Adaptive TORC detection for MC-CDMA wireless systems

3460 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 11, NOVEMBER 2009

Adaptive TORC Detection forMC-CDMA Wireless Systems

Barbara M. Masini, Member, IEEE, and Andrea Conti, Member, IEEE

Abstract—Multi carrier code-division multiple access (MC-CDMA) is considered for beyond third generation wirelesssystems for its effectiveness in the presence of both multipathfading and interference. This paper analyzes MC-CDMA systemsadopting an adaptive detection technique based on thresholdorthogonality restoring combining (TORC). The mathematicalframework here developed allows the evaluation of both thebit error probability and the bit error outage in downlink withperfect and imperfect channel state information and the deriva-tion of the TORC threshold that optimizes the performance.The optimal threshold is analytically derived as a function ofthe number of subcarriers, the number of active users, and themean signal-to-noise ratio. This enables an adaptive variation ofthe threshold following slow processes fluctuations. Numericalresults show very good performance of TORC with optimalthreshold comparing with others combining techniques withperfect and imperfect channel estimation. Results are shown bothin uncorrelated Rayleigh fading as well as in time and frequencycorrelated fading channels.

Index Terms—MC-CDMA, TORC detector, performance eval-uation, fading channels, adaptive system.

I. INTRODUCTION

MULTI-CARRIER code division multiple access (MC-CDMA) technique is gaining interest for beyond third

generation (B3G) mobile radio systems due to its efficiency insupporting high data rate mobile communication against bothfrequency selective fading and multi-user interference [1]–[3]. The suitability of MC-CDMA as multiple-access schemederives from the combination of orthogonal frequency-divisionmultiplexing (OFDM) and spreading. In this work we considerthe system architecture presented in [2] and [4], where thespreading is performed in the frequency-domain and Walsh-Hadamard (W-H) codes are used with spreading factor equal tothe number of subcarriers. The transmitter and receiver blockschemes are depicted in Fig. 1.1

The main impairment of this multiplexing technique is givenby the multiple-access interference (MAI), which occurs inthe presence of multipath fading propagation due to loss oforthogonality among the received spreading codes in spite of

Paper approved by G. E. Corazza, the Editor for Spread Spectrum of theIEEE Communications Society. Manuscript received July 28, 2006; revisedFebruary 13, 2008 and December 19, 2008.

B. Masini is with WiLAB, IEIIT/CNR at the University of Bologna, Italy(e-mail: [email protected]).

A. Conti is with ENDIF, WiLAB, CNIT, University of Ferrara, Italy (e-mail: [email protected]).

This work has been inspired and supported in part by the FP7 EuropeanNetwork of Excellence NEWCom++.

Digital Object Identifier 10.1109/TCOMM.2009.11.0604391Note that these schemes are useful for the analysis, although typical

implementation uses inverse and direct fast Fourier transform.

the use of W-H codes. Typically, the MAI mitigation is accom-plished at the receiver using single-user or multiuser detectionschemes; thus, the choice of the combining technique becomescritical and drastically affects the system performance. Severalcombining techniques with different complexities have beenstudied in the literature (see, e.g., [5]–[7]). In this paper, weaddress low complexity schemes such as linear combining,since in the downlink the detection is performed at the userterminal and we aim at proposing a simple adaptive combiningtechnique.

Within the family of linear combining techniques, severalschemes based on channel state information (CSI) are knownin the literature (see, e.g., [8]–[10]), for which signals comingfrom different subcarriers are weighted by suitable coefficients𝑔𝑚 (𝑚 being the subcarriers index) and summed. The equalgain combining (EGC) technique consists in equally weight-ing each subchannel contribution and compensating only thephases with 𝑔𝑚 = ℎ∗

𝑚/∣ℎ𝑚∣, where ℎ𝑚 is the 𝑚th channelcoefficient (notation ∗ stands for complex conjugation). If thesystem is noise-limited, (i.e., the number of active users isnegligible with respect to the number of subcarriers), maximalratio combining (MRC), for which 𝑔𝑚 = ℎ∗

𝑚, represents thebest choice. On the other hand, this choice totally destroysthe orthogonality between the codes in case of multiple users.For this reason, when the system is interference-limited, agood choice is given by restoring the orthogonality betweenthe sequences as with the orthogonality restoring combining(ORC) technique, for which 𝑔𝑚 = ℎ∗

𝑚/∣ℎ𝑚∣2. This implies atotal cancellation of the multiuser interference but emphasizesthe noise. This is corrected with threshold ORC (TORC),where a threshold is introduced to cancel the contributionsof those subchannels highly corrupted by the noise, allowinggood performance despite the low complexity (see [11]–[13]).Indeed, while EGC is desirable for its simplicity and MRCfor its noise counteracting capability, neither of these tech-niques directly address the interference and the exploitationof the spreading codes. B3G communication systems intendto multiplex as many users as possible sharing the sameresources, they are typically interference-limited; this is thereason why TORC is assuming an increasing interest. Sincethe orthogonality of the users is encoded in the phase of eachsubcarriers, TORC is mainly beneficial in the downlink wherephase distortion may be more easily corrected rather than inthe uplink [2].

Earlier works investigated the performance of MC-CDMAsystems varying the detector and some of these involve TORCtechnique: in [4] and [13] the performance was studied bysimulation or through semi-analytical models. In particular,

0090-6778/09$25.00 c⃝ 2009 IEEE

MASINI and CONTI: ADAPTIVE TORC DETECTION FOR MC-CDMA WIRELESS SYSTEMS 3461

(a) Transmitter (b) Receiver

Fig. 1. Transmitter and receiver block schemes.

the bit error probability (BEP) was approximated as the sumof the binomial distribution of the number of subcarriers abovethe threshold and may result in a cumbersome expression tobe computed in closed form or implemented in upper layersimulators. In [12] the bit error rate (BER) was obtainedby simulation. In [11] and [14] an analytical framework wasproposed to derive the downlink BEP with TORC detector forsynchronous users: in particular, in [11] decision-directed MC-CDMA detectors with and without explicit channel estimationare investigated; the decision-directed and pilot aided ap-proaches are also compared and the trade-off between channelestimation overhead and performance explored to verify howthe channel estimation process affects the performance of MC-CDMA detectors with Kalman filter to track the channel.

In this paper, we mathematically derive a simple BEP ap-proximation for downlink frequency selective fading channelsadopting a TORC detector with both perfect and imperfectCSI. In addition to the BEP, our framework enables the inves-tigation of the bit error outage (BEO), that is the BEP-basedoutage probability and represents a significant performancefigure when small-scale fading is superimposed to large-scale fading [15]. We also analytically evaluate the optimalthreshold and its relation with slow varying processes, suchas the number of active users, the number of subcarriers, andthe mean signal-to-noise ratio (SNR) averaged over small-scale fading. Imperfect CSI is also considered to analyze howthe optimum threshold changes with respect to the case ofperfect channel estimation. Note that, the number of activeusers, the number of subcarriers, and the mean SNR representslow varying processes with respect to the variations in theperformance perceived by the user, allowing an optimized andadaptive use of the threshold that can be changed trackingslow processes variation (for a discussion on slow and fastprocesses affecting the performance of adaptive and non-adaptive wireless communications see, e.g., [16], [17]). Itwill be shown that the proposed adaptive TORC significantlyimproves the performance with respect to others linear com-bining techniques. Analytical results will be confirmed bysimulations.The rest of the paper is organized as follows: system modeland methodology are described in Sec. II, the decision variableis given in Sec. III, and the BEP in Sec. IV. The optimalthreshold is evaluated in Sec. V and the BEO in Sec. VI.

Finally, numerical results and our conclusions are derived inSec. VII and Sec. VIII, respectively.

II. SYSTEM MODEL AND METHODOLOGY

In the considered MC-CDMA architecture, each data-symbol is copied over all subcarriers, and multiplied by thechip assigned to each particular subcarrier. We consider W-H orthogonal codes and binary phase shift keying (BPSK)modulation. Hence, the signal transmitted to the 𝑘th user canbe written as (see Figg. 1(a)):

𝑠(𝑘)(𝑡) =

√2𝐸b

𝑀

+∞∑𝑖=−∞

𝑀−1∑𝑚=0

𝑐(𝑘)𝑚 𝑎(𝑘)[𝑖]𝑔(𝑡 − 𝑖𝑇b) (1)

× cos(2𝜋𝑓𝑚𝑡 + 𝜙𝑚),

where 𝐸b is the energy per bit, 𝑀 is the number of subcar-riers, 𝑚 is the subcarrier index, 𝑐𝑚 is the 𝑚th chip (takingvalues ±1), 𝑎(𝑘)[𝑖] is the data-symbol transmitted during thesymbol-time 𝑖, 𝑔(𝑡) is the pulse with rectangular waveform,duration [0, 𝑇 ], and unitary energy, 𝑇b is the bit-time, 𝑓𝑚 =𝑓0 + (𝑚 − 𝑀/2)Δ𝑓 is the 𝑚th subcarrier-frequency (withΔ𝑓 𝑇 and 𝑓0 𝑇 integers to have orthogonality in frequency),and 𝜙𝑚 is the random phase uniformly distributed within[−𝜋, 𝜋). In particular, 𝑇b = 𝑇 +𝑇g is the total OFDM symbolduration, increased with respect to 𝑇 by a time-guard 𝑇g

(usually inserted between consecutive symbols to eliminatethe residual inter symbol interference, ISI, due to the channeldelay spread).

