Error Floor-Like Behavior in Maximum-Likelihood

Detection of Single Carrier FDMA

Mark Geles∗, Ofer Amrani, Amir Averbuch, Fellow, IEEE †and Doron Ezri‡

Abstract

Single carrier FDMA (SC-FDMA) plays an important role in modern wireless commu-

nication as a low peak-to-average power ratio (PAPR) alternative to OFDM. Due to its

advantages, this transmission scheme was chosen for the long term evolution (LTE) up-

link. As with other SC transmission schemes, the SC-FDMA signal arrives at the receiver

with significant inter symbol interference (ISI) in the presence of multipath. This makes

the regular single tap frequency domain equalization suboptimal. The optimal maximum

likelihood (ML) detector in SC-FDMA is in most cases prohibitively complex, making it

useful mostly as a performance bound for suboptimal detection schemes. In this paper,

the performance of the optimal decoder applied to SC-FDMA is analyzed. Closed form

bounds on the bit error rate (BER) are provided for the low and high SNR regimes, in

correlated and uncorrelated Rayleigh fading channels. The bounds show that the diversity

order at high SNR is significantly smaller than that in low SNR. Moreover, error floor-like

behavior is demonstrated under optimum detection of SC-FDMA.

Maximum likelihood, single carrier FDMA.

∗M. Geles and O. Amrani are with the Department of Systems Electrical Engineering, Tel Aviv University,

Tel Aviv 69978 Israel†A. Averbuch is with the Department of Computer Science, Tel Aviv University, Tel Aviv 69978 Israel‡D. Ezri is with Greenair Wireless, 47 Herut St., Ramat Gan, Israel

1

1 Introduction

Single carrier FDMA (SC-FDMA), also known as DFT spread OFDM, is a lower PAPR vari-

ant of OFDM. As in OFDM, the SC FDMA transmitter uses different orthogonal frequencies

(sub-carriers) to transmit information symbols. However, the data symbols undergo a DFT

operation prior to the allocation to sub-carriers. This arrangement creates a single carrier like

waveform and reduces considerably the envelope fluctuations in the time domain signal. There-

fore, SC-FDMA signals have inherently lower PAPR than OFDM signals. However, this PAPR

advantage does not come without a cost. The SC-FDMA signal arrives at the receiver with

significant ISI in the presence of multipath propagation even with delay spread smaller than

the cyclic prefix.

A practical SC-FDMA detector usually involves a single tap frequency domain equal-

izer (FDE) followed by an IDFT operation. Frequency domain minimum mean square error

(MMSE) is most commonly employed while other linear detectors as zero forcing (ZF) are also

possible. However, for distributed SC-FDMA (or localized SC-FDMA in severe multi-path en-

vironment), linear detectors are suboptimal. For evaluating their effectiveness, one would like

to be able to compare the performance of linear detectors with the optimum maximum likeli-

hood detector (MLD). Simulating the MLD is of exponential complexity and hence prohibitive

for a sufficiently large DFT size. This motivates the derivation of analytical expressions for

evaluating the expected performance of the MLD scheme.

There are several fairly recent contributions that analyze the performance of several de-

tection schemes applied to SC-FDMA transmission for various channel models. Performance

analysis of ZF and MMSE linear detectors, was given by Wang et al. [1]. In addition to the

exact ZF and MMSE bit error rate (BER) formulae, an upper bound on MLD packet error rate

(PER) is given in a paper by Nisar et al. [2]. Therein, a packet means a group of modulated

2

symbols allocated to the DFT block (per user). Since an erroneous packet may include any

number of erroneous bits, BER derivation from PER is not trivial. In both papers [1, 2], the

expressions are functions of the fading channel realization and not functions of a fading channel

statistics, so the average error rate is not provided.

In this work , we present simple yet tight lower bounds on the bit error rates of ML detected

SC-FDMA transmission assuming Rayleigh-fading channel model. The bounds are calculated

by means of integration of the selected error-vector probabilities w.r.t. the fading channel

distribution function. The closed form expressions derived herein depend only on the size of

the DFT. The lower bounds reveal an unusual error floor-like behavior in ML detection of

SC-FDMA. Specifically, the diversity order at high SNR is significantly lower than in low SNR.

The paper is organized as follows. In Section 2 we discuss SC-FDMA detection and promi-

nent detection schemes. In Section 3 we derive closed form bounds on the BER in ML detected

SC-FDMA. In Section 4 we give simulation results corroborating our analysis. Section 5 con-

tains discussion and conclusions.

