Signal Detection for OFDM/OQAM System Using Cyclostationary Signatures Haijian ZHANG, Didier LE RUYET, Michel TERRE Electronics and Communications Laboratory, CNAM Paris, France
[email protected],
[email protected],
[email protected]
Abstract—The purpose of this paper is to analyze Orthogonal Frequency Division Multiplexing carrying Offset QAM (OFDM/OQAM) signal detection using cyclostationary signatures, which are artificially embedded into digital modulated signals. The spectral correlation characterization of OFDM/OQAM signal can be described by a special Linear Periodic Time-Variant (LPTV) system. Using this description we have derived the explicit theoretical formulas of Cyclic Autocorrelation Function (CAF) and Spectral Correlation Function (SCF) for OQAM signal. A low-complexity cyclostationary signature detector can be utilized for OQAM signal detection, and experimental results show the advantages of this signature detector. 1
I. I NTRODUCTION Multicarrier modulation techniques offer some advantages compared to single carrier modulation techniques because of their high data rate and channel distortion resistance. Much of the attention in the present literature emphasizes on the use of conventional Orthogonal Frequency Division Multiplexing. However, OFDM system sacrifices data transmission rate because of the insertion of a cyclic prefix (CP). Besides, FFT frame misalignment causes serious spectrum leakage problem. For this purpose, an efficient OFDM scheme based on offset quadrature amplitude modulation (OFDM/OQAM) system has been developed [1]∼[4]. OFDM/OQAM in [1] can achieve smaller intersymbol interference (ISI) and interchannel interference (ICI) without using the cyclic prefix by utilizing well designed pulse shapes that satisfy the perfect reconstruction conditions. Saltzberg in [2] showed that by designing a transmit pulse-shape in a multichannel QAM system, and by introducing a half symbol space delay between the in-phase and quadrature components of QAM symbols, it is possible to achieve a baud-rate spacing between adjacent subcarrier channels and still recover the information symbol free of ISI and ICI. Further progress was made by Hirosaki [3], who showed that the transmitter and receiver part of this modulation method could be implemented efficiently in a polyphase Discrete Fourier Transform (DFT) structure. Some new developments about OFDM/OQAM system can be found in [4]. A particularly convenient method for calculating the CAF and SCF for many types of modulated signals is to model the signal as a purely stationary waveform transformed by a Linear Periodically Time-Variant (LPTV) transformation [5], 1 Part
of this work has been supported by PHYDYAS UE FP7 project.
978-1-4244-2644-7/08/$25.00 © 2008 IEEE
in this case explicit formulas of the CAF and SCF for various types of single carrier modulated signals are derived in [6][7]. OFDM/OQAM signal as a type of multicarrier modulated signal can be regarded as a special model with the multiinput transformed by LPTV transformation and one scalar output. By modeling OQAM signal into a LPTV system it is convenient to analyze OQAM signal using the LPTV theory. Herein we derive the explicit formulas for cyclic autocorrelation function and spectral correlation function of OQAM signal, which are very helpful for blind OQAM signal processing. Cognitive Radio has been proposed as a possible solution to improve spectrum utilization via dynamic spectrum access, and spectrum sensing has also been identified as a key enabling functionality to ensure that cognitive radios would not interference with primary users. In this paper we study the spectral detection of OFDM/OQAM signal considering the AWGN and Rayleigh fading environment by using an induced cyclostationary scheme [8], which is realized by intentionally embedding some cyclostationary signatures. Cyclostationary signature has been shown to be a powerful tool in overcoming the challenge of the distributed coordination of operating frequencies and bandwidths between co-existing systems [9]. The paper is organized as follows: Section II presents a LPTV system for spectral correlation analysis of OQAM signal, derives the theoretical formulas of cyclic autocorrelation and spectral correlation functions. Section III introduces a simple technique to embed cyclostationary signature in OQAM signal. A signature detector and some experimental results are mentioned in Section IV. Finally, conclusions are presented in Section V. II. S PECTRAL C ORRELATION OF OQAM S IGNAL The principle of OFDM/OQAM multicarrier modulation system is to divide the transmission into M independent transmissions using M subcarriers. Instead of the rectangular shape filter, a more normal filter bank p(t) is used. Subcarrier bands are spaced by the symbol rate 1/T0 (T0 is one OQAM symbol period). An introduced orthogonality condition between subcarriers guarantees that the transmitted symbols arrive at the receiver free of ISI and ICI, which is achieved through time staggering the in-phase and quadrature components of the subcarrier symbols by half a symbol period
f) = G(t,
∞
−∞
t − τ )e−j2πf τ dτ h(t,
(6)
which can be also represented by a Fourier series: ∞
f) = G(t,
n (f + n/T0 )ej2πnt/T0 G
(7)
n=−∞
where: n (f ) = G
∞
n=−∞
n (τ )e−j2πf τ dτ g
(8)
So the cyclic autocorrelation and cyclic spectrum of the (t) and output y(t) of the LPTV system are related by input x the formulas [5]: Fig. 1.