We assume a frequency-domain channel model in which thetransfer function is considered constant in the bandwidth ofeach subcarrier and is given by

ℎ𝑚 = 𝛼𝑚𝑒𝑗𝜓𝑚 , (2)

where 𝛼𝑚 and 𝜓𝑚 are the 𝑚th amplitude and phase coeffi-cients, respectively. Each ℎ𝑚 is assumed independent identi-cally distributed (i.i.d.) complex zero-mean Gaussian randomvariable (RV) with variance per dimension 𝜎2

h (typically equalto 1/2 so that 𝔼

{𝛼2𝑚

}= 1, for all 𝑚).2

Assuming the coherence bandwidth smaller than the distancebetween two adjacent subcarriers (owing, for instance, tofrequency interleaving), each 𝛼𝑚 and 𝜓𝑚 are considered

2𝔼 {⋅} denotes the expectation.


independent. We assume that each subcarrier experiences flatfading. This represents a realistic case when the subcarriersare sufficiently spaced in frequency or when only a subsetof the amount of subcarriers is used for the transmissionof a symbol. Starting from these common assumptions, weanalytically evaluate the performance in terms of BEP and wecarefully investigate the performance dependencies on systemparameters, then also allowing comparison with previousresults appeared in the literature [3], [18], [19].

In the downlink, the system is assumed synchronous and thesignal components associated to different users experiment thesame fading. Furthermore, since the orthogonality of the usersis encoded in the phase of the subcarriers, the TORC techniqueis primarily beneficial in the downlink where phase distortionfor all users can be more easily corrected [2].3 Hence, thereceived signal can be written as:

𝑟(𝑡) =

√2𝐸b

𝑀

𝑁u−1∑𝑘=0

+∞∑𝑖=−∞

𝑀−1∑𝑚=0

𝛼𝑚𝑐(𝑛)𝑚 𝑎(𝑘)[𝑖]𝑔(𝑡 − 𝑖𝑇b)

× cos(2𝜋𝑓𝑚𝑡 +

𝜗𝑚︷︸︸︷𝜙𝑚 + 𝜓𝑚) + 𝑛(𝑡), (3)

where summations are on users, symbols, and subcarriers,respectively, 𝑔(𝑡) is the response of the channel to the pulse𝑔(𝑡) with unitary energy and duration 𝑇 ≜ 𝑇 + 𝑇d, being𝑇d the pulse enlargement due to the overall distortion ofthe channel, 𝑛(𝑡) is the additive white Gaussian noise withtwo-side power spectral density 𝑁0/2, and 𝜗𝑚 ≜ 𝜙𝑚 + 𝜓𝑚

uniformly distributed in [−𝜋, 𝜋).To analyze the performance with imperfect CSI, we assume

a channel estimation error [20] on each subcarrier, 𝑒𝑚, com-plex Gaussian distributed with zero-mean and variance 2𝜎2

e ,thus 𝑒𝑚 ∼ 𝒞𝒩 (0, 2𝜎2

e ), and we define the normalized channelestimation error variance as 𝜀 ≜ 𝜎2

e /𝜎2h . The 𝑚th estimated

complex channel coefficient is given by:

ℎ𝑚 = ℎ𝑚 + 𝑒𝑚 , (4)

where, ℎ𝑚 = ��𝑚𝑒𝑗𝜓𝑚 .By observing the receiver structure depicted in Fig. 1(b) andfocusing on the 𝑙th subcarrier for the 𝑛th user (being inthe downlink, perfect synchronization and phase tracking areassumed), the correlation output at the 𝑗th instant is given by:

𝑧(𝑛)𝑙 [𝑗] =

1√𝑇

∫ 𝑗𝑇b+𝑇

𝑗𝑇b

𝑟(𝑡) 𝑐(𝑛)𝑙

√2 cos(2𝜋𝑓𝑙𝑡 + 𝜗𝑙)𝑑𝑡 . (5)

3Note that, in the downlink, power efficient mobile receivers with lowcomplexity are required, whereas for the uplink, a high complexity multi-user detector can be used to cope with the presence of several users.

Hence, substituting (3) in (5), the term 𝑧(𝑛)𝑙 [𝑗] results as in

(6), where:

𝑤𝑚,𝑙[𝑗] =

{cos(𝜗𝑙 − 𝜗𝑙) 𝑚 = 𝑙0 𝑚 ∕= 𝑙 .

Since ISI is avoided, we will remove the time-index 𝑗 in thefollowing. After some manipulation we can write:

𝑧(𝑛)𝑙 =

√𝐸b𝛿d

𝑀𝛼𝑙 cos(𝜗𝑙 − 𝜗𝑙)𝑎

(𝑛) (7)

+

√𝐸b𝛿d

𝑀𝑐(𝑛)𝑙 𝛼𝑙 cos(𝜗𝑙 − 𝜗𝑙)

𝑁u−1∑𝑘=0,𝑘 ∕=𝑛

𝑐(𝑘)𝑙 𝑎(𝑘) + 𝑛𝑙 ,

where 𝛿d ≜ 1/(1+𝑇d/𝑇 ) represents the loss of energy causedby the time-spreading, 𝑇d, of the received pulse.From the received signal expression before the combiner, wecan now proceed to evaluate the performance in terms ofBEP and BEO. The BEP averaged over small-scale fadingis a common performance figure for applications in whichthe user’s perceived quality of service (QoS) is related to theerror rate observed in a time interval of several symbol times(see e.g., [17]). Thus, the variability of processes affectingthe useful signal during this interval has to be carefully takeninto account (e.g., the mean SNR, 𝛾, also varies due to theshadowing, but slowly with respect to the interval consideredfor the BEP evolution). For this reason, the BEO, that is theprobability that the BEP exceeds a maximum tolerable level, isalso an appropriate figure of merit for digital communicationsystems where small-scale fading is superimposed on large-scale fading, such as shadowing [15].

III. DECISION VARIABLE

The decision variable, 𝑣(𝑛), is obtained by a linear combi-nation of the weighted signals from each subcarrier as follows:

𝑣(𝑛) =

𝑀−1∑𝑙=0

∣𝑔𝑙∣𝑧(𝑛)𝑙 . (8)

The TORC detector consists in choosing the complex weightreferring to the 𝑙th subcarrier as:

∣𝑔𝑙∣ = 𝑢(∣ℎ𝑙∣ − 𝜌TH)/∣ℎ𝑙∣ = ��−1𝑙 𝑢(��𝑙 − 𝜌TH) . (9)

where 𝑢(⋅) is the unitary-step function and 𝜌TH is the positivethreshold introduced to cancel the contributions of those sub-channels highly corrupted by the noise. Hence, by substituting

𝑧(𝑛)𝑙 [𝑗] = 2

√𝐸b

𝑀𝑇

+∞∑𝑖=−∞

∫ 𝑗𝑇b+𝑇

𝑗𝑇b

𝑁u−1∑𝑘=0

𝑀−1∑𝑚=0

𝛼𝑚𝑐(𝑘)𝑚 𝑐(𝑛)𝑙 𝑎(𝑘)[𝑖]𝑔(𝑡− 𝑖𝑇b) cos(2𝜋𝑓𝑚𝑡+ 𝜗𝑚) cos(2𝜋𝑓𝑙𝑡+ 𝜗𝑙)𝑑𝑡 (6)

+

𝑛𝑙[𝑗]︷︸︸︷∫ 𝑗𝑇b+𝑇

𝑗𝑇b

√2𝑐(𝑛)𝑙√𝑇𝑛(𝑡) cos(2𝜋𝑓𝑙𝑡+ 𝜗𝑙)𝑑𝑡 =

√𝐸b

𝑀

𝑇

𝑇

𝑁u−1∑𝑘=0

𝑀−1∑𝑚=0

𝑤𝑚,𝑙[𝑗]︷︸︸︷1

𝑇

∫ 𝑗𝑇b+𝑇

𝑗𝑇b

2 cos(2𝜋𝑓𝑚𝑡+ 𝜗𝑚) cos(2𝜋𝑓𝑙𝑡+ 𝜗𝑙)𝑑𝑡

× 𝛼𝑚𝑐(𝑘)𝑚 𝑐(𝑛)𝑙 𝑎(𝑘)[𝑗] + 𝑛𝑙[𝑗] =

√𝐸b

𝑀

𝑇

𝑇

⎧⎨⎩

𝑁u−1∑𝑘=0

𝛼𝑙𝑐(𝑘)𝑙 𝑐

(𝑛)𝑙 𝑎(𝑘)[𝑗]𝑢𝑙,𝑙[𝑗] +

𝑀−1∑𝑚=0,𝑚∕=𝑙

𝑁u−1∑𝑘=0

𝛼𝑚𝑐(𝑘)𝑚 𝑐(𝑛)𝑙 𝑎(𝑘)[𝑗]𝑢𝑚,𝑙[𝑗]

