An Improved Detection Technique for Cyclic ... - Semantic Scholar
JOURNAL OF NETWORKS, VOL. 5, NO. 7, JULY 2010
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An Improved Detection Technique for Cyclic-Prefixed OFDM Giovanni Garbo, Stefano Mangione Dipartimento di Ingegneria Elettrica, Elettronica e delle Telecomunicazioni (DIEET) Università degli Studi di Palermo Palermo, Italy Email: {giovanni.garbo, stefano.mangione}@tti.unipa.it
Abstract—A novel Orthogonal Frequency Division Multiplexing detection technique compatible to standard (e.g. Wireless LAN) transmitters is proposed. It features enhanced error-rate performance with flexible computational complexity and robustness to imperfect channel estimation. It is based on exploitation of the redundancy available in the cyclic prefix after cancellation of interference from the preceding block. In order to show the effectiveness of our proposal, an analysis of computational complexity and a number of comparisons to the standard per-subcarrier receiver and a previously existing method in terms of error rates are reported. Index terms—Orthogonal Frequency Division Multiplexing; Frequency-selective channels; Maximum-Likelihood Detection; Linear Detection; Interference Cancellation
I. INTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) [1, 2] is a signaling technique particularly suited for transmission over linear distortion channels. The key feature of OFDM is the fact that both the transmitter and the receiver can be based on the well-known and efficient Fast Fourier Transform (FFT) [3]. The simplicity of the FFT-based receiver structure has led to its inclusion in many existing and future wireless standards, such as IEEE 802.11 [4] and 3GPP LTE [5]. It is well known that a cyclic prefix is needed in the OFDM signal in order to preserve the orthogonality between the subcarriers. The cyclic prefix reveals its usefulness also for synchronization purposes. Much less investigated is the use of the cyclic prefix as an aid to enhance data detection by virtue of its intrinsic redundancy. To the best of our knowledge, a technique present in the literature capable of exploiting part of the redundancy of the cyclic prefix has been presented in [6]. It starts from the assumption that the cyclic prefix itself is occasionally over-dimensioned with respect to the actual channel memory, and it consists in using the cyclic prefix segment not affected by interference from the preceding block in order to obtain an over determined system of equations. The mathematical approach presented in [6] cleverly exploits the circulant nature of the equivalent channel matrix and results in a simple detection scheme which, unfortunately, gives rise to a small performance gain.
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A preliminary version of this work was presented in [7], where we suggested to cancel the interference arising in the cyclic prefix from the preceding block in order to take advantage of all its redundancy. In this work we will first report, for the sake of completeness, the derivation of the model described in [7], followed by a detailed description of a linear detection strategy and a computational complexity analysis. We will then show the robustness of the proposed receiver to imperfect channel knowledge and data detection, which could be reasonably questioned about being susceptible to error propagation, compare its performance to that of relevant existing techniques, and conclude showing how the proposed detection scheme may be employed in soft-output detection for coded transmissions. The paper is structured as follows: section II presents the standard and the proposed OFDM model, section III presents detection strategies based on the proposed model, section IV reports comparison between receivers based on our proposal and other receivers with perfect and imperfect channel state knowledge and/or coded transmission. Section V draws conclusions. II. SYSTEM MODEL A. OFDM with a cyclic prefix We hereby consider a baseband OFDM modulation. The i-th data block consists of K independent and i identically distributed symbols d k[ ] which may be put in vector form as T
i i i i d[ ] = ⎡⎣ d1[ ] , d 2[ ] ,… , d K[ ] ⎤⎦ .
(1)
The symbols d k[ ] are drawn from a two-dimensional constellation (e.g. PSK or QAM) having zero mean and σ d2 variance. i
The K symbols are arranged in an N-elements vector i a[ ] as follows: T
i i i i i i a[ ] = ⎡⎣0, d1[ ] , d 2[ ] ,… , d K[ ]/ 2 , 0, 0,… , 0, d K[ ]/ 2 +1 ,… , d K[ ] ⎤⎦ . (2)
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The null terms in a[ ] correspond to the so-called “virtual subcarriers”, in agreement to standardized OFDM (see e.g. [4]). The virtual subcarriers are needed in order to limit the effects of imperfect phase synchronization.
where wm are complex-valued circular Gaussian i.i.d. random variables having zero mean and 2N 0 variance and hl are the components of
Note that vector a[ ] may be obtained from d[ ] by
h ( t ) = ∫ γ (τ ) g ( t − τ ) dτ
i
i
i
∞
(9)
−∞
a[ ] = Kd[ ] i
i
(3)
In what follows, we assume that hl ≠ 0 only for 0 ≤ l ≤ L , so that the discrete-time equivalent channel can be modeled as a Finite Impulse Response (FIR) filter with memory L.
where K is the block matrix ⎡ 01× K ⎤ ⎢I ⎥ 0 K /2 ⎥ ⎢ K /2 ⎢ 0( N − K −1)× K ⎥ ⎢ ⎥ ⎣⎢ 0 K / 2 I K / 2 ⎦⎥
K
(4)
while I n and 0n are the n-order identity and square null matrix. The vector a[ ] is processed via a normalized Inverse Discrete Fourier Transform (IDFT) obtaining the sequence i
N −1
1
∑ a[ ] e N
xn[ ] = i
k =0
i k
j
2π kn N
with respect to the chosen orthonormal basis.
,
(5)
Now, it is well known (e.g. from [2]) that if the cyclic i prefix length N P ≥ L the subsequence zm[ ] of ym zm[ ] i
with 0 ≤ m ≤ N − 1 , only depends on data symbols from the i-th block. Moreover, thanks to the cyclic prefix periodicity, the internal summation in (8) is equivalent (for the terms appearing in (10)) to a discrete-time circular convolution, so that the following identity between the normalized Discrete Fourier Transform (DFT) of the i i sequences zm[ ] and xn[ ] holds:
where ak[ ] is the k-th component of a[ ] , which is intrinsically periodic with respect to the index n. i
i
The i-th ( N + N P ) -elements OFDM symbol is then constructed by taking the sequence elements having indexes − N P ≤ n ≤ N − 1 . The cyclic prefix is the vector of elements having indexes − N P ≤ n ≤ −1 . The sequence xn[ ] is used to modulate a band-limited i
pulse g ( t ) obtaining the baseband signal v (t ) = ∑ i
N −1
∑
n =− N P
x g ( t − nT − i ( N + N P ) T ) , [i ] n
γ ( t ) = ∑ ρ vδ ( t − τν ) .