Baseband OFDM/OQAM transmitter
n,m=−∞
T0 /2. The typical baseband OFDM/OQAM transmitter system is shown in Fig. 1. Supposing the complex input symbols of OQAM system are: xlk
=
alk
+
jblk
(1)
where alk and blk are respectively the real and imaginary parts of the kth subcarrier of the lth symbol. The complexvalues baseband OFDM-OQAM signal is defined as:
y(t) =
M −1
∞ alk p(t − lT0 ) + jblk p(t − lT0 − T0 /2)
∞
Syα (f ) =
2πt π jk( T + 2 ) 0
(2)
A. LPTV System LPTV transformation is defined as follows [5]: ∞ u) h(t, x(u)du y(t) =
(3)
is a L-element column vector input (L is any where x non-zero positive integer) and y(t) is a scalar response. u) = h(t+T h(t, 0 , u+T0 ) is the periodically time-variant (Lelement row) vector of impulse response functions that specify the transformation. The function h(t+τ, t) is periodic in t with a period T0 for each τ represented by the Fourier series: + τ, t) = h(t
n (τ )ej2πnt/T0 g
(4)
n=−∞
where: n (τ ) = g
1 T0
T0 2
T − 20
+ τ, t)e−j2πnt/T0 dt h(t
(9)
x n (f + α )S G 2
f − [n + m]/2T0
T (f − α )∗ (10) ·G m 2 where ” ⊗ ” denotes convolution operation, the superscript symbol ”T ” denotes matrix transposition and ” ∗ ” denotes β is the matrix of cyclic cross correlation of conjugation. R x (t): the elements of the vector x β (τ ) = Lim R x
1 T →∞ T
T 2
T −2
(t+τ /2) x xT (t−τ /2)e−j2πβt dt (11)
and rα nm is the matrix of finite cyclic cross correlation:
−∞
∞
⊗ rα (−τ ) nm n−m α− T 0
n,m=−∞
k=0 l=−∞
·e
α− n−m jπ(n+m)τ x T0 (τ ) · e− T0 trace [R ]
∞
Ryα (τ ) =
(5)
+ τ, t) is defined as The Fourier transform of function h(t a system function:
rα nm (τ ) =
∞
−∞
∗ nT (t + τ /2) gm (t − τ /2)e−j2παt dt g
(12)
The formulas (9) and (10) reveal that the cyclic autocorrelation and spectra of a modulated signal are each selfdeterminant characteristics under an LPTV transformation. B. Spectral Analysis of OFDM/OQAM Signal using LPTV From (2) and Fig. 1 we can see that OFDM/OQAM signal (t) transformed by LPTV is a special model with M-input x transformation h(t) and one scalar output y(t). The baseband OQAM signal (2) can also be expressed as a sum of M single carrier signals like (3): y(t) =
M −1
xk (t)hk (t)
(13)
k=0
where xk (t) is the element of the input vector of LPTV system and hk (t) is the element of impulse response of LPTV:
⎤ ⎡ ⎤ δ(τ ) ··· 0 gn1 π .. ⎥ ⎢ . ⎥ ⎢ n (τ ) = ⎣ ... ⎦ = ⎣ .. g ⎦ . ej 2 k δ(τ ) π gnM −1 0 ... ej 2 (M −1) δ(τ ) ⎤ ⎡ 1 ⎤ ⎡ 1 ··· 0 Gn π .. ⎥ . ⎥ ⎢. n (f ) = ⎢ G (22) ⎦ ⎣ .. ⎦ = ⎣ .. ej 2 k . π M −1 (M −1) j Gn 0 ... e 2 ⎡
∞ ak (lT0 )p(t − lT0 )+
xk (t) =
l=−∞
jbk (lT0 )p(t − lT0 − 2πt π jk( T + 2 ) 0 ,
hk (t) = e
,
T0 2 )
k = 0, 1, . . . , M − 1 (14)
k = 0, 1, . . . , M − 1
(15)
for which ak and bk are the purely stationary data, T0 is one OQAM symbol duration, p(t) is the prototype filter bank pulse function, and hk (t) can be regarded as the periodic function in t with the period T0 for k = 0, 1, . . . , M − 1. xk (t) also can be regarded as a two-element vector LPTV transformation of input data ak and bk with the time-invariant filters p(t) and p(t − T0 /2): xk (t) = a0(t) ⊗ p(t) + b0(t) ⊗ p(t − T0 /2)
(16)
where: a0(t) = b0(t) =
