ISSN 01464116, Automatic Control and Computer Sciences, 2010, Vol. 44, No. 3, pp. 160–170. © Allerton Press, Inc., 2010. Original Russian Text © A.I. Aboltins, D.I. Klavins, 2010, published in Avtomatika i Vychislitel’naya Tekhnika, 2010, No. 3, pp. 59–71.
Synchronization and Correction of Channel Parameters for an OFDMbased Communication System A. I. Aboltins and D. I. Klavins Riga Technical University, ul. Azenes 12, Riga, LV1048 Latvia email:
[email protected] Received February 2, 2010
Abstract—A review and description of synchronization problems that occur in communication sys tems based on OFDM (orthogonal frequency division multiplexing) are presented. Solutions for sym bol, frame, and frequency synchronization are covered. An account is given and results are presented of the simulation of devices for the estimation of the channel parameters. Results on the simulation of communication systems employing different methods for synchronization and estimation of the chan nel parameters are presented. A review of the influence of additive noise, multipath propagation, and Rayleigh fading on the synchronization is given. Key words: communication system, estimation, synchronization, time DOI: 10.3103/S0146411610030077
1. INTRODUCTION In the development of a new type of communication system, we took orthogonal frequency division multiplexing (OFDM) as the basis. Inasmuch as rapid progress in this method took place in the late 1990s, at the moment, a rich variety of descriptions, models, and completed solutions based on this technology are available. We had to investigate and test numerous methods for the timing and frequency synchroni zation, as well as schemes for the estimation of the channel parameters, in order to find among them the implementations that are the most efficient and best suited to our use. The aim of the present study is to review numerous methods of synchronization and to separate out the most efficient ones among them. The results of the simulations we conducted, which are presented in Section 7, are given to demonstrate the capabilities and restrictions of the chosen methods from the standpoint of their practical implemen tation. A similar review can be found in [1]. This PhD thesis is dedicated to algorithms for signal processing applied in OFDM. The chapter devoted to synchronization offers but a brief review of the methods and does not contain the details necessary for the construction of practical implementations. This paper includes a great number of useful references. Another review of this kind is available in [2]. Article [3] pro vides a good practical basis to begin the construction of OFDMs but does not include any mathematical substantiation for the algorithms used. Books [4] and [5] contain only a description of the basic methods for the synchronization and estimation of the channel parameters. OFDM has found wide utility in wireless local communication networks (IEEE 802.11a, g, j, and n) [6], IEEE 802.16d (WiMAX) [7], television broadcasting (DVBT, DVBH, TDMB, and ISDBT) [8], radio broadcasting (EUREKA 147, Digital Radio Mondiale, HD Radio, TDMB, and ISDBTSB), and mobile communications (3GPP, 4G, and MCCDMA) [9]. Under the name discrete multitone modula tion (DMT), OFDM is used in cable systems, such as xDSL and PLC. Unfortunately, the sustenance of the subcarriers’ mutual orthogonality, as well as the application of coherent modulation types, demands accurate timing and frequency synchronization between the receiver and transmitter. In fact, the synchronization proves to be a key factor in providing the high capac ity of a communication system. 160
SYNCHRONIZATION AND CORRECTION OF CHANNEL PARAMETERS
161
2. THE BASICS OF OFDM The OFDM technology is based on the transmission of information in blocks. A block of N informa n j2πk N
in a balanced way, where tion symbols modulates N discrete mutually orthogonal carriers j(n, k) = e n is the index of a subcarrier (in the frequency domain), and k is the index of a discrete sample (in the time domain). The process of the information symbols’ modulation and transformation to the time domain is represented in the form N–1
1 s ( k ) = d ( n )ϕ ( n, k ) , Nn = 0
∑
k = 0, 1, …, N.