1
i
(7)
The channel is also affected by zero-mean additive white Gaussian noise with two-sided power spectral density N 0 / 2 . Assuming a receiver employing an orthonormal bandlimited basis expansion (for example Root-Raised-Cosine filtering), the following baseband discrete-time model for the received signal is obtained:
i m
m=0
ym = ∑ i
∑
n =− N P
xn[ ] hm − n − i ( N + NP ) + wm , i
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2π km N
L
H k = ∑ hl e
= H k ak[ ] + Wk[ ] , i
i
(11)
−j
2π kl N
(12)
l =0
is the unnormalized DFT of the equivalent channel impulse response and Wk[ ] =
N −1
1 N
∑ wm+i( N + NP ) e
−j
2π km N
(13)
m =0
has, by virtue of the orthonormality of the DFT, the same statistics of wm . Equation (11) leads to the simple per-subcarrier detection scheme which is one of the main reasons for the widespread adoption of OFDM in many wireless standards. B. Interference cancellation in the cyclic prefix Let us consider the subsequence pm[ ] of ym defined as i
pm[ ] i
ym + i ( N + N P )
(14)
with − N P ≤ m ≤ −1 . Owing to the hypothesis L ≤ N P , pm[ ] depends only on data symbols from the i -th and i
N −1
−j
where
(6)
ν
N −1
∑ z[ ]e N
Z k[ ] =
i
which is then transmitted over a quasi-static multipath frequency selective channel whose baseband equivalent impulse response is
(10)
ym + i ( N + N P )
(8)
( i − 1) -th
OFDM symbol.
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The following identity is easily derived from (8): pm[ ] = i
m + N P +1
∑ l =0
hl xm[ + N] P −l + i −1
m + NP
∑ h x[ ]
previous block
i m −l
l
l =0
+ wm + i ( N + NP ) . (15)
current block
Assuming that a reliable channel estimate hˆl = hl + ηl is available, where ηl is the estimation error, and symbolby-symbol Forward Error Correction (FEC) is employed, the interference term contribution from the preceding symbol can be reconstructed and canceled with a high i i reliability. Let pm[ ] denote the subsequence pm[ ] after interference cancellation: [i ] m
[i ] m
p =p −
m + N P +1
∑ l =0
hˆl xˆ
,
∑
pm =
+
(h x
[i −1]
[i −1]
− hˆl xˆm + NP − l
m + NP −l
l
l =0
m + NP
∑ hx
m −l
l
l =0
[i ]
, and the
pm[ ] = i
∑ l =0
ηl xm[i −+1N]
P
−l
+
pm =
∑ h x[ ] l =0
i
m + NP
i m −l
l
m + N P +1
∑ l =0
∑ h x[ ] l =0
l
)
(17)
+ wm + i ( N + NP ) , (18)
[i −1]
ηl xm + N
P
−l
,
(19)
i m −l
+ wm[ ] . i
(20)
C. Matrix model for the detection strategy It is useful to arrange the above defined sequences in vector form. For ease of notation, we will hereby drop the OFDM symbol index superscript [i ] since, after cancellation, the preceding symbol contribution does not appear explicitly in any equation. Let
Z
[ Z1 , Z 2 ,… , Z K / 2 , Z N − K / 2 , Z N − K / 2+1 ,…, Z N −1 ]
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l =0
l
j
2π k ( m −l ) N
+ wm ,
(23)
T
(21)
(24)
may be put in matrix form as (25)
where F ∈ NP × N is a sparse matrix whose elements are defined as follows:
Fml
⎧0 ⎪ ⎨hl + m ⎪0 ⎩
0 ≤ l < −m −m ≤ l ≤ min ( N P , L − m ) , min ( N P , L − m ) < l < N
(26)
(note that the index m spans from − N P in the first row to −1 in the last) and the matrix Φ is the normalized N-points DFT matrix whose elements are: Φ ln =
1
N
e
−j
2π ln N
.
(27)
Recalling (3) we finally obtain the following matrix model for the detection (after cancellation of interference in the cyclic prefix) of the i-th OFDM symbol:
leading to the virtually interference-free cyclic prefix term pm[ ] =
k =0
k
T
and finally, the residual term from the first summation in (18) can be taken into account as an equivalent noise term as follows: wm = wm + i ( N + NP ) +
∑a ∑ he N
p = ⎡⎣ p− NP , p− NP +1 ,… , p−1 ⎤⎦ ,
+ wm + i ( N + NP ) .
m + NP
[i ]
m + NP
N −1
1
p = FΦa + w ,
In the limit for high signal-to-noise ratios, the i −1 estimated sequence xˆn[ ] is error-free, so that eq. (17) may be simplified to m + N P +1
(22)
Taking now into account the relationship (5) between xn and ak , we get from (20) the following
(16) i −1]
i −1
m + N P +1
Z = Hd + W .
which, denoting the interference-free cyclic prefix with [i −1] m + NP −l
where xˆn[ ] are the already detected xn[ following identity results: [i ]
denote the K-elements vector collecting the data-related terms from (11), and in the same way H diag ( H k ) and W denote the channel matrix and noise vector whose elements are defined in (12) and (13) respectively, so that (11) may be rewritten as
⎡p ⎤ ⎡w⎤ ⎢ Z ⎥ = Γd + ⎢ W ⎥ , ⎣ ⎦ ⎣ ⎦
(28)
⎡ FΦK ⎤ Γ=⎢ ⎥. ⎣ H ⎦
(29)
where
We observe that the noise vectors w and W are uncorrelated since they are relative to disjoint received sequences. Note that the model (28) is the standard noisy observation vector subspace model typical in the detection of linearly modulated signals over additive Gaussian channels. We will show that, thanks to the redundancy intrinsic in an over-determined linear system with unknowns, even simple linear detection techniques exhibit a significant
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improvement both over the standard per-subcarrier OFDM detection and previously developed techniques such as the one presented in [6]. Moreover, the matrix Γ is a block matrix built from highly structured or sparse components, and its computation (if needed) is straightforward. III. DETECTION STRATEGY In this section we first describe the channel estimation strategy which was adopted in numerical simulations, then briefly discuss a maximum likelihood detection strategy which has been implemented in order to lower bound the error-rate performance of any receiver. Subsequently, we will present a linear detection strategy along with a complexity analysis, and conclude the section with a suboptimal log-likelihood computation strategy whose effectiveness will be shown by means of numerical simulations in the next section.
Multiplying by ak[ ] H both sides of (11), taking i
expectations and recalling that Wk[ ] is a zero-mean i
[i ]
random variable uncorrelated with ak , the following identity is obtained: [i ]
[i ]
ak Z k = H k ak
2
.
