC4BI-algorithm for noisy multisine signal over time . . . 63

40 RMSD of RSC4BI-algorithm for sine wave over time . . . 64

41 RMSD of RSC4BI-algorithm for noisy sine wave signal over time . . . 64

44 RMSD of RSC4BI-algorithm over time depending onK withN = 20. . . 66

45 RMSD of RSC4BI-algorithm over time depending onK withN = 40. . . 66

46 Comparison of the dierent methods depending on lter length if K= 20 . . . 67

47 Comparison of the dierent methods depending on lter length if K= 30 . . . 67

48 Runtime of complex-valued RSC4BI on computer . . . 76

49 RMSD of RSC4BI-algorithm for 16-QAM signal over time . . . 77

50 RMSD of RSC4BI-algorithm for noisy 16-QAM signal over time . . . 77

51 RMSD of RSC4BI-algorithm for 256-QAM signal over time . . . 78

52 RMSD of RSC4BI-algorithm for noisy 256-QAM signal over time . . . 78

53 RMSD of RSC4BI-algorithm for complex Gaussian white noise with µ= 0.1 over time . . 79

54 RMSD of RSC4BI-algorithm for complex Gaussian white noise with µ= 0.01over time . 79 55 RMSD of outliers for complex Gaussian white noise withµ= 0.1over time . . . 80

56 RMSD of a single outlier for complex Gaussian white noise withµ= 0.1over time . . . . 80

57 Comparison of the dierent methods depending on lter length if complex signals applied 81 58 RMSD of RSC4BI if the imaginary part of the main diagonal not removed . . . 82

B1 Addition of two complex numbers in Cartesian form . . . 87

B2 Multiplication of two complex numbers in Cartesian form . . . 88

B3 Multiplication of two complex numbers by Karatsuba-algorithm . . . 88

B4 Multiplication of two complex numbers in exponential form . . . 88

D5 Block diagram of lter design by DurbinLevinson-algorithm . . . 91

List of tables

2 Possible combinations of GF and EGF . . . 24

3 Parameters of real-valued Golay lter . . . 29

4 Parameters of complex-valued Golay lter . . . 30

5 Computational requirements for updatingRby permutation . . . 52

6 Computational requirements for updatingpby permutation . . . 52

7 Number of additions to update the correlation matrices . . . 53

8 Number of multiplications to update the correlation matrices . . . 53

9 Complexity evolution of matrix inversion . . . 56

10 Computational requirements to updateR⁻¹ in general case . . . 57

11 Computational requirements to updateR⁻¹ in optimised case . . . 59

Abbreviations

3G Third Generation Technology Standard for Broadband Cellular Networks 4G Forth Generation Technology Standard for Broadband Cellular Networks 5G Fifth Generation Technology Standard for Broadband Cellular Networks 6G SIxth Generation Technology Standard for Broadband Cellular Networks ADC Analog-to-Digital Converter

ASK Amplitude Shift Keying

AWGN Additive White Gaussian Noise

BER Bit Error Ratio

BPSK Binary Phase-Shift Keying CDMA Code-Divison Multiple Access

CEF Channel Estimation Field

CFRE Channel Frequency Response Estimation

CIR Channel Impulse Response

CIRE Channel Impulse Response Estimation DAC Digital-to-Analog Converter

DFT Discrete Fourier Transform

EGF Ecient Golay Filter

FBMC Filter Bank Multicarrier

FFT Fast Fourier Transform

FIR Finite Impulse Response

FPGA Field-Programmable Gate Array

FSK Frequency Shift Keying

FZC Frank-Zado-Chu

GF Golay Filter

GPS Global Positioning System

HDFT Hopping Discrete Fourier Transform IDFT Inverse Discrete Fourier Transform

IQ signal Complex signal having a real and an imaginary component ISI Intersymbol Interference

LFSR Linear Feedback Shift Register

LPF Low-Pass Filter

LS Least Square

LTI Linear Time-Invariant

LU Lower-Upper

MIMO Multiple-Input and Multiple-Output

MMSE Minimum Mean-Square Error

MMSEC Minimum Mean-Square Error Compensation OFDM Orthogonal Frequency-Division Multiplexing

OGF Optimised Golay Filter

oSDFT Observer-Based Sliding Discrete Fourier Transform

PN Pseudo-noise

QAM Quadrature Amplitude Modulation

RAM Random Access Memory

RDFT Recursive Discrete Fourier Transform

RMSD Root Mean Square Deviation

RSC4BI Recursion with Splitting the Correlation matrix into 4 Blocks for Inversion SDFT Sliding Discrete Fourier Transform

SISO Single-Input and Single-Output

SNR Signal-to-Noise Ratio

STF Short Training Field

SWF Sliding Wiener Filter

WF Wiener Filter

WH Wiener-Hopf

ZFC Zero Forcing Compensation

C Set of complex numbers

N Set of natural numbers

N⁺ Set of positive natural numbers

R Set of real numbers

0 Zero matrix

A Temporary matrix to simplify the notation ofR[n]

B Temporary matrix to simplify the notation ofR[n+ 1]

d Vector of a discrete reference signal

e Identity vector

H Hadamard matrix

h Coecients vector of a discrete lter

hopt Optimal settings of a estimated discrete lter coecients

I Identity matrix

Pρ Matrix of aρpermutation

p Cross-correlation vector

R Auto-correlation matrix

U Temporary matrix to simplify the notation ofR⁻¹[n]

