Filter Bank Multi-Carrier Modulation

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Filter Bank Multi-Carrier Modulation

FR ´ ED ´ ERIC DE PORET

Master’s Degree Project

Stockholm, Sweden 2015

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Acknowledgements

The work presented in this thesis have been realized with the radio access technology department at Ericsson research and the Communication Theory Department of the Electrical Engineering School of the Royal Institute of Technology from September 2014 to March 2015.

First of all I would like to thanks my supervisor, Ather Gattami, for his help and his encouragements during the thesis work.

I would like to give a big thanks to Lars Rasmussen without whom I would not have done this thesis.

I thank the Radio Access Technology Department of Ericsson Research at Kista for their welcome and all the good time I spend there and particularly Fredrik Lindqvist, Robert Baldemair, G¨oran Klang, Ning He who help me in my work

Hiou, Ali, Maxym, Naffis, Igor with whom I spend really nice time during the thesis.

Finally, many thanks to my friends and my family who support me in my life and

my work.

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Abstract

During the last years, multi-carrier modulations have raised a particular interest due to their high spectral efficiency and the possible assumption of flat frequency fading.

The most broadly used multi-carrier modulation is the CP-OFDM. It allows very simple equalizations methods and MIMO transcoders which keep orthogonality between carrier waveforms. However, it does not allow any waveform flexibility and is not fully spectrally efficient. Due to the low waveform flexibility its performance is quite limited in scenarios like the frequency division multiple accesses. In order to improve its performance in this case, it is important to have a multi-carrier modulation with the possibility to have a waveform well localized in time and frequency. Grey analysis shows that the only way to get both full spectral efficiency and waveforms well localized in both time and frequency domain is to give up the orthogonality in the complex field.

Using filter bank multi-carrier (FBMC) with Offset-QAM (OQAM) is one combination which achieves this task. In this thesis, we study this modulation, how it is possible to efficiently modulate and demodulate it but also the transcoder (pre-coder, equalizer or both) that can be used when transmitting through multi-tap and MIMO channels.

Another modulation, based on FBMC with OQAM, cyclic offset-QAM (COQAM)

tries to make a tradeoff between spectral efficiency and simplicity of the equalization

and transcoding methods. In this thesis, FBMC based modulation schemes are tested

through different scenario: unsynchronized multi-users, unsynchronized uplink, multi-

taps channels, SIMO, MISO and MIMO channels. COQAM is tested with some of

these scenario when it is considered as relevant to test it.

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Sammanfattning

Under senare ˚ar har intresset för multi-carrier modulering ökat p˚a grund av dess höga spektrumeffektivitet under antagandet om slät frekvensfäding. Den vanligaste multi- carrier moduleringen i praktiken är CP-OFDM. Denna modulering till˚ater användning av enkla equalizeringsmetoder och MIMO transkodare vilket beh˚aller ortogonalitet mellan v˚agformerna. Däremot till˚ater den inte n˚agon flexibilitet i val av v˚agform och är inte heller helt spektraleffektiv. P˚a grund av inflexibilitet i val av v˚agform är dess pre- standa begränsad under till exempel frekvensmultiplexing (FDMA). För att förbättra prestandan hos CP-OFDM är det viktigt att ha en modulering med möjlighet att en v˚agform begränsad i b˚ade tid och frekvens. Grey analys visar att det enda sättet att uppn˚a full spektrumeffektivitet, i kombination med v˚agformer som är väl begränsade i b˚ade tid och frekvens, är att ge upp ortogonaliteten i komplexa fältet. Filter Bank Multi-Carrier (FBMC) med Offset-QAM (OQAM) är ett alternativ som uppn˚ar detta.

I den här avhandlingen studeras just denna modulation, hur det är möjligt att göra ef-

fektiv modulation och demodulation, men ocks˚a hur dess equalizer, pre-kodare och

transkoder anv¨ands vid multi-tap och MIMO-kanaler. En annan modulation, baserad

p˚a FBMC med OQAM, är cyclisk offset-QAM (COQAM). COQAM försöker göra

avv¨agning mellan spektral effektivitet och enkelhet av utj¨amnings- och transkodrar

metoder. I den h¨ar avhandlingen studeras b˚ade FBMC-baserade modulationer och

COQAM i en rad olika scenarios: osynkroniserade multianv¨andare, osynkroniserad

uppl¨ank, flera v¨agar kanaler, SIMO, MISO och MIMO-kanaler.

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List of Figures

2.1 General modulation scheme of the FBMC modulation . . . . 6

2.2 General modulation scheme of the FBMC demodulation in the Hermi- tian case . . . . 6

2.3 General modulation scheme of the FBMC demodulation in the Eu- clidean case . . . . 6

2.4 Implementation, K = 4 . . . . 8

2.5 Polyphase network modulation scheme . . . . 9

2.6 The H k filter . . . . 10

2.7 Global modulation scheme using the polyphase network implementation 10 2.8 Implementation of the receiver, K = 4 . . . . 11

2.9 Polyphase network demodulation scheme . . . . 12

2.10 Comparison of the complexity of the different modulation methods . . 13

2.11 Transmission scheme . . . . 13

2.12 Equalization type 1 . . . . 15

2.13 Equalization type 2 . . . . 15

2.14 Equalization type 3 . . . . 15

2.15 Transmission model . . . . 16

2.16 MIMO transmission model . . . . 24

3.1 Two first modulation steps . . . . 39

3.2 Process to add the cyclic prefix and the window . . . . 39

3.3 Computational complexity . . . . 41

4.1 Power spectrum density using KM = 4096 . . . . 46

4.2 SIR as a function of the sub-channel for an unsynchronized multi-user scenario, equal power between users . . . . 47

4.3 SIR as a function of the sub-channel for an unsynchronized multi-user scenario, power of the interferer 10 ⁴ times more powerful than the main user . . . . 47

4.4 SINR as a function of the delay between the received signal and the demodulator for 20dB SNR . . . . 49

4.5 SINR as a function of the delay between the received signal and the demodulator for 80dB SNR . . . . 49

4.6 Capacity as a function of the SNR for an ”epa” channel with different equalizers . . . . 51

4.7 SINR as a function of the SNR for an ”epa” channel with different equalizers . . . . 51

4.8 Capacity as a function of the SNR for an ”etu” channel with different

equalizers . . . . 52

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4.9 SINR as a function of the SNR for an ”etu” channel with different equalizers . . . . 52 4.10 SINR as a function of the sub-carrier for a particular ”etu” channel for

different equalizers with 20 dB SNR . . . . 53 4.11 SINR as a function of the sub-carrier for a particular ”etu” channel for

different equalizers with 80 dB SNR . . . . 53 4.12 Capacity as a function of the SNR for SIMO and SISO ”etu” channels 54 4.13 SINR as a function of the SNR for SIMO and SISO ”etu” channels . . 54 4.14 Capacity as a function β for a MISO channel (2 × 1) for different SNR 56 4.15 Capacity as a function of the SNR for a MISO channel(2 × 1) and the