(1)
where d(n) is the transmitted information symbol. Note that transformation (1) in fact is an inverse dis crete Fourier transform (DFT). For continuous time expression, (1) becomes N–1
N–1
1 1 s ( t ) = d ( n )ϕ ( n, t ) = d ( n )e Nn = 0 Nn = 0
∑
∑
j2πk n ⎛ t – T ⎞ T ⎝ 2⎠
,
(2)
where T is the duration of one OFDM symbol. Designate a time domain signal at the receiver’s input distorted in the transmission along the commu nication channel as r(k) = F [s(k)]. To obtain information symbols, the receiver needs to perform the transformation of the received signal to the starting frequency domain: N–1
1 dˆ ( k ) = r ( k )γ ( n, k ) , Nk = 0
∑
n = 0, 1, …, N,
(3)
k – j2π n N
where γ(n, k) = e are the basis functions for the inverse transformation. Note that (3) can be easily implemented with the aid of DFT. When signal (1) is transmitted along a channel with timing dispersion, the partial superposition of adja cent OFDM symbols, i.e., intersymbol interference (ISI), can occur. Peled and Ruiz in [11] suggested using cyclic continuation of OFDM symbols—a cyclic prefix (CP)—for solving this problem. The whole OFDM symbol with a CP of length L is represented in the form N–1
1 s ( k ) = d ( n )ϕ ( n, k – L ) , Nn = 0
∑
k = 0, 1, …, N + L – 1 .
(4)
The cyclic continuation of a signal preserves its amplitude spectrum and just shifts the signal’s phase spectrum. If the length of the CP is greater than the pulse response of the channel, then the intersymbol interference is completely resolved by the proper choice of the time for the beginning of the OFDM sym bol, i.e., by symbol synchronization. Actual OFDM systems include intermediatefrequency and radiofrequency sections for the transla tion of a signal to the required frequency and back. The discrepancy between the sampling and carrier fre quencies is also a powerful source of errors in data transmission. In addition, in mobile communication systems, there occurs a problem of the frequency shift due to the Doppler effect. The frequency synchro nization in OFDM also ranks among the synchronization problems. Most of the modern OFDM systems employ framing of the OFDM symbols’ flow. Overhead informa tion and pilot symbols for the quick estimation of the channel’s parameters are commonly located in the beginning of a frame. Frame synchronization is required in order that the receiver be able to determine the beginning of the next frame. 3. SYMBOL SYNCHRONIZATION In OFDM, a signal consists of N orthogonal carriers modulated by N parallel data flows (1). In theory, OFDM symbols (3) can be received using a bank of matched filters. However, in actual practice, another scheme of demodulation based on DFT is used. The successful demodulation of a signal by means of DFT requires the correct marking of the bounds of the individual OFDM symbols. In a nonscattering medium, AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 3
2010
162
ABOLTINS, KLAVINS
the samples of the transmitted signal s(k) from (1) and the samples of the received signal r(k) from (3) are related by the expression r ( k ) = s ( k – θ )e
j2πεk
+ ξ ( k ),
(5)
where θ is the sought integer delay, ε is the difference in the carrier frequencies of the receiver and trans mitter, and ξ is the additive Gaussian noise. To mark the beginning of the next symbol, a special starting pulse can be transmitted at the beginning of each OFDM symbol, but this would lead to a decrease in the capacity of the system. A much more efficient means of symbol synchronization is based on the use of the redundance of the OFDM symbols, i.e., on a cyclic prefix (CP). There are several ways of using a CP for determining the bounds of symbols. In study [12], the authors suggest using the difference between the sig nal and its copy delayed by N samples: m+L–1
∑
v(m) =
r(k) – r(k + N) ,
m ∈ { 0, …, θ, …, N + L – 1 }.
(6)
k=m
Then, the sought estimate for the delay of the symbols is represented as θˆ = arg min { v ( θ ) }.
(7)
θ
Other authors in a more recent study [13] suggested using autocorrelation: m+L–1
y(m) =
∑
r ( k )r* ( k + N ),
m ∈ { 0, …, θ, …, N + L – 1 }.