V Temporary matrix to simplify the notation ofR⁻¹[n+ 1]

X Observation matrix of a discrete input signal y Vector of a discrete output signal

B Bandwidth

CM M SE Discrete Fourier transform of a minimum mean-square error compensation signal C_ZF Discrete Fourier transform of a zero forcing compensation signal

c_{M M SE} Discrete compensation signal using minimum mean-square error compensation cZF Discrete compensation signal using zero forcing compensation

d Discrete reference signal

fc Carrier frequency

Hch Discrete frequency response of a transfer channel

j Immaginary unit j=√

−1

K Coecients number of a lter

k Filter coecient index

L Number of estimations

l Estimation index

m Discrete frequency index

N Samples number of a discrete signal

n Discrete time index

P_s Signal power

P_n Noise power

Xrec Discrete Fourier transform of a received signal Xtr Discrete Fourier transform of a transmitted signal

x Discrete input signal

xrec Discrete received signal x_tr Discrete transmitted signal

y Discrete output signal

δ Discrete Dirac delta function

ε Discrete error signal

η Signal-to-noise ratio

µ Expected value

Ξch Discrete Fourier transform of stochastic noise of transmission channel

ξ Stochastic noise

ξ_ch Stochastic noise of transmission channel

ρ Descriptor of a permutation

ρk Delay element of a Golay lter

σ Standard deviation

σ² Variance

GA Golay complementary sequences

G_A^{DF T} Discrete Fourier transform of Golay complementary sequences N µ, σ²

Normal distribution withµexpected value andσ² variance N² µ, σ²

Bivariate normal distribution withµexpected value andσ² variance Operators

∗ Convolution

|·| Absolute value of the parameter [·] Discrete signal indexing

(·) Complex conjugate of the parameter c(·) Estimated value of the parameter

∂f(·)

∂(·) Derivative of the functionf with respect to the parameter (·)⁻¹ Multiplicative inverse of the parameter

(·)^H Hermitian (or conjugate) transpose of the parameter (·)I Imaginary part of the parameter

(·)ij Element in thei-th row andj-th column of the parameter (·)R Real part of the parameter

(·)^T Transpose of the parameter det(·) Determinant of the parameter E{·} Expected value of the parameter C(·) Cost function of the parameter L_(·) L-function of the parameter

O(·) Computational complexity of the operation

R_(·) Non-periodic auto-correlation function of the parameter U_(·) U-function of the parameter

X_(·) Correlation function of the parameter calculated by Golay correlator

1 Introduction

1.1 Motivation

Nowadays, in the rst quarter of the 21st century, one of the most researched eld of the electrical engineering is the wireless communication based on electromagnetic waves. Firstly, during the 20th cen- tury, they had been used for audio broadcast, which was later expanded for video transmission as well.

Since the Millennium, the telecommunication industry has been developing and growing exponentially, and now the fth generation technology standard for broadband cellular networks (5G) is the currently introduced technology, but the sixth generation technology standard (6G) is already envisioned and de- signed as well. The 5G networks are able to handle 10 times more devices than the forth generation technology standard (4G); therefore, it allows to connect sensors, cars, household devices to the network forming the Internet of Things (IoT). Furthermore, the high number of the connectable sensors and their secured communication protocols enables them to use these networks e.g. in healthcare or in industry leading to the Fourth Industrial Revolution or the so called Industry 4.0.

Unfortunately, the wireless transmissions are less robust than the wired ones. In the later case, as long as the wire remains intact, usually only the superposed noise has to be taken into account during transmission. Controversy to that, if there is wireless connection between the devices, further distortions may appear as indirect transmission paths leading to multipath propagation, which can be modelled as follows. There are many paths between the transmitter and the receiver: a direct one and some others that are reected. These paths have dierent lengths; thus, the propagation times of the waves are also dierent causing time delay between the waves at the receiver side. Furthermore, the attenuation of the paths is dierent; therefore, the received waves have various magnitudes. These phenomena can be described by an impulse response where the amplitude of the coecients are proportional to the magnitudes, while the phases of the coecients are proportional to the time delays.

Considering these circumstances, the transmission channel has to be determined to compensate the eects of the multipath propagation, which requires system identication performed generally as follows.

The observed system is excited by a signal, which is commonly a composition of sine waves with random phases (so called multisine signal) or noise. Using these signals and measuring the system response, the frequency response can be estimated. An other approach is the time domain identication: in this case, applying a right exciting signal, the system impulse response can the observed. Having a wireless transmission for which the channel has to be identied, these conventional methods can not be applied any more. The distance between the transmitter and the receiver can grow up to hundreds of kilometers as well; thus, the excitation and the response of the system can not be observed at the same time. In this case, however, if the receiver knows the transmitted signal, then it can identify the transmission channel.

Considering that the above mentioned signals have random parameters (noise, multisine signal), such sequences have be transferred, properties of which are similar to the previously mentioned signals. These sequences can be considered as pseudorandom signals, but they are predened; therefore, the receiver knows the excitation; thus, channel identication can be performed. These sequences have many variants, and they are standardized for each type of networking. In this thesis, the Golay sequences are introduced and analysed for telecommunication applications.