SISO channels taken independently . . . . 57 4.16 SINR as a function of the SNR for a MISO channel(2 × 1) and the

SISO channels taken independently . . . . 57 4.17 Capacity as a function of the SNR for MIMO channels with different

equalizers with 2 streams . . . . 58 4.18 SINR as a function of the SNR for MIMO channels with different

equalizers with 2 streams . . . . 58 4.19 Capacity as a function of the SNR for MIMO channels with different

equalizers with 4 streams . . . . 59 4.20 SINR as a function of the SNR for MIMO channels with different

equalizers with 4 streams . . . . 59

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Chapter 1

Introduction

1.1 Background

For years, cellular networks have been in great expansion. Today, available bandwidth is a limiting factor. The performance of a radiowave system depends on the modulation used. It determines the symbol density, the power spectrum but also the robustness of the channel link. These properties influence the capacity of the transmission for each user. A well shaped power spectrum can also allow smaller guard band between users without too much interferences enhancement. Shannon and Nyquist have shown the theoretical limit in therms of capacity of a channel with a limited bandwidth. However, this is a theoretical model which consider asymptotic systems and does not consider the problem of computation complexity. Multi-carrier modulation schemes allow high spectrum efficiency with reasonable complexity for both modulation and equalization.

However, according to the Balian-Low theorem [9], it is not possible to have the orthogonality in the complex field, flexibility in the pulse shape and to transmit at the Nyquist rate at the same time. The existing multi-carrier modulations make choices between these properties. For example, CP-OFDM has the orthogonality in the complex field but do not transmit at the Nyquist rate and has no waveform flexibility. In order to improve the performances of a system in terms of data rate and frequency division multiple access, these two last properties are important. Filter bank multi-carrier with OQAM use the orthogonality in the real field instead of the complex field and keeps the two other properties.

1.2 Previous work

Two simple ways to generalize the CP-OFDM modulation in order to get more freedom

in the waveform are the application of a window (WCP-OFDM) or of a filter (filtered

CP-OFDM). Both technics can improve the spectral leakage of the pulse shape but

have a cost in terms of spectral efficiency. Moreover, due to the demodulation used for

OFDM, these technics necessitate a filter at the receiver side to suppress the interfer-

ence from unsynchronized users. Due to this filtering large guard bands are necessary

between users. Another approach is the one of FBMC with QAM. In that case more

freedom in the waveform design is possible but it has a cost in terms of spectral effi-

ciency. For example, if the waveform used is a root raised cosine with coefficient α,

the spectral efficiency will be at most _1+α ¹ . It is then possible to reduce the guard band

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between unsynchronized users to 1 sub-band. According to the Balian-Low theorem [9], the only way to have a pulse shape well localized in time and in frequency, see Appendix Equation 5.3, and to transmit at the Nyquist rate is to reduce the constraint on the orthogonality. FBMC with Offset-QAM uses orthogonality in the real field instead of the complex field. It has been first introduced by R. W. Chang [6] and B. R.

Saltzberg [14] during the mid-1960s. In 1974, an efficient way to modulate and demodulate FBMC has been developed by M. Bellanger [5]. More recently, OFDM modulations have been broadly studied. Many simple equalization and pre-coding approaches adapted to CP-OFDM appeared. These techniques result in perfect equalization in the particular case of CP-OFDM. They can also be used with near perfect equalization with other multi-carrier modulation if the channel can be assumed flat frequency fading for each sub-band. During the last few year, the necessity to improve the performances of the modulation used, mainly in terms of frequency division multiple access, appeared.

Studies about FBMC, which appear as one of the solutions, started in order to see its capabilities in different communication contexts. One of these studies is the PHY- DYAS project ([4], [3], [2]). Some reports from this project present some equalization methods for FBMC transmitted through multi-taps channels or MIMO channels. The performances of FBMC are always limited because of the intrinsic interference and the non-perfect orthogonality. In order to suppress these problems new modulations such as cyclic offset-QAM has been proposed in [13].

1.3 Project Purpose and Goal

The FBMC modulation aims to replace OFDM in some cases. In such context, it is important to estimate its performances and when it could improve OFDM or not. When it can improve the OFDM modulation, a question in terms of cost of this improvement is raised. The purpose of this thesis is to evaluate this improvement and this cost. More precisely, it is to study FBMC and different equalization models and test them under simple channel models relevant of a cellular network. Among them the unsynchronized multi-user scenario, the unsynchronized uplink scenario, more or less difficult multi- taps channels, MISO, SIMO and MIMO channels. A comparison is done between equalizer and precoding and between FBMC and OFDM. By this way, it is possible to compare the performance and the cost of FBMC and OFDM. Another important field of comparison is the channel estimates and the computations necessary to obtain the equalizer.

1.4 Outline

The rest of the thesis is organized as follow

Chapter 2 presents the FBMC modulation with OQAM, the practical implementations of modulation and the demodulation as well as investigates different linear equalization and application to MIMO systems.

Chapter 3 presents a possible evolution of FBMC, the COQAM modulation and its simple equalizations methods for both multi-taps and MIMO channels

Chapter 4 investigates the performance of both FBMC and COQAM modulations

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and compare them with OFDM for different properties or scenario: Power spectrum density, unsynchronized multi-users, unsynchronized uplink, multi-taps channels, MIMO channels. There is also a comparison of the different pre-coders and equalizers of FBMC

Chapter 5 Summarizes the thesis, concludes the work and gives some ideas of possible future work

The appendix presents the inner products (Euclidean and Hermitian one) over L ² (C), the Balian-Low theorem, the derivation of the linear minimum mean square error filter, the maximum likelihood and minimum mean square error estimation in the case of a single tap channel and, finally, the derivation of the orthogonality of the set of waveforms used for FBMC with OQAM.

