Goodput oriented power allocation with adaptive modulation and coding algorithm for cognitive BIC-OFDM wireless systems

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Master’s Thesis in Telecommunication Engineering

Francesco Pierini

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Abstract

Cognitive Radio technology has big importance in our days, because of the increas-ing requests of availability of frequency spectrum, due to the large use of wireless devices. This thesis studies the problem of resources allocation in a cognitive wire-less system transmitting over multiple parallel channels. The main importance is attributed to the power allocation which goal is the maximization of the total system goodput, or, the offered layer 3 data rate, keeping at the same time the interference caused to the licensed users, which transmit over the same bands of the cognitive device, under the prescribed threshold. This optimization problem results to be convex and then solvable with conventional numerical methods. An opti-mal iterative algorithm is exposed. Extensive simulation results successfully verify the accuracy of the proposed analysis under operating conditions, thereby demon-strating the effectiveness in improving the computational efficiency of conventional simulation-based-only performance evaluation procedures.

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List of Tables

2.1 Supported MCS without bit loading. . . 22

2.2 Puncturing matrices. . . 24

2.3 Main Parameters of the OFDM system simulated. . . 25

2.4 Testcase - Channel Profile values. . . 26

4.1 Pseudo-Code for Sub-Optimal Power Allocation . . . 65

4.2 Pseudo-Code for Low Complexity Adaptive Power, Modulation and Coding Algorithm . . . 72

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List of Figures

1.1 Cognitive Scenario . . . 12

2.1 SISO-BIC-OFDM System Model. . . 14

2.2 Adaptive BICM Paradigm - Mode selection. . . 23

2.3 Testcase - Channel Profile. . . 26

2.4 Static modes - goodput performance. . . 27

3.1 Structure of a generic link performance prediction model. . . 33

3.2 Link-to-System mapping procedure. . . 34

3.3 κESM quality model structure. . . 42

4.1 Waveform Gaussian channel. . . 52

4.2 Colored noise - PSD. . . 53

4.3 Equivalent system composed of N parallel Gaussian channels. . . 54

4.4 Water-filling. . . 57

4.5 Water-filling - bad channel. . . 58

4.6 Waterfilling interpretation . . . 62

4.7 Feasible power coefficients domain . . . 66

4.8 Optimal point at 1st step . . . 67

4.9 Reduced set at next steps . . . 69

5.1 Simulation Scenario . . . 76

5.2 Suboptimal PA with underlay PUs at 250 and 300 meters from radius 77

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5.5 Suboptimal PA Simulation comparison at various distances . . . 78

5.6 SSR with AMC and static cases comparison . . . 79

5.7 SSR with underlay PUs at 250 and 300 meters from radius . . . 80

5.10 SSR with interweave PUs at 50, 80 and 100 meters from STx . . . . 81

5.11 SSR comparison at various underlay PUs distances . . . 82

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Chapter 1

Introduction

4G (also known as Beyond 3G), an abbreviation for Fourth-Generation, is a term used to describe the next complete evolution in wireless communications. A 4G system will be able to provide a comprehensive IP solution where voice, data and streamed multimedia can be given to users on an ”Anytime, Anywhere” basis [1].

There is no formal definition for what 4G is, yet there are certain objectives that are projected for 4G:

• a spectrally efficient system (in bits/s/Hz and bits/s/Hz/site); • high network capacity: more simultaneous users per cell;

• a nominal data rate of 100 Mbit/s while the client physically moves at high speeds relative to the station (outdoor traffic), and 1 Gbit/s while client and

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station are in relatively fixed positions as defined by the ITU-R, and in indoor scenarios;

• a data rate of at least 100 Mbit/s between any two points in the world; • smooth handoff across heterogeneous networks;

• seamless connectivity and global roaming across multiple networks;

• high quality of service for next generation multimedia support (real time audio, high speed data, HDTV video content, mobile TV, etc);

• interoperability with existing wireless standards; • an all IP, packet switched network [14], [1].

However, mainly due to harsh multipath propagation conditions, experienced at high data rates in both indoor and outdoor scenarios, and nowadays to the short-age of continuous large segments of bandwidth, an efficient use of the available resources, and a reliable and adaptive transmission scheme, becomes a crucial task for the design of an access network capable of guaranteeing the ambitious goals displayed above.

The recent standardization bodies of WiMAX (Worldwide Interoperability for Microwave Access) and 3GPP/LTE (Long Term Evolution) make use of radio tech-nology such as multicarrier (MC) modulation and multiple antenna based access scheme. In particular, MC modulation, in the form of Orthogonal Frequency Di-vision Multiplexing (OFDM), efficiently uses the available spectrum and is very strong to typical mobile radio multipath environment.

In order to further increase the system robustness against the trouble arising from the wireless propagation channels, an efficient modulation scheme has to be combined with a powerful and efficient channel coding technique. This is the case of BICM, which was first proposed in 1992 by Zehavi as a pragmatic coding scheme

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for bandwidth-efficient communications [66] and later, theoretically characterized by Caire in [68].

BICM technique is based on the insertion of a bit-interleaver between the channel encoder and the modulator and by using an appropriate soft-decision bit metric as an input to the Viterbi decoder. Thus, the code diversity can increase, from the smallest number of channel symbols along any error event, to the smallest number of distinct bits along any error event.

Also, the seminal works of and Tarokh [50], Foschini [62] and Telatar [51] proved that the spectral efficiency, as well as the reliability of the wireless link, can be dra-matically improved by employing Multiple-Input-Multiple-Output (MIMO) tech-niques. Signaling schemes exploiting also the spatial domain, to improve both relia-bility and throughput of wireless systems, are commonly named Space-Time Codes (STC) [50].

The STC performance improvement can be essentially accomplished by:

• exploiting the diversity in space, for instance through Space-Time Block Cod-ing (STBC) or Space-Frequency MultiplexCod-ing CodCod-ing (SFMC), so as to en-hance link reliability against the variations in signal level induced by fading; • simultaneously transmitting parallel data streams, the so-called Spatial

Mul-tiplexing (SM), in order to boost the data throughput and to increase the spectral efficiency.

The above-mentioned concepts can also be successfully combined together leading to MIMO BIC-OFDM systems.

Finally, for a flexible and efficient use of the available bandwidth, besides the aforementioned modulation and coding techniques, MAC and PHY layer must be able to support IP transmission. In order to guarantee a reliable packet trans-mission, error handling and recovery policies, such as Chase Combining (CC) or Incremental Redundancy (IR) HARQ techniques, must be employed and an appro-priate figure of merit, the goodput—briefly described at the end of this chapter and

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formally defined in the sequel of this work—has to be taken as landmark of the project.

1.1 Overall Radio Access Processing

This section discusses the main elements of the considered radio access process-ing together with the relevant physical (PHY) layer, shown in figure 1. Accordprocess-ing to the packet-centric vision for next generation wireless systems, the radio access processing is split into two main steps.

• Packet Processing, including channel coding and the retransmission protocol. For each frame, the packet processing outputs a sequence of coded packets, referred to as medium access control (MAC) protocol data unit (PDU). In this work, we do not account for the presence of a multiuser scheduler, but rather, we try to optimize the performance for a given user.

• Frame Processing, including multiplexing and modulation steps and mapping the basic physical resource. Generalized Multi-Carrier (GMC) based trans-mission implies that the basic physical resource can be expressed as a unit in the time/frequency grid.