⎫⎬⎭ + 𝑛𝑙[𝑗]


(7) and (9) in (8) we can write:

𝑣(𝑛) =

𝑈︷︸︸︷√𝐸b𝛿d

𝑀

𝑀−1∑𝑙=0

𝑢(��𝑙 − 𝜌TH)𝛼𝑙��−1𝑙 cos(𝜗𝑙 − 𝜗𝑙) 𝑎(𝑛)

+

𝐼︷︸︸︷√𝐸b𝛿d

𝑀

𝑀−1∑𝑙=0

𝑐(𝑛)𝑙 𝑢(��𝑙 − 𝜌TH)𝛼𝑙��

−1𝑙 cos(𝜗𝑙 − 𝜗𝑙)

×

in 𝐼︷︸︸︷𝑁u−1∑

𝑘=0,𝑘 ∕=𝑛

𝑐(𝑘)𝑙 𝑎(𝑘) +

𝑁︷︸︸︷𝑀−1∑𝑙=0

𝑛𝑙��−1𝑙 𝑢(��𝑙 − 𝜌TH) , (10)

where the three terms 𝑈 , 𝐼 , and 𝑁 represent the useful,interfering, and noise term, respectively.By defining

Θ𝑙(𝜌TH) ≜ 𝑢(��𝑙 − 𝜌TH)𝛼𝑙��−1𝑙 cos(𝜗𝑙 − 𝜗𝑙) , (11)

we can write (10) as in the following

𝑣(𝑛) = 𝑈 𝑎(𝑛) + 𝐼 + 𝑁 , (12)

where

𝑈 =

√𝐸b𝛿d

𝑀

𝑀−1∑𝑙=0

Θ𝑙(𝜌TH) (13)

𝐼 =

√𝐸b𝛿d

𝑀

𝑀−1∑𝑙=0

𝑐(𝑛)𝑙 Θ𝑙(𝜌TH)

𝑁u−1∑𝑘=0,𝑘 ∕=𝑛

𝑐(𝑘)𝑙 𝑎(𝑘) (14)

𝑁 =

𝑀−1∑𝑙=0

𝑛𝑙��−1𝑙 𝑢(��𝑙 − 𝜌TH) . (15)

Hence, in order to derive the BEP, we need the statisticdistribution of 𝑣(𝑛) that can be obtained by studying thestatistics of 𝑈 , 𝐼 , and 𝑁 . By considering that actual andfuture multi carrier systems generally have a high number ofsubcarriers (e.g., 2k or 8k for digital video broadcasting), wewill adopt the central limit theorem (CLT) and the law of largenumber (LLN) to approximate the BEP (the goodness of theapproximation will also be verified by simulation).

A. Useful Term

By applying the CLT,4 the useful term 𝑈 in (10) resultsdistributed as5

𝑈 ∼ 𝒩 (𝜇𝑈 , 𝜎2

𝑈

). (16)

The mean and variance of 𝑈 are respectively given by𝜇𝑈 =

√𝐸b𝛿d𝑀 𝔼 {Θ(𝜌TH)} and 𝜎2

𝑈 = 𝐸b𝛿d[𝔼{Θ2(𝜌TH)

}−(𝔼 {Θ(𝜌TH)})2].6

4CLT has been applied to analyze MC-CDMA systems also, e.g., in [2]and [4].

5Notation 𝒩 (𝜇, 𝜎2) stands for Gaussian distribution with mean 𝜇 andvariance 𝜎2.

6Note that 𝔼 {Θ(𝜌TH)} is independent on index 𝑙.

B. Interference Term

By partitioning orthogonal codes as in [4], after some mathe-matical manipulation we derive the interference as:

𝐼 =

√𝐸b𝛿d

𝑀

𝑁u−1∑𝑘=0,𝑘 ∕=𝑛

𝑎(𝑘) (17)

×

⎡⎢⎢⎢⎢⎢⎣

Λ1︷︸︸︷𝑀2∑

ℎ=1

Θ𝑥ℎ(𝜌TH)−

Λ2︷︸︸︷𝑀2∑

ℎ=1

Θ𝑦ℎ(𝜌TH)

⎤⎥⎥⎥⎥⎥⎦ ,

where indexes 𝑥ℎ and 𝑦ℎ define the following partitions:

𝑐(𝑛)𝑥ℎ𝑐(𝑘)𝑥ℎ

= 1

𝑐(𝑛)𝑦ℎ𝑐(𝑘)𝑦ℎ

= −1

{𝑥ℎ} ∪ {𝑦ℎ} = 0, 1, ...., 𝑀 − 1 .

For large values of 𝑀 , we apply the CLT to terms Λ1 and Λ2

in (17), consequently we obtain:

Λ1,Λ2 ∼ 𝒩(√

𝑀

2𝔼 {Θ(𝜌TH)} ,

𝑀

2𝜁𝛼(Θ)

), (18)

where 𝜁𝛼(Θ) = 𝔼{Θ2(𝜌TH)

} − 𝔼 {Θ(𝜌TH)}2. Hence, byexploiting the symmetry of the Gaussian probability densityfunction (p.d.f.) and the property of the sum of uncorrelatedGaussian RVs, we have:

𝑎(𝑘) (Λ1 − Λ2) ∼ 𝒩 (0, 𝑀𝜁𝛼(Θ)) .

Then, the general interference term results distributed as:

𝐼 ∼ 𝒩 (0, 𝜎2

I

), (19)

where 𝜎2I = 𝐸b𝛿d(𝑁u − 1)

[𝔼{Θ2(𝜌TH)

}− 𝔼 {Θ(𝜌TH)}2].

C. Noise Term

Since terms ��𝑙 and 𝑛𝑙 in 𝑁 are independent and 𝑛𝑙 has meanequal to zero, the total noise term consists in a sum of i.i.d.zero mean RVs with variance (𝑁0/2) 𝔼

{(��−1𝑢(�� − 𝜌TH))

2}.

By applying the CLT, the noise term results distributed as:

𝑁 ∼ 𝒩 (0, 𝜎2

N

), (20)

with 𝜎2N = 𝑀 (𝑁0/2) 𝔼

{(��−1𝑢(�� − 𝜌TH))

2}.

The expressions of 𝔼 {Θ(𝜌TH)}, 𝔼{

��−2𝑢2(�� − 𝜌TH)}, and

𝔼{Θ2(𝜌TH)

}are derived in the Appendix, and result in:

𝔼 {Θ(𝜌TH)} = 𝑒− 𝜌2TH

2𝜎2h (1+𝜀) − 2𝜀 𝑒

− 𝜌2TH(1+3𝜀/4)

2𝜎2h (1+𝜀)

(1− 𝜀2)(4 + 3𝜀)

𝔼{

��−2𝑢2(�� − 𝜌TH)}=

1

2𝜎2h (1 + 𝜀)

Γ

[0,

𝜌2TH

2𝜎2h (1 + 𝜀)

]

𝔼{Θ2(𝜌TH)

}= 𝑒

− 𝜌2TH2𝜎2

h (1+𝜀) − 4𝜀 𝑒− 𝜌2TH(1+3𝜀/4)

2𝜎2h (1+𝜀)

(1− 𝜀2)(4 + 3𝜀)

+ 𝐶(𝐵1 + 𝐵2 + 𝐵3 + 𝐵4) , (21)

where the expressions of 𝐶, 𝐵1, 𝐵2, 𝐵3, and 𝐵4 are given inthe Appendix.


D. Independence between terms

Since 𝑎(𝑘) is zero mean and statistically independent on 𝛼𝑙,(Λ1−Λ2), and 𝑛𝑙, we can write that 𝔼 {𝐼𝑁} = 𝔼 {𝐼𝑈} = 0.Since 𝑛𝑙 and 𝛼𝑙 are statistically independent, then 𝔼 {𝑁𝑈} =0. Moreover 𝐼 , 𝑁 , and 𝑈 are uncorrelated Gaussian RVs, thusalso statistically independent.