1.5 Notations

Table 1.1 Mathematical notations

E Denotes the expectation operator X ˆ X denotes the estimate of X ˆ

C ^n×m The space of complex matrices of size n × m O n,m Null matrix of size n × m

I n Identity matrix of size n × n

<(·) <(x) is the real part of x

=(·) =(x) is the imaginary part of x

≡ n ≡ m(k) mean that n − m is an integer multiple of k M ^∗ M ^∗ is the conjugate transpose matrix of M

M ^| M ^| is the transpose matrix of M

h·, ·i H ha, bi H is the Hermitian product between a and b h·, ·i _E ha, bi _E is the Euclidean product between a and b arg(·) arg(x) is the argument of x

[[a, b]] [[a, b]] is the set of the integer higher or equal to a and lower or equal to b b·c bxc denotes the largest integer less than or equal to x

iR x ∈ iR means real(x) = 0.

∗ Linear convolution

~ Circular convolution

CN (m, σ ² ) Complex normal distribution with mean m and variance σ ² δ(t) Distribution of Dirac

j j is the complex variable: j ² = −1

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1.6 Acronyms

Table 1.2 Acronyms

FBMC Filter bank multi-carrier

OFDM Orthogonal frequency division multiplexing QAM Quadrature amplitude modulation

OQAM Offset QAM COQAM Cyclic offset QAM MSE Mean square error

MMSE Minimum mean square error FFT Fast Fourier transform IFFT Inverse fast Fourier transform SNR Signal to noise ratio

SIR Signal to interference ratio SINR Signal to interference-noise ratio SISO Single inputs single outputs SIMO Single input multiple outputs MISO Multiple inputs single inputs MIMO Multiple inputs multiple outputs PPN Polyphase Network Method

FF Refers to the modulation method using a filtering in the frequency domain SVD Singular values decomposition

IIR Infinite impulse response filter

FIR Finite impulse response filter

NPR Near perfect reconstruction

PR Perfect reconstruction

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Chapter 2

Filter Bank Multi-Carrier

Modulation with Offset QAM

2.1 FBMC modulation

The FBMC modulation is a modulation which multiplexes several discrete time signals into one continuous signal. Let M be the number of discrete time signals, T the symbol time spacing, s ^l _n [i] the low rate signal for n ∈ [[1, M ]]. Each signal is modulated using a certain pulse shape. Let h n (t) be the pulse shape for the discrete time signal n. We define

s ^l _n (t) = X

i∈Z

s ^l _n [i]δ(t − iT )

s ^h (t) =

M

X

n=1

s ^l _n (t) ∗ h _n (t)

(2.1)

The block scheme of this modulation is presented in Figure 2.1. In order to demodulate the signal we should be able to reconstruct all the discrete time signals. Let ˆ s ^l _n [i]

be the estimate of s ^l _n [i]. ˆ s ^l _n [i] is given by ˆ

s ^l _n [i] = hs ^h (t), h _n (t − iT )i

=

M

X

m=1

hs ^l _m (t) ∗ h m (t), h n (t − iT )i

=

M

X

m=1

X

i∈Z

s ^l _n [k]hh _m (t − kT ), h _n (t − iT )i

(2.2)

If the Hermitian product is used, ˆ s ^l _n [i], i ∈ Z is the result of the filtering of s ^h (t) using the matched filter of h n (t) followed by the sampling at the frequency _T ¹ . If it is the Euclidean product, we must take the real part after the matched filtering. In order to have ˆ s ^l _n [i] = s ^l _n [i], it is important that the different filters satisfy some properties. First of all,∀n ∈ [[1, M ]], h n (t) should satisfy the Nyquist ISI criterion for a sampling frequency f s = _T ¹ in order to avoid inter-symbols interference. Then

∀n, m ∈ [[1, M ]], n 6= m, i ∈ Z, hh m (t − iT ), h _n (t)i = 0 in order to avoid inter-

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Figure 2.1: General modulation scheme of the FBMC modulation

Figure 2.2: General modulation scheme of the FBMC demodulation in the Hermitian case

Figure 2.3: General modulation scheme of the FBMC demodulation in the Euclidean

case

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carrier interference. Finally we must have hh n (t), h _n (t)i = 1 for the normalization.

We can consider here several categories in the FBMC modulation

• the inner product used is the Hermitian one and the criterions are perfectly satisfied

• the inner product used is the Euclidean one and the criterions are perfectly satisfied also called perfect reconstruction (PR)

• the inner product used is the Euclidean one and the criterions are almost satisfied (Usually the absolute value of the inner products are lower than 10 ⁻³ ) also called near perfect reconstruction (NPR)

OFDM is one particular case of the first category of FBMC. However this category is particularly limiting considering the waveform design if we want to reach full spectral efficiency. We will only deal with OQAM-FBMC with NPR which is a subcategory of the third category. This particular modulation is interesting because of the existence of a computationally efficient implementation algorithm, it is spectrally efficient and allows some freedom in the choice of the filters. According to the Balian-Low theorem [9], it is a necessity to have only the Euclidean orthogonality and not the Hermitian one if we want both spectral efficiency and freedom in the design of the pulse shape in order to have a good low pass and time limited filter. OQAM-FBMC with NPR, the bank of filters is generated from a filter, called the prototype filter. Let h, h(t), and H(f ) be the filter, its time impulse, and frequency response. h must be half Nyquist (which means that h∗h must satisfy the Nyquist ISI criterion) for a sampling frequency f s = _T ¹ , be frequency limited within a bandwidth smaller than _T ² , real, and symmetric.

Under these conditions, we can generate the bank of 2M filters h I,n (t) = h(t)e ^jn ( ^2π T t+ ^π ₂ )

h Q,n (t) = jh I,n

t − T

2 = jh

t − T

2 e ^jn ( ^2π _T ( ^t− ^T ₂ ) ⁺ ^π ₂ )

∀n ∈ [[0, M − 1]]

(2.3)

All these filters are orthogonal in the Euclidean sense (see Appendix). We are in a case of perfect reconstruction. However, limited bandwidth means infinite time length.

In order to avoid too much complexity and too much transition time, we need limited time filters. Let T h be the length of the prototype filter, we define the overlapping factor, noted K, K = ^T _T ^h . Due to this limitation in the time domain, some frequency leakage appears and the Nyquist ISI criterion is not anymore perfectly satisfied. Consequently, intrinsic interference appear. The bigger K, the less powerful are these interference.

For example, the PhyDyas filter [4] as an intrinsic SIR of −65dB when K = 4.

2.1.1 Practical implementation algorithms

The implementation is done in discrete time. The sampling time is T s = _M ^T . There

are several implementation algorithms. One of them consists of the calculation of

the frequency representation of each block of symbols, the use of the IFFT algorithm

and the overlap-sum method. Another algorithm uses the polyphase network method

(PPN). That latter algorithm is the most efficient one but is not compatible with some

equalization and pre-coding methods.