(30)
Substituting time averages instead of expectations, the following unbiased estimator for H k is readily obtained: i −1
i Hˆ k[ ] =
C. Least Squares Estimate We briefly discuss the effect of adding the cyclicprefix related equations to the sufficient statistic for the detection of the i-th OFDM symbol. Since the problem obtained by neglecting the noise contribution in (28) is over-determined, the Least Squares (LS) estimate for the data vector c is given by −1 −1 ⎡p ⎤ ⎡w⎤ dˆ = ( Γ H Γ ) Γ H ⎢ ⎥ = d + ( Γ H Γ ) Γ H ⎢ ⎥ , ⎣Z ⎦ ⎣W⎦
λ i ak[0] H Z k[0] + ∑ λ i − j aˆk[ j ] H Z k[ j ] 2
j =1 i −1
λ i ak[0] + ∑ λ i − j aˆk[ j ]
2
(31)
j =1
where 0 < λ ≤ 1 is a forgetting factor depending on the expected channel variability. The estimate is bootstrapped by means of a pilot symbol ( i = 0 ). For subsequent blocks, data estimates are used. Note that the transferences related to virtual (zero) subcarriers cannot be estimated since they are not observable. The time-domain estimate hˆl , needed for the i interference cancellation (16), was obtained from Hˆ [ ] k
through Least Squares estimation. Note that the timedomain channel impulse response estimate is always needed as an aid to OFDM symbol timing estimation (see e.g. [8]). B. Maximum Likelihood detection In order to show the theoretical performance gain, i.e. with unbounded complexity, of a receiver based on the proposed model, we compare in the next section the Maximum-Likelihood performance of a standard persubcarrier receiver based on the model (11) and a symbolby-symbol receiver based on the model (28). The former was implemented with a per-subcarrier Least Squares © 2010 ACADEMY PUBLISHER
(32)
so the mean squared error is given by tr dˆ − d
A. Channel estimation
[i ] H
Equalizer (LSE) followed by AWGN detection, while the latter was implemented with a Sphere Decoder (SD) using Schnorr-Euchner enumeration (see e.g. [9]) which uses a linear Least Squares estimate as a preprocessing step.
2
H ⎡ ⎤ −1 −1 ⎡w⎤⎡w⎤ = tr ⎢( Γ H Γ ) Γ H ⎢ ⎥ ⎢ ⎥ Γ ( Γ H Γ ) ⎥ (33) ⎢ ⎥ ⎣W⎦ ⎣W⎦ ⎣ ⎦
which, assuming correct detection and reliable channel estimates (so that w exhibits the same statistics of W ), simplifies to tr dˆ − d
2
= 2 N 0 tr ( Γ H Γ ) ≤ 2 N 0 tr ( H H H ) −1
−1
(34)
where the inequality is easily proven for any F in (29) with the Woodbury identity. Since the standard per-subcarrier receiver is clearly obtained from (28) by letting F = 0 , the Least Squares estimate obtained by using the proposed model must, at least in the high signal-to-noise ratio regime, be more accurate than the one obtained with the standard persubcarrier receiver. From the implementation complexity point of view, the LS estimate (32) requires for each OFDM symbol to first compute the projection ⎡p ⎤ α = ΓH ⎢ ⎥ = K H ΦH FH p + H H Z , ⎣Z ⎦
(35)
which requires K multiplications for the term H H Z , the multiplication of the sparse matrix F H with about N P2 / 2 non-zero elements with the vector p , and N P N-points IDFT’s. After the projection, the LS estimate is obtained by solving the linear system Γ H Γdˆ = α ,
(36)
whose solution, if the QR factorization of the matrix Γ = QR is available, can be computed by means of backsubstitution with K 2 complex multiplications:
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β = R − H α, dˆ = R −1β.
(37)
Assuming a quasi-static channel, the QR factorization will be computed once per received packet, so its complexity will be neglected. This analysis shows that the implementation complexity of a LS detector is O ( K 2 ) , which rules out applicability of the proposed method for applications with a high number of subcarriers, such as DVB-T and OFDMA (thousands of subcarriers). Conversely, in the small number of subcarriers application scenario such as the WLAN (64 subcarriers [4] or 128 in IEEE 802.11n Draft Proposals), the proposed method leads to attractive results which trade off with the increased detection complexity. Indeed, a noteworthy feature of our proposal is its flexibility since it is not mandatory to use all of the equations (23) in the detection process.
D. Log-Likelihood Ratios One of the advantages of the standard per-subcarrier receiver is in its suitability for simple computation of the Log-Likelihood Ratios (LLR) needed in effective softinput decoding of Forward Error Correction codes:
Lck ,i = log
Pr {ck ,i = 0 | y} Pr {ck ,i = 1 | y}
= log
( (Z
) (38) − Hˆ a )
∑
fW Z k − Hˆ k ak
∑
fW
ak :ck ,i = 0
ak :ck ,i =1
k
k
IV. PERFORMANCE ANALYSIS We present numerical simulation results since, while an exact analysis of error-rate performance for such a system is simple in the ideal channel estimation and data detection scenario, a realistic analysis is cumbersome. In order to obtain reliable performance figures, each error probability estimate has been obtained running the simulation until 10 4 independent error events occurred.
A. Comparison to previous art The method of Tarighat et al [6] exploits only partially the redundancy available in the cyclic prefix. It is based on using the part of the cyclic prefix which is not affected by interference from the previous block. Its strength lies in the surprisingly simple detection strategy arising from the circulant nature of the matrix involved in their model. Our approach mainly differs from the one used in [6] since we obtain a virtually interference-free cyclic prefix by means of interference cancellation and can thus fully take advantage from its redundancy. From a practical point of view, their approach suffers from the nature of the discrete-time channel model obtained by sampling the continuous-time impulse response (9), since in the case of band-limited signals the number of non-zero terms is (in rigorous terms) infinite and is approximated with a FIR filter response. 0
10
k
where ck ,i is the i-th coded bit transmitted on the k-th of the noise samples in (11). The computation (38) is simple because of the orthogonality of the equations (11). On the contrary, the proposed model (28) describes a non-orthogonal relationship between equations, so that a simple equation such as (38) is not available. We propose a sub-optimal approach for the computation of LLR’s, based on the output from the Least Squares estimator (32)
Lck ,i ≅ log
∑
fW ′ ( aˆk − ak )
∑
fW ′ ( aˆk − ak )
ak :ck ,i = 0
ak :ck ,i =1
Frame Error Rate
subcarrier and fW ( w ) is the probability density function -1
10
per subcarrier ML MMSE (from [6]) block MMSE (x16) block MMSE (x12) block MMSE (x8) block MMSE (x4)
-2
10
5
(39)
where fW ′ ( w ) is the (still Gaussian) probability density function of the filtered noise having variance −1 −1 2 Wk′ = 2 N 0 ⎡⎢( Γ H Γ ) ⎤⎥ = 2 N 0 ⎡⎢( R H R ) ⎤⎥ . (40) ⎣ ⎦ kk ⎣ ⎦ kk
These variances may also be computed once per received packet, so the complexity associated to their computation will be neglected.
10 15 20 Signal to Noise Ratio (Eb/N0, dB)
25
Figure 1. Minimum Mean Squared Error receivers based on the proposed model for a variable number of redundancy equations (23). Performance of the MMSE receiver from [6] is also shown.