2 SC-FDMA Detection

By considering an SC-FDMA system with N tones and sufficiently long cyclic prefix (longer

than the maximal delay spread), the received signal on the n−th tone is

yn = hn xn + ρwn, (1)

where hn represents the channel on the n−th sub-carrier, xn is the n−th element of the DFT

output transmitted on the n−th sub-carrier and ρwn is an additive white Gaussian noise with

zero mean and variance ρ2. By aggregating the received signals on all sub-carriers we have

y = Hx + ρ w, (2)

3

where y = [y1, . . . , yN ]T , H = diag{[h1 . . . , hN ]T

}, w = [w1, . . . , wN ]T and x = [x1, . . . , xN ]T

is the DFT output vector

x = Fs, (3)

where F is the unitary N × N DFT matrix, and s is N × 1 vector of symbols taken from a

(normalized) constellation. By using the expression for x, the received signal vector takes the

form

y = HF︸︷︷︸G

s + ρ w. (4)

We first consider the simple case where the channel magnitude |hi| = |h|, i = 1, . . . , N , is con-

stant within the occupied bandwidth. This corresponds, for example, to localized transmission

where the occupied bandwidth is significantly smaller than the coherence bandwidth of the

channel, or to a strong line of sight scenario. Here we have H∗H = |h|2I and the composite

matrix G satisfies

G∗G = (HF )∗(HF ) = |h|2F ∗F = |h|2I, (5)

which means that the columns of G are orthogonal. In this case, to which case linear equaliza-

tion (e.g., ZF) is optimal [4].

We continue with the case where the channel magnitude varies within the occupied band-

width. This corresponds for example to distributed SC-FDMA or to localized transmission

occupying a bandwidth larger than (or of the order of) the coherence bandwidth. Here, in

contrast to OFDM, the columns of the composite matrix G are not orthogonal. Hence, linear

detection is not optimal. The optimum solution is given by

sML = argmins∈C

‖y −Gs‖2, (6)

where C is the space of length-N vectors whose alphabet are the symbols drawn from the

constellation employed. Although the MLD approach in Eq. (6) amounts to the optimum

4

solution, its calculation is impractical since implementation complexity grows exponentially

fast with N .

More practical, yet typically suboptimal, MMSE receiver computes the MMSE estimator

sMMSE = G∗ (GG∗ + ρ2I

)−1y

= F ∗H∗ (HH∗ + ρ2I

)−1y, (7)

where (·)∗ denotes conjugate transposition, and I is the identity matrix of appropriate dimen-

sions. The elements of the MMSE estimator are then sliced, i.e. hard detected, to obtain an

estimate for the transmitted symbols. Using the fact that H is a diagonal matrix and F is the

unitary DFT matrix, the MMSE estimator simplifies to

sMMSE = IDFT{[x1, . . . , xN ]T

}, (8)

where xn is frequency domain MMSE estimator

xn =h∗n

|hn|2 + ρ2yn. (9)

This means that the MMSE receiver can be viewed as a single tap FDE (similar to OFDM)

followed by an IDFT operation, corresponding to the additional DFT at the transmitter.

More advanced SC-FDMA detection methods with performance lying between those of

MMSE and MLD have been proposed. Turbo equalization (TEQ) [5] and frequency domain

TEQ (FD-TEQ) [6] are natural approaches to SC-FDMA detection [7, 8] when channel coding

is employed. In TEQ, the equalizer and the channel decoder are considered as soft input/soft

output blocks, separated by bit interleavers. Information produced by one of these blocks is

treated as apriori information for the other, leading to an iterative receiver structure.

A different approach to SC-FDMA detection, which combines MMSE and ML, has been

recently proposed [9]. The receiver adopts the group interference suppression (GIS) technique

5

in which groups of symbols, which correspond to highly correlated columns of G, are consid-

ered. Each group undergoes joint detection following a linear pre-filtering procedure aiming at

suppressing the interference from all other symbols.

The performance of the optimal detector analyzed herein will provide a bound for the

aforementioned advanced methods and allows us to evaluate how much room for improvement

still exists.

3 Closed-Form Bounds on the ML Error Probability

In this section, we derive approximated lower-bounds for the BER performance of ML detected

SC-FDMA.