∞
ak (lT0 )δ(t − lT0 )
l=−∞ ∞
jbk (lT0 )δ(t − lT0 )
(17)
l=−∞
Assuming E[al,k a∗l,k ] = E[bl,k b∗l,k ] = σ 2 , each entity of α (τ ) and S α (f ) in (9) and (10) reduces to: matrices R x x
=
σ2 T0
=
2
σ T0
α · rp1 (τ )(1 + e−jπαT0 ),
Sxαk (f ) =
σ2 T0
α α · Sp1 (f ) + Sp2 (f )
=
σ2 T0
α · Sp1 (f )(1 + e−jπαT0 ),
Rxαk (τ )
· δ(τ ) ⊗
α rp1 (τ )
+ δ(τ ) ⊗
integer T0
(18)
α rp2 (τ )
= =
∞
−∞ ∞
α=
integer T0
(19)
p(t + τ /2)p(t − τ /2)e−j2παt dt
p(t + τ2 − T20 )p(t −∞ α (τ )e−jπαT0 rp1
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
πM τ sin( T ) 0 πτ sin( T )
jπ(M −1)τ T0
α=
· (1 + e−jπαT0 ),
2·integer , |τ | T0
0,
α (f ) = Soqam ⎧ σ2 M −1 P (f + ⎪ ⎪ ⎪ T0 n=0 ⎨
⎪ ⎪ ⎪ ⎩
0
−
·e
0,
α =
α 2
−
n ∗ T0 )P (f
−
α 2
−
< KT0 ;
2·integer ; T0
(23)
n T0 )(1
+ e−jπαT0 ), α = 2·integer ; T0
α =
2·integer ; T0
where KT0 is the time length of the prototype filter bank, α (τ ) is described P (f ) is the Fourier transform of p(t) and rp1 as (20). III. C YCLOSTATIONARY S IGNATURE FOR OQAM S IGNAL
α α (f ) is the Fourier Transform of rp1 (τ ) and: where Sp1 α (τ ) = rp1
α (τ ) = Roqam ⎧ ⎪ σ2 α ⎪ ⎪ ⎪ T0 · rp1 (τ ) · ⎪ ⎨
(24)
α rp2 (τ )
α=
Substitution of (13)∼ (22) into (9) and (10), the cyclic autocorrelation and cyclic spectra of OQAM signal is transformed into:
−
τ 2
−
T0 −j2παt dt 2 )e
(20)
Other terms corresponding to the LPTV system can be similarly calculated: j2πt π u) = δ(t − u), je T0 δ(t − u), · · · , ej 2 (M −1) h(t, j2π(M −1)t T0 ·e δ(t − u) j2π(M −1)t j2πt π f ) = 1, je T0 , · · · , ej 2 (M −1) e T0 G(t, (21)
Multicarrier signals like OFDM contain inherent cyclostationary features due to the underlying periodicities properties [9]. The magnitude of spectral correlation function (SCF) of OQAM signal without signature is drawn in graphical terms as the heights of surfaces above a bi-frequency plane in Fig. 2, where M = 8, a reference filter bank is designed using the method in [10]. We unfortunately found that OQAM signal has very poor inherent cyclostationary property when the cyclic frequency is not equal to zero, which can be interpreted by (24), where the value of cross product ”P (f + α2 )P ∗ (f − α2 )” ” due to the low sidelobes tends to zero when ”α = 2·integer T0 property of OQAM signal. The poor inherent cyclostationarity is unsuitable for practically applications in the Cognitive Radio for dynamic spectrum access. Here is an alternative approach using cyclostationary signature (CS). A cyclostationary signature is a feature, intentionally embedded in the physical properties of a digital communication signal. CSs are effectively applied to overcome the limitations associated with the use of inherent cyclostationary features for signal detection and analysis with minimal additional complexity for existing transmitter architectures. Detection
and analysis of CS may be also achieved using low-complexity receiver architectures and short observation durations. CS provides a robust mechanism for signal detection, network identification and signal frequency acquisition. A CS is easily created by mapping a set of subcarriers onto a second set as: γn,l = γn+p,l
n∈N
(25)