IV. BIT ERROR PROBABILITY EVALUATION

From the statistical distribution of the terms 𝐼 and 𝑁 ,the total disturbance (interference plus noise) results to beGaussian distributed as:

𝐼 + 𝑁 ∼ 𝒩 (0, 𝜎2

I + 𝜎2N

). (22)

This enables us to derive the BEP approximation as a functionof the mean SNR 𝛾, the TORC threshold 𝜌TH, the number ofsubcarriers 𝑀 , the number of active users 𝑁u, and the channelestimation error related to 𝜀.The BEP conditioned to the RV 𝑈 is given by

𝑃b∣𝑈 =1

2erfc

{𝑈√

2(𝜎2I + 𝜎2

N)

}, (23)

where erfc is the complementary error function. By applyingthe LLN, that is by approximating

∑𝑀−1𝑙=0 Θ𝑙(𝜌TH) with

𝑀𝔼 {Θ(𝜌TH)}, we derive the unconditioned BEP as

𝑃b ≃ 1

2erfc

(√𝐸b𝛿d𝑀 𝔼 {Θ(𝜌TH)}√

2(𝜎2N + 𝜎2

I )

). (24)

From the expressions of 𝜎2I , 𝜎2

N, and 𝔼 {Θ(𝜌TH)} derivedabove, the BEP can be written as a function of the signal-to-interference-plus-noise ratio (SINR) as

𝑃b(

𝜁) ≃ 1

2erfc

(√𝜁

), (25)

where 𝜁 is the mean SINR averaged over small-scale fadingand is given by:

𝜁 =𝛾 𝑒

− 𝜌2TH𝜎2

h (1+𝜀) 𝑓1(𝜀, 𝜌TH)

1(1+𝜀)Γ[0,

𝜌2TH

2𝜎2h (1+𝜀)

] + 2𝑁u−1𝑀 𝑓2(𝜀, 𝜌TH)𝛾

. (26)

In (26), 𝛾 ≜ 𝐸b𝛿d2𝜎2h /𝑁0 is the mean SNR averaged over

small-scale fading and

𝑓1(𝜀, 𝜌TH) =

(1− 2𝜀

(1− 𝜀2)(4 + 3𝜀)𝑒− 𝜌2TH

8𝜎2h (1+𝜀)

3𝜀

)2

(27a)

𝑓2(𝜀, 𝜌TH) = 𝑒− 𝜌2TH

2𝜎2h (1+𝜀)

⎡⎢⎣1− 4𝜀𝑒

− 𝜌2TH8𝜎2

h (1+𝜀)3𝜀

(1− 𝜀2)(4 + 3𝜀)

⎤⎥⎦

+ 𝑒𝜌2TH

2𝜎2h (1+𝜀) 𝐶(𝐵1 + 𝐵2 + 𝐵3 + 𝐵4)

− 𝑒− 𝜌2TH

𝜎2h (1+𝜀)

⎡⎢⎣1− 2𝜀𝑒

−𝜌2TH8𝜎2

h (1+𝜀)3𝜀

(4 + 3𝜀)(1− 𝜀2)

⎤⎥⎦2

. (27b)

Remark: Note that the main objective of this work is not onlyto assess a tight approximation for the BEP, but mostly toderive the value of the TORC threshold, 𝜌TH, which minimizes

the BEP depending on the other system parameters. Fromthis point of view, we will show a good agreement betweensimulative and analytical results in Sec. VII.

V. TORC THRESHOLD OPTIMIZATION

We aim at finding the optimum value of the threshold,𝜌(opt)TH , defined as the value that minimizes the BEP (hence that

maximizes the SINR since the BEP in (25) is monotonicallydecreasing with the SINR):

𝜌(opt)TH = argmin

𝜌TH

{𝑃b} = argmax𝜌TH

{𝜁}

. (28)

A. Optimal TORC with Non-Ideal Channel Estimation

By forcing to zero the derivative of the argument in (28) andremembering that 𝛾 > 0, Γ[0, 1/[2𝜎2

h (1+ 𝜀)]] > 0, and 𝜌TH ≥0, we obtain:

(− 2𝜌TH

𝜎2h (1 + 𝜀)

𝑓1(𝜀, 𝜌TH) + 𝑓 ′1(𝜀, 𝜌TH)

)(29)

×⎛⎝Γ

[0,

𝜌2TH

2𝜎2h (1+𝜀)

](1 + 𝜀)

+ 2𝑁u − 1

𝑀𝑓2(𝜀, 𝜌TH)𝛾

⎞⎠

= 𝑓1(𝜀, 𝜌TH)

⎡⎣Γ′

[0,

𝜌2TH

2𝜎2h (1+𝜀)

](1 + 𝜀)

+ 2𝑁u − 1

𝑀𝑓 ′2(𝜀, 𝜌TH)𝛾

⎤⎦ ,

where 𝑓 ′(⋅) stands for the first derivative of 𝑓(⋅) evaluatedwith respect to 𝜌TH. Let us now define the system load as

𝜂 ≜ 𝑁u − 1

𝑀(30)

and the parameter 𝜉 as

𝜉 ≜ 2𝛾𝜂 . (31)

Since

Γ′[0,

𝜌2TH

2𝜎2h (1 + 𝜀)

]= − 2

𝜌TH𝑒−𝜌2

TH/(2(1+𝜀)𝜎2h ) ,

the (29) becomes as in (32) (bottom of the next page), where

𝑓 ′1(𝜀, 𝜌TH) =

𝜌TH3𝜀2

𝜎2h (1 + 𝜀)(1− 𝜀2)(4 + 3𝜀)

𝑒− 𝜌2TH

8𝜎2h

3𝜀(1+𝜀)

×[1− 2𝜀

(1 − 𝜀2)(4 + 3𝜀)𝑒− 𝜌2TH

8𝜎2h

3𝜀(1+𝜀)

](33)


and

𝑓 ′2(𝜀, 𝜌TH) =

𝜌TH

𝜎2h (1 + 𝜀)

𝑒− 𝜌2TH

2𝜎2h(1+𝜀)

⎡⎢⎣𝜀𝑒

− 𝜌2TH8𝜎2

h (1+𝜀)3𝜀

(1 − 𝜀2)− 1

⎤⎥⎦

+𝜌TH

𝜎2h (1 + 𝜀)

𝑒𝜌2TH2𝜎2

h(1+𝜀)

𝐶 (𝐵1 + 𝐵2 + 𝐵3 + 𝐵4)

+2𝜌TH

𝜎2h (1 + 𝜀)

𝑒− 𝜌2TH

𝜎2h(1+𝜀)

⎛⎜⎝1− 2𝜀𝑒

− 𝜌2TH8𝜎2

h (1+𝜀)3𝜀

(1− 𝜀2)(4 + 3𝜀)

⎞⎟⎠

2

− 𝜌TH

𝜎2h (1 + 𝜀)

𝑒− 𝜌2TH

2𝜎2h(1+𝜀)

×

⎡⎢⎣ 3𝜀2𝑒

− 𝜌2TH8𝜎2

h (1+𝜀)3𝜀

(1 − 𝜀2)(4 + 3𝜀)

⎛⎜⎝1− 2𝜀𝑒

− 𝜌2TH8𝜎2

h (1+𝜀)3𝜀

(1− 𝜀2)(4 + 3𝜀)

⎞⎟⎠⎤⎥⎦ . (34)

It is important to note that the parameter 𝜉 quantifies how thesystem is noise-limited (low values) or interference-limited(high values) and represents the implicit solution for theproblem of finding the optimum value of 𝜌TH for all possiblevalues of SNR, number of subcarriers, and number of activeusers, with non-ideal channel estimation. Thus, for each com-bination of the system parameters, there is a value of 𝜌TH thatminimizes the BEP. In addition, it is important to observe that𝜌TH only depends, through 𝜉, on processes slowly varying withrespect to the performance perceived by the users. Hence, wepropose and investigate an adaptive TORC detection in which𝜌TH is slowly adapted to the optimum value for the currentset of parameters 𝛾, 𝑀 , and 𝑁u, for a given 𝜀.

B. Optimal TORC with Ideal Channel Estimation

In the case of ideal CSI (𝑒𝑚 = 0 in (4) and 𝜀 = 0) equation(26) becomes

𝜁 =𝛾 𝑒

− 𝜌2TH𝜎2

h

2𝑁u−1𝑀

(𝑒− 𝜌2TH

2𝜎2h − 𝑒

− 𝜌2TH𝜎2

h

)𝛾 + Γ

[0,

𝜌2TH

2𝜎2h

] . (35)

For what concern the minimization of the BEP, the solvingequation giving the optimum 𝜌TH results

𝜉 = 2

(𝜎2

h

𝜌2TH

− 𝑒𝜌2TH2𝜎2

h Γ

[0,

𝜌2TH

2𝜎2h

]). (36)

As can be observed, the expression of 𝜉 as a function of 𝜌TH

given by (32) is rather complex with respect to (36), thusthe evaluation of the optimum threshold is more complicatedwhen errors on the channel estimation occur. However, as willbe verified in Sec. VII-A, for several situations of interest,the optimum threshold is almost the same in both cases, with

perfect and imperfect CSI. This means that a system designercan adopt the value of the threshold in perfect CSI conditionsalso in presence of channel estimation errors.

VI. BIT ERROR OUTAGE

In digital wireless communications where small-scale fadingis superimposed to large-scale fading (i.e., shadowing) [21],another important performance metric is given by the BEO,defined as the probability that the BEP exceeds a maximumtolerable level (i.e., the target BEP, 𝑃 ★

b ) [15]:

𝑃o ≜ ℙ{𝑃b > 𝑃 ★b } = ℙ{𝛾dB < 𝛾★

dB} , (37)

where 𝛾dB = 10 log10 𝛾 and 𝛾★dB is the SNR (in dB) giving

the BEP equal to 𝑃 ★b (i.e., 𝑃b(𝛾

★) = 𝑃 ★b ).