A deviation vector e , s − s is defined to be equal to the source vector subtracted from

the estimated vector. The error probability, corresponding to a deviation vector e given the

channel matrix H , is

Pr {e|H} = Q

(‖HFe‖√2ρ

), (10)

where

Q(x) =1√2π

∫ ∞

x

exp

(−y2

2

)dy. (11)

The performance of the classical flat fading OFDM setting can be derived from Eq. (10)

assuming that F and H are 1× 1 matrices. In this case, F = 1 and H is a scalar denoted by

h; hence, the error probability using normalized QPSK is simply

Pr {e|h} = Q

(‖h · 1 · e‖√2ρ

)= Q

(√SNR

), (12)

where SNR , 1/ρ2. Note that Eq. (12) constitutes the well known result for the flat fading

OFDM case.

6

Denoting the number of errors in the N × 1 deviation vector e as ne (0 ≤ ne ≤ N), then

the exact BER conditioned on H is

Pb(H) =1

NE {ne|H} =

1

N

N∑ne=1

ne · Pr {ne|H} , (13)

where Pr {ne|H} = Pr {⋃{e : number of errors is ne}|H} is the probability that the deviation

vector e contains exactly ne non-zero elements. By identifying the most probable deviation

vector eo, we can derive from Eqs. (10) and (13) the following lower bound

Pb(H) ≥ 1

Nne(eo) ·Q

(‖HFeo‖√2ρ

), (14)

where eo = argmine∈E

{‖HFe‖} and E is the set of all possible deviation vectors. Note that the

bound (14) is difficult to calculate due to the fact that there are 4N − 1 different deviation

vectors. Therefore, this bound can be used only for cases where the DFT size is sufficiently

small.

In order to obtain a bound on the average BER, the expression (14) has to be averaged

using the joint pdf of the channel vector h = [h1, . . . , hN ]T . We adopt the classical correlated

Rayleigh model [4, 10], which corresponds to a multipath environment. Specifically, we assume

that h is a complex normal random vector with zero mean and covariance matrix C with

det(C) 6= 0.

Following this approach, the average error probability is

Pr {e} =

∫Pr {e|H}Pr {H} dH (15)

=

∫Q

(‖HFe‖√2ρ

)1

πNdet(C)exp{−h∗C−1h}dh.

Using the following upper bound Q(x) ≤ 1

2exp

(−x2

2

)on the Q-function, we get

Pr {e} ≤ 1

2

∫exp

{−‖HFe‖2

4ρ2

}

· 1

πNdet(C)exp{−h∗C−1h}dh. (16)

7

Since the matrix H is diagonal, the expression ‖HFe‖2 can be rewritten as

‖HFe‖2 = (Fe)∗diag{h}∗diag{h}(Fe)

= h∗diag{Fe}∗diag{Fe}h

= h∗Mh, (17)

where f 1, .., fN are the rows of the DFT matrix F and M , diag{|f 1e|2, .., |fNe|2}. Substi-

tuting Eq. (17) into Eq. (16) we get

Pr {e} ≤ 1

2

∫exp

{−h∗

M

4ρ2h

}exp{−h∗C−1h}

πNdet(C)dh

=1

2πNdet(C)

∫exp

{−h∗

(M

4ρ2+ C−1

)h

}dh. (18)

Since the pdf of a complex Gaussian vector x of length L with covariance V takes the form

1

πLdet(V )exp{x∗V −1x} and its integral equals 1, then Eq. (18) is rewritten as

Pr {e} ≤ 1

2det(C)

[det

(M

4ρ2+ C−1

)]−1

=1

2

[det

(M

4ρ2C + I

)]−1

. (19)

The error probability bound in Eq. (19) depends on the deviation vector e.

Denote by bn the specific bit position n in the vector s. The error probability in bit bn

satisfies

Pb(n) = Pr {En} ≥ maxe∈En

Pr {e} , (20)

where En is the subset of E in which all the deviation vectors contain an error in the n-th

position.

There are cases for which we can explicitly calculate the lower bound defined in Eq. (20).

Next, we derive closed form expressions for some of these cases.

8

3.1 Uncorrelated Fading Channel

We shall now examine the particular case of an uncorrelated Rayleigh channel. In this case,

C = σ2I and Eq. (19) becomes

Pr {e} ≤ 1

2

[det

(σ2

4ρ2M + I

)]−1

=1

2

[det

(σ2

4ρ2diag{|f 1e|2, .., |fNe|2}+ I

)]−1

=

(2

(σ2

4ρ2|f 1e|2 + 1

)· · ·

(σ2

4ρ2|fNe|2 + 1

))−1

=1

2

N∏n=1

(1

4|fne|2SNR + 1

)−1

. (21)

Based on Eq. (21), we shall derive bounds on Pb for the high and low SNR regimes.