where γn,k is the lth independent and identically distributed message at nth subcarrier frequency, N is the set of subcarrier values to be mapped and p is the number of subcarriers between mapped subcarriers. So a correlation pattern is created and a cyclostationary feature is embedded in the signal by redundantly transmitting message symbols. According to (9) (10) (23) (24) (25), we can rewrite the cyclic autocorrelation and spectral correlation formulas of OQAM signal with cyclostationary signatures: α (τ ) = Roqam·cs ⎧ πM τ jπ(M −1)τ sin( T ) − ⎪ α 0 ⎪ σ2 · rp1 T0 (τ ) · ·e · (1 + e−jπαT0 ), ⎪ πτ T ⎪ 0 sin( ) ⎪ T 0 ⎪ ⎪ ⎪ ⎪ α = 2·integer , 2 · integer = p, |τ | < KT0 ; ⎪ T0 ⎪ ⎪ ⎪ ⎨ p jπ(2n+p)τ α− T − 2σ 2 0 ⎪ T0 · r (τ ) · e , ⎪ p1 n∈N T ⎪ 0 ⎪ ⎪ p ⎪ α = ⎪ T0 , |τ | < KT0 ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, α = 2·integer , α = Tp0 ; T0 (26) α Soqam·cs (f ) = ⎧ 2 M −1 σ α n α n ∗ −jπαT0 ⎪ ), ⎪ n=0 P (f + 2 − T0 )P (f − 2 − T0 )(1 + e T0 ⎪ ⎪ 2·integer ⎪ ⎪ α = , 2 · integer = p; ⎪ T0 ⎪ ⎪ ⎪ ⎪ ⎨ 2 n+p 2σ α α n ∗ n∈N P (f + 2 − T0 )P (f − 2 − T0 ), T0 ⎪ ⎪ ⎪ α = Tp0 ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, α = 2·integer , α = Tp0 ; T0 (27)
where N is the set of subcarriers to be mapped and p ∈ P (P = ±2 · i, i = 1, 2, 3, 4, · · · ). The magnitude of spectral correlation function of OQAM signal with CS is drawn in Fig. 3, where two cyclostationary signatures are embedded corresponding to two different values of p (choosing p = 2 and p = 4). We can see that for the OQAM signal with CS the strong cyclostationary features appear at the cyclic frequency ”α = 2/T0 ” and ”α = 4/T0 ”. IV. S IGNATURE D ETECTOR The signature detector in [9] can be used for efficient OFDM/OQAM signal detection. Cyclostationary features generated by subcarriers set mapping can be successfully detected
Fig. 2.
Spectral Correlation Function for OQAM signal without CS
Fig. 3. Spectral Correlation Function for OQAM signal with two CSs at cyclic frequencies α = 2/T0 and α = 4/T0
using spectral resolution (subcarrier spacing f0 ). So the lowcomplexity signature detector can be designed by sliding a window W with the width N0 · f0 (N0 is the number of subcarriers in the mapped set) around estimated SCF: Tα = max m
Sxα (n)W (m − n)
(28)
n
The performances of signatures with different set sizes are described in [9]. In order to evaluate the performance of OFDM-OQAM signal detection using CS for practical CR systems, we have simulated a 512-subcarriers OQAM signal with assumed conditions and methods as follows: 1) CR system with a bandwidth of 5M Hz, and assuming signals are transmitted at 2.4GHz. 2) We consider the Additive White Gaussian Noise (AWGN) and frequency selective channels, respectively. A typical urban channel is used with a total ten taps and spread delay τ ≈ 2μs. 3) Subcarriers are modulated using OQAM. Three and nine subcarriers are respectively used as the mapping subcarrier sets at one single cyclic frequency. 4) For detection statistic Tα , the entry Sxα (f ) is estimated using a time-smoothed cyclic cross periodogram [5]. For simplicity, a rectangular sliding window is chosen.