The area, where the radio waves spread and the conditions of the transmission change permanently: the buildings have always the same position, but the vehicles and the people can move if it is windy, then the reection from trees changes, and so on. Therefore, the result of the channel identication shows only the

temporary conditions, it can not be considered as stationary. In this case, if the channel estimation does not give satisfactory results over longer time, it has to be periodically estimated for each transmission, in each seconds, milliseconds, etc. This means that the computation time of channel identication should be reduced to milliseconds or even to microseconds. This reduction can be reached by two approaches:

either the computation is accelerated being shorter than the transmission frame, or the computational load is distributed over the time, and the current results are evaluated from the previous ones.

•

Nowadays, the signicance of the IQ modulation increases because of the higher data rate and the better spectral eciency. Having two independent signal sources, they can be superposed into a single wave using cosine and sine wave as carriers. If the signal sources are considered as real and imaginary part of a complex number, then this model can be described by complex-valued arithmetics. In this case, the conventional channel identication methods may be extended to handle complex-valued signals as well. Furthermore, the computational time becomes more crucial, deducing the complex-valued compu- tations to real-valued arithmetic units requires longer runtime. Therefore, in this thesis, the presented and investigated identication methods are extended for complex-valued signals as well.

The structure of the thesis is the following. Three methods are presented: transfer channel estimation using Golay sequences (Sec. 2), evaluation of Wiener-Hopf equation in sliding window manner using recursion with splitting the correlation matrix (Sec. 3), and its extension for complex-valued signals (Sec. 4). In each section, a general description is given about the presented method. After that, complexity analysis is performed, then the theoretical results are proved by simulations. At the end of each section, a brief summary concludes the results, and it introduces the new scientic theses.

1.2 Overview of special sequences

In this section, a brief and selected overview is given about the dierent sequences, which are commonly used for communication purposes. The complementary Golay sequences are not described in this section because they are one of the main topic of this thesis; therefore, they are detailed introduced in Sec. 2.

1.2.1 Pseudo-Noise sequences

Pseudo-noise (PN) sequences are semi-random periodic series of0and1bits in which the appearance of the bits seems to be random, but they may contain repeating patterns. Without knowing the code, the pattern, or the generation rule, the received sequences seem to be noise. The PN sequences should satisfy the following properties [1]:

• Balance property: the number of 0and1 bits have to be equal.

• Run property: the length of the same consecutive bits has to fulll given limitations during the construction.

• Correlation property: the auto-correlation function of the sequence has to ensure high sidelobe suppression.

• Shift and Add: after shifting a PN sequence, the XOR connection of the shifted and the original ones gives back the original sequence with another shift.

1.2.1.1 M-sequences:

Maximal length sequences or M-sequences are such PN sequences that are generated by Linear Feedback Shift Registers (LFSR). These sequences can be described by a characteristic polynomial p(x) =PM

i=0pixⁱwherepiis the feedback connection coecients [2]. Having bad cross-correlation prop- erties, the usage of these sequences are nowadays limited in multi-user environments.

1.2.1.2 Gold sequences:

Gold sequences were introduced by Gold in 1967 [3], and they can be generated based on M-sequences as follows: the so called preferred pair of sequences are chosen from the set of M-sequences. After that, a modulo 2 addition is performed over the pair resulting a Gold sequence. Because of the good cross- correlation properties, these sequences are used in asynchronous Code-Division Multiple Access (CDMA), in satellite communications, or in Global Positioning System (GPS).

1.2.1.3 Kasami sequences:

Kasami sequences were introduced by Kasami in 1966 [4]. They can be generated similarly to the Gold sequences; however, not only a modulo 2 addition is performed over the pairs, but these sequences are decimated as well. These Kasami sequences make it possible for large number of users to be present on the same network; thus, they are widely used in third generation networks (3G).

1.2.1.4 Barker sequences:

Barker sequences are aperiodic binary sequences containing+1 and −1 values, which was rstly de- scribed by Barker in 1950s [5]. Despite the limited length of these sequences, they are widely used for communication purposes (for example: channel estimation in IEEE 802.11 standard). Considering that the implementation of the required algorithms is very simple, these sequences are very suitable for low cost microcontrollers and Field-Programmable Gate Arrays (FPGAs).

1.2.1.5 Walsh-Hadamard sequences:

Walsh-Hadamard sequences are binary sequences that are based on the Hadamard matrices. If aH1= (1) 1×1 Hadamard matrix is given, then the HN matrices can be constructed through the following recursion,

HN = H_N/2 H_N/2 HN/2 −HN/2

!

. (11)

Using these matrices, the Walsh-Hadamard sequences can be given by extracting the rows of the matrices, which sequences are orthogonal to each other. Although they do not have optimal auto-correlation prop- erties (e.g., there are multiplex peaks), they are often used in CDMA systems because of their orthogonal properties.

1.2.2 Frank-Zado-Chu sequences

Frank-Zado-Chu (FZC) sequences are the improvement of the Walsh-Hadamard sequences [6, 7], and they are complex-valued polyphase sequences with ideal correlation properties and constant amplitude.

AnN-long FZC sequence is dened as follows

xFZC[n] =





 e

j2πr N

n2 2 +qn

forN ≡0 (mod 2), e^j2πrN (ⁿ⁽ⁿ⁺¹⁾2 +qn) forN ≡1 (mod 2)

(12)

whereqandr are arbitrary construction integers that are relative primes ofN. Considering their corre- lation properties and their complex values, they are widely used in such networks where IQ transmission is used, like in 4G.