In this embodiment, the key design guideline for efficient adaptive modulation and coding (AMC) schemes is the optimization toward the transport of the packets, which represent the fundamental pieces of information to be communicated over the radio interface. In order to satisfy such a packet-centric paradigm, the packet should be kept as an integral unit as far down in the protocol stack as possible. Con-sequently a one-to-one mapping of packets to RLC-PDUs will be assumed, wherein one RLC-PDU is characterized by a sequence number, the corresponding payload and the cyclic redundancy check (CRC), and it is exactly mapped into one FEC block. This peculiar structure enables a link resource adaptation (LRA) strategy aimed at optimizing the delivery of packets since we are interested in the entire packet and not in the reception of a part of it.

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1.2. BACKGROUND OF ADAPTIVE TECHNIQUES

1.2 Background of Adaptive techniques

Link adaptation to varying radio channel conditions is nowadays one of the key technologies for fulfilling the purpose of a 4G communication system, by efficiently making use of the available resort of the terminals.

When a system does not adapt the transmission parameters to the actual channel conditions, the designer must consider a fixed link margin to maintain acceptable performance in the worst case channel conditions. As apparent, the more channel state get better, this strategy leads to a very inefficient utilization of the available resources.

Of course, the basic premise for a proper link adaptation is the simultaneous knowl-edge of some sort of Channel State Information (CSI), by both the transmitter and the receiver, and this can be accomplished, as we will see in the sequel, with an estimate of the channel at the receiver and a feed back to the transmitter, of some link quality information based on that estimate.

There are many parameters that can be adapted, according to the current channel status, such as:

• data rate;

• coding rate/scheme;

• power distribution among the subchannels; • space-time coding.

In order to further improve the link performance, these adaptive techniques can be combined together, to design most powerful hybrid techniques which jointly adapt multiple system parameters.

1.2.1 Adaptive modulation

In variable-rate modulation, the data rate is varied with respect to the channel gain. This can be done by setting the symbol rate of the modulation and by using

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multiple modulation schemes or constellation sizes. In this work we consider vari-able rate QAM transmission, where the number of QAM levels is varied according to the channel status and the quality criteria which we intend to optimize [45].

1.2.2 Power allocation

Another issue to face in adaptive transmitter design is how to distribute the available power across a set of subchannels. Among the diverse strategies that can be invoked, a very popular criterion is based on the maximization of the input-output mutual information. In the ideal case of parallel Gaussian channels, with Gaussian input distribution, the solution is given by the well-known water-filling policy [13]. Since practical systems usually operate far from the theoretical capacity, or when the target of a certain communication system does not involve directly the maximization of the mutual information, various alternative criteria could be used instead.

1.2.3 Adaptive coding

In adaptive coding different channel codes are used to provide different amounts of coding gain to the transmitted bits. A stronger error correction code may be used for harsh propagation conditions, while a weaker code could be more suitable for favorable channel conditions. The implementation of adaptive coding used in the system considered in this work, is called Rate-Compatible Punctured Convolutional (RCPC) codes [49], and it consists of a family of convolutional codes at different code rates r. The basic premise of RCPC codes is to have a single encoder and decoder, whose error correction capability can be modified by not transmitting certain coded bits (e.g. by puncturing the code). Moreover, RCPC codes have a rate-compatibility constraint, so that, the coded bits associated with a high-rate (weaker protection) code are also used by all lower-rate (stronger protection) codes. Thus, in order to increase the error correction capability of the code, the coded bits

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1.2. BACKGROUND OF ADAPTIVE TECHNIQUES

of the weakest code are transmitted along with additional coded bits to achieve the desired level of error correction. The rate compatibility makes it very easy to adapt the error protection of the code, since the same encoder and decoder are used for all codes in the RCPC family, with puncturing at the transmitter to achieve the desired error correction.

Decoding is performed by a Viterbi algorithm, operating on the trellis associated with the lowest rate code, with the puncturing incorporated into the branch metrics. Adaptive coding, combined with adaptive modulation as a hybrid technique, takes the name of Adaptive Modulation and Coding (AMC).

1.2.4 Adaptive Space-Time coding

Multiple antenna approach is a powerful way to improve the wireless system performance. Multiple antennas were traditionally used to combat fading impair-ments by exploiting multiple independent paths between transmitting and receiving antennas.

In general, MIMO systems can supply two types of gain:

• diversity gain, the improvement in link reliability obtained by receiving more replicas of the information signal through independently fading links. This type of diversity is clearly related to the random nature of the channel and is closely connected to the specific channel statistics. The basic idea is that, if the various received path are uncorrelated, with high probability, at least one or more of these links will not be in a fade at any given instant. Thus, the use of multiple dimensions reduces the fluctuations of the received signal and eliminates the deep fades. This concept is suited for wireless communications where fading exists due to multipath effects;

• spatial multiplexing gain, the improvement in data rate, achieved by transmit-ting independent information streams in parallel, through the spatial channels. The price is a poor error rate performance, but in the high SNR regime, this effect is neglected.

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Every MIMO channel can provide both gains simultaneously and hence, there is a fundamental tradeoff between how much of each type of gain can be extract by any coding scheme. In a block fading channel, an expression of optimum diver-sity gain as function of spatial multiplexing gain is provided in [52] as optimum tradeoff curve. This allows to create a series of intermediate space-time coding con-figurations limited by two extreme points: full diversity and full multiplexing gain respectively. Every configuration or MIMO option represents the best solution in a particular channel condition. Therefore, adaptive space/time coding techniques change antenna use, to trade off diversity and multiplexing gain based on channel conditions [54], [55], [56].

Thus, an additional degree of freedom in AMC can be represented by the space dimension, by making use of multiple antennas at the transmitter and/or at the receiver.

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1.3. OUTLINE OF THE GOODPUT FIGURE OF MERIT

1.3 Outline of the Goodput figure of merit

In heterogeneous systems, different service classes are supported, so the first step to design an efficient LRA algorithm consists in identifying a significative figure of merit (e.g. objective function) capable of accommodating the requirement of each class.

Moreover, when applications employ hybrid automatic repeat request (HARQ) pro-tocols, only error-free RLC-PDUs are kept by the receiver, while those corrupted have to be retransmitted. This specific feature clearly acts upon the structure of LRA, leading to a couple of classes of strategies:

1. Delay Oriented algorithms (DO ), aimed at avoiding frequent retransmission, by keeping the packet error rate (PER) below a given threshold;

2. Goodput Oriented algorithms (GO ): aimed at maximizing the amount of ceived error-free data per unit of time—or goodput—irrespective of the re-transmission rate.

Each of these strategy can be suitable for a particular service class.

The most common application can be divided into four service classes, characterized by different Quality of Service (QoS) requirements [57], as listed below.

• Unsolicited Grant Service (USG) supports constraint bit rate of fixed through-put connections such as voice over IP (VoIP). This service provides guarantees on throughput and latency;

• Real-time Polling Service (rtPS ) provides guarantess on throughput and la-tency, but with greater tolerance on latency relative to UGS. This class sup-ports services such as MPEG video conferencing and video streaming;

• Non-real-time Polling Service (nrtPS ) provides guarantees in terms of through-put only and it is therefore suitable for applications like File Transfer Protocol (FTP);

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• Best Effort (BE ) service provides no guarantees on delay or throughput, and it is used for Hypertext Transport Protocol (HTTP) and electronic mail (e-mail).

Let us note that the last two classes do not require strict delay constraint. For these applications, the users are only interested in one quantity: the number of data bits delivered in error-free packets per unit of time (e.g. the offered layer 3 data rate), which is exactly what the goodput criterion expresses. In this thesis, we make use of the goodput metric as originally proposed in [63] and [64], as objective function around which some bit loading strategy have been derived, to be included into a general LRA algorithm suitable for 4G systems.

When a wireless station is ready to transmit a data packet, its expected goodput is defined as the ratio of the data payload to be delivered, to the expected transmis-sion time, assuming that the signal will experience the current channel conditions known at the transmitter. Clearly, depending on the data payload length and the wireless channel conditions, the expected goodput varies with different transmission strategies. The more robust the transmission strategy, the more likely the frame will be delivered successfully within the frame retry limit, however, with less spectral efficiency.