V. C ONCLUSION In this paper we have derived the explicit cyclic autocorrelation and spectral correlation functions formulas for OFDM/OQAM signal using a LPTV model, which is convenient to analyze the cyclostationary properties of OQAM signal. A signature detector is introduced as an effective tool for OQAM signal detection. Although we can improve detection performance with increased subcarriers mapping set size, it causes a reduction in overall data rate because of the increased overhead. In practice we have to make a compromise between the numbers of mapping set size and the desired detection performance. Future work will focus on the utilization of pilots for generating cyclostationary signatures. R EFERENCES
[3] B. Hirosaki, ”An Orthogonally Multiplexed QAM System Using the Discrete Fourier Transform”, IEEE Trans. On Comm. Tech, Vol. 29, Issue 7, Jul 1981, pp. 982-989. [4] P. Siohan, C. Siclet and N. Lacaille, ”Analysis and design of OFDM/OQAM systems based on filterbank theory”, IEEE Trans. Signal Processing, Vol. 50, May 2002, pp. 1170-1183. [5] William A. Gardner, ”The spectral correlation theory of cyclostationary time-series”, Signal Processing Vol. 11. pp. 13-36, July 1986. [6] William A. Gardner, ”Spectral correlation of modulated signal, Part IAnalog modulation”, Communications, IEEE Transactions Vol. COM35, pp. 584-594, June 1987. [7] W. A. Gardner, W. A. Brown, and C.-K. Chen, ”Spectral correlation of modulated signals, Part II-Digital modulation”, Communications, IEEE Transactions Vol. COM-35, pp. 595-601, June 1987. [8] M. K. Tsatsanis, G. B. Giannakis, ”Transmitter induced cyclostationarity for blind channel equalization”, IEEE Transaction on Signal Processing, Vol. 45, NO. 7, pp. 1785-1794, 1997. [9] Paul D. Sutton, Keith E. Nolan, and Linda E. Doyle, ”Cyclostationary Signatures in Practical Cognitive Radio Application”, IEEE Journal on selected areas in Communications, Vol. 26, NO. 1. January 2008, pp. 13-24. [10] M. Bellanger, ”Specification and design of a prototype filter for filter bank based multicarrier transmission”, Proceedings of IEEE ICCASP, Vol. 1, pp. 2417-2420, 2001.
1 0.9
Detection Probability
0.8 0.7 0.6 0.5 0.4
0.2 0.1 0
Fig. 4.
0
0.2
N=10 N=30 N=20 N=10 N=30 N=20 N=10
0.4 0.6 False Alarm Probability
SNR=0dB SNR=−5dB SNR=−5dB SNR=−5dB SNR=−10dB SNR=−10dB SNR=−10dB
0.8
1
Receiver Operating Characteristic performance for AWGN channel
1 0.9 0.8 0.7 0.6 0.5 0.4 Multipath Multipath Multipath Multipath Multipath Multipath Multipath
0.3 0.2 0.1 0
[1] R. W. Chang, ”Synthesis of Band-Limited Orthogonal Signals for Multicarrier Data Transmission”, Bell. Syst. Tech. J. Vol. 45, Dec 1966, pp. 1775-1796. [2] B.R. Saltzberg, ”Performance of an efficient parallel data transmission system”, IEEE Trans. On Comm. Tech, Vol. 15, no. 6, Dec 1967, pp. 805-811.
AWGN AWGN AWGN AWGN AWGN AWGN AWGN
0.3
Detection Probability
Receiver operating characteristic (ROC) curves are drawn in Fig. 4 and Fig. 5 by averaging 500 Monte Carlo simulations. Fig. 4 gives the simulation results for an AWGN channel at different SNR levels (0dB, -5dB, and -10dB). ROC curves for the cases with different lengths (10, 20, and 30 OQAM symbols) of observation time are also shown. Fig. 4 shows that the performance of the signature detector improves when longer observation times are used. Desired detection performance can be achieved even at low SNR level (-5dB), and almost 100% detection rate can be achieved when using more than 10 symbols at the SNR level more than 0dB. However, the results deteriorate when more realistic time variant multipath channels are considered. Effects of multipath are illustrated in Fig. 5 for a fixed SNR level (-5dB). The signature detection performance is estimated for a 240Hz Doppler frequency, which corresponds to a moving speed of 30m/s. Compared with Fig. 4, it can be seen that multipath channels affect detection performance significantly. As the decrease of observation times, the detection performance deteriorates. In order to achieve higher detection reliability, longer observation time or more mapping subcarriers are needed. Fig. 5 indicates that perfect detection performance can be achieved if nine or more subcarriers mapping are used. Simulations show that the good performances are achieved even for the short time observation. Reliable detection may be performed at a low SNR for a small time length 10T0 (equivalent to the duration of 10 OFDM/OQAM symbols). Detection reliability can be seriously impacted by time-variant multipath fading, which can be overcome through the use of longer observation time and more mapping subcarriers. Besides, these improvements do not come at the cost of increased detector complexity, and a simple signature detector structure can be applied.
0
0.2
0.4 0.6 False Alarm Probability
N=50 N=40 N=50 N=40 N=30 N=20 N=10
Subcarriers=9 Subcarriers=9 Subcarriers=3 Subcarriers=3 Subcarriers=3 Subcarriers=3 Subcarriers=3 0.8
1
Fig. 5. Receiver Operating Characteristic performance for Multipath channel at a fixed SNR (-5dB)