We consider the case of a shadowing environment in which𝛾 is log-normal distributed with parameters 𝜇dB and 𝜎2

dB (i.e.,𝛾dB is a Gaussian RV with mean 𝜇dB and variance 𝜎2

dB). Hence,the BEO results in

𝑃o =1

2erfc

{𝜇dB − 𝛾★

𝑑𝐵√2𝜎dB

}. (38)

By inverting (25), the target SINR giving a BEP equal to thetarget BEP 𝑃 ★

b is

𝜁★=(erfc−1{2𝑃 ★

b })2

, (39)

where erfc−1 is the inverse complementary error function.Hence, we derive the required SNR, 𝛾★, as a function of 𝜁

★,

𝜌TH, and 𝜂, as given by

𝛾★ =𝜁★Γ[0,

𝜌2TH

2𝜎2h (1+𝜀)

](1 + 𝜀)

[𝑓1(𝜀, 𝜌TH)𝑒

− 𝜌TH𝜎2

h (1+𝜀) − 2𝜂𝜁★𝑓2(𝜀, 𝜌TH)

] (40)

that, in case of perfect CSI, reduces to

𝛾★ =𝜁★Γ[0,

𝜌2TH

2𝜎2h

]𝑒− 𝜌2TH

𝜎2h − 2𝜂𝜁

★

(𝑒− 𝜌2TH

2𝜎2h − 𝑒

− 𝜌2TH𝜎2

h

) . (41)

The equation (40) (or (41) with perfect CSI) enables thederivation of the optimal 𝜌TH which minimizes the requiredSINR for a target 𝑃 ★

b and a given system load 𝜂. In addition,given the target BEP and BEO, from (40) (or (41) with perfectCSI) and (38) we obtain the required value of 𝜇dB that is themedian value of the SNR; this is useful for wireless digitalcommunication systems design, since it is strictly related tothe link budget when the path-loss law is known.By fixing the target performance in terms of 𝑃 ★

b , starting from(26) and remembering the expressions of 𝜂, we can derive therelation between 𝜂, 𝛾★, 𝜌TH, and 𝜁

★as given by

𝜂 =𝛾★𝑓1(𝜀, 𝜌TH)𝑒

− 𝜌2TH𝜎2

h (1+𝜀) − 𝜁★

1+𝜀Γ[0,

𝜌2TH

2𝜎2h (1+𝜀)

]2𝜁

★𝛾★𝑓2(𝜀, 𝜌TH)

, (42)

𝜉 =

11+𝜀

(2𝜌TH

𝜎2h (1+𝜀)

𝑓1(𝜀, 𝜌TH)Γ[0,𝜌2

TH2𝜎2

h (1+𝜀)]− 2𝑓1(𝜀,𝜌TH)

𝜌TH𝑒− 𝜌2TH

2𝜎2h (1+𝜀) − 𝑓 ′

1(𝜀, 𝜌TH)Γ[0,𝜌2

TH2𝜎2

h (1+𝜀)]

)

𝑓 ′1(𝜀, 𝜌TH)𝑓2(𝜀, 𝜌TH)− 𝑓1(𝜀, 𝜌TH)𝑓 ′

2(𝜀, 𝜌TH)− 2𝜌TH

𝜎2h (1+𝜀)

𝑓1(𝜀, 𝜌TH)𝑓2(𝜀, 𝜌TH), (32)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1ρTH

10−2

10−1

100

Pb

γ=10dBγ=5dB

ε=0,10%,30%

ε=0,10%,30%

Fig. 2. BEP vs. 𝜌TH varying the mean SNR 𝛾, and the variance of thechannel estimation errors for 𝑀 = 𝑁u = 1024.

0 5 10 15 20γ [dB]

10−3

10−2

10−1

100

Pb

MRCEGCTORC, ρTH=0.2

ε=0,5%,10%,20%,30%

MRC

EGC

Fig. 3. BEP vs. 𝛾 in dB varying the variance of the channel estimation errorfor 𝑀 = 𝑁u = 1024 with 𝜌TH = 0.2. A comparison with MRC and EGCtechniques with perfect CSI is also reported.

that, in case of perfect CSI reduces to:

𝜂 =𝛾★ 𝑒

− 𝜌2TH𝜎2

h − 𝜁★Γ[0,

𝜌2TH

2𝜎2h]

2𝜁★𝛾★

(𝑒− 𝜌2TH

2𝜎2h − 𝑒

− 𝜌2TH𝜎2

h

) . (43)

VII. NUMERICAL RESULTS

In this section, numerical results related to the BEP and theBEO for the downlink of MC-CDMA systems with optimizedTORC detection are given and compared with those of othercombining techniques. Both ideal and non-ideal channel esti-mation are considered in uncorrelated Rayleigh fading as wellas in time and frequency correlated channels. The value of 𝜎2

hwill be considered equal to 1/2.

8 10 12 14 16 18 20γ [dB]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ρTH

NU=256NU=512NU=M=1024

ε=0,10%

ε=0,10%

ε=0,10%

Fig. 4. Optimum value of 𝜌TH vs. 𝛾 in dB giving 𝑃 ★b = 10−2 when varying

the system load and the variance of the channel estimation error.

A. Non-Ideal Channel Estimation

We now present analytical results in the presence of channelestimation error and we compare the BEP obtained withTORC detector with the BEP using MRC and EGC.

In Fig. 2, the BEP given by (25) as a function of the TORCthreshold 𝜌TH, is presented when the system is fully loadedwith 𝑁u = 𝑀 = 1024 as a function of the mean SNR 𝛾. Theoptimum value of 𝜌TH minimizing the BEP can be observed.Moreover, the effect of channel estimation errors is shown andcompared with the case of perfect CSI (𝜀 = 0). In particular,a normalized variance of channel estimation errors, 𝜀 = 0,10%, and 30% is considered. Note that 𝜌TH strongly affectsthe BEP: in fact, focusing on the curves related to 𝛾 = 10 dBand 𝜀 = 10%, a 𝑃b ≃ 7.3 ⋅ 10−2 is given by adopting a valueof 𝜌TH = 0 (ORC), while 𝑃b ≃ 1.7 ⋅ 10−2 with 𝜌TH = 0.2.It is also important to observe that, in spite of the presence ofchannel estimation errors, the optimum value of the thresholdminimizing the BEP does not significantly change with 𝜀 fora given value of the mean SNR. Hence, the optimum value of𝜌TH for perfect CSI in (36) represents a good solution also inthe case of channel estimation errors.

In Fig. 3, the BEP as a function of the mean SNR 𝛾, isshown when 𝜌TH = 0.2 varying the normalized variance ofthe channel estimation errors 𝜀. The system is assumed fullyloaded with 𝑁u = 𝑀 = 1024. A comparison with MRCand EGC techniques (these with perfect CSI) is presented. Ascan be observed, TORC detection always outperforms MRCalso when 𝜀 is not zero. On the other hand, EGC outperformsTORC detection for low values of the mean SNR due to asub-optimum value of the threshold. In fact, as we can seefrom Fig. 2, 𝜌TH = 0.2 is optimum when 𝛾 = 10 dB, but theoptimum value changes with the mean SNR. As 𝛾 increases,TORC outperforms EGC.

In Fig. 4, the value of the threshold 𝜌TH giving 𝑃b = 𝑃 ★b =

10−2 (which is a typical value for uncoded systems) is plottedas a function of the mean SNR 𝛾 for fully loaded (𝑁u = 𝑀 =1024), half loaded (𝑁u = 512) and a quarter loaded (𝑁u =


−25 −15 −5 5 15 25 35ξ [dB]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ρTH

(opt)

analysissimulationε=0, 5%, 30%

Fig. 5. Optimum 𝜌TH as a function of 𝜉: analytical and simulative results.Simulations are for the case of perfect channel estimation.

256) system, when varying 𝜀. The strong impact of the choiceof 𝜌TH on the required SNR can be observed; this motivatesthe fact that, to achieve the target BEP, a suitable value for𝜌TH has to be chosen to avoid an unnecessary increasing of𝛾. Moreover, the low impact of the channel estimation erroron the choice of 𝜌TH with respect to the case of perfect CSIcan be appreciated.

From (32) and (36), for the case of ideal and non-idealCSI, respectively, the evaluation of the optimum value of 𝜌TH

is enabled for different combinations of 𝛾, 𝑀 , and 𝑁u, that isfor different values of 𝜉, as reported in Fig. 5. In particular,the figure states that if the system is noise-limited (low 𝜉),an optimum TORC detector requires a value for 𝜌TH closeto 1, while, if the system is interference-limited (high 𝜉), achoice close to ORC (i.e., 𝜌TH = 0) is required. Monte Carlosimulation results are also provided for the case of perfectCSI, showing a perfect agreement on the optimum 𝜌TH and,thus, validating our analysis.

B. Ideal Channel Estimation

In Fig. 6 we show the analytical BEP for TORC detection asa function of 𝜌TH for different values of 𝛾 and 𝑀 = 𝑁u = 64.Simulation results are also given showing the goodness of thepresented approach even for a moderate number of subcarriers.Analysis and simulation are in good agreement especiallyfor what concern the value of 𝜌TH providing the minimumBEP. Moreover, it can be noticed that the choice of theoptimum value of 𝜌TH guarantees a significant performanceimprovement (which appears more relevant as the mean SNRincreases) confirming the literature [11], [12], [14], [18]. Notealso that, by increasing the mean SNR, 𝛾, the effect ofnoise becomes negligible, thus, the optimum value of thethreshold minimizing the BEP approaches zero (remember, infact, that the threshold is chosen to neglect those subchannelscontribution highly corrupted by the noise).