We report in figure 1 the frame-error-rate performance of a Minimum Mean Squared Error (MMSE) receiver based on the proposed model for a variable number of equations when the channel has a flat power-delay profile and memory L = 8. The parameters used in the simulations reported in this and the next subsection are N = 64, K = 52, N P = 16 and uncoded QPSK modulation.
As expected from the above argument, it is seen that our proposal and [6] align when the number of used
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equations equals the number of interference-free terms in the cyclic prefix, while when employing an higher number of equations, our proposal outperforms the previously existing technique.
0
10
0
Frame Error Rate
Frame Error Rate
10
-1
10
-2
10
per-subcarrier block-LS block-ML 5
L=4 L=8 L=16 per-subcarrier ML MMSE (from [6]) block-MMSE
-2
10
5
10 15 20 Signal to Noise Ratio (Eb/N0, dB)
10 15 20 Signal to Noise Ratio (Eb/N0, dB)
25
Figure 4. Detectors of figure 3 with imperfect channel knowledge.
25
Figure 2. Prior existing and proposed receivers for various channel memory lengths (flat power-delay profile). Downward arrows denote the proposed detector, upward arrows denote the standard and [6] receivers.
The distinguishing feature of our proposal is the fact that the error-rate performance is enhanced as the channel selectivity increases, i.e. when the fraction of cyclic prefix free from interference is reduced. Conversely, when the channel memory reaches the size of the cyclic prefix the receiver of Tarighat et al degenerates to the standard persubcarrier receiver and no gain is observed. This shows that our detection scheme is substantially different from the one by Tarighat et al, as reported in figure 2.
B. Maximum Likelihood detection Figures 3 and 4 show the error-rate performance of a Maximum Likelihood receiver (‘block-ML’ in the following) based on the model (28) which was implemented by means of Sphere decoding (see e.g. [9]) with an enumeration strategy that uses the Least Squares estimate (32) as starting point.
Figure 3 is relative to the performance of the standard per-subcarrier detector, the Least Squares detector (32) and the high complexity (roughly O ( K 3 ) for high signalto-noise ratios) block-ML. Channel memory was set to L = 16 with a flat power-delay profile. This figure shows that a significant gain in error-rate is already achieved by a linear receiver, and only a marginal gain is obtained by employing the higher complexity block-ML. In order to show the performance in a more realistic scenario, we also ran simulations without assuming perfect channel knowledge. Figure 4 reports the error-rate performance obtained with imperfect channel estimation.
0
10
Each simulation run consisted in transmission of five OFDM symbols over a randomly generated static channel ( λ = 1 ), and the error-rates were evaluated on the fifth block, in order to take into account the effects of error propagation on channel estimation, interference cancellation and detection. Figure 4 shows how a significant performance gain is still observed even with imperfect channel knowledge and uncoded transmission, so we feel safe in asserting that error propagation effects have a negligible impact on the performance of the proposed detection strategy.
-1
10 Frame Error Rate
-1
10
-2
10
per-subcarrier block-LS block-ML
-3
10
5
10 15 20 Signal to Noise Ratio (Eb/N0, dB)
25
Figure 3. Maximum-Likelihood (ML) detection frame error rates for a receiver based on the per-subcarrier standard model (11) and a block-by-block receiver based on (28).
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V. CONCLUSIONS
0
10
We presented a novel detection architecture for OFDM with a cyclic prefix which is based on cancellation of interference arising in the cyclic prefix itself. In order to demonstrate our proposal, we described a linear detection strategy and reported its performance in the case of perfect and imperfect channel knowledge and both uncoded and coded transmission.
-1
Frame Error Rate
10
-2
The error-rate performance of the proposed receiver favorably compares to prior existing receiver structures and its complexity is limited and scalable by simply using less redundancy than available. Extension to MultipleInput/Multiple-Output channels is trivial and not detailed here.
10
-3
10
-4
10
5
per-subcarrier (Hard) block LS (Hard) per-subcarrier (Soft) block LS (Soft) 10 15 Signal to Noise Ratio (Eb/N0, dB)
20
Figure 5. Coded OFDM receivers based on the standard per-subcarrier and proposed model.The Forward Error Correction code is a parallel concatenated turbo code with nominal rate 7/8 and a random interleaver.
C. Coded transmissions We conclude this section reporting in figure 5 the error-rate performance of the proposed detection scheme versus the standard detector in the coded OFDM scenario. We used a parallel concatenated convolutional turbo-code (for a description of turbo-codes see e.g. [10]) having rate 7/8 and randomly generated interleaver. The parameters of the system are (as with the other reported simulation results) N = 64, K = 52, N P = 16, QPSK modulation and channel memory L = 16 . We used the approximated softoutput detection strategy (39), which is based on the output from the block-LS linear detector (32).
As evident from inspection of figure 5, the performance of soft- and hard-decision decoding for a receiver based on our proposal favorably compares to the performance of the standard per-subcarrier receiver. A significant gain in signal-to-noise ratio is observed for any target frame error rate.
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REFERENCES [1]
R. W. Chang, “Synthesis of band-limited orthogonal signals for multi-channel data transmission”, Bell System Technical Journal 46, pp. 1775-1796, 1966. [2] R. van Nee, R. Prasad, OFDM for Wireless Multimedia Communications, Artech House Publishers, 1999. [3] J. G. Proakis, D. K. Manolakis, Digital Signal Processing, Prentice Hall, 2007. [4] IEEE 802.11-2007 “IEEE Standard for Information technology Telecommunications and information exchange between systems Local and metropolitan area networks - Specific requirements Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications”, June 12, 2007. [5] 3GPP TSG-RAN E-UTRA Technical Specification 36.211 “Physical Channels and Modulation”, http://www.3gpp.org/ [6] A. Tarighat, A. H. Sayed, “An Optimum OFDM Receiver Exploiting Cyclic Prefix for Improved Data Estimation”, Proceedings of ICASSP 2003, Vol. IV, pp. 217-220. [7] G. Garbo, S. Mangione, “An OFDM Receiver Exploiting Multipath Diversity”, MIC-CCA 2008, pp. 65-70, 8-10 August 2008. [8] J. Liu, J. Li, “Parameter Estimation and Error Reduction for OFDM-Based WLANs,” IEEE Transactions on Mobile Computing, Vol. 3, No. 2, pp. 152-163, April-June 2004. [9] E. Agrell, T. Eriksson, E. Vardy and K. Zeger, “Closest Point Search in Lattices,” IEEE Transactions on Information Theory, Vol. 48, No. 8, pp. 2201-2214, August 2002. [10] C. Berrou, A. Glavieux, P. Thitimajshima, “Near Shannon-Limit Error-Correcting Coding and Decoding: Turbo-Codes,” Proceedings of IEEE ICC93, Vol. 2, pp.1064-1070, 23-26 May 1993.