So, the goodput provides a tradeoff between transmitted data rate and reliability of the received data, and the key idea, behind our LRA strategy, is make use of it in order to select the most appropriate transmission strategy, such that ”the packet” could be successfully delivered in the shortest possible transmission time.

This is what the goodput helps to achieve, anyway its analytical coverage is presented in Chapter 3, after the derivation of a Link Quality Metric (LQM) which will allow the goodput to be efficiently employed in practical transmission scheme.

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1.4. COGNITIVE SCENARIO

1.4 Cognitive scenario

Cognitive Radio (CR) technology [65] is a good solution for a better use of electromagnetic spectrum, that is becoming a precious resource; it relies on the idea that an unlicensed device (Secondary User, SU) can transmit over the same frequencies belonging to other licensed users (Primary Users, PUs), with the con-straint of keeping the interference caused to them under a prescribed limit, named interference threshold. According to a well-known model adopted in literature [31], the secondary transmitter causes two different kind of interference to the PUs, as depicted in fig. 1.1.

The underlay paradigm

The first type is an in-band interference, due to the fact that the SU transmits at the same time and over the same bands of the PUs. This case, named underlay paradigm, can be described as follows. Let assume, in the most general case, a set of J = {1, ..., J} PUs transmitting over J subbands of width Bj. If the secondary transmitter is geographically set at a certain distance with respect to these PUs, it is allowed to transmit over those subbands, so that Btot =PJ_j=1Bj becomes the total available bandwidth in transmission for the SU, that it is assumed to be composed of Ns subcarriers. Let define the mapping function φ(n) = j, meaning that subcarrier n belongs to the j-th subband. PUs have an interference-free zone of radius R, in which any secondary transmission is completely avoided. The SU must guarantee that the interference created on the edge of the generic j-th zone is kept below a prescribed threshold, named interference temperature, eTj. If Θj is the path-loss between the SU and the edge of the j-th zone, the maximum allocable power for the secondary transmitter on the j-th subband results therefore Tj = eTjΘj.

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Figure 1.1: Cognitive Scenario

The interweave paradigm

The second type of interference is an out-of-band interference. In fact, it cor-responds to the case in which SUs and PUs are co-located. With respect to these PUs, the STx transmits only over frequency bands that are near or adjacent to their ones. Anyway, the side lobes of the signal transmitted by the STx causes an interference over the PUs bands. In particular, assuming as set of L = {1, ..., L} interweave PUs, each receiving over a band of width Wl, l = 1, ..., L, and defining ψ(·) a function depending on the power spectral density of the transmitted signal and the fading gain between the STx and generic PRx, the interference caused by the i-th subcarrier of the STx on the l-th PU band results

I_i(l) = pi

di,l_Z+Wl₂

di,l−Wl₂

ψ(f )df = piK_i(l), (1.1)

where di,l is the spectral distance between subcarrier i and the l-th PU band and piis the power allocated over the above mentioned subcarrier. Thus, denoting with I_th(l) the highest interference allowed over the l-th PU band, it follows that

PN n=1I (l) i = PN n=1piKi(l)≤ I (l) th ∀l, l ∈ L (1.2)

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Chapter 2

BIC-OFDM System description

This chapter is focused on the description of the intertwinement of the technolo-gies before introduced, and which are located in the MAC and PHY layers of the base stations (BSs) and/or of the mobile terminals (MTs).

In Section 2.1 the block diagram of a SISO-BIC-OFDM transmission scheme has been showed, together with the detailed description of the processing performed over the signals. Such a framework represents the subject matter for which some optimization policies will be elaborated in the follows chapters of this thesis. Then, in Section 2.3 a possible adaptive paradigm, exploiting the system just described, is presented, with a simulation example that will introduce and justify the need of ”adaptation”.

2.1 SISO System model

With reference to the block diagram depicted in Figure 2.1 and according to the radio access processing outlined in Section 1.1, each RLC-PDU of Nb information bits um (including the RLC-PDU overhead), is transmitted through L consecutive OFDM symbols which form an OFDM frame (or block ). All the OFDM symbols belonging to the same frame experience the same fading realization.

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certain code rate r. This process generates, for each RLC-PDU encoded, Nc = N∆ b/r coded binary symbols (cbs) bk, which are subsequently randomly interleaved.

Figure 2.1: SISO-BIC-OFDM System Model.

The bit-level interleaver randomly maps the generic cbs bk, onto one of the label bits carried by the symbols of the OFDM subcarriers, according to the follow notation:

bk→ cΠ(k), (2.1)

where Π(k) =∆ {lk, nk, ik} is the interleaver law, which maps the index k of the cbs into a set of three coordinates:

1. lk, the position of the OFDM symbol within the frame; 2. nk, the OFDM subcarrier number;

3. ik, the position of the cbs within the label of the QAM symbol on a certain subcarrier.

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2.1. SISO SYSTEM MODEL

The interleaver is assumed to be fully random, so that the probability of mapping the generic bk (taken out of the available Nc) into the ith label bit of the QAM symbol transmitted on the nth subcarrier into the lth OFDM block, denoted as cl,n,i, with l = 1, . . . , L, n = 1, . . . , N and i = 1, . . . , m(n), is

Pr_{bk→ cl,n,i}=∆ 1 Nc

, (2.2)

where m(n)_{denotes the number of cbs allocated on the nth subcarrier.}

With respect to Figure 2.1, the cbs outbounding the interleaver are well ordered by the Subcarrier Parser and ready to be Gray mapped onto M-QAM symbols, with M = 2∆ m and m∈ {0, 2, 4, 6}. The ”0” means no transmission.

So, denoting with χ(n) the M-QAM modulation selected for the nth subcarrier and with x(n)_l ∈ χ(n)_{the transmitted symbol, the lth OFDM symbol will bear the follow} QAM symbol vector:

x_l=∆nx(n)_l oN

n=1. (2.3)

The total available power for transmitting an OFDM symbol is N P , where P represents the amount of power allocated on a certain subcarrier in case of uniform distribution.

At this point the data-bearing QAM symbols are transmitted, after having been subject to the digital OFDM processing, including:

• IFFT;

• addiction of cyclic prefix (CP); • Digital to Analog (D/A) conversion. • root-raised-cosine (RRC) filtering. • High power amplifier (SSPA)

The resulting OFDM signal experiences a frequency-selective fading channel which will be assumed stationary for whole frame duration (Block fading [27], [28], [29] and [30]).

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At the receiver side, the signal will go through the whole inverse OFDM process-ing in order to be coherently demodulated subchannel per subchannel, exploitprocess-ing the channel estimation, which will be assumed perfect.

From this channel estimation, the post-processing SNRs are derived, and besides that for the BICM metric evaluation and for the Viterbi decoding, they are also used for making Link Adaptation at the next packet transmission.

Finally, the soft-demodulated metrics are deinterlived and the Log-Likelihood Ratios (LLRs) will feed a Viterbi decoder in order to obtain a ML estimation of the information bits, ˆuk, composing the RLC-PDU.

2.1.1 Post-processing SNRs computation

In this section, we take advantage of the fact that the OFDM system can be rep-resented as a set of parallel gaussian subchannels to state an analysis of it, which will be helpful for a further definition of a comprehensive Link Quality Metric (LQM). So, the post-processing SNRs for each subchannels are computed.