1.3 Overview of IQ transmission model

Before the introduction of the IQ transmission, a brief overview is given about the conventional transmissions, which can be seen in Fig. 1. After the data generation, such modulation is applied e.g., frequency shift keying (FSK) or amplitude shift keying (ASK) symbols of which are purely real in the

Preamble and

data generation Modulation

(e.g., FSK) ×

× LPF Demodulation Receiver

bits

cos (2πfc)

cos (2πfc)

bits Transmitter

Receiver

transmitted signal

received signal

Figure 1: Structure of real-valued data modulation and transmission (fcis the frequency of the carrier signal)

Preamble and

data generation Modulation (e.g., QAM)

×

×

+

×

×

LPF

LPF

+ Demodulation Receiver bits

real part

imaginary part

cos (2πfc)

sin (2πf_c)

cos (2πfc)

sin (2πfc)

real part

imaginary part

bits Transmitter

Receiver

transmitted signal

received signal

Figure 2: Structure of IQ data modulation and transmission (fcis the frequency of the carrier signal)

modulation alphabet; thus, a R → R mapping is performed. The modulated signal is generated by a carrier wave, which is usually a sine/cosine wave with frequency fc). This radio signal is transmitted either over wired or over wireless connection.

The receiver converts the radio signal down to the baseband, and it reduces the disturbing components and noise by a low-pass lter (LPF), thereafter the demodulator generates the received bits. These bits may be identical to the transmitted ones, but if the channel distortions can not be compensated completely by the modulation, then bit errors may occur. To reduce the bit error ratio (BER), dierent coding schemes can be applied to increase the fault tolerance of the transmission.

These main principles are the similar in the case of IQ transmission as well (Fig. 2), but the modulation diers: instead of R→R, a complex-valued modulation alphabet is applied with a mapping R→Cas in the case of quadrature amplitude modulation (QAM). This signal is separated to real and imaginary parts, which are modulated by two carrier waves. Their phase dierence is set to get orthogonal signals allowing it to add them and transmit them as a single signal. Due to this orthogonality, the receiver can separate the two waves. Therefore, the real and the imaginary parts can be used to form a complex-valued signal after down-conversion and ltering. After these steps, this signal can be demodulated. If bit errors occur after demodulation, then the quality of the transmission can be increased by adding coding schemes to reduce BER.

Comparing these methods, the IQ transmission has 2-times higher spectral eciency than the real- valued transmission. Having a baseband signal with bandwidthB, after up-conversion, the radio channel is located aroundfc with the same bandwidthB. In the case of real-valued transmission, the channel is used only by a cosine wave. On the contrary, if IQ transmission is applied, then theB-wide part of the spectrum is used for communication not only by a cosine wave but by a sine wave as well. Due to their orthogonality, they can be located at the same frequency of the spectrum; thus, the spectral eciency increases.

1.4 List of publications

This thesis is the results of the work that was published in the following journals and selected confer- ence contributions.

1.4.1 Journal papers

J1. B. Csuka and Zs. Kollár. Software and Hardware Solutions for Channel Estimation based on Cyclic Golay Sequences. Radioengineering, 25(4):801807, December 2016. Impact factor: 0.944

J2. B. Csuka and Zs. Kollár. R-DFT-based Parameter Estimation for WiGig. Periodica Polytechnica - Electrical Engineering and Computer Science, 61(2):224230, May 2017

J3. A. Kraker, B. Csuka, and Zs. Kollár. Sliding Window Evaluation of the Wiener-Hopf Equation.

Radioengineering, 29(2):365375, June 2020. Impact factor: 1.077 1.4.2 Conference papers

C1. B. Csuka and Zs. Kollár. R-DFT-based Channel Estimation in 802.11ad Systems (Original title in Hungarian: R-DFT alapú csatornabecslés a 802.11ad rendszerekben). In Mesterpróba 2015, pages 814, 2015

C2. B. Csuka and Zs. Kollár. Parameter Estimation in 802.11ad Systems (Original title in Hungarian:

Paraméterbecslés 802.11ad rendszerekben). In HTE MediaNet 2015 Konferencia szemle: Diákszek- ció, pages 814, 2015

C3. B. Csuka, I. Kollár, Zs. Kollár, and M. Kovács. Comparison of Signal Processing Methods for Calculating Point-by-point Discrete Fourier Transforms. In 26th International Conference Ra- dioelektronika, pages 5255, 2017

Independently cited by [14, 15].

C4. M. Kovács, B. Csuka, and Zs. Kollár. Eects of Quantization on Golay Sequence based Channel Estimation. In 27th International Conference Radioelektronika, pages 5255, 2017

2 Complementary Golay sequences

Wireless transmission systems using high data rates call for fast and accurate parameter estimation for compensation of the imparities in the transceiver chain. Devices require algorithms with low complexity and high precision for reduced power consumption and elevated data rates.

The focus of the section is the application of the complementary Golay sequences [17] for baseband signal processing used in digital transmissions. These sequences have advantageous properties as that their auto-correlation function being zero except for one point, or their being evenly distributed along the available spectrum. Based on these advantageous properties, powerful algorithms can be developed for the receivers to estimate and compensate imperfections [18, 19], or channel impulse response (CIR) [20, 21, 22] can be estimated and compensated as well. This conventional solution calculates correlations of the sequences, and the sum of these correlations provides the CIR in time domain.

In this chapter, the complex CIR estimation techniques are presented based on complex-valued com- plementary Golay sequences, and the corresponding Golay lter is discussed in detail. As an alternative solution, the frequency domain approach for CIR estimation is presented as well using the discrete Fourier transform, which provides similar results as the Golay lter.