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An Improved Detection Technique for Cyclic-Prefixed OFDM Giovanni Garbo, Stefano Mangione Dipartimento di Ingegneria Elettrica, Elettronica e delle Telecomunicazioni (DIEET) Università degli Studi di Palermo Palermo, Italy Email: {giovanni.garbo, stefano.mangione}@tti.unipa.it
Abstract—A novel Orthogonal Frequency Division Multiplexing detection technique compatible to standard (e.g. Wireless LAN) transmitters is proposed. It features enhanced error-rate performance with flexible computational complexity and robustness to imperfect channel estimation. It is based on exploitation of the redundancy available in the cyclic prefix after cancellation of interference from the preceding block. In order to show the effectiveness of our proposal, an analysis of computational complexity and a number of comparisons to the standard per-subcarrier receiver and a previously existing method in terms of error rates are reported. Index terms—Orthogonal Frequency Division Multiplexing; Frequency-selective channels; Maximum-Likelihood Detection; Linear Detection; Interference Cancellation
I. INTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) [1, 2] is a signaling technique particularly suited for transmission over linear distortion channels. The key feature of OFDM is the fact that both the transmitter and the receiver can be based on the well-known and efficient Fast Fourier Transform (FFT) [3]. The simplicity of the FFT-based receiver structure has led to its inclusion in many existing and future wireless standards, such as IEEE 802.11 [4] and 3GPP LTE [5]. It is well known that a cyclic prefix is needed in the OFDM signal in order to preserve the orthogonality between the subcarriers. The cyclic prefix reveals its usefulness also for synchronization purposes. Much less investigated is the use of the cyclic prefix as an aid to enhance data detection by virtue of its intrinsic redundancy. To the best of our knowledge, a technique present in the literature capable of exploiting part of the redundancy of the cyclic prefix has been presented in [6]. It starts from the assumption that the cyclic prefix itself is occasionally over-dimensioned with respect to the actual channel memory, and it consists in using the cyclic prefix segment not affected by interference from the preceding block in order to obtain an over determined system of equations. The mathematical approach presented in [6] cleverly exploits the circulant nature of the equivalent channel matrix and results in a simple detection scheme which, unfortunately, gives rise to a small performance gain.
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A preliminary version of this work was presented in [7], where we suggested to cancel the interference arising in the cyclic prefix from the preceding block in order to take advantage of all its redundancy. In this work we will first report, for the sake of completeness, the derivation of the model described in [7], followed by a detailed description of a linear detection strategy and a computational complexity analysis. We will then show the robustness of the proposed receiver to imperfect channel knowledge and data detection, which could be reasonably questioned about being susceptible to error propagation, compare its performance to that of relevant existing techniques, and conclude showing how the proposed detection scheme may be employed in soft-output detection for coded transmissions. The paper is structured as follows: section II presents the standard and the proposed OFDM model, section III presents detection strategies based on the proposed model, section IV reports comparison between receivers based on our proposal and other receivers with perfect and imperfect channel state knowledge and/or coded transmission. Section V draws conclusions. II. SYSTEM MODEL A. OFDM with a cyclic prefix We hereby consider a baseband OFDM modulation. The i-th data block consists of K independent and i identically distributed symbols d k[ ] which may be put in vector form as T
i i i i d[ ] = ⎡⎣ d1[ ] , d 2[ ] ,… , d K[ ] ⎤⎦ .
(1)
The symbols d k[ ] are drawn from a two-dimensional constellation (e.g. PSK or QAM) having zero mean and σ d2 variance. i
The K symbols are arranged in an N-elements vector i a[ ] as follows: T
i i i i i i a[ ] = ⎡⎣0, d1[ ] , d 2[ ] ,… , d K[ ]/ 2 , 0, 0,… , 0, d K[ ]/ 2 +1 ,… , d K[ ] ⎤⎦ . (2)
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The null terms in a[ ] correspond to the so-called “virtual subcarriers”, in agreement to standardized OFDM (see e.g. [4]). The virtual subcarriers are needed in order to limit the effects of imperfect phase synchronization.
where wm are complex-valued circular Gaussian i.i.d. random variables having zero mean and 2N 0 variance and hl are the components of
Note that vector a[ ] may be obtained from d[ ] by
h ( t ) = ∫ γ (τ ) g ( t − τ ) dτ
i
i
i
∞
(9)
−∞
a[ ] = Kd[ ] i
i
(3)
In what follows, we assume that hl ≠ 0 only for 0 ≤ l ≤ L , so that the discrete-time equivalent channel can be modeled as a Finite Impulse Response (FIR) filter with memory L.
where K is the block matrix ⎡ 01× K ⎤ ⎢I ⎥ 0 K /2 ⎥ ⎢ K /2 ⎢ 0( N − K −1)× K ⎥ ⎢ ⎥ ⎣⎢ 0 K / 2 I K / 2 ⎦⎥
K
(4)
while I n and 0n are the n-order identity and square null matrix. The vector a[ ] is processed via a normalized Inverse Discrete Fourier Transform (IDFT) obtaining the sequence i
N −1
1
∑ a[ ] e N
xn[ ] = i
k =0
i k
j
2π kn N
with respect to the chosen orthonormal basis.
,
(5)
Now, it is well known (e.g. from [2]) that if the cyclic i prefix length N P ≥ L the subsequence zm[ ] of ym zm[ ] i
with 0 ≤ m ≤ N − 1 , only depends on data symbols from the i-th block. Moreover, thanks to the cyclic prefix periodicity, the internal summation in (8) is equivalent (for the terms appearing in (10)) to a discrete-time circular convolution, so that the following identity between the normalized Discrete Fourier Transform (DFT) of the i i sequences zm[ ] and xn[ ] holds:
where ak[ ] is the k-th component of a[ ] , which is intrinsically periodic with respect to the index n. i
i
The i-th ( N + N P ) -elements OFDM symbol is then constructed by taking the sequence elements having indexes − N P ≤ n ≤ N − 1 . The cyclic prefix is the vector of elements having indexes − N P ≤ n ≤ −1 . The sequence xn[ ] is used to modulate a band-limited i
pulse g ( t ) obtaining the baseband signal v (t ) = ∑ i
N −1
∑
n =− N P
x g ( t − nT − i ( N + N P ) T ) , [i ] n
γ ( t ) = ∑ ρ vδ ( t − τν ) .