(8)
k=m
In this case, an estimate for the delay of the symbols is represented as θˆ = arg min { y ( θ ) }.
(9)
θ
Moreover, the phase of the abovementioned statistic at the instant m = θˆ allows one to find the fre quency shift of the subcarrier: 1 ∠y ( θ ), εˆ = – 2π
(10)
where ∠ signifies the argument of a complex number. The estimation of the timing and frequency shift using the algorithm of maximum likelihood (ML) was proposed by the authors of [16]. This method is built upon statistic (8) in conjunction with the additional statistic 1 u(m) = 2
m+L–1
∑
2
2
r(k) + r(k + N) .
(11)
k=m
The combined estimate for the delay of the symbols is represented in the form θˆ = arg max { y ( θ ) – ρu ( θ ) },
(12)
θ
ML
where ρ is the coefficient of correlation depending on the signaltonoise ratio at the receiver’s input. For mula for the frequency shift (10) also remains valid for estimate (12). If the medium in which a signal propagates is scattering, i.e., if multipath propagation takes place, then (5) turns into the expression M
r(k) =
∑ α s ( k – τ – θ )e l
l
j2πεk
+ ξ ( k ),
(13)
l=1
where M is the number of propagation paths, and αl and τl are the fading and delay introduced by the lth path. For a signal of this kind, the methods described above yield only approximate positions of the OFDM symbols. In channels with multipath propagation, the part with a CP is badly distorted by the pre vious symbol. As a rule, this results in a wrong estimate for the beginning of a symbol. The simplest solu tion of this problem is to use continuous averaging of θ. Such an approach is suggested in [17]. AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 3
2010
SYNCHRONIZATION AND CORRECTION OF CHANNEL PARAMETERS
163
differential
10 5 0
autocorrelation 50
0 0
maximum likelihood
100 200 0.5
frequency shift
0 –0.5 0
50
100
150 Time
200
250
300
Fig. 1. Determination of the OFDM symbols’ bounds by different methods. The dashed line shows the result of averaging over eight symbols.
Implicit averaging of the symbol synchronization is also used in model [18]. In the same model, the derivative of a linearly approximated estimate for the phasefrequency response of the channel is used for the accurate symbol timing synchronization within a fraction of one sample. The method for the estimation of the symbol timing described in [19] is based on the observation that, in a channel with scattering, the function of correlation (11) is actually the sum of the individual correla tion functions taken along the paths Pl. In the study in question, it is shown that the correct instant for the beginning of the next OFDM symbol is at the point where the combined correlation function starts to decline. In studies [20–22], the estimation of the delay of the symbols and the turning of the phases using the algorithm of expectation maximization (EM) [23] is considered. The algorithm affords fast convergence of the unknown parameters to the values from the ML estimation. The iterative approach to finding the delays and phases is used in the algorithm. Article [25] suggests an extension of the model described in [16] to a more general case with the use of pulse shaping and consideration of the interference from adjacent symbols. In the described OFDM sys tem, the first and the last KSH samples are multiplied by a raised cosine. The results of the simulations pre sented in [25] show that, with the use of pulse shaping, the quality of the synchronization can be preserved or even improved (approximately by 10% in channels with fading). The results of our own simulations for different ways of estimation of the symbol synchronization are presented in Fig. 1. A beneficial effect of averaging the resulting estimates for the synchronization can be seen from the figure. 4. ESTIMATION OF THE CARRIER FREQUENCY OFDM systems are sensitive to a shift of the carrier frequency. Small shifts of the frequency result in the slow rotation of the phase of the received information symbols. These phase shifts can be easily com pensated for by a device for the correction of the channel parameters (a frequency domain equalizer). However, large discrepancies between the carrier frequencies lead to a shift of the OFDM subcarriers and a loss of the subcarriers’ mutual orthogonality. AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 3
2010
164
ABOLTINS, KLAVINS
The classical methods for the estimation of the carrier frequency involve measuring the phase’s speed of rotation for pilot symbols. To take one example, such a method is described in [3]. The suggested scheme of the frequency synchronization for a wireless LAN is based on the correlation of the timing functions of two adjacent identical pilot symbols for the estimation of the channel parameters. A pilot symbol (an OFDM symbol with all its subcarriers d(n) occupied by pilot signals) is transformed into the time domain with the aid of the inverse DFT presented in expression (2). Denote the period of the pilot symbols’ recurrence by T, and the shift of the carrier frequency by Δfc. Represent the total phase shift over the time of one symbol in the form φ = Δf c T.