Following the estimation of the CIR the channel equalization can be performed. Two equalization methods are presented: the zero forcing and the minimum mean square error channel equalization. The theoretical methods for CIR estimation are validated in simulations using real- and complex-valued chan- nel models.

The chapter is organized as follows. Section 2.1 gives a description about the Golay sequences, their generation and correlation techniques are presented in Sec. 2.2. Section 2.3 presents the channel estima- tion and compensation techniques using time and frequency domain approaches, and their complexity are discussed in Sec. 2.4. Finally, the simulation results are presented and evaluated in Sec. 2.5, then Section 2.6 draws the conclusions, and it summarizes the new scientic results.

2.1 General description of the Golay sequences

2.1.1 Denitions

Golay created three denitions for his sequences [17], which are presented below. It can be proven that they are equivalent to each other [23]; therefore, any of them can be used depending on the application eld of the sequences.

2.1.1.1 Using separated pairs

For an N-long Golay sequence GA[n] = (GA[0],GA[1], . . . ,GA[N−1]), let dene the function UA[i]

that gives the number of unlike pairs inG_A as follows

UA[i] =

N−i−1

X

n=0

(GA[n] ! =GA[n+i]) (21)

where the operator! =returns0or1 as

(G_A[n] ! =G_A[n+i]) =







0 forGA[n] =GA[n+i], 1 forGA[n]6=GA[n+i].

(22)

FurthermoreLA[i] can be dened similarly to get the number of like pairs inGAas follows

LA[i] =

N−i−1

X

n=0

(GA[n] ==GA[n+i]) (23)

where the operator==returns0 or1as

(GA[n] ==GA[n+i]) =







1 forG_A[n] =G_A[n+i],

0 forG_A[n]6=G_A[n+i]. (24) If GB[n] = (GB[0],GB[1], . . . ,GB[N−1]) is also given, then UB and LB are similarly dened as UA

(21) andLA (23). Using theseU- andL-functions,GAandGB are complementary Golay sequences to each other ifUA[i] =LB[i]wherei∈(0;N). Considering thatUA[i] +LA[i] =N−i,LA[i] =UB[i]also fullls.

For example, the following sequences are Golay complement to each other

GA[n] = (+1,+1,−1,+1), (25)

GB[n] = (+1,+1,+1,−1). (26)

To show that they are Golay complement, theU- andL-functions have to be evaluated as it is presented in Tab. 1.

Table 1: Number of like and unlike pairs inG_AandG_B

i GA GB UA[i] LB[i] LA[i] UB[i]

1 (+1,+1); (+1,−1); (−1,+1) (+1,+1); (+1,+1); (+1,−1) 2 2 1 1 2 (+1,−1); (+1,+1) (+1,+1); (+1,−1) 1 1 1 1

3 (+1,+1) (+1,−1) 0 0 1 1

2.1.1.2 Using auto-correlation function

If a GA[n] = (GA[0],GA[1], . . . ,GA[N−1]) sequence is given, then its non-periodic auto-correlation function can be evaluated as follows

RA[n] =

N−1−n

X

i=0

GA[i]· GA[i+n] =GA[n]∗ GA[n] ; (27) furthermore,RB can be calculated similarly

RB[n] =

N−1−n

X

i=0

GB[i]· GB[i+n] =GB[n]∗ GB[n]. (28) Using (27) and (28),GA andGB are Golay complementary sequences if

R_A[n] +R_B[n] =







2N forn= 0, 0 for0< n < N.

(29)

For example, for the sequences, given in (25) and (26), the following auto-correlation functions can be calculated as

RA[n] = (+4,−1,0,+1), (210)

RB[n] = (+4,+1,0,−1) (211)

for which to sum satises the condition of (29), thus

RA[n] +RB[n] = (+8,0,0,0). (212)

2.1.1.3 Using polynomials

In this case, letGAbe considered as aC→Cpolynomial of the complex-valued variablez:GA(z)where the coecients are the corresponding values from sequenceGA. Now create the product ofGA(z)GA z⁻¹, and after its expansion (see details in Appendix A), it can be seen that the values of R_A appear as coecients

G_A(z)G_A z⁻¹

=R_A[0] +R_A[1] z¹+z⁻¹

+· · ·+R_A[N−1]

z^N−1+z^−(N−1)

. (213)

ForGB, the same polynomials and expressions can be written as forGA; therefore using (29) GAand GB are Golay complementary sequences if the following equation fulls

GA(z)GA z⁻¹

+GB(z)GB z⁻¹

= 2N. (214)

For example, for the sequences, given in (25) and (26), the following polynomials can be written using (210) and (211) as well,

GA(z)GA z⁻¹

= 4− z¹+z⁻¹

+ 0· z²+z⁻² +

z^N−1+z^−(N−1)

, (215)

GB(z)GB z⁻¹

= 4 + z¹+z⁻¹

+ 0· z²+z⁻²

−

z^N−1+z^−(N−1)

. (216)

Summing (215) and (216), it can be seen that∀z∈C, the sum satises the condition of (214):

G_A(z)G_A z⁻¹

+G_B(z)G_B z⁻¹

= 8. (217)

2.1.2 Operations on Golay sequences

If the sequencesGAandGBare given, then further complementary sequences can be constructed from them through using the following operations for one of them or for both of them:

• multiplication by a constant c:GA→(cGA[0], cGA[1], . . . , cGA[N−1]);

• reversing the sequence:G_A→(G_A[N−1],G_A[N−2], . . . ,G_A[0]);

• altering the sequence:GA→

(−1)⁰GA[0],(−1)¹GA[1], . . . ,(−1)^N−1GA[N−1]

. Two sequences with length ofN can be combined to each other resulting2N long sequences:

• through concatenation:(GA;GB)→(GA[0],GA[1], . . . ,GA[N−1],GB[0], . . . ,GB[N−1]), or

• through interleaving:(GA;GB)→(GA[0],GB[0],GA[1],GB[1], . . . ,GA[N−1],GB[N−1]). Turyn showed [24] that the tensor product of the sequences results in Golay complementary sequence as well; furthermore, their lengths do not have to be the same. Let GA with length of N and GB with length ofM Golay complementary sequences, then its tensor product with length ofM N is the following, (GA;GB)→(GA[0]GB[0],GA[0]GB[1], . . . ,GA[0]GB[N−1],GA[1]GB[0], . . . ,GA[N−1]GB[N−1]).

(218) These operations allow to create new sequences from existing ones, and longer versions can also be generated through using some basic, short sequences. However, there are many limitations about the possible length of sequences, which are described in the following section (Sec. 2.1.3).

2.1.3 Lengths of the sequences

Currently, there exists no exact rule for the possible lengths of Golay sequences, just some restrictions are known; however, the existence of Golay sequences have been known for more than 50 years. The following list contains the currently known limitations about the possible or impossible lengths.

• Originally, the following conditions and methods were given by Golay [17]:

N should be even, thusN≡0 (mod 2),

N should be the sum of two squares, thusN =p²+q²; however, one of them can be0², a construction method was given for length ofN = 2^p,

nally, the existence of length of10and26was given by handmade calculations.

• Golay had gave another construction method to get2N- and2M N-long sequences from the existing ones with length of M or N. This procedure was developed by Turyn [24] to generate M N-long sequences; thus, the sequences with any length of the form2^p10^q26^rcan be created.

• Grin showed [25] thatN can not be in this form:N = 2·3^p ∀p∈N⁺.

• Eliahou, Kervaire, and Saari proved [26] that the length N can not have such prime factor pfor that p≡3 (mod 4).

Using these restrictions and construction methods, the following lengths of Golay sequences may exists up to100:N = (1,2,4,8,10,16,20,26,32,34,40,50,52,58,64,68,74,80,82), but in parallel with these re- sults, new researches were also done, and the nonexistence of the following lengths were proven:(34,50,58) by Andres [27] and 68 by James [28]. Borwein and Ferguson give a complete collection of the existing Golay sequences up to100[29]

N = (1,2,4,8,10,16,20,26,32,40,52,64,80),

which means that they proved the nonexistence forN = (74,82)as well. Over 100, almost all cases are still open.

x[n]

z^{−ρ1 ω1}

+

+ z^{−ρ2 ω2}

+

+ z^{−ρK ωK}

+

+ XA[n]

XB[n]

− − −

Figure 3: Structure of Golay lter given by Budi²in

x[n]

z^−ρ1

ω1

+

+

z^−ρ2

ω2

+

+

z^−ρK

ωK

+

+ XA[n]

X_B[n]

− − −

Figure 4: Structure of ecient Golay lter given by Popovi¢

2.1.4 Application elds

The Golay sequences were originally given for multislit spectrometry [17]. Since that, they are applied for many other elds like for ultrasound, acoustic measurements, radar pulse compression. In this thesis, they are used for communication purposes to estimate and compensate the channel impulse response.

For this purpose, Golay sequences can be found in Wi-Fi (802.11ad), CDMA networks, or in orthogonal frequency-division multiplexing (OFDM) systems.

2.2 Generation and correlation of Golay sequences

As it is presented in Sec. 2.1.3, it is not obvious to choose the length of the Golay sequences for an application because their existence for a givenN has to be proven. Furthermore, if anN is already given, it is also not trivial to get the right sequences with length ofN [29]. For practical purposes, ifN = 2^K, the following recursive construction method can be applied that was given by Golay [17] as

XA,0[n] =XB,0[n] =δ[n], (219)

XA,k[n] =X_A,k−1[n] +X_B,k−1

n−2^k−1

, (220)

XB,k[n] =XA,k−1[n]− XB,k−1

n−2^k−1

. (221)

Using this generation procedure, after K stages of recursion,XA,K[n] = GA[n] and XB,K[n] = GB[n]. Sivaswamy generalized this solution to polyphase sequences [30] that method was further developed by Budi²in (Fig. 3) [31, 32],

XA,0[n] =δ[n], (222)

XB,0[n] =δ[n], (223)

XA,k[n] =XA,k−1[n] +ω_kXB,k−1[n−ρ_k], (224) X_B,k[n] =X_A,k−1[n]−ω_kX_B,k−1[n−ρ_k] (225) where ρ is any permutation of powers of 2 from 2⁰ to 2^K−1 while ωk are the coecients of this Golay lter (GF) considering that ∀ |ω_k|= 1. Budi²in proved [31] that this method gives the same results as

(27) and (28). To reduce the latency of the GF, Popovi¢ updated Budi²in's structure, and he presented the ecient Golay lter (EGF, see in Fig. 4) as follows [33]

XA,k[n] =X_A,k−1[n−ρk] +ωkX_B,k−1[n], (226) XB,k[n] =XA,k−1[n−ρk]−ωkXB,k−1[n]. (227) The sequences generated by GF (GGF) or EGF (GEGF), are equivalent to each other if the same preset forρandωis applied. However, their directions are reversed to each other, thus

(GGF[0],GGF[1], . . .GGF[N−1]) = (GEGF[N−1], . . . ,GEGF[1],GEGF[0]).