1
i
(7)
The channel is also affected by zero-mean additive white Gaussian noise with two-sided power spectral density N 0 / 2 . Assuming a receiver employing an orthonormal bandlimited basis expansion (for example Root-Raised-Cosine filtering), the following baseband discrete-time model for the received signal is obtained:
i m
m=0
ym = ∑ i
∑
n =− N P
xn[ ] hm − n − i ( N + NP ) + wm , i
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2π km N
L
H k = ∑ hl e
= H k ak[ ] + Wk[ ] , i
i
(11)
−j
2π kl N
(12)
l =0
is the unnormalized DFT of the equivalent channel impulse response and Wk[ ] =
N −1
1 N
∑ wm+i( N + NP ) e
−j
2π km N
(13)
m =0
has, by virtue of the orthonormality of the DFT, the same statistics of wm . Equation (11) leads to the simple per-subcarrier detection scheme which is one of the main reasons for the widespread adoption of OFDM in many wireless standards. B. Interference cancellation in the cyclic prefix Let us consider the subsequence pm[ ] of ym defined as i
pm[ ] i
ym + i ( N + N P )
(14)
with − N P ≤ m ≤ −1 . Owing to the hypothesis L ≤ N P , pm[ ] depends only on data symbols from the i -th and i
N −1
−j
where
(6)
ν
N −1
∑ z[ ]e N
Z k[ ] =
i
which is then transmitted over a quasi-static multipath frequency selective channel whose baseband equivalent impulse response is
(10)
ym + i ( N + N P )
(8)
( i − 1) -th
OFDM symbol.
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The following identity is easily derived from (8): pm[ ] = i
m + N P +1
∑ l =0
hl xm[ + N] P −l + i −1
m + NP
∑ h x[ ]
previous block
i m −l
l
l =0
+ wm + i ( N + NP ) . (15)
current block
Assuming that a reliable channel estimate hˆl = hl + ηl is available, where ηl is the estimation error, and symbolby-symbol Forward Error Correction (FEC) is employed, the interference term contribution from the preceding symbol can be reconstructed and canceled with a high i i reliability. Let pm[ ] denote the subsequence pm[ ] after interference cancellation: [i ] m
[i ] m
p =p −
m + N P +1
∑ l =0
hˆl xˆ
,
∑
pm =
+
(h x
[i −1]
[i −1]
− hˆl xˆm + NP − l
m + NP −l
l
l =0
m + NP
∑ hx
m −l
l
l =0
[i ]
, and the
pm[ ] = i
∑ l =0
ηl xm[i −+1N]
P
−l
+
pm =
∑ h x[ ] l =0
i
m + NP
i m −l
l
m + N P +1
∑ l =0
∑ h x[ ] l =0
l
)
(17)
+ wm + i ( N + NP ) , (18)
[i −1]
ηl xm + N
P
−l
,
(19)
i m −l
+ wm[ ] . i
(20)
C. Matrix model for the detection strategy It is useful to arrange the above defined sequences in vector form. For ease of notation, we will hereby drop the OFDM symbol index superscript [i ] since, after cancellation, the preceding symbol contribution does not appear explicitly in any equation. Let
Z
[ Z1 , Z 2 ,… , Z K / 2 , Z N − K / 2 , Z N − K / 2+1 ,…, Z N −1 ]
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l =0
l
j
2π k ( m −l ) N
+ wm ,
(23)
T
(21)
(24)
may be put in matrix form as (25)
where F ∈ NP × N is a sparse matrix whose elements are defined as follows:
Fml
⎧0 ⎪ ⎨hl + m ⎪0 ⎩
0 ≤ l < −m −m ≤ l ≤ min ( N P , L − m ) , min ( N P , L − m ) < l < N
(26)
(note that the index m spans from − N P in the first row to −1 in the last) and the matrix Φ is the normalized N-points DFT matrix whose elements are: Φ ln =
1
N
e
−j
2π ln N
.
(27)
Recalling (3) we finally obtain the following matrix model for the detection (after cancellation of interference in the cyclic prefix) of the i-th OFDM symbol:
leading to the virtually interference-free cyclic prefix term pm[ ] =
k =0
k
T
and finally, the residual term from the first summation in (18) can be taken into account as an equivalent noise term as follows: wm = wm + i ( N + NP ) +
∑a ∑ he N
p = ⎡⎣ p− NP , p− NP +1 ,… , p−1 ⎤⎦ ,
+ wm + i ( N + NP ) .
m + NP
[i ]
m + NP
N −1
1
p = FΦa + w ,
In the limit for high signal-to-noise ratios, the i −1 estimated sequence xˆn[ ] is error-free, so that eq. (17) may be simplified to m + N P +1
(22)
Taking now into account the relationship (5) between xn and ak , we get from (20) the following
(16) i −1]
i −1
m + N P +1
Z = Hd + W .
which, denoting the interference-free cyclic prefix with [i −1] m + NP −l
where xˆn[ ] are the already detected xn[ following identity results: [i ]
denote the K-elements vector collecting the data-related terms from (11), and in the same way H diag ( H k ) and W denote the channel matrix and noise vector whose elements are defined in (12) and (13) respectively, so that (11) may be rewritten as
⎡p ⎤ ⎡w⎤ ⎢ Z ⎥ = Γd + ⎢ W ⎥ , ⎣ ⎦ ⎣ ⎦
(28)
⎡ FΦK ⎤ Γ=⎢ ⎥. ⎣ H ⎦
(29)
where
We observe that the noise vectors w and W are uncorrelated since they are relative to disjoint received sequences. Note that the model (28) is the standard noisy observation vector subspace model typical in the detection of linearly modulated signals over additive Gaussian channels. We will show that, thanks to the redundancy intrinsic in an over-determined linear system with unknowns, even simple linear detection techniques exhibit a significant
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improvement both over the standard per-subcarrier OFDM detection and previously developed techniques such as the one presented in [6]. Moreover, the matrix Γ is a block matrix built from highly structured or sparse components, and its computation (if needed) is straightforward. III. DETECTION STRATEGY In this section we first describe the channel estimation strategy which was adopted in numerical simulations, then briefly discuss a maximum likelihood detection strategy which has been implemented in order to lower bound the error-rate performance of any receiver. Subsequently, we will present a linear detection strategy along with a complexity analysis, and conclude the section with a suboptimal log-likelihood computation strategy whose effectiveness will be shown by means of numerical simulations in the next section.
Multiplying by ak[ ] H both sides of (11), taking i
expectations and recalling that Wk[ ] is a zero-mean i
[i ]
random variable uncorrelated with ak , the following identity is obtained: [i ]
[i ]
ak Z k = H k ak
2
.