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Figure 2.4: Implementation, K = 4

Filtering in the frequency domain We want to calculate

s ^h (t) =

M

X

n=1

X

i∈Z

s ^l _I,n [i]h I,n (t − iT ) + s ^l _Q,n [i]h Q,n (t − iT ) (2.4)

One solution is to calculate s I [i](t) = P M

n=1 s ^l _I,n [i]h _I,n (t − iT ) and s Q [i](t) = P M

n=1 s ^l _Q,n [i]h Q,n (t − iT ) for each i and then sum up over i. The same process is used to calculate s I [i](t) and s Q [i](t) for each value of i so it will only be described for s I [0](t). This signal is non-zero only during the duration of a symbol T h = KM T _s . The idea of this algorithm is to modulate the signal in the frequency domain and then apply the IFFT algorithm. The representation of s I [0](t) in the frequency domain is

S I [0](f ) =

M

X

n=1

s ^l _I,n [0]H I,n (f ) (2.5) As h has a bandwidth smaller than _T ² , H I,n (f ) has only 2K − 1 non-zero coefficients. One of them is 1. By symmetry H I,n (f ) has only K different coefficients.

Consequently, this step requires only M (K − 1) real multiplications. Figure 2.4 il- lustrates these summations in the particular case of K = 4. One should notice that H I,n (f ) = e ^j ^π ² H I,n+1 (f + K).

The polyphase network method

This method has been introduced by Maurice Bellanger [5] and reduce considerably the computationnal cost of the modulation. The M filters used for the modulation have for impulse responses

h I,n (t) = h(t)e ^jn ( ^2π M t+ ^π ₂ ), ∀n ∈ [[0, M − 1]] (2.6)

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Figure 2.5: Polyphase network modulation scheme

In the Z-domain

H I,n (Z) =

KM −1

X

t=0

h(t)e ^jn ( ^2π _M ^t+ ^π ₂ )Z ^−t

=

KM −1

X

t=0

h(t)e ^jn ( ^2π _M ^t+ ^π ₂ )Z ^−t

=

M −1

X

k=0 K−1

X

t=0

h(tM + k)e ^jn ( ^2π M (tM +k)+ ^π ₂ )Z ^{−(tM +k)}

=

M −1

X

k=0

Z ^−k

K−1

X

t=0

h(tM + k)e ^jn ( ^2π(t+ _M ^k ⁾⁺ ^π ₂ )Z ^−tM

=

M −1

X

k=0

Z ^−k e ^jn ( ^2π _M ^k ⁺ ^π ₂ )

K−1

X

t=0

h(tM + k)Z ^−tM

!

(2.7)

Let H _k (Z ^M ) be P K−1

t=0 h(tM + k)Z ^−tM . We get H _I,n (Z) =

M −1

X

k=0

Z ^−k e ^jn ( ^2π _M ^k )e ^jn ^π ² H _k (Z ^M )

(2.8)

It is important to see that only the factor Z ^−k e ^j ( ^2π ^kn M )e ^jn ^π ² that depends on the frequency. Moreover, ∀k ∈ [[0, M − 1]], the k ^th term of the summation contains exactly all the terms of degree u, so that u ≡ k(M ), in the variable Z. This particular decomposition shows that the modulation can be decomposed into three steps. The first one is the pre-coding due to the OQAM modulation (for each frequency n, a multiplication with the coefficient e ^jn ^π ² to get alternatively one real and one pure imaginary number).

The second is an IFFT of size M : P M −1

k=0 Z ^−k e ^j ( ^2π ^kn _M ) correspond to the modulation of the current data at the frequency n over the discrete exponential n. Finally, to calculate the output at a delay u so that u ≡ k(M ), we apply the filter H k (Z ^M ). This last filtering can be seen as the filtering of the k ^th output of the IFFT with the filter H k (Z) before the parallel to serial step. This filter has only K non-zero coefficients.

This process and the filters H k (Z) are represented in Figures 2.5 and 2.6.

However, we have only modulated half of the symbols. The symbols modulated

using the filters h Q,n have not been considered yet. In fact, the same implementation

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Figure 2.6: The H k filter

Figure 2.7: Global modulation scheme using the polyphase network implementation

can be used in that case, it is only important to include the multiplication with j in the pre-coding and a delay of ^M ₂ samples. The complete modulation scheme is represented in Figure 2.7

The receiver

Both practical implementations of the transmitter have their corresponding receiver implementation. They are built using the same principle as the transmitter. Moreover, the filter h is real and symmetric. Consequently, its matched filter is also h. Figure 2.8 shows the implementation of the decoding for the in-phase component when the filtering is done in the frequency domain. For the quadrature components, one should take the imaginary part instead of the real part of the sum. Figure 2.9 shows the implementation of the decoding for the in-phase component using the polyphase network method.

The post-processing step suppresses the phase which has been introduced during the

pre-processing and takes the real part of the result. For the quadrature components,

one should delay the signal of − ^M ₂ samples and takes the imaginary part instead of

the real part during the post-processing. The received signal has a total delay of KT .

One should notice that for both modulations it can be interesting to keep both real and

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Figure 2.8: Implementation of the receiver, K = 4

imaginary parts. In fact, the imaginary parts when decoding the in-phase component at the time n correspond to the quadrature component at the time n − ¹ ₂ . It is then easy to get the low rate signals at twice the original rate with a limited complexity cost.

2.1.2 Complexity

The computation complexity is, here, calculated for of a block of M real symbols in terms of real multiplications. No equalization is considered here and the complexity of the receiver is the same as the one of the transmitter. The computational complexities are calculated for either the receiver of the transmitter.

For the first modulation method

• filtering: (K − 1) × M

• IFFT: 2KM log ₂ (KM ) For the second modulation method

• IFFT: 2M (log ₂ (M ) − 1)

• filtering: 4 × K × M

The overall computational complexity in terms of real multiplications for one modulated complex symbol for either the transmitter or the receiver is

• Filtering in the frequency domain: 2K(1 + 2 log ₂ (KM )) − 2

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Figure 2.9: Polyphase network demodulation scheme

• Polyphase network method: 4 log ₂ (M ) + 8K − 4

We can compare it to the computational complexity of OFDM: 2 log ₂ (M ). One should notice that additional calculations are necessary when the channel is non-ideal.

Figure 2.10 represents the evolution of the computational complexity per complex symbol in terms of real multiplication for OFDM, FBMC when K = 4 and FBMC for K = 8. We can see that the complexity of the first method is much higher and is much more sensible with K than the second one. For example, for K = 4 and M = 2048, the complexity of the first method is 4.86 times the one of OFDM. With the same parameters, the complexity of the second method is 1.64 times the one of OFDM.