Focusing on the transmission of the generic bk, according to the notation outlined beforehand, this cbs has been conveyed over the QAM symbol x(n)_l . So, it can be recovered from the complex sample relevant to the nth subcarrier, within the lth OFDM symbol, at the output of the OFDM receiver, whose expression is

z_l(n)= H(n)y_l(n)+ w(n)_l , (2.4)

with H(n), the complex-valued channel gain experienced on the nth subchannel and w(n)_l a zero-mean, σ2_w-variance, gaussian RV, representing the noise contribu-tion.

The instantaneous post-processing SNR relevant to the generic nth subchannel is then defined as

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2.2. THE RADIO-MOBILE CHANNEL MODEL γ(n) ∆= H(n) σw(n) !2 (2.5)

and we will refer at the set of all the post-processing SNRs as the vector

γ=nγ(n)oN

n=1. (2.6)

2.2 The radio-mobile Channel model

In a radio-mobile scenario, signals experience several paths due to reflection, scattering and, in general, to the high number of obstructs between transmitter and receiver: this phenomenon is called multipath fading and is represented by multiple versions of the received signal with different attenuation, phase and delays arrive. Fading can be either a good or a bad effect:

• in a Line-of-Sight (LOS) context, it can be a disadvantage since the signal replies can cause a destructive interference to the direct signal;

• in Non Line-of-Sight (NLOS) condition, reflections are essential and exploited to correctly receive the signal.

The characterization of a radio-mobile channel can be given by some parameters: the number of paths N (t) at time t, the amplitude αn(t), the delay τn(t) and the phase φn(t) of the n-th path at time t (these are modeled as independent random process), the transmission frequency fc and the possible presence of a path in LOS. The channel is thus modeled as a time-variant linear system:

c(t, τ ) = N (t)_X n=0

αn(t)e−jφn(t)δ(τ − τn(t)) (2.7) where τ represents the temporal interval between the applied impulse and the generic observation instant t.

An important parameter to take into account is the delay spread, which measure the temporal distance between the first arriving path and the arriving last. This

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delay indeed produces a extension of the signal transmitted that can cause inter-symbolic interference. Moreover, due to Doppler effect, the channel cause a spread also in frequency: relying on the relative speed between transmitter and receiver, each tone is affected by frequency translation, called Doppler bandwidth:

βf = v

λcos α (2.8)

where v is the mobile speed, λ is the wavelength and α is the angle between the wave propagation and the terminal shift. We can now give a general expression for the transmitted signal s(t) and the received signal r(t), in absence of noise:

r(t) = Re (" N (t)_P n=1 αn(t)u(t− τn(t)) # ej(2πfc(t−τn(t))+φDn) ) = Re (" N (t)_P n=1 αn(t)e−jφn(t)u(t− τn(t)) # e−j2πfct ) (2.9)

φDn is the phase shift due to Doppler effect and then the phase shift for each path due to Doppler effect and delay τn(t) is:

φn(t) = 2πfcτn(t)− φDn (2.10)

All this events that affect a radio-mobile signal can cause fading. In fact the phases φn(t) of the multiple versions of the signal can produce constructive or destructive interferences which are unpredictable.

Fading can be classified as:

• Long-term fading: the signal amplitude fluctuates relatively slowly, due to shadowing of the LOS path and obstructs, when the terminal movements are bigger than the wavelength λ;

• Short-term fading: due to multipath, the signal amplitude varies quickly around is its mean value for shift on the order of the wavelength λ;

• Slow fading: the channel characteristics varies after several signaling interval; • Fast fading: the channel characteristics varies within a signaling interval;

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2.2. THE RADIO-MOBILE CHANNEL MODEL

• Frequency flat fading: the channel delay spread is negligible respect to the signaling interval;

• Frequency selective fading: the channel delay spread is not negligible respect to the signaling interval.

2.2.1 Static Channel Model

If the channel characteristics do not vary quickly in time, the channel is not selective in time and the model results:

c(τ ) = N X n=1 αne−jφnδ(τ − τn) = N X n=1 αnδ(τ − τn) (2.11) Let consider the channel frequency response:

C(f ) = N X n=1

αne−j2πf τn (2.12)

and the delay spread:

Tm = max

n |τn− ¯τ| (2.13)

with ¯τ the average delay. If all the delays were equal (τn= ¯τ ∀n), |C(f)| would be constant: C(f ) = e−j2πf τ N X n=1 αn (2.14) |C(f)| = | N X n=1 αn| (2.15)

Thus, we can define the coherence bandwith of the channel the maximum frequency interval in which |C(f)| does not vary significantly, and it is:

Bc∝ 1 Tm

(2.16)

A signal transmitted on channel _{|C(f)| is distorted if bandwith B of the former} is bigger than the coherence bandwith Bc of the latter. We can now differentiate multipath channels in two classes:

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1. Frequency selective: if B > Bc (T > Tm) and then the frequency components will experience different magnitudes of fading;

2. Frequency flat: if B < Bc (T << Tm) and then the frequency components will experience the same magnitude of fading.

where T is the signaling interval.

The effect of the static channel can be represented by a complex coefficient:

a = ρejφ = e−j2πfcτ

N X n=1

αn (2.17)

which means that the signal is affected by an attenuation and a phase shift but that it is not distorted, r(t) = Re_{as(t)ej2πfct_}.

2.2.2 Time Varying Channel Model

In the previous section, we assumed an and τn static in time. This assumption though is not valid in a radio-mobile scenario in which attenuation and delays change with the mobile position, becoming function of time.

A simple model for the parameter a(t) is achievable supposing a high number of path N and exploiting the central limit theorem. In this way, a(t) can be considered a gaussian process with spectral density depending on the mobile speed. Clarke model describes a(t) as a zero mean gaussian process with independent real and imaginary components. The spectral density results:

Sac(f ) = Sas(f ) =      σ2 2πfD 1 √ 1−(f /fD)2 for − fD 6 f 6 fD 0 otherwise (2.18)

Analyzing eqn. 2.18, the inverse of the Doppler frequency fD results to be the coherence time of the channel, which is to say the time over which the frequency response of the channel do not vary significantly. Thus, if the signal has a signaling time T , the number of signals in the coherence time is 1/fDT and the channel produces:

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2.3. ADAPTIVE BICM PARADIGM WITHOUT BIT LOADING

1. slow fading: if 1/fDT >> 1, or, the channel amplitude response does not vary for several signaling interval;

2. fast fading: if 1/fDT << 1.

2.2.3 Block Fading Channel Model

In packet transmission simulations, a slow fading channel model is usually as-sumed; we can suppose constant a(t) for several signaling intervals or for all the entire packet transmission. It is equivalent to approximate a(t) with a step function and, in each step, a can be modeled as in eqn. 2.17, where ρ and φ are two random variables.

• ρ is a Rayleigh random variable if there is no LOS, otherwise is modeled as a Ricean random variable.

• φ is supposed to be uniformly distributed in [0,2π].

The channel assumed in this thesis is indeed the block fading one described above. The received signal can thus be expressed by:

r(t) = N X n=1

ans(t− nT ) (2.19)

where T is the signaling interval of the transmitted signal s(t), nT is the delay with whom the ray reach the receiver and an are the channel coefficients described in eqn. 2.17. This model is also referred as Tapped-delay-line model.

Usually, the channel profile is given providing the number of the path N , the delay and the variance σ_n2 = E∆ _|an|2 of the distortion coefficients on each path. These profile are standard and refer to typical environments as Typical Urban (TU), Hilly Terrain (HT), Rural Area (RA), etc.

2.3 Adaptive BICM paradigm without bit loading

When bit loading is not adopted by an adaptive GMC transmission scheme, the same modulation haS to be transmitted over all the subcarriers, with the same

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power allocation as well. Anyway, for each packet transmission, based on the channel feedback, it can be possible to select a different modulation (e.g.M-QAM ) and a different code rate r. The combination of thess two parameters is called Modulation and Coding scheme (MCS), or mode for short. The modes considered in our example are summarized in Table 2.1.