(14)
If there is no frequency shift, then the correlation between the samples of two neighboring symbols is found from the formula N–1
J0 =
N–1
∑ r ( k )r* ( k + N ) = ∑ r ( k )
k=0
2
(15)
,
k=0
where r(k) are the received samples of the pilot symbols. In fact, the operation in question resembles the foregoing correlation of a CP (8). If there is a frequency shift, then, at the output of the correlator, we have the signal J = J0 e
– j2πφ
(16)
,
where the phase shift is determined by the expression 1 J* φ = – ∠⎛ ⎞ , 2π ⎝ J ⎠
(17)
where ∠ signifies the argument of a complex number. The capabilities of the described method are restricted by the condition φ < 1. The authors of [3] suggest correlating short training symbols of the frame synchronization for the rough estimation of the frequency shift: N – 1 4
G =
∑ r ( k )r* ⎛⎝ k + 4⎞⎠
N = e
N – 1 – j2π φ 4 4
k=0
∑ r(k)
2
(18)
,
k=0
φ 1 ⎛ G* = ∠ ⎞ . 4 2π ⎝ G ⎠
(19)
Since the correlator uses signals that are four times shorter than the OFDM information symbols, no uncertainty occurs in finding φ even for large (over 100 ppm) discrepancies between the frequencies. For the same reason, the method in question is less accurate. The same effect can be achieved by the correlation of the cyclic prefix and the end of any OFDM sym bol. According to expression (10), the frequency shift can be determined from the argument of the com plex output signal of the CP correlator at the instants when the absolute value of this signal reaches its maximum. The required range and accuracy of the frequency synchronization are attained by combining the rough and exact estimates: 4 G* 1 ⎛ J* φ = ∠⎛ ⎞ + ∠ ⎞ . ⎝ ⎠ 2π G 2π ⎝ J ⎠
(20)
5. FRAME SYNCHRONIZATION The most obvious way to provide timing synchronization is to insert special marking signals into the transmitted signal. In IEEE Standard 802.11a wireless networks [6], special symbols (short training sym bols) with sharply defined correlation properties and limited spectrums are used for the frame and fre quency synchronization (19). The receiver continuously correlates the incoming signal with a sample short symbol. When a pair of short symbols occurs at the input, ten peaks spaced 16 samples apart appear AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 3
2010
SYNCHRONIZATION AND CORRECTION OF CHANNEL PARAMETERS
165
at the output of the correlator. This image, in conjunction with a plateau 80 samples in length at the output of the CP autocorrelator, is used as a signal for the beginning of the next OFDM frame. In article [26], an extended scheme for the frame and symbol synchronization is proposed. The authors of the article suggest using an additional burst frame containing a distinctive guard space and pilot symbols to provide quick and accurate frame synchronization during the course of one OFDM symbol. According to the results of the simulations performed by the authors of the article, in channels with Rayleigh fading and multipath propagation, the scheme in question is far superior to classical methods of synchronization. In a widely cited article by Schmidl and Cox [27], the authors suggest using two adjacent training sym bols. The first symbol in the time domain consists of two identical parts. It is generated by way of filling even subcarriers of an OFDM symbol with a pseudonoise sequence and setting the values of the odd sub carriers equal to zero. The symmetry of the OFDM symbol renders it tolerant to detuning of the carrier frequency and allows one to determine the beginning of a frame quickly. Moreover, the values of the even subcarriers can be used in the estimation of the channel’s parameters. The second symbol consists of two other pseudonoise sequences, which make it possible to perform the estimation of the channel’s parame ters at the remaining frequencies, as well as to compute the detuning of the carrier frequency. 