(30)
Substituting time averages instead of expectations, the following unbiased estimator for H k is readily obtained: i −1
i Hˆ k[ ] =
C. Least Squares Estimate We briefly discuss the effect of adding the cyclicprefix related equations to the sufficient statistic for the detection of the i-th OFDM symbol. Since the problem obtained by neglecting the noise contribution in (28) is over-determined, the Least Squares (LS) estimate for the data vector c is given by −1 −1 ⎡p ⎤ ⎡w⎤ dˆ = ( Γ H Γ ) Γ H ⎢ ⎥ = d + ( Γ H Γ ) Γ H ⎢ ⎥ , ⎣Z ⎦ ⎣W⎦
λ i ak[0] H Z k[0] + ∑ λ i − j aˆk[ j ] H Z k[ j ] 2
j =1 i −1
λ i ak[0] + ∑ λ i − j aˆk[ j ]
2
(31)
j =1
where 0 < λ ≤ 1 is a forgetting factor depending on the expected channel variability. The estimate is bootstrapped by means of a pilot symbol ( i = 0 ). For subsequent blocks, data estimates are used. Note that the transferences related to virtual (zero) subcarriers cannot be estimated since they are not observable. The time-domain estimate hˆl , needed for the i interference cancellation (16), was obtained from Hˆ [ ] k
through Least Squares estimation. Note that the timedomain channel impulse response estimate is always needed as an aid to OFDM symbol timing estimation (see e.g. [8]). B. Maximum Likelihood detection In order to show the theoretical performance gain, i.e. with unbounded complexity, of a receiver based on the proposed model, we compare in the next section the Maximum-Likelihood performance of a standard persubcarrier receiver based on the model (11) and a symbolby-symbol receiver based on the model (28). The former was implemented with a per-subcarrier Least Squares © 2010 ACADEMY PUBLISHER
(32)
so the mean squared error is given by tr dˆ − d
A. Channel estimation
[i ] H
Equalizer (LSE) followed by AWGN detection, while the latter was implemented with a Sphere Decoder (SD) using Schnorr-Euchner enumeration (see e.g. [9]) which uses a linear Least Squares estimate as a preprocessing step.
2
H ⎡ ⎤ −1 −1 ⎡w⎤⎡w⎤ = tr ⎢( Γ H Γ ) Γ H ⎢ ⎥ ⎢ ⎥ Γ ( Γ H Γ ) ⎥ (33) ⎢ ⎥ ⎣W⎦ ⎣W⎦ ⎣ ⎦
which, assuming correct detection and reliable channel estimates (so that w exhibits the same statistics of W ), simplifies to tr dˆ − d
2
= 2 N 0 tr ( Γ H Γ ) ≤ 2 N 0 tr ( H H H ) −1
−1
(34)
where the inequality is easily proven for any F in (29) with the Woodbury identity. Since the standard per-subcarrier receiver is clearly obtained from (28) by letting F = 0 , the Least Squares estimate obtained by using the proposed model must, at least in the high signal-to-noise ratio regime, be more accurate than the one obtained with the standard persubcarrier receiver. From the implementation complexity point of view, the LS estimate (32) requires for each OFDM symbol to first compute the projection ⎡p ⎤ α = ΓH ⎢ ⎥ = K H ΦH FH p + H H Z , ⎣Z ⎦
(35)
which requires K multiplications for the term H H Z , the multiplication of the sparse matrix F H with about N P2 / 2 non-zero elements with the vector p , and N P N-points IDFT’s. After the projection, the LS estimate is obtained by solving the linear system Γ H Γdˆ = α ,
(36)
whose solution, if the QR factorization of the matrix Γ = QR is available, can be computed by means of backsubstitution with K 2 complex multiplications:
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β = R − H α, dˆ = R −1β.
(37)
Assuming a quasi-static channel, the QR factorization will be computed once per received packet, so its complexity will be neglected. This analysis shows that the implementation complexity of a LS detector is O ( K 2 ) , which rules out applicability of the proposed method for applications with a high number of subcarriers, such as DVB-T and OFDMA (thousands of subcarriers). Conversely, in the small number of subcarriers application scenario such as the WLAN (64 subcarriers [4] or 128 in IEEE 802.11n Draft Proposals), the proposed method leads to attractive results which trade off with the increased detection complexity. Indeed, a noteworthy feature of our proposal is its flexibility since it is not mandatory to use all of the equations (23) in the detection process.
D. Log-Likelihood Ratios One of the advantages of the standard per-subcarrier receiver is in its suitability for simple computation of the Log-Likelihood Ratios (LLR) needed in effective softinput decoding of Forward Error Correction codes:
Lck ,i = log
Pr {ck ,i = 0 | y} Pr {ck ,i = 1 | y}
= log
( (Z
) (38) − Hˆ a )
∑
fW Z k − Hˆ k ak
∑
fW
ak :ck ,i = 0
ak :ck ,i =1
k
k
IV. PERFORMANCE ANALYSIS We present numerical simulation results since, while an exact analysis of error-rate performance for such a system is simple in the ideal channel estimation and data detection scenario, a realistic analysis is cumbersome. In order to obtain reliable performance figures, each error probability estimate has been obtained running the simulation until 10 4 independent error events occurred.
A. Comparison to previous art The method of Tarighat et al [6] exploits only partially the redundancy available in the cyclic prefix. It is based on using the part of the cyclic prefix which is not affected by interference from the previous block. Its strength lies in the surprisingly simple detection strategy arising from the circulant nature of the matrix involved in their model. Our approach mainly differs from the one used in [6] since we obtain a virtually interference-free cyclic prefix by means of interference cancellation and can thus fully take advantage from its redundancy. From a practical point of view, their approach suffers from the nature of the discrete-time channel model obtained by sampling the continuous-time impulse response (9), since in the case of band-limited signals the number of non-zero terms is (in rigorous terms) infinite and is approximated with a FIR filter response. 0
10
k
where ck ,i is the i-th coded bit transmitted on the k-th of the noise samples in (11). The computation (38) is simple because of the orthogonality of the equations (11). On the contrary, the proposed model (28) describes a non-orthogonal relationship between equations, so that a simple equation such as (38) is not available. We propose a sub-optimal approach for the computation of LLR’s, based on the output from the Least Squares estimator (32)
Lck ,i ≅ log
∑
fW ′ ( aˆk − ak )
∑
fW ′ ( aˆk − ak )
ak :ck ,i = 0
ak :ck ,i =1
Frame Error Rate
subcarrier and fW ( w ) is the probability density function -1
10
per subcarrier ML MMSE (from [6]) block MMSE (x16) block MMSE (x12) block MMSE (x8) block MMSE (x4)
-2
10
5
(39)
where fW ′ ( w ) is the (still Gaussian) probability density function of the filtered noise having variance −1 −1 2 Wk′ = 2 N 0 ⎡⎢( Γ H Γ ) ⎤⎥ = 2 N 0 ⎡⎢( R H R ) ⎤⎥ . (40) ⎣ ⎦ kk ⎣ ⎦ kk
These variances may also be computed once per received packet, so the complexity associated to their computation will be neglected.
10 15 20 Signal to Noise Ratio (Eb/N0, dB)
25
Figure 1. Minimum Mean Squared Error receivers based on the proposed model for a variable number of redundancy equations (23). Performance of the MMSE receiver from [6] is also shown.