Mode Modulation Code Rate

1 4-QAM 1/2 2 4-QAM 2/3 3 4-QAM 3/4 4 4-QAM 5/6 5 16-QAM 1/2 6 16-QAM 2/3 7 16-QAM 3/4 8 16-QAM 5/6 9 64-QAM 1/2 10 64-QAM 2/3 11 64-QAM 3/4 12 64-QAM 5/6

Table 2.1: Supported MCS without bit loading.

With reference to Figure 2.2, within this example, the FEC block can be con-catenated with a single M-QAM mapper, through a bit-level interleaver, being the modulation constrained to be the same for each subchannel. Nonetheless, the more general scheme depicted in Figure 2.1 is still valid. M-QAM modulation with Gray mapping is used in order to minimize the number of bit errors in every channel symbol error event.

The considered system operates with a feedforward, punctured, 64-state convo-lutional encoder, with mother code rate r = 1/2 and df ree= 10, whose polynomial

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Figure 2.2: Adaptive BICM Paradigm - Mode selection.

generators are

g1 = 1338 = 1011011

g2= 1718 = 1111001. (2.20)

Different code rates have been obtained from the mother convolutional code, by making use of the puncturing patterns shown in Table 2.2.

For the termination of convolutional code, there are three possibilities:

• Truncation: encoding is terminated with the last bit of the message and the encoder terminates in a state which is not known at the receiver side. This option provides a bad protection of the last bits of the message, leading to an error floor, and should therefore be avoided.

• Zero-termination: after encoding of the message, some tails bits are appended which drive the encoder state to zero. Here, we have the same protection for all the bits of the message, but, some extra bits, for driving the encoder state to zero, are required.

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Code rate Puncturing matrix df ree 1/2 P =   1 1   10 2/3 P =   1 1 1 0   6 3/4 P =   1 1 0 1 0 1   5 5/6 P =   1 1 0 1 0 1 0 1 0 1   4

Table 2.2: Puncturing matrices.

• Tail-biting: the initial state is chosen such that it coincides with the final state. This is the chosen solution, because it combines the advantages of the two previous methods, that is, all the bits are protected equally and the code rate is been maintained.

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2.3.1 Simulation example

The aim of this example is to show how the performance of an OFDM varies when a different MCS is employed, and to provide, based on these achievements, an introduction over how an adaptive scheme could actually operate.

We consider an OFDM system, whose main parameters are displayed on Table 2.3.

Testcase

System Channel Bandwidth (MHz) 20 Number of used subcarrier 64 Subcarrier spacing (kHz) 312.5

Cyclic prefix (µs) 0.8

OFDM Symbol duration (µs) 4

Table 2.3: Main Parameters of the OFDM system simulated.

We focus on the average goodput per subcarrier performance even though a for-mal mathematical definition of the goodput has not yet been provided (it will be the topic of the next chapter). Nevertheless, remembering that the goodput expresses the expected number of bits belonging to error-free packets per unit of time, from the graph depicted in Figure 2.4, it will not be difficult extrapolating some infor-mation, useful to justify the introduction of an adaptive paradigm.

Each of the 12 curves is obtained by transmitting a frame made of 10000 pack-ets,each made of Nb = 1024 payload bits and Nh= 32 header bits, with the respec-tive MCS. Each multipath channel realization is modeled as a 6-taps independent Rayleigh RVs and will be assumed stationary for the whole frame duration. Looking at the channel profile in Figure 2.3, we notice that its maximally path delayed is at 0.6µs, so the delay spread introduced by the channel over the transmitted signal will be entirely within the cyclic prefix of 0.8µs, leading the hypothesis of parallel gaussian channels to be well grounded.

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σr 2 [dB] 600 500 400 300 200 100 0 τ [ns] 0 -2 -4 -6 -8 -10

Testcase - Channel Profile

Figure 2.3: Testcase - Channel Profile.

Path # Delay (µs) Relative Power (dB)

1 0 -3.3 2 50 0 3 100 -0.99 4 200 -1.5 5 250 -3 6 600 -9.40

Table 2.4: Testcase - Channel Profile values.

Let us note that the curves stabilize on a determined level after a certain SNR value, depending on each particular mode and on the RLC-PDU header length. However, after reading Chapter 3, and coming back to this part, it will be every-thing clear.

Moreover, for low average SNRs, the modes with a low complex modulation and a stronger code protection achieve the highest goodput performance. On the other hand, the higher the SNR, the more complex the MCS tends to achieve the best

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2.3. ADAPTIVE BICM PARADIGM WITHOUT BIT LOADING 5 4 3 2 1 0 GP (bit/subcarrier) 40 30 20 10 0 E_s/N₀(dB) 'Mode 1' 'Mode 2' 'Mode 3' 'Mode 4' 'Mode 5' 'Mode 6' 'Mode 7' 'Mode 8' 'Mode 9' 'Mode 10' 'Mode 11' 'Mode 12'

Figure 2.4: Static modes - goodput performance.

performance.

In a wireless mobile scenario it is typical for the signal experiencing different channel propagation condition, due to plenty of factors, as pathloss, shadowing, fast fading, intercell-interference, etc. [79], and a non-adaptive transmission scheme is constrained to certain fixed performance, which could be represented by one of the curves of Figure 2.4. On the contrary, an adaptive scheme, for example the one considered above, thanks to its capability of varying its transmission parameters, can be able to achieve the best performance for every channel conditions.

In our example the transmitter will vary its mode, based on some feedback infor-mation, addressed as LQM in Figure 2.1. The particular LQM exploited within this work will be exposed in the future.

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enough to obtain a full adaptation, so the entire post-processing SNR vector might be sent at the transmitter.

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2.4. CONCLUDING REMARKS

2.4 Concluding remarks

In the first section of the chapter, we have defined the framework of the trans-mission scheme which will be progressively optimized, thanks to adaptive techniques outlined in Section 2.3.

Link adaptation will be possible exploiting some feedback information, which have been briefly introduced and represented in the transmission schemes. This informa-tion essentially concerns the channel quality, and the transmitter will use them in order to adapt its parameters and to obtain a certain system performance.

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Chapter 3

Link Quality Metric and

Goodput analysis

Adaptive modulation and coding (AMC) enables robust and spectrally-efficient transmission over time-frequency selective channels by making the best use of avail-able resource. In fact, when a system does not adapt the transmission parameters to the actual channel conditions, the designer must consider a fixed link margin to maintain acceptable performance in the worst case channel condition.

The AMC design problem is essentially an optimization task with objective functions and constraints properly defined so as to accommodate specific system scenarios. Each AMC strategy is characterized by an adaptation criterion (e.g. objective func-tion), the target Quality of Service (QoS) parameters (e.g. the constraints of the problem), the available information on the channel status, and the resulting output, which is represented by a set of optimized transmission parameters. The feasibility of this method, however, requires that the transmitter can perform precise and sim-ple evaluation of the actual link performance, in particular an accurate prediction of the packet error probability (PER).

It is well known that PER link performance of a given coded digital communication over a stationary additive white Gaussian noise (AWGN) channel can be expressed as a function of the signal to noise ratio (SNR) [32]. In multi-carrier systems such as

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orthogonal frequency division multiplexing (OFDM), however, the frequency selec-tive fading over the transmission channel introduces large SNR variations across the subcarriers, thus making link PER prediction a demanding task. The need for an accurate yet manageable link-level performance figure has led to the effective SNR mapping (ESM) concept [33], where the vector of the received SNRs across the sub-carriers is ?compressed? into a single SNR value, which in turn can be mapped into a PER value [81]. An overview of the link level performance prediction method, in particular of the κESM concept is presented Section 3.2 devoted to the derivation and the analysis of the ”κESM ”.