6. ESTIMATION OF THE CHANNEL PARAMETERS In an actual communication system, the amplitudes and phases of the received OFDM signals are badly distorted and can change fast. There are two approaches to providing the correct decoding of binary information transmitted using OFDM subcarriers. First, one can use differential modulation, for instance, DQAM or DPSK, and measure the relative variations of the adjacent symbols. This technique makes it possible to do without the correction of the channel’s parameters but, at the same time, results in the reduced capacity of the communication channel. The second approach is to use coherent detection. In such an event, the information on the current reference amplitude and phase, which is required to determine the bounds of the constellations for the individual subchannels, can be obtained through the estimation of the channel’s parameters. Owing to the narrowness of the bands of the individual OFDM subcarriers, their frequency responses can be considered flat. This significantly simplifies the estimation and correction of the channel’s param eters, because the problem concerning the estimation of the channel’s parameters is reduced to finding the complex scalar fadings of the subcarriers. To perform the estimation of the channel, one needs to transmit pilot signals at some subcarriers. The modulation technique that employs such an approach is called pilot symbol assisted modulation (PSAM). There are several ways to insert pilot symbols: • Pilot symbols can be inserted into particular subcarriers of all OFDM symbols. When the symbol syn chronization is attained, the receiver extracts these subcarriers and measures their complex fading. The results are interpolated with respect to the frequency to the neighboring information subcarriers and sent to a unit for the correction of the channel parameters—the equalizer of the frequency domain. • The selected OFDM symbols (pilot symbols) with all their subcarriers occupied by pilot signals are used. After the attainment of the frame synchronization, the OFDM receiver periodically finds an esti mate of the fading and the delay at all the subcarriers. The complex fading of the information symbols located between the pilot symbols is obtained by way of interpolation with respect to the time of the results for the pilot symbols. • The pilot symbols can be scattered, i.e., appear in particular subcarriers of particular symbols accord ing to a specified law. In such an event, the unit for the estimation of the channel’s parameters needs to perform twodimensional interpolation with respect to the frequency and time simultaneously. Such an approach based on the use of finite impulse response Wiener filters is discussed in article [26]. A discrete signal transmitted with the aid of N subcarriers is described by Eq. (1). The DFT for the dis crete impulse transient response of the channel is represented as Δ
k
– j2πn Δ 1 H ( n ) = h ( k )e , Δk = 0
∑
AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
(21)
No. 3
2010
166
ABOLTINS, KLAVINS
where Δ is the length of the channel’s transient impulse response. The signal received at the subcarrier n is described by the expression N–1
1 r ( k ) = s ( k ) · h ( k ) = H ( n )d ( n )ϕ ( n, k ). Nn = 0
∑
(22)
One can notice that Eqs. (1) and (22) are almost identical; the only difference is in the fact that, in (22), the subcarrier n modulates H(n)d(n) instead of d(n). If no account is taken of the noise in the channel, the received samples p(n) assume the form p(n) = H(n)d(n). (23) Denote the transmitted and received pilot subcarriers by dp(n) and pp(n), respectively. The complex fadings of the channel are computed directly with the use of the expression pp ( n ) ˆ ( n ) = H . dp ( n )
(24)
The resulting complex fadings at the frequencies corresponding to the pilot carriers are used in finding the estimates of the channel’s parameters at the remaining frequencies by way of interpolation. The obtained estimates for the complex fadings of the channel are passed to a unit for the correction of the amplitude–frequency and phase–frequency responses of the channel—the frequency domain equalizer. Here, the direct compensation for distortions introduced by the communication channel and the restora tion of the transmitted information symbols are performed with the aid of the zero forcing algorithm according to the equation ˆ ( n )* H p ( n ) = (25) d ( n ) = p ( n ). ˆ (n) ˆ (n) 2 H H We performed simulations of communication systems using this approach. The results of these simu lations are given in Fig. 2. Different strategies can be employed to minimize the error in the estimation of the channel parameters. In study [28], descriptions of devices for the estimation of the channel parameters using DFT and singular value decomposition (SVD) are given. An algorithm for finding the optimum estimate