Furthermore,x[n] =GA[n] =⇒ XA[n] =RA[n] orx[n] =GB[n] =⇒ XB[n] =RB[n]; thus, these lters are able not only to generate the Golay sequences, but their non-periodic auto-correlation functions can be evaulated by using GF or EGF as well. The possible combinations of generators and correlators can be seen in Tab. 2.

Table 2: Possible combinations of GF and EGF

Generation Correlation Remark

GF GF Order of sequences has to be reversed before correlation.

GF EGF

EGF GF

EGF EGF Order of sequences has to be reversed before correlation.

There exists such applications of Golay sequences as well, where they are transmitted not sequentially but in parallel (Fig. 5). In this case, two lters have to be used parallel to each other to get the correlations (RA and RB) simultaneously. To reduce the doubled complexity, Donato proposed the following opti- mised Golay lter (OGF) to evaluateR_AandR_B parallel if multiple-input and multiple-output (MIMO) transmission is applied (Fig. 6) [34], thus

XA,0[n] =x_A[n], (228)

X_B,0[n] =x_B[n], (229)

XA,k[n] =X_A,k−1[n−ρk] +X_B,k−1[n−ρk], (230) XB,k[n] =ωk(X_A,k−1[n]− X_B,k−1[n]). (231)

Golay

generator DAC channel ADC Golay

correlator G_A

GB

GA+ξch

GB+ξch

Golay

generator DAC channel ADC Golay

correlator GA

GB

GA+ξch

GB+ξch

Figure 5: Application of Golay sequences in SISO and MIMO transmissions (ξch: noise of channel)

xA[n]

xB[n]

+

+

z^−ρ1

ω1

+

+

z^−ρ2

ω2

+

+

z^−ρK

ωK

XA[n]

X_B[n]

+ X[n]

− − −

Figure 6: Structure of optimised Golay lter given by Donato

2.3 Channel estimation based on Golay sequences

In the previous section, the usage of Golay generators and correlators were presented. These sequences are widely used for dierent electrical engineering applications such as signal detection in noisy environ- ment. In this section, an other usage is presented and analysed: channel estimation based on Golay sequences in time [20, 21, 22] and in frequency domain [8].

As it is presented in Sec. 2.2, there are many solutions for the generation of the Golay sequences and for the calculation of their correlation. Furthermore, they can be structured into larger structures to construct longer sequences for dierent purposes. In this thesis, the methods and structures of the 802.11ad standard are applied for analysis and simulation [35, 36].

2.3.1 Time domain channel estimation based on impulse response

Let a transfer channel be given, which has the following impulse response: hch[n], and xtr[n] be a discrete transmitted signal. In this case, the received signal can be considered as

xrec[n] =hch[n]∗xtr[n] +ξch[n] (232) whereξch[n]denotes the noise of the transmission channel. To retrieve the correct transmitted information on the receiver side, the eects of the transfer channel have to be compensated; thus, it has to be identied.

Let us have a noiseless transmission, thusξ_ch[n] = 0 ∀n∈N, and if the transmitted signalx_tr is a G_A sequence, then the received signalxrec can be expressed using (232) (and respectively forGB) as follows x_tr[n] =GA[n] =⇒x_rec,A[n] =h_ch[n]∗ GA[n], (233) xtr[n] =GB[n] =⇒xrec,B[n] =hch[n]∗ GB[n]. (234) If these signals are correlated by Golay lters as Fig. 5 shows, then the following results are given through XA[n] =xrec,A[n]∗ GA[n] =hch[n]∗ GA[n]∗ GA[n], (235) XB[n] =x_rec,B[n]∗ GB[n] =h_ch[n]∗ GB[n]∗ GB[n]. (236) Adding (236) to (235) and using (27)-(28) results in

y[n] =XA[n] +XB[n] =hch[n]∗(GA[n]∗ GA[n] +GB[n]∗ GB[n]) =hch[n]∗(RA[n] +RB[n]). (237)

-0.5 0 0.5 1 1.5

Correlation

Correlations with sequence A and B

-32 -28 -24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28 32

Time index

Golay A Golay B

-0.5 0 0.5 1 1.5

Sum of correlations

Sum of correlations

Impulse response

-32 -28 -24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28 32

Time index

Figure 7: Normalized correlations and their sum over time with ideal transfer channel

-0.5 0 0.5 1 1.5

Correlation

Correlations with sequence A and B

-32 -28 -24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28 32

Time index

Golay A Golay B

-0.5 0 0.5 1 1.5

Sum of correlations

Sum of correlations

Impulse response

-32 -28 -24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28 32

Time index

Figure 8: Normalized correlations and their sum over time with multipath propagation

STF CEF

GA GA −GA GU GV −GB

−GB −GA GB −GA −GB GA −GB −GA

Figure 9: Structure of the applied preamble in IEEE 802.11ad standard

Considering (29), the impulse response of the transfer channel can be expressed as

y[n] =hch[n]∗2N δ[n] =⇒hch[n] = y[n]

2N . (238)

If the transfer channel is distorted by noise, then it can not be exactly identied, just an estimationbhch, the Channel Impulse Response Estimation (CIRE) can be given through

y⁰[n] =hch[n]∗(RA[n] +RB[n]) +ξ⁰_ch[n] =hch[n]∗2N δ[n] +ξ_ch⁰ [n] (239) wherey⁰ is the noisy output of the Golay lter containing noiseξ⁰_ch. After evaluation of the convolution, the estimation can be expressed as

hch[n]≈bhch[n] = y⁰[n]

2N . (240)

For this estimation method, an example is presented in Fig. 78 for two dierent scenarios. Firstly, an ideal channel is applied withh_ch= 1, after that, the second channel impulse response contains six taps.