LQM and goodput have been presented in the same chapter because, as we will widely see, they become strongly related, once a fast and precise estimation of the latter is required at the transmitter in order to determine the parameters (i.e. constellation sizes, code rate, powers, etc.) for actually achieving the best performance possible.

3.1 Link Level Performance Prediction

Traditionally, the performance of radio links have been evaluated in terms of packet error rate (PER) as a function of the signal-to-interference plus noise ratio (SINR) averaged over all the channel realizations of one specific channel model. PER versus average SINR performance has therefore been widely used as the inter-face between the Link-level and System-level simulators [79], but unfortunately, this model became useless when we want an estimation of the instantaneous link level performance. In fact, as shown in [80], the specific channel realization encountered may result in an performance which is significantly different compared to the one predicted by the previous model.

Consequently, the performance assessment of fast link adaptation in system level simulations requires a more accurate link performance model relying on the instan-taneous channel and interference conditions. Naturally, so as to obtain an as general

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3.1. LINK LEVEL PERFORMANCE PREDICTION

as possible model, the effect that LRA could provide at the instantaneous perfor-mance must be taken into account, as depicted in Figure 3.1.

Figure 3.1: Structure of a generic link performance prediction model.

The first step is the extraction of a set of Quality Measures (QMp, p = 1, . . . , P ), once for each resource element involved in the packet of interest [79]. Since the num-ber of resource elements is generally very large (i.e. the numnum-ber of subcarrier of an OFDM system), a compression over these amounts is required, massively reducing the number of quality measure to typically one: a scalar LQM. Eventually, this scalar is mapped to a PER value.

However, from the many proposals compatible with this approach found in the literature, we will focus on the Effective SNR Mapping (ESM) [78],[74]. The ob-jective of such a method is to find a compression function that maps a sequence of varying SNRs into a single value that is strongly correlated with the actual PER. The basic principle of ESM is to be able to go from an instantaneous channel state, such as the instantaneous SNR for each subcarrier in the case of OFDM, to a corre-sponding PER, through a compressed value called effective SNR and tagged in the

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sequel as γef f. This is used to find an estimate of the PER from a basic AWGN link level performance.

Figure 3.2: Link-to-System mapping procedure.

For instance, given the set of the instantaneous post-processing SNRs experi-enced by the subchannels

γ =∆ hγ(c0)_{, γ}(c1)_{, ..., γ}(c|Ψ|−1)iT_, _(3.1) with γ(c) _{from (2.5), Ψ the set of all the subchannel indexes and c}_{= (n, q)}∆ _{∈ Ψ a} generic element, we want achieve

P ERAW GN(γef f) = P ER(γ), (3.2)

where P ERAW GN and P ER refer to the AWGN equivalent system and the one to be modelled, respectively.

In literature, different ESM methods [79] based on different mapping functions have been proposed and a couple of them have been shortly described below.

• Exponential ESM or EESM, based on the PEP Chernhoff bound for the case of binary signalling. This mapping function can be expressed in a simply closed

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3.2. THE κESM METHODOLOGY

form, but a generalization for high order modulations does not exist. A fine tuning factor is necessary for adjusting the PER estimate and reaching good accuracy for each modulation and coding scheme, but an accurate prediction when different modulation are used among the subcarriers is a tough task. • Mutual Information Based ESM or MIESM, includes two separates model, one

for the modulation and the other for the coding, providing good performance prediction for the mixed-modulation as well. Unfortunately, a closed expres-sion for calculating the mutual information does not exist, so a polynomial approximation is essential, as proposed in [40].

3.2 The κESM methodology

Differently from the conventional ESM methods, the κESM relies on an accurate evaluation of the PEP figure through the statistical description of the BIC log-likelihood metrics, thus offering an efficient accuracy versus manageability tradeoff. Recalling the system model description outlined in Section 2.1, the discrete-time signal relevant to the subchannel (n, q) in the lth OFDM block can be equivalently expressed as z_l(n,q)= q γ(n,q)_y(n,q) l + n (n,q) l , (3.3)

where γ(n,q) is the post-processing SNR experienced on the subchannel (n, q) and n(n,q)_l is a Gaussian RV with variance σ2_n = 1 representing the equivalent noise contribution. Resorting to a vectorial notation, the I/O relationship relevant to the nth subcarrier can be expressed as

z(n)= Υ(n)1/2 y(n)+ n(n), (3.4)

where Υ(n) _{is a diagonal matrix defined as Υ}(n) ∆

= D(γn,1_,_{· · · , γ}n,Ns_{), while the} noise vector is nn= [n∆ (n,1),_{· · · , n}(n,Ns)_]T_.

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3.2.1 PEP Analysis

Let first consider two distinct codewords, i.e., two sequences of coded binary symbols which originate from the same state of the code and merge after d trellis steps, and let denote them as two d-dimensional arrays b and b′, whose kth elements are bkand b′k, respectively. We also denote with ˆba codeword at the decoder output. The aim of this sub-section is to evaluate the Pairwise Error Probability (PEP), defined as

P EP = Pr∆ nbˆ = b′_{|b, Υ}o, (3.5)

where Υ is a block diagonal matrix defined as Υ = D(Υ∆ (1)_,_{· · · , Υ}(N )_{). In case of} ideal CSI, i.e., assuming that the actual value of the post processing SNR matrix Υ is known, the BICM log-likelihood metric for the kth coded binary symbol at the decoder input can expressed as

Lk= log P ˜ x∈χ(ik,nk,qk) b′ k pz(nk,qk) lk |x (nk,qk) lk = ˜x, Υ P ˜ x∈χ(ik,nk,qk) bk pz(nk,qk) lk |x (nk,qk) lk = ˜x, Υ , (3.6) where: pz(nk,qk) lk |x (nk,qk) lk = ˜x, Υ ∝ exp   − z(nk,qk) lk − p γ(nk,qk)x˜ 2 σ2 n    (3.7)

is the Gaussian-shaped Probability Density Function (p.d.f.) of the received sample value, conditioned on the transmitted symbol ˜x and on the post-processing SNRs Υ, while χ(i,n,q)a represents the subset of all the M-QAM symbols belonging to χ(n,q) whose ith label bit is equal to a. By replacing (3.7) into (3.6) (recalling that σ2

n= 1), the log-likelihood metric _Lk can be rewritten as

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3.2. THE κESM METHODOLOGY Lk= log P ˜ x∈χ(ik,nk,qk) b′ k exp −z(nk,qk) lk − p γ(nk,qk)x˜ 2 P ˜ x∈χ(_bkik,nk,qk)exp −z(nk,qk) lk − p γ(nk,qk).˜x 2 . (3.8)

Under the assumption of ideal interleaving, the BICM subcarriers behave as mem-oryless BIOS channels and the PEP can be computed as the tail probability [70]

P EP = Pr ( _d X k=1 Lk> 0 ) = Pr{Θ > 0} , (3.9) where we defined Θ=∆ Pd_k=1Lk.