2.2 Application with Multi-Path and MIMO channels

The interference due to near perfect reconstruction will be considered as a part of the noise.

2.2.1 Multi-Path Channels

We consider a linear and time invariant channel g with additive white Gaussian noise.

Let g(t) be the impulse response of this channel. We consider this channel as a time limited channel with a maximum delay T g . Let s ^h _t , s ^h _r and η be the signal sent by the transmitter, the signal received by the receiver and the noise. The transmission model without equalization is represented Figure 2.11.

s ^h _r (t) = g(t) ∗ s ^h _t (t) + η (2.9) There are several ways to equalize the signal. The first possibility is to equalize after the demodulation. In that case, the processing including the modulation, the transmission and the demodulation is a black box. After all these steps, there is inter- symbols interference and crosstalk between channels. The equalization step tries to estimate the original symbols using the outputs of the demodulation. The demodulation may process at a higher rate than the original symbol rate. This is illustrated in Figure 2.12.

The second possibility is to equalize s ^h _r . In other words, to find a filter w so that ˆ

s ^h _t (t) = w(t) ∗ s ^h _r (t) ≈ s ^h _t (t). This is illustrated in Figure 2.13.

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10 ⁰ 10 ¹ 10 ² 10 ³ 10 ⁴ 0

50 100 150 200 250 300

Complexity per complex symbol at the receiver side.

M Number of real multiplication per complex symbol

FBMC, method 1, K=4 FBMC, method 2, K=4 FBMC, method 1, K=8 FBMC, method 2, K=8 OFDM

Figure 2.10: Comparison of the complexity of the different modulation methods

Figure 2.11: Transmission scheme

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The last possibility is to exchange, for each channel n, the matched filter h ^∗ _n (−t) used during the demodulation with a filter which is adapted to the channel. This is illustrated in Figure 2.14.

Equalization after demodulation

Effect of a multi-path channel over the sub-channels. We need to know the structure of the ”black box” which represents the transmission: Structure of the noise, inter- symbols interference and crosstalk between channels. During the post processing, we suppress the phase which has been introduced during the preprocessing and keep only the real or the imaginary part according to the channel we are demodulating (I or Q).

As it has been noticed previously, it is possible to keep both real and imaginary part when we demodulate each channel. Then, the imaginary part resulting of the demodulation of the symbol from the I channel at the time i, correspond to the Q channel at the time i − ¹ ₂ . Respectively, the real part resulting of the demodulation of the symbol from the Q channel at the time i, correspond to the I channel at the time i + ¹ ₂ . The result of the demodulation is then 2M real channels at twice the original rate. For each sub-carrier n, i ∈ ¹ ₂ Z, the first channel is < hs ^h (t), h _I,n (t + iT )i _H , the second one is = hs ^h (t), h I,n (t + iT )i H . Let s ^l _I,n,r and s ^l _Q,n,r be these channels. We will estimate the influence of the channel g over the different sub-channels. We can consider n = 0 as it is eventually possible to make the transformation of the channel g _n (t) = g(t)e ^2jπt ^T ⁿ and then treat the problem as if we were considering the baseband channels. As the filter h is frequency limited and the channel is linear and time invariant, there is only crosstalk between adjacent channel. Consequently, ˆ s ^l _I,0 and ˆ s ^l _Q,0 are functions of s ^l _I,0,t , s ^l _Q,0,t , s ^l _I,1,t , s ^l _Q,1,t , s ^l _I,−1,t and s ^l _Q,−1,t . Our model is represented by the scheme in Figure 2.2.1. The filter g I,n,0 , g Q,n,0 , g N,0 are the channel filter resulting of all the transmission steps. They can be represented as in Figure 2.15.

Let L c , L w and L tot be the maximum of the lengths of the sub-channels, the length of the equalizer filters, and L _tot = ^(L ^w ₂ ⁺¹⁾ + L _c . We define

• G I,n,0 and G Q,n,0 , n ∈ {−1, 0, 1}, the convolution matrices of size (L w + 1) × L tot corresponding to the sub-channels g I,n→0 and g Q,n→0

• Γ 0 the convolution matrix of size (L w +1)×(KM +(L w −1) ^M ₂ ) corresponding to g N →0 and η the noise vector (of length KM + (L w − 1) ^M ₂ )

• S I,n,t and S Q,n,t , n ∈ {−1, 0, 1} the vectors of length L tot of the transmitted signal (S I,n,t =

s I,n (i − L c ) s I,n (i − L c + 1) . . . s I,n (i + ^L ^w ₂ ⁻¹ ) )

• S I,r and S Q,r the vectors of length (L w + 1) of the received signals: S I,t =

s _I,0,r (i) s _I,0,r (i − ¹ ₂ ) . . . s _I,0,r (i + ^L ^w ₂ ⁺¹ ) .

The equation of the received signals can be written as

S I,r = <(G I,−1,0 S I,−1,t + G Q,−1,0 S Q,−1,t + G I,0,0 S I,0,t + G Q,0,0 S Q,0,t

+G I,1,0 S I,1,t + G Q,1,0 S Q,1,t + Γ 0 η) S _Q,r = =(G _I,−1,0 S _I,−1,t + G _Q,−1,0 S _Q,−1,t + G _I,0,0 S _I,0,t + G _Q,0,0 S _Q,0,t +G I,1,0 S I,1,t + G Q,1,0 S Q,1,t + Γ 0 η)

(2.10)

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Figure 2.12: Equalization type 1

Figure 2.13: Equalization type 2

Figure 2.14: Equalization type 3

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Figure 2.15: Transmission model

1-tap equalizer. The goal of this equalizer is to be simple. We consider that for each frequency the channel introduces only a phase and a gain. Let c n be the complex representing this phase and this gain for the frequency n

s I,n,r (i) = <(c n )<(hs ^h _t (t) + η(t), h n (t − iT )i) − =(c n )=(hs ^h _t (t) + η(t), h n (t − iT )i) s _Q,n,r (i) = =(c _n )<(hs ^h _t (t) + η(t), h _n (t − iT )i) + <(c _n )=(hs ^h _t (t) + η(t), h _n (t − iT )i)

(2.11) If i ∈ Z, then <(hs ^h (t) + η(t), h _i (t − iT )i) ≈ s _I,n,t (i) and =(hs ^h _t (t), h _n (t − iT )i) is a term of interference so

ˆ

s I,n,t (i) = 1

<(c n ) ² + =(c n ) ² (<(c n )s I,n,r (i) + =(c n )s Q,n,r (i))

= < 1

c _n hs ^h _t (t) + η(t), h _n (t − iT )i

(2.12)