in terms of the minimum meansquared error ˆ – H 2 , is given in article [29]. The described method also uses (MMSE), i.e., for the minimization of E H SVD, along with the theory of optimal rank reduction. The method in question is of purely theoretical sig nificance, since its practical implementation is extremely complicated. Simplified (and more suitable for practical implementation) approaches based on the method described above are given in studies [30] and [31]. It is significant that the number of pilot subcarriers is to be chosen in accordance with the widely known Nyquist criterion, which asserts that the interval between the samples of a signal is to be smaller than the reciprocal width of the signal’s duplex frequency band. Thus, when computing the timing interval between pilot symbols, one needs to allow for the scattering of the Doppler frequency shift Bd so that Tt < 1/Bd. On the other hand, when computing the frequency interval between pilot subcarriers, it is necessary to take into account the scattering of the signal delay Bτ so that Tf < 1/Bτ. As one example, in wireless networks [6], pilot signals are transmitted continuously at four of 52 subcarriers at 4.375 MHz intervals. For the given scheme of pilot signals, the maximum admissible scattering of the signal delay is 228 nsec, while the scattering of the Doppler effect is unlimited. Paper [32] is dedicated to the comparative mathematical analysis of diverse schemes for the allocation of pilot signals. For a communication system with the pulse response of length Δ and the number of sub carriers N, the following implications hold. • In the absence of noise, any Δ of N pilot signals would suffice to restore the channel parameters com pletely. • If the channel parameters are timeindependent, then the scheme with periodic pilot symbols and the scheme with continuous pilot subcarriers are equivalent. • n a channel with time variant parameters, the scheme with continuous transmission of the pilot sub carriers is superior to the scheme with periodically transmitted pilot symbols. AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 3
2010
AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 3
2010
15
20 25 SNR, dB
30
35
without an equalizer with an equalizer with an equalizer and frame synchronization
BER
BER
10–5
10–4
10–3
10–2
10–1
100
5
10
15
20 SNR, dB
25
30
an ideal AWGN channel a f requencyselective channel with an equalizer a frequencyselective channel without an equalizer
Fig. 2. Comparison of the productivity for different communication systems with correction of the channel parameters based on pilot symbols (left) and pilot subcarriers (right).
10–5 10
10–4
10–3
10–2
10–1
100
SYNCHRONIZATION AND CORRECTION OF CHANNEL PARAMETERS 167
168
ABOLTINS, KLAVINS
In addition to the foregoing, it is important to say that, in OFDM systems operating in a packet mode, for instance, in wireless networks, there is a need for instant and relatively accurate estimation of the chan nel parameters at the very beginning of a packet. To ensure such an estimation, one or several pilot symbols consisting only of pilot subcarriers are transmitted at the beginning of every packet. 7. RESULTS OF SIMULATIONS Since we are working on the practical implementation of an OFDM system, we give much attention to the construction of models both for individual devices and for entire communication systems. Numerous models of our construction operate in the Simulink environment developed by the MathWorks company. For reasons of space, we present only a selection of the most interesting results of simulations. As a basis for all the models of OFDM systems, we took the IEEE 802.11a standard for a wireless LAN [6]. The model of a communication system consists of the transmitter, the communication channel, and the receiver. For simplicity’s sake, the transmitter and the receiver are taken to contain only lowfrequency sections. The model of a communication channel is constructed on the basis of a finite impulse response filter and a source of white noise. The system employs OFDM frames consisting of 24 symbols, two of which were frame synchronization symbols (short training symbols) and the other two, pilot symbols (long training symbols). The OFDM information symbols comprised 64 information samples and a CP consist ing of 16 samples. For the estimation of the channel