3.1.1 High SNR Regime

First, we present a lemma that will be proved useful for the derivation of our main theorem.

Let us define the following condition:

Condition 1. A vector e is said to satisfy Condition 1, if DFT(e) = Fe has a single non-zero

element.

Lemma 3.1. For any DFT of size N (a multiple of 4) there are exactly 8·2N possible transmitted

QPSK vectors such that there exists a valid deviation vector, which satisfies Condition 1, that

contains an error in a specific bit b0.

Proof. The rows of the DFT matrix(fk = e−jθn, θ = 2π

Nk, n = 0, ..., N − 1

)are the basis for

the N -th length vector space. The deviation vector e can be represented as e =N−1∑

k=0

(fke∗)fk.

Therefore, the deviation vector e satisfying Condition 1 can be only one of the DFT matrix

rows multiplied by a constant. Only four rows of F are relevant. They are

9

1 1 1 1 ... 1 1,

1 j −1 −j ... −1 −j,

1 −1 1 −1 ... 1 −1,

1 −j −1 j ... −1 j.

(22)

corresponding to θ = 0, θ = π2, θ = π and θ = 3π

2, respectively.

For every deviation vector above and the bit b0 there can be 2 · 2N vectors out of 4N possible

QPSK vectors. 2 stands for two values of b0, and 2N stands for the second bit in the QPSK

symbols. Hence, in total, there are 8 · 2N different vectors which satisfy condition 1.

Theorem 3.2. A tight lower bound on ML detected SC-FDMA for uncoded QPSK is given by

limSNR→∞

P(QPSK,MLD)b =

4

N2N SNR, (23)

and the diversity order is 1.

Proof. For high SNR, the deviation vector with the smallest diversity order (DO) dominates

the error probability [11]. The DO is defined in [11] as

DO , − limSNR→∞

loge Pr {error}loge SNR

. (24)

The DO, which is associated with Pr {e}, is computed by using Eqs. (24) and (21)

DO{Pr {e}} = − limSNR→∞

1

loge SNRloge Pr {e}

≥ limSNR→∞

1

loge SNRloge 2

N∏n=1

(1

4|fne|2SNR + 1

)

≥ limSNR→∞

1

loge SNRloge 2

∏

n:fne 6=0

1

4|fne|2

+ limSNR→∞

loge SNRK

loge SNR= K, (25)

10

where K is the number of non-zero elements in Fe. Therefore, from Eq. (25) we conclude that

the DO, which is associated with Pr {e}, is greater or equal to the number of non-zero elements

in Fe.

Specifically, a deviation vector ex, which satisfies Condition 1, results with the following

error probability

Pr {ex} =

∫Pr {ex|H}Pr {H} dH

=

∫Q

(‖HFex‖√2ρ

)P (H)dH

=

∫Q

(√N SNRy

)2y exp(−y2)dh, (26)

where y = |h| is a Rayleigh distributed scalar. Equation (26) is a standard Q-function integral

w.r.t. Rayleigh distribution function that can be found in [12]

∫ ∞

0

Q(ay)PRayleigh(y) dy =1

2

1− 1√

1 + 2a2

≈ 1

2a2.

Plugging the argument a =√

N SNR we get

Pr {ex} =1

2N SNR. (27)

The DO associated with Eq. (27) is

DO{Pr {ex}} = limSNR→∞

loge 2N SNR

loge SNR= 1. (28)

From Eqs. (28) and (25) it follows that ex, which satisfies Condition 1, produces the error

probability associated with the smallest DO. Therefore, it dominates the error probability in

the high SNR regime. This is true because if we compare two probabilities P1 and P2 associated

with diversity orders DO1 < DO2 when SNR tends to infinity, we get

limSNR→∞

(log P1 − log P2) = − limSNR→∞

(DO1 −DO2) log SNR = ∞

11

However, in order to obtain an explicit lower bound on the bit error probability, one must take

into account that not all the possible deviation vectors are valid for a given transmitted vector

s. For example, all the elements of the deviation vector e may be equal to 2 with non-zero

probability, only if all the elements of the transmitted vector s are equal to -1. Lemma (3.1)

states that there are exactly 8 · 2N possible length N QPSK vectors, where N is a multiple

of 4, which satisfy Condition 1. In other words, the probability of transmitting a vector with

DO=1 is8

2N. Multiplying Eq. (27) by this factor, we get that the MLD bit-error probability,

for SC-FDMA in an uncorrelated Rayleigh fading channel, is lower bounded by

P(QPSK,MLD)b (high SNR) ≥ 8

2N

1

2N SNR=

4

N2N SNR.