If n ∈ Z+ ¹ ₂ , =(hs ^h (t)+η(t), h _n (t−iT )i) ≈ s _Q,n,t (i) and <(hs ^h _t (t)+η(t), h _n (t−

iT )i) is a term of interference so

ˆ

s Q,n,t (i) = 1

<(c n ) ² + =(c n ) ² (−=(c n )s I,n,r (i) + <(c n )s Q,n,r (i))

= = 1

c _n hs ^h _t (t) + η(t), h _n (t − iT )i

(2.13)

MMSE equalizer Let ν be the delay of the equalizer. Here we want to estimate

s I,0,t (i + ν) or s Q,0,t (i + ν) based on the vectors S I,r and S Q,r with the MMSE

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equalizer. Let G 0 be the matrix G 0 =

G I,−1,0 G Q,−1,0 G I,0,0 G Q,0,0 G I,1,0 G Q,1,0

(2.14) The equation of the received signal can be written as

S I,r

S Q,r

=

<(G 0 )

=(G 0 )







S I,−1,t

S Q,−1,t

S I,0,t

S Q,0,t

S I,1,t

S Q,1,t





 +

<(Γ 0 ) −=(Γ 0 )

=(Γ 0 ) <(Γ 0 )

<(η)

=(η)

(2.15)

According to the formula derived in Appendix, when we want to estimate s I,0,t (i + ν), the linear MMSE filter is

W I = e 2L _tot +L _c +ν

<(G 0 )

=(G 0 )

<(G 0 )

=(G 0 )

<(G 0 )

=(G 0 )

|

+ σ ² _η σ ² _s

<(Γ ₀ ) −=(Γ ₀ )

=(Γ 0 ) <(Γ 0 )

<(Γ ₀ ) −=(Γ ₀ )

=(Γ 0 ) <(Γ 0 )

| ! −1

(2.16)

For s Q,0,t (i + ν), the linear MMSE filter is

W Q = e 3L _tot +L _c +ν

<(G 0 )

=(G 0 )

<(G 0 )

=(G 0 )

<(G 0 )

=(G 0 )

|

+ σ ² _η σ ² _s

<(Γ ₀ ) −=(Γ ₀ )

=(Γ 0 ) <(Γ 0 )

<(Γ ₀ ) −=(Γ ₀ )

=(Γ 0 ) <(Γ 0 )

| ! ⁻¹

(2.17)

Our estimates are ˆ

s I,0,t (i + ν) = W I

S I,r

S Q,r

ˆ

s Q,0,t (i + ν) = W Q

S I,r

S Q,r

(2.18)

Multiband MMSE equalizer In this case, we want to estimate the transmitted symbols through the channels at the frequency i using the received symbols corresponding to the channels at the frequency i − 1, i and i + 1. Let G −1 , G 0 and G 1 be the matrices

G ₋₁ =

G I,−2,−1 G Q,−2,−1 G I,−1,−1 G Q,−1,−1 G I,0,−1 G Q,0,−1 0 0 0 0 G ₀ =

0 0 G I,−1,0 G Q,−1,0 G I,0,0 G Q,0,0 G I,1,0 G Q,1,0 0 0 G ₁ =

0 0 0 0 G I,0,1 G Q,0,1 G I,1,1 G Q,1,1 G I,2,1 G Q,2,1

(2.19)

The received signals can be written as

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





S I,−1,r

S Q,−1,r

S I,0,r

S _Q,0,r S _I,1,r S _Q,1,r







=







<(G −1 )

=(G −1 )

<(G 0 )

=(G 0 )

<(G 1 )

=(G ₁ )













S I,−2,t

S Q,−2,t

S I,−1,t

S Q,−1,t

S I,0,t

S _Q,0,t S _I,1,t S _Q,1,t S _I,2,t S _Q,2,t





 +







<Γ −1 ) −=(Γ −1 )

=(Γ −1 ) <(Γ −1 )

<(Γ 0 ) −=(Γ 0 )

=(Γ 0 ) <(Γ 0 )

<(Γ −1 ) −=(Γ −1 )

=(Γ ₋₁ ) <(Γ ₋₁ )







<(η)

=(η)

(2.20) Let S _r , G, S _t , Γ, η be

S _r =







S _I,−1,r S _Q,−1,r S _I,0,r S _Q,0,r S _I,1,r S Q,1,r





 , G =







<(G −1 )

=(G −1 )

<(G 0 )

=(G ₀ )

<(G ₁ )

=(G 1 )





 , S _t =







S I,−2,t

S Q,−2,t

S _I,−1,t S _Q,−1,t S _I,0,t S _Q,0,t S _I,1,t S Q,1,t

S I,2,t

S Q,2,t





 ,

Γ =







<(Γ ₋₁ ) −=(Γ ₋₁ )

=(Γ ₋₁ ) <(Γ ₋₁ )

<(Γ 0 ) −=(Γ 0 )

=(Γ 0 ) <(Γ 0 )

<(Γ −1 ) −=(Γ −1 )

=(Γ −1 ) <(Γ −1 )





 , η =

<(η)

=(η)

(2.21)

The previous equation can be written as

S r = G S t + Γη (2.22)

When we want to estimate s I,0,t (i + ν), the multi-band MMSE filter is

W _I = e _4L _tot _+L _c _+ν G G G ^| + σ _η ² σ _s ² Γ Γ ^|

! ⁻¹

(2.23) The estimate is

ˆ

s I,0,t (i + ν) = W I S r (2.24)

When we want to estimate s Q,0,t (i + ν), the multi-band MMSE filter is

W Q = e 5L _tot +L _c +ν G G G ^| + σ ² _η σ ² _s Γ Γ ^|

! −1

(2.25)

The estimate is

ˆ

s Q,0,t (i + ν) = W Q S r (2.26)

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Equalization before demodulation

Equalizer based on the zero forcing. We are working at the same rate as the sampling rate. The zero forcing filter is an IIR filter and is usually not stable. We propose a filtering based on the zero forcing filter which can be implemented with an accept- able complexity without any stability problem. Let G be the convolution matrix of the channel, S ^h _t the vector of the transmitted signal and S _r ^h the vector of the received signal.

S _t ^h =







s ^h _t (−T g ) .. . s ^h _t (0) s ^h _t (1)

.. . s ^h _t (N − 1)





 , S ^h _r =







s ^h _r (0) s ^h _r (1)

.. . s ^h _r (N − 1)







G =







g T _g g T _g −1 . . . g 0 0 0 . . . 0 0 g T _g . . . g 1 g 0 0 . . . 0

.. . . . . . . . .. .