We report in figure 1 the frame-error-rate performance of a Minimum Mean Squared Error (MMSE) receiver based on the proposed model for a variable number of equations when the channel has a flat power-delay profile and memory L = 8. The parameters used in the simulations reported in this and the next subsection are N = 64, K = 52, N P = 16 and uncoded QPSK modulation.
As expected from the above argument, it is seen that our proposal and [6] align when the number of used
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equations equals the number of interference-free terms in the cyclic prefix, while when employing an higher number of equations, our proposal outperforms the previously existing technique.
0
10
0
Frame Error Rate
Frame Error Rate
10
-1
10
-2
10
per-subcarrier block-LS block-ML 5
L=4 L=8 L=16 per-subcarrier ML MMSE (from [6]) block-MMSE
-2
10
5
10 15 20 Signal to Noise Ratio (Eb/N0, dB)
10 15 20 Signal to Noise Ratio (Eb/N0, dB)
25
Figure 4. Detectors of figure 3 with imperfect channel knowledge.
25
Figure 2. Prior existing and proposed receivers for various channel memory lengths (flat power-delay profile). Downward arrows denote the proposed detector, upward arrows denote the standard and [6] receivers.
The distinguishing feature of our proposal is the fact that the error-rate performance is enhanced as the channel selectivity increases, i.e. when the fraction of cyclic prefix free from interference is reduced. Conversely, when the channel memory reaches the size of the cyclic prefix the receiver of Tarighat et al degenerates to the standard persubcarrier receiver and no gain is observed. This shows that our detection scheme is substantially different from the one by Tarighat et al, as reported in figure 2.
B. Maximum Likelihood detection Figures 3 and 4 show the error-rate performance of a Maximum Likelihood receiver (‘block-ML’ in the following) based on the model (28) which was implemented by means of Sphere decoding (see e.g. [9]) with an enumeration strategy that uses the Least Squares estimate (32) as starting point.
Figure 3 is relative to the performance of the standard per-subcarrier detector, the Least Squares detector (32) and the high complexity (roughly O ( K 3 ) for high signalto-noise ratios) block-ML. Channel memory was set to L = 16 with a flat power-delay profile. This figure shows that a significant gain in error-rate is already achieved by a linear receiver, and only a marginal gain is obtained by employing the higher complexity block-ML. In order to show the performance in a more realistic scenario, we also ran simulations without assuming perfect channel knowledge. Figure 4 reports the error-rate performance obtained with imperfect channel estimation.
0
10
Each simulation run consisted in transmission of five OFDM symbols over a randomly generated static channel ( λ = 1 ), and the error-rates were evaluated on the fifth block, in order to take into account the effects of error propagation on channel estimation, interference cancellation and detection. Figure 4 shows how a significant performance gain is still observed even with imperfect channel knowledge and uncoded transmission, so we feel safe in asserting that error propagation effects have a negligible impact on the performance of the proposed detection strategy.
-1
10 Frame Error Rate
-1
10
-2
10
per-subcarrier block-LS block-ML
-3
10
5
10 15 20 Signal to Noise Ratio (Eb/N0, dB)
25
Figure 3. Maximum-Likelihood (ML) detection frame error rates for a receiver based on the per-subcarrier standard model (11) and a block-by-block receiver based on (28).
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V. CONCLUSIONS
0
10
We presented a novel detection architecture for OFDM with a cyclic prefix which is based on cancellation of interference arising in the cyclic prefix itself. In order to demonstrate our proposal, we described a linear detection strategy and reported its performance in the case of perfect and imperfect channel knowledge and both uncoded and coded transmission.
-1
Frame Error Rate
10
-2
The error-rate performance of the proposed receiver favorably compares to prior existing receiver structures and its complexity is limited and scalable by simply using less redundancy than available. Extension to MultipleInput/Multiple-Output channels is trivial and not detailed here.
10
-3
10
-4
10
5
per-subcarrier (Hard) block LS (Hard) per-subcarrier (Soft) block LS (Soft) 10 15 Signal to Noise Ratio (Eb/N0, dB)
20
Figure 5. Coded OFDM receivers based on the standard per-subcarrier and proposed model.The Forward Error Correction code is a parallel concatenated turbo code with nominal rate 7/8 and a random interleaver.
C. Coded transmissions We conclude this section reporting in figure 5 the error-rate performance of the proposed detection scheme versus the standard detector in the coded OFDM scenario. We used a parallel concatenated convolutional turbo-code (for a description of turbo-codes see e.g. [10]) having rate 7/8 and randomly generated interleaver. The parameters of the system are (as with the other reported simulation results) N = 64, K = 52, N P = 16, QPSK modulation and channel memory L = 16 . We used the approximated softoutput detection strategy (39), which is based on the output from the block-LS linear detector (32).
As evident from inspection of figure 5, the performance of soft- and hard-decision decoding for a receiver based on our proposal favorably compares to the performance of the standard per-subcarrier receiver. A significant gain in signal-to-noise ratio is observed for any target frame error rate.
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REFERENCES [1]
R. W. Chang, “Synthesis of band-limited orthogonal signals for multi-channel data transmission”, Bell System Technical Journal 46, pp. 1775-1796, 1966. [2] R. van Nee, R. Prasad, OFDM for Wireless Multimedia Communications, Artech House Publishers, 1999. [3] J. G. Proakis, D. K. Manolakis, Digital Signal Processing, Prentice Hall, 2007. [4] IEEE 802.11-2007 “IEEE Standard for Information technology Telecommunications and information exchange between systems Local and metropolitan area networks - Specific requirements Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications”, June 12, 2007. [5] 3GPP TSG-RAN E-UTRA Technical Specification 36.211 “Physical Channels and Modulation”, http://www.3gpp.org/ [6] A. Tarighat, A. H. Sayed, “An Optimum OFDM Receiver Exploiting Cyclic Prefix for Improved Data Estimation”, Proceedings of ICASSP 2003, Vol. IV, pp. 217-220. [7] G. Garbo, S. Mangione, “An OFDM Receiver Exploiting Multipath Diversity”, MIC-CCA 2008, pp. 65-70, 8-10 August 2008. [8] J. Liu, J. Li, “Parameter Estimation and Error Reduction for OFDM-Based WLANs,” IEEE Transactions on Mobile Computing, Vol. 3, No. 2, pp. 152-163, April-June 2004. [9] E. Agrell, T. Eriksson, E. Vardy and K. Zeger, “Closest Point Search in Lattices,” IEEE Transactions on Information Theory, Vol. 48, No. 8, pp. 2201-2214, August 2002. [10] C. Berrou, A. Glavieux, P. Thitimajshima, “Near Shannon-Limit Error-Correcting Coding and Decoding: Turbo-Codes,” Proceedings of IEEE ICC93, Vol. 2, pp.1064-1070, 23-26 May 1993.