We can see from Eq. (23) that the diversity gain increases (and the error probability

curve shifts to the left) as the DFT size N increases. This is true in the setting used in this

work, i.e. uncoded transmission over an uncorrelated fading channel. In a practical system

such as OFDMA, the channel diversity will be utilized by means of interleaving, coding and

allocation techniques. Therefore, the performance difference between OFDMA and SC-FDMA

with different DFT sizes is expected to be small.

In the simpler case of BPSK, the number of rows in Equation (22) is reduced to two (only real

values) and the expression in Eq. (23) is divided by 2 (the two bits of QPSK are orthogonal),

yielding

P(BPSK,MLD)b (high SNR) ≥ 4

2N

1

4N SNR=

1

N2N SNR.

12

3.1.2 Low SNR Regime

In order to identify the dominant deviation vector for the low SNR regime, we rewrite Eq.(21)

as follows

Pr {e} ≤ 1

2

N∏n=1

(1

4|fne|2SNR + 1

)−1

≈ 1

2

(1 +

N∑n=1

1

4|fne|2SNR

)

=1

2

(1 +

1

4SNR‖e‖2

) . (29)

The error probability is upper bounded by expression (29), which obtains its maximum value for

the deviation vectors of smallest norm. Thus, the dominant deviation vectors in the low SNR

regime are the vectors having only one non-zero element. Based on the following Q-function

approximation Q(x) ≈ 1

12exp

{−x2

2

}+

1

4exp

{−4

3

x2

2

}([13]), an expression similar to (19),

as derived from (15), can be written

Pr {e} =1

12

[det

(diag{|f 1e|2, .., |fNe|2}

4ρ2C + I

)]−1

+1

4

[det

(4

3

diag{|f 1e|2, .., |fNe|2}4ρ2

C + I

)]−1

.

Substituting e = [√

2, 0, ..., 0], and f 1, .., fN into this expression, one can get an approximated

lower bound for the BER in the low SNR regime

P(QPSK,MLD)b (low SNR) ≥1

12

(1 +

1

2NSNR

)−N

+1

4

(1 +

2

3NSNR

)−N

. (30)

Note that this lower bound has a performance slope respective to diversity order N .

In the BPSK case, the distance between the constellation points is smaller by a factor of

13

√2. Therefore, the lower bound takes the form

P(BPSK,MLD)b (low SNR) ≥1

12

(1 +

1

NSNR

)−N

+1

4

(1 +

4

3NSNR

)−N

. (31)

3.2 Correlated Fading Channel

In a correlated Rayleigh channel with correlation matrix C, the deviation vector associated with

the smallest DO is the same as in the uncorrelated case (23). This is true because substituting

a deviation vector, which satisfies Condition 1, yields an expression which is linear in the SNR

and therefore DO=1. For example, by substituting e = [√

2,√

2, ...,√

2] into the determinant

in Eq. (19), we get

det

(diag{|f 1e|2, . . . , |fNe|2}

4ρ2C + I

)=

= det

N2· SNR + 1 N

2c1,2SNR ... N

2c1,NSNR

0 1 ... 0

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

0 0 ... 1

=N

2· SNR + 1. (32)

For any other non-zero deviation vector (which does not satisfy Condition 1), Eq. (32) will

become a polynomial of degree at least 2 of the SNR. Consequently, the corresponding BER

will be affected by DO of at least 2. Therefore, when det(C) 6= 0, the asymptotical lower

bound of the BER for a correlated Rayleigh channel is the same as the one for the uncorrelated

Rayleigh channel (23).