0 . . . 0 g _T _g g _T _g ₋₁ . . . g ₀ 0 0 . . . 0 0 g _T _g . . . g ₁ g ₀







(2.27)

The received signals equation is

S _r ^h = GS _t ^h + η (2.28)

In order to simplify the problem, we consider that we are in a case of a circular convolution. This can be justified by the asymptotic equivalency of toeplitz matrices and circulant matrices (see Lemma 4.2, [10]). We can include part of the interference in the noise. We define e S _t ^h , e G and η e

S e _t ^h =







s ^h _t (0) s ^h _t (1)

.. . s ^h _t (N − 1)





 , e G =







g 0 0 . . . . . . 0 g T _g g T _g −1 . . . g 1

g 1 g 0 0 . . . 0 0 g T _g . . . g 2

. . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . 0 g T _g g T _g −1 . . . g 1 g 0 0 0 . . . . . . 0 g T g g T g −1 . . . g 1 g 0







e η = η + GS ^h _t − e G e S _t ^h

(2.29) The modified equation of the received signal is

S _r ^h = e G e S _t ^h + e η (2.30) The matrix e G is a circulant matrix. Let W N be the Fourier transform matrix of size N . We can write e G = W _N ⁻¹ D

G e W N with D the diagonal matrix defined as D

G e = W N GW e _N ⁻¹ . D

G e is diagonal because of e G is a circulant matrix.

S ˆ _t ^h = W _N ⁻¹ D

G e W N S e _t ^h + W N η e D ⁻¹

G e W N S _r ^h = W N S e _t ^h + D ⁻¹

G e W N η e (2.31)

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D ⁻¹

G e W _N S _r ^h is an estimate of the Fourier transform of e S ^h _t . Let N = KM , the first demodulation method uses the Fourier transform of e S _t ^h of size KM . Consequently, it is possible to apply this equalization during the demodulation using the first algorithm.

The term e η is in fact composed of two parts. The first one corresponds to the noise and the intrinsic interference. The second part corresponds to the interference due to the channel. It is important to study the influence of these interference. Let κ be these interference

κ = W _KM ⁻¹ D ⁻¹ _˜

G W KM

GS _t ^h − e G × e S _t ^h

= W _KM ⁻¹ D ⁻¹ _˜

G W _KM







g _T _g g _T _g ₋₁ . . . g ₁ 0 g _T _g . . . g ₂ .. . . . . . . . .. . 0 . . . 0 g T _g

O KM −T _g ,T _g













s ^h _t (−T g ) − s ^h _t (KM − 1 − T g ) s ^h _t (1 − T _g ) − s ^h _t (KM − T _g )

.. .

s ^h _t (−1) − s ^h _t (KM − 1)





 (2.32) We can consider s ^h _t (i − T g ) − s ^h _t (i + KM − 1 − T g ), i ∈ [0; T g − 1] as a noise.

This noise is white only in the case where the whole bandwidth is used and the power repartition is independent of the frequency. This might not be the case but the study in the case of a white noise gives results which are general enough to be considered in the non-white case. Considering the white noise case with σ _s ² as variance for s ^h _t , s ^h _t (i − T g ) − s ^h _t (i + KM − 1 − T g ) is zero-mean and has a variance of 2σ ² s . These interference are filtered with the matched filter h n when we are demodulating. Let G n

be

G i =







h n (1)g T _g h n (1)g T _g −1 . . . h n (1)g 1

0 h n (2)g T g . . . h n (2)g 2

.. . . . . . . . .. . 0 . . . 0 h _n (T _g )g _T _g

O _{KM −T} _g _,T _g







(2.33)

The interference κ _n for the sub-channel n are zero-mean and has a variance

R _κ _n _κ _n =

1 . . . 1 W _KM ⁻¹ D ⁻¹ _˜

G W _KM G _n G ^| _n W _KM ⁻¹ D ⁻¹ _˜

G W _KM





 1 .. . 1





 (2.34) When the channel length is short compared to the length of the symbol, the matrix G i is almost equal to zero. Consequently, the interference power is small when the sub- channel gain D G ˆ is not too low compared to the gain of the whole channel. However, when the sub-channel gain is small compared to the channel gain, the interference can be powerful. One should see that, with this equalization, it is mainly the bad sub- channels which suffer from interference and not so much the good sub-channels.

MMSE. In the previous paragraph the filter used was based on the zero-forcing filter.

It is possible to use an MMSE filter. Let w, L p and L f be the MMSE filter, it length in

the past and in the future (The time 0 correspond to the sample we want to estimate).

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Here we write

S _r ^h =







s ^h _r (k − T L _p ) .. . s ^h _r (k + T _L _f )





 , S _t ^h =







s ^h _t (k − T L _p − T g ) .. .

s ^h _t (k + T _L _f )





 (2.35)

The equalization will be

ˆ s ^h _t (i) =

L p

X

k=−L _f

w k s ^h _r (i − k) (2.36)

The equation of the transmission is S _r ^h = GS _t ^h + η

ˆ s ^h _t (i) = h

w T _Lp . . . w T _−Lf

i







s ^h _r (k − T _L _p ) .. . s ^h _r (k + T _L _f )







(2.37)

The minimum mean square criterion results in the filter h w T _Lp . . . w T _−Lf

i

= e T _Lp +1 G (GR ss G ^∗ + R ηη ) ⁻¹ (2.38) Where e T _Lp +1 is the unit row vector with a 1 in position T L p + 1, R ss is the correlation matrix of transmitted signal and R ηη is the correlation matrix of the noise. If we assume a white noise, with zero mean and σ _η ² as variance: R ηη = σ _η ² I T _Lp +T g +T _Lf +1 . The value of R ss depends on the bandwidth which is used. In order to simplify the problem it is possible to consider R ss = σ ² _s I T _Lp +T _g +T _Lf +1 with σ _s ² the power of the signal. By this way, we obtain the MMSE filter. This filter must be long enough in order to reduce the interference however this filtering may increase too much the computational complexity. In order to avoid such problem, it is possible to use filter in the frequency domain when we are applying the demodulation. However, during the demodulation we also apply the matched filter which is of length KM . The resulting filter (MMSE followed by the matched filter) will have a total length bigger than KM . Consequently, we should either do a Fourier transform with a bigger length or keep the same length and consider new interference. This interference is the terms which are consequence of the transformation of the linear convolution in a cyclic convolution. Increasing the length of the Fourier transform might increase too much the complexity of the demodulation. Added to this interference, we should remember that this filter minimize the mean square error of the main signal and not the one of each channel taken independently. This has for consequence channel estimates which have important bias. All these problems are solved with the next estimator.