In Fig. 7 the performance improvement of an adaptiveTORC detector with respect to classical EGC and fixed TORC(with threshold 𝜌TH = 0.25 and 𝜌TH = 0.4) is shown for

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1ρTH

10−3

10−2

10−1

100

Pb

simulationanalysis

γ=5dB

γ=10dB

γ=12.5dB

Fig. 6. Analytical and simulated BEP as a function of the threshold 𝜌TH fordifferent values of the mean SNR 𝛾, and 𝑀 = 𝑁u = 64.

0 5 10 15 20γ [dB]

10−4

10−3

10−2

10−1

100

Pb EGCTORC, ρTH=0.4TORC, ρTH=0.25PC, β(opt)

TORC, ρTH

(opt)

MMSE

Fig. 7. BEP as a function of the mean SNR 𝛾 in dB for a fully loaded system(𝑁u = 𝑀 = 1024), when EGC, TORC detector with optimum threshold,TORC with threshold fixed to 0.25, and 0.4, optimum partial combining, andMMSE are investigated.

a fully-loaded system (𝑁u = 𝑀 = 1024) at various SNRs.The BEP for MMSE detection from [23] is also plotted forfurther comparison. Note that MMSE represents the optimumamong the linear combining techniques always providing thebetter performance and it is less than 1 dB away from whatobtained using TORC with the optimum 𝜌TH. Hence, anadaptive TORC detector allows to obtain performance veryclose to the optimum MMSE despite the lower complexity.For further comparison, the BEP is also plotted for the partialcombining (PC) technique presented in [19] with optimumparameter 𝛽 and it can be observed that TORC outperformsPC as the SNR increases, when the optimal threshold is used.

The case of time and frequency correlated fading channelsis considered in Fig. 8, where the simulated BER is plottedas a function of 𝜌TH when 𝑁𝑢 = 𝑀 = 64 for the SUI-1three-rays channel model [24] [25] (on the three rays: delays


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1ρTH

10−3

10−2

10−1

BER

γ=5 dB

γ=10 dB

γ=12.5 dB

Fig. 8. Simulated BER vs. 𝜌TH in time and frequency correlated SUI-1channel varying the mean SNR 𝛾 when 𝑁𝑢 = 𝑀 = 64.

0, 0.4, and 0.9 𝜇𝑠; attenuation 0, 15, and 20 dB referredto omnidirectional antennas; Ricean factor 4, 0, 0; Dopplerspectrum 0.4, 0.3, 0.5 Hz, respectively). Different values ofmean the SNR, 𝛾, are considered. As can be observed, theBER achieved in this kind of correlated fading channel isinferior to that of uncorrelated fading presented in Fig. 6.Moreover, there is still an optimum value for the thresholdgiving the minimum BER and this value almost agree withthe optimum one in uncorrelated fading. This is an importantaspect which highlights how the proposed framework enablesto obtain a good value of the TORC threshold also in thesekinds of channel.In Fig. 9, the BEO as a function of the median SNR, 𝜇dB, isreported in uncorrelated fading channel for target BEP 𝑃 ★

b =10−2 and different system loads when TORC with optimum𝜌TH or EGC are adopted in log-normal shadowing with 𝜎dB =8. Fully loaded EGC is not considered since it can not reachthe target performance in the considered range for 𝜇dB. In fact𝜂 = 67% is the maximum tolerable load which guarantees𝑃 ★

b = 10−2 with EGC (by adopting MRC, 𝑃 ★b = 10−2 can

be reached with a maximum system load 𝜂 = 18%). It canalso be observed that, while the BEO is quite sensitive tothe system load for EGC, it is quite insensitive adopting theoptimum threshold (i.e., adaptively changing 𝜌TH to 𝜇dB and𝜂 variations).

By fixing both BEP and BEO, our framework enables thederivation of the required median SNR for various systemloads. As an example, in Fig. 10, the value of 𝜇dB requiredby different combining solutions to obtain a target BEP equalto 10−2 and a target BEO equal to 1% and 5%, is plottedversus 𝜂 for 𝜎dB = 8. These results show a strong impactof the 𝜌TH choice on the required 𝜇dB. In particular, when aTORC detector is adopted, the use of the optimum value of𝜌TH, following the system load variations, always provides thebetter performance with respect to a fixed values of 𝜌TH suchas 0.15 or 0.25. The EGC provides the better performanceonly for low values of 𝜂 and it is significantly outperformedby adaptive TORC when the system load increases.

5 15 25 35μdB [dB]

10−3

10−2

10−1

100

Po

EGC,η=67%EGC,η=50%ρTH

(opt),η=100%

ρTH

(opt),η=67%

ρTH

(opt),η=50%

Fig. 9. 𝑃o vs. 𝜇dB for 𝑃 ★b = 10−2, and 𝜎dB = 8 varying the system load

𝜂: comparison among TORC with optimum use of 𝜌TH and EGC.

0 0.2 0.4 0.6 0.8 1η

16

18

20

22

24

26

28

30

μ dB [d

B]

ρTH=0.15ρTH=0.25ρTH

(opt)

EGC

Po=1%

Po=5%

Fig. 10. 𝜇dB vs. 𝜂 for 𝑃 ★b = 10−2 and 𝑃o = 1% and 5%: comparison

among TORC with optimum use of 𝜌TH, TORC with fixed 𝜌TH equal to 0.15and 0.25, and EGC.

VIII. CONCLUSION

In this paper we analyzed the downlink of a MC-CDMAsystem with TORC detector at the receiver side and evaluatedthe performance, in terms of both BEP and BEO, with idealand non-ideal channel estimation. We derived the value of theTORC threshold that optimizes the BEP, showing a perfor-mance improvement with respect to both the traditional TORCdetection with fixed threshold and other linear combiningtechniques, such as EGC and MRC. We evaluated the optimalthreshold as a function of other system parameters, as thenumber of subcarriers, the number of active users, and themean SNR averaged over small-scale fading. This opensthe way to an adaptive variation of the threshold followingslow processes variations. Adaptive TORC significantly im-proves the performance in small-scale fading and log-normalshadowing conditions. Moreover, the analytical frameworkdeveloped allows to assert that the optimum value of the


TORC threshold does not significatively change consideringperfect or imperfect CSI, meaning that a system designercan adopt the optimum value of the threshold in perfect CSIconditions also with channel estimation errors. The results arealso compared with simulations confirming the validity of theanalytical framework. Both uncorrelated and correlated fadingchannel models have been considered.

APPENDIX

TEST STATISTICS WITH CHANNEL ESTIMATION ERROR

We evaluate the following expressions: (i) 𝔼 {Θ(𝜌TH)}; (ii)𝔼{

��−2𝑢2(�� − 𝜌TH)}; (iii) 𝔼

{Θ2(𝜌TH)

}; where Θ(𝜌TH) =

𝑢(�� − 𝜌TH)𝛼��−1 cos(𝜗 − 𝜗) and �� is Rayleigh distributed:

𝑝��(𝑥) =𝑥

𝜎2h + 𝜎2

e𝑒− 𝑥2

2(𝜎2h +𝜎2

e ) . (44)

The 𝑚th estimated complex channel coefficient is ℎ𝑚 = ℎ𝑚+𝑒𝑚. By decomposing in the real and imaginary parts (ℎ𝑚 =𝑋𝑚+𝑗𝑌𝑚 and 𝑒𝑚 = 𝑋𝑒𝑚+𝑗𝑌𝑒𝑚 , with 𝑗 imaginary unit), andsince ℎ𝑚 and 𝑒𝑚 are zero mean complex Gaussian distributed,then 𝑋𝑚 and 𝑌𝑚, 𝑋𝑒𝑚 and 𝑌𝑒𝑚 result zero mean Gaussiandistributed with variance 𝜎2

h and 𝜎2e , respectively.