Unfortunately, the computation of (3.9) by the p.d.f. of Θ reveals too involved. We resort then to the moment generating function (m.g.f.) of the RV Θ, which is given by

MΘ(s)

∆

= EΘ{exp (sΘ)} = [ML(s)]d, (3.10)

where ML(s) = E∆ Lk{exp (sLk)} is the m.g.f. of the RV Lk, and we exploited the fact that the LLRs_Lkare independent and identically distributed (i.i.d.) RVs. By this way, we can evaluate the PEP by using the following integral1 _[67]

P EP = 1 2π Z σ+∞ σ−∞ MΘ(s) ds s = 1 2π Z σ+∞ σ−∞ [ML(s)]d ds s . (3.11)

From (3.8), the m.g.f. to be used in (3.11), turns out

ML(s) = Ek               P ˜ x∈χ(ik,nk,qk) b′_k exp −z(nk,qk) lk − p γ(nk,qk)x˜ 2 P ˜ x∈χ(_bkik,nk,qk)exp −z(nk,qk) lk − p γ(nk,qk)x˜ 2      s         , (3.12)

1_{In(3.11), σ is any real number which ensures that the contour path lies in the region of}

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where the statistical expectation is to be taken with respect to all those parameters that depend on the index k. Substituting the expression of the subcarrier output (3.3), and under the assumption of ideal CSI, we have

ML(s) = Ek               P ˜ x∈χ(ik,nk,qk) b′ k exp −pγ(nk,qk) x(nk,qk) lk − ˜x + n(nk,qk) lk 2 P ˜ x∈χ(ik,nk,qk) bk exp −pγ(nk,qk) x(nk,qk) lk − ˜x + n(nk,qk) lk 2      s         . (3.13)

Similarly to the approach suggested in [70], the m.g.f. can be upper-bounded and, it has also been demonstrated that, at high SNRs, the bound is dominated by the term relevant to the nearest neighbor (in the sense of Euclidean distance) of x(nk,qk)

lk in the complementary subset χ(ik,nk,qk)

b′

k . As a consequence, we can apply the Dominated

Convergence Theorem (DCT) [70], [58], obtaining

ML(s)≃ Ek n exp_−γ(nk,qk)_d2_x(nk,qk) lk , x s_{− s}2o, (3.14)

where x is the nearest neighbor of x(nk,qk)

lk in the complementary subset χ

(ik,nk,qk)

b′

k ,

and d(y, w) is the Euclidean distance between the complex-valued symbols y and w. Let us note that, for Gray mapping rule, the distance d2(x(nk,qk)

lk , x) can be expressed as d2x(nk,qk) lk , x _∆ =∆(nk,qk)_d(nk,qk) min 2 , (3.15) where d(nk,qk)

min is the minimum Euclidean distance between the symbols in the com-plete QAM set χ(nk,qk)_{associated with the n}

kth subcarrier, and ∆(nk,qk)is a positive integer coefficient. The statistical expectation in (3.14) must be evaluated with re-spect to all the possible values of the transmitted symbol x(nk,qk)

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possible positions of the coded binary symbol bk into the label of the QAM symbols of the complementary subset.

Now, let us note that, for each subchannel c = (n, q), there are m∆ (c) label bits, and each label bit has 2m(c)/2 symbols on its complementary subset, so that, the number of terms to be averaged results m(c)_{· 2}m(c)−1. However, it can be easily verified that there are only m(c)_{/2 distinct values.}

For example, assuming for the sake of simplicity a 16-QAM modulation, we have 32 terms, 24 of which are at distance d(_minc)2, and 8 at distance 2d(_minc)2. Thus, by averaging with respect to the coded binary symbol index k, we end up with

ML(s)≃ X c∈Ψ Pr (c) m(c)₂m(c)₋₁ K X ∆=1 ψ(c)(∆)_{· e}−γ(c) ∆·d(_minc) 2(s−s2₎ , (3.16)

where Ψ is the subspace containing all the possible values of the pair c = (n, q), with:

• 1 ≤ n ≤ N; • 1 ≤ q ≤ Ns; • Pr (c)=∆ _Mm(c)

T OT, the probability that a codeword bit is sent through the sub-channel c in the case of ideal random interleaving;

• MT OT =∆ P_u_∈Ψm(u), the total number of bits transmitted during an OFDM symbol period;

• ψ(c)(∆), the number of symbols at distance ∆_{· d}(c)_min in the complementary subset.

3.2.2 Gaussian Approximation

A simple way to estimate the PEP integral (3.11) is the so called Gaussian approximation, expressed by [70]

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P EP ≃ Qp−2d κL(ˆs)

, (3.17)

where κL(·) is the cumulant-generating function (c.g.f.), defined as

κL(s)

∆

= log ML(s) (3.18)

and ˆs is the so-called ”saddlepoint”, which is defined [71] as that value for which κ′

L(ˆs) = 0. In the case of BIOS channels [72], we have ˆs = 1/2, so that

κL(ˆs)≃ log  X c∈Ψ 1 MT OT2m(c)−1 K X ∆=1 ψ(c)(∆)_{· e}− γ(c) ∆·d(_minc) 2 4   . (3.19)

Let us notice that the approximation (3.17) is actually the zeroth order of the Lugannani-Rice asymptotic series [69], and it corresponds to the PEP of an equiv-alent system with binary modulation (labelled as Equivequiv-alent Binary Modulation, EBM) that experiences a simple AWGN channel with SNR

SN REBM =∆ −κL(ˆs) (3.20)

According to the results above, the Gaussian approximation can be rewritten as

P EP _{≃ Q}p2d SN REBM = Q       v u u u u t−2d logX c∈Ψ K X ∆=1 ψ(c)_(∆)e− γ(c) ∆·d(_minc) 2 4 MT OT2m(c)−1      . (3.21)

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3.2.3 κESM definition

Recalling that the (3.21) corresponds to the PEP of an equivalent system with bi-nary modulation that experiences a simple AWGN channel, for maximum-likelihood decoding, the packet error probability (PER) of linear binary codes over BIOS chan-nels can be estimated by using the union-bound [73]. Thus, explicitly indicating the dependence of the PEP on the distance d and on the post-processing SNR matrix Υ, the PER can be upper-bounded by

P ER (Υ)≤ Nc X d=df ree ω (d) P EP (d, Υ) , (3.22) where:

• ω (d) is the weight of all the error events at Hamming distance d; • df ree is the minimum distance between two codewords.

Thus, the P ER performance can be estimated by computing the P EP . These considerations suggest the choice of a LQM based on the c.g.f. for our BIC-OFDM system. Figure 3.3 depicts the proposed link quality model structure carried out from the previous analysis, showing two separate models for modulation and coding: • The modulation model gives the m.g.f. evaluated at the saddlepoint relevant to a certain modulation scheme over an AWGN channel with SNR γ. The m.g.f. is estimated by a weighted sum of exponential functions, and for this reason, in the sequel, we tagged the modulation model as ”Exponential Modulation Model” (EMM), whose expressions is:

EM M(c) =∆ K X ∆=1 ψ(c)(∆) 2m(c)₋₁ · e − γ(c) ∆·d(_minc) 2 4 . (3.23)

• The coding model is obtained by firstly collecting and normalizing the mo-ment generating functions, and eventually taking the natural logarithm. The resulting effective SNR value is given by

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Figure 3.3: κESM quality model structure. γκESM ∆ =_{− log} 1 MT OT X c∈Ψ EM M(c) ! . (3.24)

The γκESM is then mapped into a PER number by means of a look-up table (LUT).

Let us remark, that being separated the modulation and the coding scheme, the κESM allows an accurate PER prediction also in the case of mixed modulation. For these reasons, the κESM is the LQM, which LRA techniques and particularly bit loading algorithms within the BIC-OFDM system analyzed before, rely on.

3.3 The Goodput Criterion

In the previous sections the tools used by the AMC strategy to evaluate the LQM have been investigated. Now, let us focus on the objective function that should be optimized .

Heterogeneous wireless systems can support different service classes with differ-ent QoS constraints, as outlined in Section 1.3, leading to differdiffer-ent link adaptation strategies. For instances:

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3.3. THE GOODPUT CRITERION

• when delay-constrained applications (i.e. voice, video, gaming) are consid-ered, the adaptive modulation should minimize outage probability (objective function) for a fixed data rate (constraint) or maximize the spectral efficiency (objective function) for a fixed error probability (constraint);

• when non-real-time polling service and best effort service are considered, there are not strict delay constraints. For this kind of applications, the user is only interested in the average number of information bits received in error-free packets by unit of time [89].