Equalization during demodulation

Per sub-channel MMSE. In the previous paragraph, the MMSE filter was done in

order to minimize the mean square error between the transmit signal and the received-

filtered signal. However this filtering does not consider the particular structure of the

signal and what is the final goal of the processing: the estimation of the symbols which

are transmitted through each sub-channel. It is possible to have an MMSE filter for

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each sub-channel. Let G n be the convolution matrix corresponding to the cascade of the oversampling, the filtering using the pulse shape of the sub-channel n and the channel impulse response.







s ^h _r (t − T L _p ) .. . s ^h _r (t + T L _f )





 = G n s ^l _n (i) + G







s ^h _t (t − T L _p − T g ) .. .

s ^h _t (t + T L _f )





 − G n s ^l _n (i) + η

ˆ

s ^l _n (i) = h

w T _Lp . . . w T _−Lf

i







s ^h _r (t − T L _p ) .. . s ^h _r (t + T L _f )







(2.39)

ˆ

s ^l _n is complex. From it, we can easily extract ˆ s ^l _I,n or ˆ s ^l _Q,n using the real or the imaginary part. The calculation of the MMSE gives

h w _T _Lp . . . w _T

−Lf

i

= G n (GR ss G ^∗ + R ηη ) ⁻¹ (2.40) Here, the filter includes the matched filtering. Consequently, the filter w must be at least as long as h. A good length for this filter is KM in order to avoid problems when filtering in the frequency domain during the demodulation. Let R s _n s _n and σ s _n be the correlation matrix and the variance of the signal s ^l _n . The residual MSE, noted Ω, is

Ω = R s _n s _n (1 − G ^| _n (GR s G ^| R η ) ⁻¹ G n R s _n s _n )

= σ ² _s _n (1 − G ^| _n ( σ _s ²

σ ² _sn GG ^| + σ _η ² σ _s ²

n

I KM ) ⁻¹ G n )

SINR = 1

(1 − G ^| n ( _σ ^σ ₂ ^s ²

sn

GG ^| + _σ ^σ ₂ ² ^η

sn I KM ) ⁻¹ G _n )

(2.41)

In the case where all the frequencies are used and an equal repartition of the power over each channel σ ² _η = σ ² _s _n and

SINR = 1

(1 − G ^| n (GG ^| + _SNR ¹ I KM ) ⁻¹ G n ) (2.42) The calculation can be done in real using the knowledge that the symbol is real. It will result two filters, one for the real part of the signal and one for the imaginary part.

If we can expect an improvement of the results, it would increase a lot the demodulation complexity.

Complexity

The complexity is given in terms of real multiplications for the modulation of 1 complex symbol.

• 1-tap

– Demodulation: 4 log 2 (M ) + 8K − 4 (PPN) – Equalization: 4

– Total complexity: 4 log ₂ (M ) + 8K

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• MMSE

– Demodulation: 4 log 2 (M ) + 8K − 4 (PPN) – Equalization: 4 × L w

– Total complexity: 4(log ₂ (M ) + 8K − 4 + L _w )

• Multiband MMSE

– Demodulation: 4 log ₂ (M ) + 8K − 4 (PPN) – Equalization: 12 × L w

– Total complexity: 4(log ₂ (M ) + 8K − 4 + 3L w )

• Zero Forcing Based Equalizer

– Demodulation: 2K(1 + 2 log 2 (KM )) − 2 (FF) – Equalization: 4 × (2K − 1)

– Total complexity: 2K(5 + 2 log 2 (KM )) − 6

• Per sub-carrier MMSE

– Demodulation: 2K(1 + 2 log ₂ (KM )) − 2 (FF) – Equalization: 4 × (2K − 1)

– Total complexity: 2K(5 + 2 log ₂ (KM )) − 6

2.2.2 MIMO channel

Let K t , K r , g u,v be the number of transmitting antenna, number of receiving antenna and the impulse response of the channel from the transmitting antenna v to the receiving antenna u.

SIMO and MIMO without transmitter precoding

With a SIMO configuration, each receiving antenna receives one transmitted signal which passed through different channels and with different additive noises. It is possible to use, for each received signal, an equalization method presented in the previous section. Then, a simple diversity method can be used to obtain our estimate of the symbols: maximum ratio combining, equal gain combining, selection combining, and so on. However, if the noise is independent from a received signal to another, the interference is not. Consequently, even the maximum ratio combining will be suboptimal.

The first approach can be improved by using all the received signals in the equalization method. This second approach can also be used with a MIMO configuration if there is no pre-coding. Let s I,n,r,u and s Q,n,r,u be the signals at the frequency band n received by the antenna u. We consider the same transmission model as the one presented in Figure 2.2.1 for each transmitted and received signals. We redefine the filters g _I,n,m,u,v , g _I,n,m,u,v and g _N,m,u as shown in Figure 2.16.

• S I,m,r,u (respectively S Q,m,r,u ) the vector of the received signal at the sub- channel I (respectively Q), at the frequency band m, received by antenna u.

• S I,n,t,v (respectively S Q,n,t,v ) the vector of the sent signal at the sub-channel I

(respectively Q), at the frequency band n, sent by antenna v.

(33)

Figure 2.16: MIMO transmission model

• G _I,n,m,u,v (respectively G _Q,n,m,u,v ) the convolution matrix corresponding to the filter g _I,n,m,u,v (respectively g _Q,n,m,u,v )

• Γ m the convolution matrix corresponding to the filter g N,m

• η u the noise received by antenna u.

The equations of the received signals are

S I,m,r,u =<

n+1

X

m=n−1 K _t

X

v=1

(G I,n,m,u,v S I,n,t,v + G Q,n,m,u,v S Q,n,t,v ) + Γ m η u

!

S _Q,m,r,u ==

n+1

X

m=n−1 K t

X

v=1

(G _I,n,m,u,v S _I,n,t,v + G _Q,n,m,u,v S _Q,n,t,v ) + Γ _m η _u

!

∀u ∈ [[1, K r ]]

(2.43) MMSE equalizer. As in the SISO case, we equalize the baseband sub-channel. Let ν and v 0 be the delay of the equalizer and the index of the transmitted signal we want to equalize. We want to estimate s I,0,t,v ₀ (i + ν) or s Q,0,t,v ₀ (i + ν) based on the vectors S I,0,r,u and S Q,0,r,u , u ∈ [[1, K r ]] with the MMSE equalizer. Let G 0,u,v , G 0,u , G 0 , Γ 0 , S r,0,u , S t,v , S t be the matrices

Filter Bank Multi-Carrier Modulation