It can be seen that the correlations are calculated and added to each other, and the resulting signal is zero except for the coecients of the impulse response.

2.3.2 Time domain channel estimation using frame structure of 802.11ad

If a single carrier transmission is used, then the following preamble is sent before the data in 802.11ad to estimate transfer parameters: Short Training Field (STF) whereG_A is repeated is for synchronization and frequency oset estimation while CEF represents the Channel Estimation Field (Fig. 9). This eld contains two longer sequences,GU andGV having length of4N; furthermore, it consist ofGAandGB.

The incoming signal is correlated by a Golay lter (Fig. 10) resulting X_A (235) and X_B (236) respectively. Considering the fact that the lter delay isN −1, it requires2N−1 samples to ll it up with GA (or GB), and then remove it from the lter. Therefore, due to these overlapping, the previous sequences distort the correlation of the current sequences as well. The initialization transient is the same

x[n]

z^−ρ1

ω1 +

+ z^−ρ2

ω2 +

+ z^−ρK

ωK +

+ X_A[n]

XB[n]

− − −

Figure 10: Structure of Golay lter used in IEEE 802.11ad standard

Figure 11: Example for pair of correlations to the time domain channel estimation using Golay lter

as the decaying transient; thus, the result is symmetric ifXA[0] = 1, thenXA[n] =XA[−n]. Therefore, XB can be reversed to match it exactly toXA.

Suppose that the transfer channel is ideal, the outputs of the Golay lter are presented in Fig. 11, where the matched pairs are denoted by yellow background. The steps of the estimation is the following.

The green part ofXAis the correlation of the incoming sequences to GA, which can be described as

XA[n] =−GA[n]∗ GA[n] +GA[n]∗ GB[n] (241) because the previous−GA sequence (denoted by purple) distorts the result, it is not removed perfectly from the lter yet. For the green part ofXB, the similar equation can be written due to the same distortion XB[n] =−GB[n]∗ GB[n] + (−GA[n])∗ GB[n]. (242) Adding (241) to (242), the following result can be obtained

XA[n] +XB[n] =−GA[n]∗ GA[n] +GA[n]∗ GB[n]− GB[n]∗ GB[n]− GA[n]∗ GB[n] =

=−(GA[n]∗ GA[n] +GB[n]∗ GB[n]) (243)

yielding the channel impulse response (cf. Sec. 2.3.1), which is the discrete Dirac delta function in this case after correcting the sign.

GU GV

XA:

XB:

−GB −GA GB −GA −GB GA −GB −GA

−GB −GA GB −GA −GB GA −GB −GA

G_U G_V

Figure 12: Parts ofX_A andX_B used for channel estimation

GU GV −GB

X_A:

XB:

−G_B −G_A G_B −G_A −G_B G_A −G_B −G_A −G_B

−GB −GA GB −GA −GB GA −GB −GA −GB

GU GV −GB

Figure 13: Parts ofXAandXB for extended channel estimation

In practice, using the frame structure of CEF, two pairs are applied for channel estimation purpose as Figure 12 shows. This combination results is two estimations for the impulse response, which are averaged to get the nal CIRE. In this setup, unused sequences are located between the used ones. This arrangement protects the estimator sequences from the intersymbol interference (ISI) to counter act the interference of the estimations.

As long as the ISI is shorter than the applied Golay sequences, the number of the pairs can be extended as Figure 13 shows. In this setup, the number of the CIREs is increased to 4 resulting better noise suppression during the channel estimation process. However, if there is no noise on the channel, then these extra pairs do not give any further information compared to the conventional solution (Fig. 12).

2.3.2.1 Coecients of Golay lter

In case of 802.11ad, the values of the Golay lter parameters (ω_kandρ_k) are determined in the standard [36]. The lengths of the sequences areN = (16,32,64,128). For those, Table 3 shows the corresponding parameters.

Table 3: Parameters of real-valued Golay lter

N K= log₂N ωk ρk

16 4 (+1,+1,+1,−1) (8,1,2,4)

32 5 (−1,+1,−1,+1,−1) (1,4,8,2,16)

64 6 (+1,+1,−1,−1,+1,−1) (2,1,4,8,16,32) 128 7 (−1,−1,−1,−1,+1,−1,−1) (1,8,2,4,16,32,64)

Nowadays, complex-valued IQ digital transmission is commonly used. Applying such a transmission, the channel model has to be also considered as complex-valued. To achieve complex channel estimation, the estimator sequences have to be complex-valued as well. The Golay lter parameters is given for real- valued scenario in Tab. 3, but it can be expanded to a complex-valued case as well. The idea originates from the 802.11ad standard [36], where a π/2-rotation is applied. It can be utilised by multiplying the