For what concern the evaluation of 𝔼 {Θ(𝜌TH)}, it can bewritten as in (45) at the bottom of this page. By defining nowthe variables �� = 𝑋 +𝑋𝑒 and 𝑌 = 𝑌 +𝑌𝑒, they are the sumof independent zero mean Gaussian RVs, thus distributed as

��, 𝑌 ∼ 𝒩 (0, 𝜎2

h + 𝜎2e

), (46)

then, 𝔼{

��𝑌}

= 0, 𝔼{

��𝑋𝑒

}= 𝜎2

e and 𝔼

{𝑌 𝑌𝑒

}= 𝜎2

e .Hence, (45) becomes

𝔼 {Θ(𝜌TH)} = 𝐴1 − 𝐴2 ,

where (𝜀 = 𝜎2e /𝜎2

h )

𝐴1 = 𝔼 {𝑢(�� − 𝜌TH)} =

∫ ∞

𝜌TH

𝑝��(𝑥)𝑑𝑥 = 𝑒− 𝜌2TH

2𝜎2h (1+𝜀) . (47)

For what concern the term 𝐴2, by passing from cartesian topolar coordinates, thus by defining 𝑟 ≜

√��2 + 𝑌 2, 𝜙′ =

arctg(𝑌 /��), 𝑟𝑒 ≜√

𝑋2𝑒 + 𝑌 2

𝑒 and 𝜙𝑒 = arctg(𝑌𝑒/𝑋𝑒), thenit can be written as in (48). By considering that∫ 2𝜋

0

∫ 2𝜋

0

cos(𝜙 − 𝜙𝑒)𝑒𝑟𝑟𝑒 cos(𝜙−𝜙𝑒)

2𝜎2h 𝑑𝜙𝑑𝜙𝑒 = 4𝜋2𝐼1

(𝑟𝑟𝑒2𝜎2

h

)with 𝐼1(⋅) being the modified Bessel function of the first kindand order one, then

𝐴2 =

∫ +∞

𝜌TH

𝑒− 𝑟2

2𝜎2h

∫ +∞

𝜌TH

𝑟2𝑒𝑒−(

1

2𝜎2h+ 1

2𝜎2e

)𝑟2𝑒

𝜎2e (𝜎

2h − 𝜎2

e )𝐼1

(𝑟𝑟𝑒2𝜎2

h

)𝑑𝑟𝑒𝑑𝑟

≃∫ +∞

𝜌TH

𝑒− 𝑟2

2𝜎2h

⎛⎜⎝ 𝜎2

e 𝜎2h ⋅ 𝑟 ⋅ 𝑒

𝑟2𝜎2e

8𝜎2h (𝜎2

h +𝜎2e )

2(𝜎2h − 𝜎2

e )(𝜎2h + 𝜎2

e )2

⎞⎟⎠ 𝑑𝑟

=4𝜎4

h 𝜎2e 𝑒

− 𝜌2TH(4𝜎2h +3𝜎2

e )

8𝜎2h (𝜎2

h +𝜎2e )

2(𝜎2h − 𝜎2

e )(𝜎2h + 𝜎2

e )(4𝜎2h + 3𝜎2

e ). (49)

Finally, we can write

𝔼 {Θ(𝜌TH)} = 𝑒− 𝜌2TH

2𝜎2h (1+𝜀) − 2𝜀𝑒

−𝜌2TH(4+3𝜀)

8𝜎2h (1+𝜀)

(1 − 𝜀2)(4 + 3𝜀). (50)

For what concern the second term to be evaluated

𝔼{

��−2𝑢2(�� − 𝜌TH)}=

1

2(𝜎2h + 𝜎2

e )Γ

[0,

𝜌2TH

2(𝜎2h + 𝜎2

e )

]

=1

2𝜎2h (1 + 𝜀)

Γ

[0,

𝜌2TH

2𝜎2h (1 + 𝜀)

], (51)

being Γ[0, 𝑥] the incomplete Euler Gamma function.The term 𝔼

{Θ2(𝜌TH)

}is given in (52), where 𝐴3 = 𝐴1,

whereas 𝐴4 and 𝐴5 are evaluated in the following. Bychanging coordinates into polars, we can write

𝐴4 =

∫ +∞

𝜌TH

𝑟−1𝑒− 𝑟2

2𝜎2h (53)

×

⎛⎜⎝∫ +∞

𝜌TH

𝑟3𝑒𝑒−(

1

2𝜎2h+ 1

2𝜎2e

)𝑟2𝑒

𝜎2e (𝜎

2h − 2𝜎2

e )𝐼1

(𝑟𝑟𝑒2𝜎2

h

)𝑑𝑟𝑒

⎞⎟⎠

︸︷︷︸𝐴′

4

𝑑𝑟 ,

𝔼 {Θ(𝜌TH)} = 𝔼

{𝑢(�� − 𝜌TH)𝛼��−1 cos(𝜗 − 𝜗)

}= 𝔼

{𝑢(�� − 𝜌TH)𝛼��−1(cos𝜗 cos𝜗 + sin𝜗 sin𝜗)

}= 𝔼

{𝑢(�� − 𝜌TH)

√𝑋2 + 𝑌 2

(√(𝑋 + 𝑋𝑒)2 + (𝑌 + 𝑌𝑒)2

)−1 𝑋(𝑋 + 𝑋𝑒) + 𝑌 (𝑌 + 𝑌𝑒)√𝑋2 + 𝑌 2

√(𝑋 + 𝑋𝑒)2 + (𝑌 + 𝑌𝑒)2

}

= 𝔼{

𝑢(�� − 𝜌TH)[(𝑋 + 𝑋𝑒)2 + (𝑌 + 𝑌𝑒)

2]−1[𝑋(𝑋 + 𝑋𝑒) + 𝑌 (𝑌 + 𝑌𝑒)]}

. (45)

𝐴2 = 𝔼

{𝑢(��− 𝜌TH)(��

2 + 𝑌 2)−1(��𝑋𝑒 + 𝑌 𝑌𝑒)}=

∫ +∞

𝜌TH

∫ +∞

𝜌TH

∫ +∞

𝜌TH

∫ +∞

𝜌TH

(��2 + 𝑌 2)−1(��𝑋𝑒 + 𝑌 𝑌𝑒)

× 𝑒− 1

2[ 1

𝜎2h(��2+𝑌 2)− 1

𝜎2h(��𝑌𝑒+𝑌 𝑋𝑒)+( 1

𝜎2h+ 1

𝜎2e)(𝑋2

𝑒+𝑌 2𝑒 )]

4𝜋2𝜎2e (𝜎2

h − 𝜎2e )

𝑑��𝑑𝑋𝑒𝑑𝑌 𝑑𝑌𝑒

=

∫ +∞

𝜌TH

∫ +∞

𝜌TH

𝑟2𝑒𝑒−[

𝑟2

2𝜎2h+

(1

2𝜎2h+ 1

2𝜎2e

)𝑟2𝑒

]

4𝜋2𝜎2e (𝜎2

h − 𝜎2e )

(∫ 2𝜋

0

∫ 2𝜋

0

cos(𝜙− 𝜙𝑒)𝑒𝑟𝑟𝑒 cos(𝜙−𝜙𝑒)

2𝜎2h 𝑑𝜙𝑑𝜙𝑒

)𝑑𝑟𝑑𝑟𝑒 . (48)


where

𝐴′4 ≃

√𝜋𝜎2

e 𝑟𝑒𝜎2

e 𝑟2

16𝜎2h (𝜎2

h +𝜎2e )

4√

12𝜎2

h+ 1

2𝜎2e(2𝜎2

h + 2𝜎2e )

3

×{12𝜎4

h + 12𝜎2h 𝜎2

e + 𝜎2e 𝑟2

𝜎2h − 𝜎2

e⋅ 𝐼0

[𝜎2

e 𝑟2

16𝜎2h (𝜎

2h + 𝜎2

e )

]

+4𝜎2

h + 4𝜎2h 𝜎2

e + 𝜎2e 𝑟2

𝜎2h − 𝜎2

e⋅ 𝐼1

[𝜎2

e 𝑟2

16𝜎2h (𝜎

2h + 𝜎2

e )

]}.

Then 𝐴4 results in (54), with

𝐶 =

√𝜋 𝜀√

1+𝜀2𝜎2

e4𝜎4

h (1 − 𝜀)(1 + 𝜀)2

𝐵1 ≃ 3𝜎3h

2

√2

𝜋𝐾

(𝜀

4(1 + 𝜀)

)

= 3𝜎2h

√𝜋

(8 + 7𝜀

𝜎2h (1 + 𝜀)

)−1/2

2𝐹1

[1

4,3

4, 1,

𝜀2

(8 + 7𝜀)2

]

𝐵2 ≃ 𝜀

(1 + 𝜀)

2√

𝜋 2𝐹1

[34 , 5

4 , 1, 𝜀2

(8+7𝜀)2

](

8+7𝜀𝜎2

h (1+𝜀)

)3/2

𝐵3 ≃√

𝜋𝜎2e 𝜎h

4(8 + 7𝜀)

√1 + 𝜀

8 + 7𝜀2𝐹1

[3

4,5

4, 2,

𝜀2

(8 + 7𝜀)2

]

𝐵4 ≃ 3√

𝜋𝜀2

2(1 + 𝜀)

2𝐹1

[54 , 7

4 , 2, 𝜀2

(8+7𝜀)2

](8 + 7𝜀)

(8+7𝜀

𝜎2h (1+𝜀)

)3/2 ,

being 𝐼0 the modified Bessel function of the first kind andorder zero, 𝐾(𝑚) the complete elliptic integral of the firstkind, and 2𝐹1[𝑎, 𝑏, 𝑐, 𝑧] the hypergeometric function.