parameters, four continuously transmitted pilot car riers were used. In the first experiment, we compared three devices for the estimation of the symbol timing—a differ ential one (6), an autocorrelation one (8), and the maximum likelihood one (12). The results of the sim ulations can be seen in Fig. 1. From the given oscillograms, it follows that the method of maximum like lihood yields the most distinct maximums at the bounds of the OFDM symbols. The averaging of the results all the more improves the robustness and accuracy of the estimation. In another succession of simulations, we tried to compute the beneficial effect of applying the correc tion of the channel parameters, as well as the influence of the synchronization on the number of errors in the data transmission. In the first model, we applied the correction of the channel parameters based on pilot symbols. The bit error rates (BERs) were compared for three OFDM systems—one without correction of the channel parameters, one with correction of the channel parameters, and one with correction of the channel parameters and frame synchronization. In the first two cases, implicit frame synchronization based on the agreed start up of the transmitter’s and receiver’s models was used. The communication channel has a very long (80 samples) random pulse response and is subject to additive noise. The results of the simulations, which are shown in Fig. 2 (left), clearly demonstrate the beneficial effect of applying the estimation and correction of the channel parameters. At the same time, in the third plot, we observe a reduction in pro ductivity of the system due to a loss of the frame synchronization for a signaltonoise ratio of less than 17 dB. In the second model, we employed the estimation and correction of the channel parameters based on pilot subcarriers. In the experiment in question, a channel with a pulse response of four samples and an amplitude–frequency response with two deep notches was used. The results of the simulations displayed in Fig. 2 (right) also attest that the correction of the channel parameters is integral with providing an error rate sufficient for the operation of the errorcorrecting codes and dramatically illustrate the efficiency of the OFDM concept as a whole. We presented the results of the aforementioned and some other simulations at conference [33]. 8. CONCLUSIONS When comparing the numerous methods for synchronization and estimation of channel parameters, as well as the results of our simulations, we arrived at the following conclusions: • The most suitable method for rough symbol and frequency synchronization is the maximum likeli hood autocorrelation of a CP (12). • For accurate symbol synchronization, one can use the derivative of an estimate for the phase–fre quency response of the channel, as, for instance, in model [18]. • Accurate frequency synchronization can be provided by measuring the phase at the output of the pilot symbols’ correlator (17). AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 3
2010
SYNCHRONIZATION AND CORRECTION OF CHANNEL PARAMETERS
169
• Frame synchronization requires using special synchronizing pulses, which can also be applied in rough tuning of the carrier frequency. • Periodically transmitted pilot symbols can be used for the estimation of the parameters for a channel with slowly varying characteristics. The same symbols are to be used in the beginning of each packet if the transmission is not conducted continuously. • It is necessary to use continuously transmitted pilot subcarriers in the estimation of the parameters for a channel with fastvarying characteristics. The mathematical and practical principles embedded in the OFDM concept are quite simple and can be simulated easily. On the other hand, the creation of actually working systems is very labor consuming, since the implementation of a multilevel synchronization system is required. In essence, it is the efficiency of the systems for the synchronization and estimation of the channel parameters that determines the pro ductivity of an OFDM communication system. To construct a complete prototype of an OFDM trans ceiver, it is necessary, in addition to synchronization systems, to introduce error correction systems, sys tems for automatic gain control, and circuits that improve the interference immunity of synchronization systems.