As a sanity check, let us examine a flat fading Rayleigh channel as a particular case for

14

which det(C) = 0. In this case, Eq. (15) no longer holds since the distribution function

contains division by det(C). Instead, we can rewrite Eq. (10) for a flat fading case H = h · I,

where h is a complex scalar with a Gaussian distribution. Hence,

Pr {e|H} = Q

(‖HFe‖√2ρ

)= Q

( |h| · ‖Fe‖√2ρ

)

= Q

( |h| · ‖e‖√2ρ

). (33)

Therefore,

Pb ≈ Pr {e : 1 non zero element|H} = Q(√

SNR)

, (34)

which is identical to the expression for the BER in OFDM (12). Since the performance of SC-

FDMA in a Rayleigh channel can only be better than those obtained in a flat fading channel,

we use the OFDM BER curve for upper bounding the BER performance of SC-FDMA.

It follows from the above derivation and from Fig. 1 that the BER curves for ML detected

SC-FDMA over Rayleigh channel are upper bounded by the OFDM BER performance Eq. (34)

and lower bounded by Eq. (23).

15

4 Simulation Results

In order to verify the approximations given by Eqs. (23) and (30), we conducted a Monte-Carlo

simulation study. We used ML detection for SC-FDMA with DFT sizes of 2,4,8 in uncorrelated

Rayleigh. 3.2 · 106 bits were transmitted for obtaining each SNR point. The empirical perfor-

mance of the MLD and the corresponding bounds are shown in Fig. 1. Evidently, the empirical

BER curves converge with the lower bound in the high SNR regime.

Figure 1: BER of ML detected SC-FDMA (DFT sizes 2,4,8) over an uncorrelated Rayleigh

channel using uncoded QPSK modulation.

The effect of frequency domain correlation on the BER is demonstrated in Fig. 2. The top

curve, which is used here as a reference, presents the BER performance for OFDM. All the

other curves correspond to the BER performance of ML-detected SC-FDMA (DFT of size 4) in

Rayleigh fading with the correlation function r[n] = exp{−n/K}, where n is difference in sub-

carrier indices and K is the fading-channel correlation parameter. We studied the correlation

function for the following values of the parameter K: 3,6,30,100, 1000. From Fig. 2, we conclude

that the BER curve for the Rayleigh fading channel, with the correlation function defined above,

16

Figure 2: BER of MLD applied to SC-FDMA (DFT size 4) over a correlated Rayleigh channel

using uncoded BPSK modulation. The correlation function is r[n] = exp{− nK}, where n is the

sub-carrier index and K is a constant.

is upper bounded by the OFDM BER curve, Eq. (34), and lower bounded by the lower bound

in the high SNR regime (Eq. (23)). Moreover, all the curves except the one that corresponds

to fully correlated fading channel, converge to the lower bound in the high SNR regime.

Employing MLD for QPSK with large DFT sizes is impractical due to its high computational

complexity. Thus, when addressing larger DFT sizes we turned to BPSK modulation and

employ a near-optimum QRM-MLD algorithm with a sufficiently large parameter M [14]. The

performance of QRM-MLD for DFT sizes 12 and 32 with 3.2 · 106 bits per SNR point is

demonstrated in Figs. 3 and 4, respectively.

The simulation results reveal that the derived bounds are with good agreement with the

performance of QRM-MLD when a sufficiently large M parameter is used for both low and high

SNR values. This in turn means that the performance of the reduced-complexity QRM-MLD

is close to the optimal ML. Moreover, the intersection point between the bounds predicts the

SNR region where the slope of the BER curve drops from N to 1.

17

Figure 3: BER of QRM-MLD (with M = 12) applied to SC-FDMA with 16 point DFT and

BPSK in uncorrelated Rayleigh.

Figure 4: BER of QRM-MLD (with M = 32) applied to SC-FDMA with 32 point DFT and

BPSK in uncorrelated Rayleigh.

5 Discussion and Conclusions

In this paper, we derived lower bounds on the bit error probability for ML detection of SC-

FDMA transmission in uncorrelated and correlated Rayleigh fading. Our results may serve as

18

performance bounds for advances SC-FDMA detection schemes. In the case of uncorrelated

Rayleigh, the bounds show that the slope of the BER curve for low SNR is significantly higher

than that of high SNR, revealing an unusual error floor-like behavior.

We also showed that the BER value in which the slope changes depends on the DFT size.

Specifically, the larger the DFT size, the smaller the BER value at which the slope changes.

The error floor-like behavior appears when the DFT size is small and the frequency domain

correlation is low (e.g., in distributed SC-FDMA). In coded systems, the error floor behavior is

expected in the conditions above at high coding rates where the working SNR is high. Bearing

in mind that real life SC-FDMA systems like LTE, employ DFT sizes as small as 12 and high

coding rates suggest that our results are of practical importance.

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19

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