By following the same procedure applied to 𝐴4 also for theterm 𝐴5, we obtain

𝐴5 =

∫ +∞

𝜌TH

𝑒− 𝑟2

2𝜎2h

∫ +∞

𝜌TH

𝑟2𝑒𝑒−(

1

2𝜎2h+ 1

2𝜎2e

)𝑟2𝑒

𝜎2e (𝜎

2h − 𝜎2

e )𝐼1

(𝑟𝑟𝑒2𝜎2

h

)𝑑𝑟𝑒𝑑𝑟

≃∫ +∞

𝜌TH

𝑒− 𝑟2

2𝜎2h

⎛⎜⎝ 𝜎2

e 𝜎2h ⋅ 𝑟 ⋅ 𝑒

𝑟2𝜎2e

8𝜎2h (𝜎2

h +𝜎2e )

2(𝜎2h − 𝜎2

e )(𝜎2h + 𝜎2

e )2

⎞⎟⎠ 𝑑𝑟

=2𝜎4

h 𝜎2e

(𝜎2h − 𝜎2

e )(𝜎2h + 𝜎2

e )(4𝜎2h + 3𝜎2

e )𝑒− 𝜌2TH

8

(3

𝜎2h+ 1

𝜎2h +𝜎2

e

)

=2𝜀

(1− 𝜀2)(4 + 3𝜀)𝑒− 𝜌2TH

8

(4+3𝜀

𝜎2h (1+𝜀)

). (55)

Hence,

𝔼{Θ2(𝜌TH)

} ≃ 𝐴3 + 𝐴4 − 2𝐴5 . (56)

ACKNOWLEDGMENTS

The authors would like to acknowledge Prof. O. Andrisanofor motivating and supporting this research activity, Proff. M.Chiani and M.Z. Win for helpful comments on adaptive tech-niques, and F. Zabini for helpful discussions on equalizationtechniques for MC-CDMA systems.

REFERENCES

[1] K. Fazel and S. Kaiser, Multi-Carrier and Spread Spectrum Systems. NewYork: Wiley, 2003.

[2] N. Yee, J.-P. Linnartz, and G. Fettweis, “Multi-carrier-CDMA in indoorwireless networks,” in Proc. PIMRC ’93, Yokohama, Sep, 1993. pp. 109-113.

[3] S. Hara and R. Prasad, “An overview of multicarrier CDMA,” in Proc.IEEE 4th International Symp. Spread Spectrum Techniques Applications,Sep. 1996, vol. 1, pp. 107-114.

𝔼{Θ2

𝑙 (𝜌TH)}= 𝔼

{𝑢2(��− 𝜌TH)𝛼

2𝑙 𝛼𝑙

−2 cos2(𝜃𝑙 − 𝜃𝑙)}= 𝔼

{𝑢2(��− 𝜌TH)(��

2 + 𝑌 2)−2[(��2 + 𝑌 2)− (��𝑋𝑒 + 𝑌 𝑌𝑒)]2}

= 𝔼

{𝑢2(��− 𝜌TH)(��

2 + 𝑌 2)−2[(��2 + 𝑌 2)2 + (��𝑋𝑒 + 𝑌 𝑌𝑒)2 − 2(��2 + 𝑌 2)(��𝑋𝑒 + 𝑌 𝑌𝑒)]

}= 𝔼

{𝑢2(��− 𝜌TH)

}︸︷︷︸𝐴3

+𝔼

{𝑢2(��− 𝜌TH)(��

2 + 𝑌 2)−2(��𝑋𝑒 + 𝑌 𝑌𝑒)2}

︸︷︷︸𝐴4

−2𝔼{𝑢2(��− 𝜌TH)(��

2 + 𝑌 2)−1(��𝑋𝑒 + 𝑌 𝑌𝑒)}

︸︷︷︸𝐴5

, (52)

𝐴4 ≃ 𝐶

∫ +∞

𝜌TH

𝑒− 𝑟2

2𝜎2h

(1− 𝜎2

e8(𝜎2

h +𝜎2e )

){(3

2𝜎2

h +𝜎2

e 𝑟2

8(𝜎2h + 𝜎2

e )

)𝐼0

[𝜎2

e 𝑟2

16𝜎2h (𝜎

2h + 𝜎2

e )

]+

(1

2𝜎2

h +𝜎2

e 𝑟2

8(𝜎2h + 𝜎2

e )

)𝐼1

[𝜎2

e 𝑟2

16𝜎2h (𝜎

2h + 𝜎2

e )

]}𝑑𝑟

= 𝐶

⎧⎨⎩

𝐵1︷︸︸︷∫ +∞

𝜌TH

3

2𝜎2

h 𝑒− 𝑟2

2𝜎2h

(1− 𝜎2

e8(𝜎2

h +𝜎2e )

)𝐼0

[𝜎2

e 𝑟2

16𝜎2h (𝜎

2h + 𝜎2

e )

]𝑑𝑟+

𝐵2︷︸︸︷∫ +∞

𝜌TH

𝜎2e 𝑟

2

8(𝜎2h + 𝜎2

e )𝑒− 𝑟2

2𝜎2h

(1− 𝜎2

e8(𝜎2

h +𝜎2e )

)𝐼0

[𝜎2

e 𝑟2

16𝜎2h (𝜎

2h + 𝜎2

e )

]𝑑𝑟

+

𝐵3︷︸︸︷∫ +∞

𝜌TH

1

2𝜎2

h 𝑒− 𝑟2

2𝜎2h

(1− 𝜎2

e8(𝜎2

h +𝜎2e )

)𝐼1

[𝜎2

e 𝑟2

16𝜎2h (𝜎

2h + 𝜎2

e )

]𝑑𝑟+

𝐵4︷︸︸︷∫ +∞

𝜌TH

𝜎2e 𝑟

2

8(𝜎2h + 𝜎2

e )𝑒− 𝑟2

2𝜎2h

(1− 𝜎2

e8(𝜎2

h +𝜎2e )

)𝐼1

[𝜎2

e 𝑟2

16𝜎2h (𝜎

2h + 𝜎2

e )

]𝑑𝑟

⎫⎬⎭

(54)


[4] N. Yee and J.-P. Linnartz, “BER of multi-carrier CDMA in an indoorRician fading channel,” in Conf. Record Twenty-Seventh Asilomar Conf.Signals, Systems Computers, 1993, Nov. 1993 pp. 426-430, vol. 1.

[5] S. Kaiser, “On the performance of different detection techniques forOFDM-CDMA in fading channels,” in Proc. Globecom ’95, Nov. 1995,pp. 2059-2063.

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Barbara M. Masini (S’02-M’05) received the Lau-rea degree (with honors) and the Ph.D. degree intelecommunications engineering from the Universityof Bologna, Italy, in 2001 and 2005, respectively.In 2002, she joined the Department of Electron-ics, Informatics and Systems at the University ofBologna, to develop his research activity in the areaof wireless communications. Since 2005, she is withthe Institute for Electronics and for Information andTelecommunications Engineering (IEIIT), ResearchUnit of Bologna, of the National Research Council

(CNR) working on wireless transmission techniques. Since 2007 she isassistant professor at the University of Bologna where she holds the courseof Telecommunication Laboratory including experimental implementation oftelecommunication systems on digital signal processing platforms.

Her research interests are mainly focused on OFDM and MC-CDMAsystems, short-range wireless communications, wireless local-area networks(WLAN), and coexistence among different wireless systems. She is also inter-ested in intelligent transportation systems (ITS) for infomobility applicationsand is coauthor of the chapter Heterogeneous wireless communications forvehicular network in the book Vehicular Networks: Techniques, Standards,and Applications, (CRC Press, 2009).

She is an IEEE member, and serves as a Reviewer for many transactionsand conferences and as a TPC Member for several conferences.

Andrea Conti (S’99-M’01) received the Dr.Ing.degree in telecommunications engineering and thePh.D. degree in electronic engineering and computerscience from the University of Bologna, Italy, in1997 and 2001, respectively. Since 2005, he isassistant professor at the University of Ferrara, Italy.Prior to joining the University of Ferrara he waswith Consorzio Nazionale Interuniversitario per leTelecomunicazioni (CNIT, 1999-2002) and Istitutodi Elettronica e di Ingegneria dell’Informazione edelle Telecomunicazioni, Consiglio Nazionale delle

Ricerche (IEIIT/CNR, 2002-2005) at the Research Unit of Bologna, Italy. Insummer 2001, he joined the Wireless Section of AT&T Research Laboratories,Middletown, NJ. Since February 2003, he has been a frequent visitor atthe Laboratory for Information and Decision Systems (LIDS), MassachusettsInstitute of Technology (MIT), Cambridge, where he is presently researchaffiliate.

His current research interests are in the area of wireless communicationsincluding localization, adaptive transmission and multichannel reception,coding in faded multiple-input multiple-output channels, wireless cooperativenetworks, and wireless sensor networks.

He is a coauthor of Wireless Sensor and Actuator Networks: Enabling Tech-nologies, Information Processing and Protocol Design (Elsevier, 2008). He isan Editor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS

and was Lead Editor for the EURASIP JASP (S.I. on Wireless CooperativeNetworks, 2008). He is TPC Vice-Chair for IEEE WCNC 2009, Co-Chairof the Wireless Comm. Symp. for IEEE GCC 2010, and has served as aReviewer and TPC member for various IEEE journals and conferences. Heis currently serving as secretary of IEEE RCC for the period 2008-2010.

Adaptive TORC detection for MC-CDMA wireless systems

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