Therefore, in order to achieve the goal of a cross-layer link adaptation design, an optimized allocation strategy has necessarily to maximize the number of trans-mitted information bits in the error-free packet by unit of time, or equivalently the offered layer 3 data rate, or goodput for short.

Assuming a block fading channel [27], [28], [29], [30], at each packet transmission, we define the expected goodput as the ratio between the data payload to be delivered and the expected transmission time, assuming that the signal will experience the current channel conditions known at the transmitter.

Clearly, depending on the data payload length and the wireless channel conditions, the expected goodput varies with different transmission strategies. The more robust the transmission strategy, the more likely the packet will be delivered successfully, though with a lower spectral efficiency.

The key idea for an efficient adaptive cross-layer design, consists in deriving a link adaptation strategy that maximizes the expected goodput as a trade-off between the probability that the packet will be delivered successfully and the shortest pos-sible transmission time.

In [64], analyzing a multiuser strategy for IEEE 802.11a wireless LANs, the max-imum expected goodput criterion is used for both the multiuser diversity (MUD) scheduling and the link adaptation. The paper shows that the combination of such scheduling strategy with the link adaptation based on the maximum goodput cri-terion is effective in improving the layer 3 data rate.

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3.3.1 Expected goodput derivation

In order to allow an as general as possible analysis, let us denote with Φi the subspace containing all the allowed transmission mode φiat the i th ARQ-PR, where, differently by the definition given in Chapter 2, here with the term transmission mode, we intend the collection of all the possible values for the adjustable parameters (i.e. power, constellation sizes, space/time coding scheme, coding rate, etc.). As introduced in Chapter 1, each RLC-PDU consists of:

• Nh RLC header bits, including the ARQ-PR identifier; • Np payload bits;

• NCRC CRC bits;

Thus, the total RLC-PDU length is given by

NP DU = Nh+ Np+ NCRC. (3.25)

Being m(c)_i the number of cbs transmitted on the subchannel c and ri the coding rate, both of them associated with the ith protocol round, the number of RLC-PDU bits transmitted during an OFDM symbol period is given by

ν( ˆφi)= r∆ i X c∈Ψi

m(c)_i , (3.26)

where ˆφi is the selected transmission mode and Ψi is the subspace containing the indexes of all the active subchannels.

Exploiting this notation and calling with TOF DM the time needed for the transmis-sion of an OFDM symbol, the duration of an RLC-PDU transmistransmis-sion is

TP DU ˆ φi _∆ = NP DU νφˆi TOF DM. (3.27)

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Relying on the LQM before introduced, which summarizes the combination of the transmission mode ˆφi with the channel condition, in an effective SNR value, γκESM

ˆ φi

, the probability of a successful frame (RLC-PDU) transmission can be calculated by PACK ˆ φi _∆ = 1_{− P ER}γκESM ˆ φi , (3.28)

where the ACK error probability has been neglected.

Considering the entire RLC-PDU delivery process, with maximum number of trans-mission attempts equal to Rmax and denoting with ˆφ∈ Φ the vector representing the entire transmission strategy, where Φ= Φ∆ 0× Φ1× . . . × ΦRmax, the probability of a successful RLC-PDU delivery within the retry limit, can be expressed as

Psucc ˆ φ= 1∆ ₋ R_Ymax i=0 h 1_{− P}ACK ˆ φi i . (3.29)

Thus, the average transmission duration for a successful transmission is given by

Dsucc ˆ φ = R_Xmax i=0  PACK ˆ φi Qi−1_j=1 1_{− P}ACK ˆ φj Psucc ˆ φ   i X µ=0 TP DU ˆ φµ + TW AIT ˆ φµ , (3.30)

where TW AIT( ˆφi) represents the waiting time before the i th transmission attempt. On the other hand, the time wasted for the transmission of unsuccessfully delivered packets is Df ail ˆ φ= R_Xmax µ=0 TP DU ˆ φµ + TW AIT ˆ φµ . (3.31)

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information data bits to the average transmission time, we have GP φˆ=∆ Np P∞ k=0 1_{− P}succ ˆ φkPsucc ˆ φ k_Df ail ˆ φ+_Dsucc ˆ φ bit s . (3.32)

To simplify the above expression, let us notice that when Rmax → ∞, Psucc( ˆφ)→ 1. As a consequence, assuming a maximum retransmission number sufficiently large, by using (3.30) we can approximate the goodput formula as

GP φˆ_≃ _P Np ∞ i=0PACK ˆ φi Qi−1j=0 h 1− PACK ˆ φj i ·Piµ=0 TP DU ˆ φµ + TW AIT ˆ φµ . (3.33)

3.3.2 Approximation and normalization of the expected goodput

The goodput is the objective function that must be optimized by the LRA (in particular Bit loading) algorithms proposed in this thesis. Looking at the just obtained expression, one could get lost before starting to work on the problem, for this, further approximation need to be made over the goodput in order to render it more friendly.

We begin by making two simplifications:

• We assume to deal with continuous transmission (multiple SAW, SR), so that TW AIT ˆ φi = 0;

• At the generic ith protocol round, we know the current channel status only, while we assume not to be able to predict the future channel conditions.

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So, we base the selection of the transmission mode under the assumption that the channel will remain the same along all the retransmission occurred on a certain packet, leading to:

ˆ

φi= φ ,∀i (3.34)

Thus, after some manipulations over (3.33) and reminding (3.28), we obtain the following, handier goodput formula2:

GP (φ) = Np TP DU(φ)PACK(φ) 1−PACK(φ) ∞ P i=0 (i + 1) [1− PACK(φ)]i+1 = Np TP DU(φ)_P_ACK1 _(φ) = = Np NP DUTOF DM · ν (φ) · [1 − P ER (γκESM (φ))] bit s . (3.35)

Within the expression above we can identify the two parameters for which the goodput represents the compromise, in fact:

• Np

NP DUTOF DMν (φ) is the average number of information bits per OFDM period; • While, [1 − P ER (γκESM(φ))] represents the reliability of a successful

trans-mission of these bits.

Let us notice, that the objective function just obtained is relevant to an OFDM sym-bol transmission period, and this result is a direct consequence of a Block fading assumption. This means, assuming that the channel and consequently the transmis-sion mode remain stationary for all the OFDM symbols within a packet transmistransmis-sion time, so that we could concentrate only over the generic OFDM symbol transmis-sion period.

2_{We have also used the series [77]:} P∞

k=0,1

kαk₌ α

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In the graph rashly seen in Section 2.3.1 and in all the following simulation results concerning the goodput, we will refer to a normalized expected goodput, defined as the average number of information bits belonging to error-free packets, carried by a certain subcarrier, that is

g GP (φ)=∆ GP (φ) N · TOF DM = (3.36) = Np NP DUN · ν (φ) · [1 − P ER (γκESM (φ))] bit subcarrier .

Anyway, for the further optimizations, we can simplify again (3.36) by including Np, NP DU and N into a single factor

C=∆ Np NP DUN

, (3.37)

which remains constant for all the packets and for all the possible transmission modes. So, reminding (3.26), we can write

g GP (φ) =C · " rX c∈Ψ m(c) # · [1 − P ER (γκESM(φ))] bit subcarrier (3.38)

3.4 Concluding remarks

The use of ESM models, in particular the κESM analyzed before, allows us to have available at the transmitter a fast and accurate instantaneous PER perfor-mance prediction, because this model makes use of simple exponential functions for determining the Quality Measures. The accuracy of the κESM has been fully proved