REFERENCES 1. Edfors, O., LowComplexity Algorithms in Digital Receivers, PhD Thesis, Luleå, Sweden: Luleå University of Technology, 1996. 2. Van de Beek, J.J., Synchronization and Channel Estimation in OFDM Systems, PhD Thesis, Luleå, Sweden: Luleå University of Technology, 1998. 3. Chiu, Y., Markovich, D., Tang, H., and Zhang, N., OFDM Receiver Design, EE225C Final Report Fall 2000, Berkley, California, USA, 2000. 4. Schulze, H., and Lüders, C., Theory and Applications of OFDM and CDMA, Wiley, 2005. 5. Meyr, H., Moeneclaey, M., and Fechtel, S.A., Digital Communication Receivers, Wiley, 1998. 6. IEEE Standard 802.11a1999: Part 11, Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications HighSpeed Physical Layer in the 5 GHz Band, 1999. 7. IEEE Standard 802.162004: Part 16: Air Interface for Fixed Broadband Wireless Access Systems, 2004. 8. http://www.dvb.org 9. Yee, N., Linnartz, J., and Fettweis, G., MultiCarrier CDMA In Indoor Wireless Radio Networks, Proc. IEEE Int. Conf. on Personal Indoor and Mobile Radio Communications (PIMRC), Yokohama, Japan, 1993, pp. 109– 113. 10. Galli, S., and Logvinov, O., Recent Developments in the Standardization of Power Line Communications within the IEEE, IEEE Commun. Magazine, 2008, vol. 46, no. 7, pp. 64–71. 11. Peled, A., and Ruiz, A., Frequency Domain Data Transmission Using Reduced Computational Complexity Algorithms, Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Denver, Colorado, USA, 1980, pp. 964–967. 12. Tourtier, P.J., Monnier, R., and Lopez, P., Multicarrier Modem for Digital HDTV Terrestrial Broadcasting, Sig nal Proc.: Image Commun.1993, vol. 5, nos. 5–6, pp. 379–403. 13. Sandell, M., van de Beek, J.–J., and Börjesson, P.O., Timing and Frequency Synchronization in OFDM Sys tems Using the Cyclic Prefix, Proc. IEEE Int. Symp. on Synchronization, Essen, Germany, 1995, pp. 16–19. 14. Lee, D., and Cheon, K., A New Symbol Timing Recovery Algorithm for OFDM Systems, IEEE Trans. Consum. Electron., 1997, vol. 43, no. 3, pp. 767–775. 15. US Patent 5602835, 1997. 16. Van de Beek, J.J., Sandell, M., and Börjesson, P.O., ML Estimation of Time and Frequency Offset in OFDM Systems, IEEE Trans. Signal Proc., 1997, vol. 45, no. 7, pp. 1800–1805. 17. Van de Beek, J.J., Sandell, M., and Börjesson, P.O., On Synchronization in OFDM Systems Using the Cyclic Prefix, Proceedings of Radiovetenskaplig Konferens (RVK), Luleå, Sweden, 1996, pp. 663–667. 18. Benson, D., MATLAB Central Carrier & Symbol Timing Recovery, http://www.mathworks.com/matlabcen tral/fileexchange/6742 19. Pan, X., Zhou, Y., Ma, S., and Ng, T.S., An Improved Derivative Method for Symbol Synchronization in OFDM Systems, Proceedings of IEEE Wireless Communications and Networking Conference (WCNC), Hong Kong, China, 2007, pp. 2463–2467. 20. Huq, A.T., Panayirci, E., and Georghiades, C.N., ML NDA Carrier Phase Recovery for OFDM Systems, Pro ceedings of IEEE International Conference on Communications (ICC), Vancouver B.C., Canada, 1999, vol.2, pp. 786–790. AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 3
2010
170
ABOLTINS, KLAVINS
21. Panayirci, E., and Georghiades, C.N., Carrier Phase Synchronization of OFDM Systems over Frequency Selective Channels via EM Algorithm, Proceedings of IEEE Vehicular Technology Conference (VTC), Houston, Texas, USA, 1999, vol. 1, pp. 675–679. 22. Panayirci, E., A New Algorithm for Joint Time and Phase Synchronization in OFDM Systems, Proceedings of the 11th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), Lon don, UK, 2000, vol. 2, pp. 819–823. 23. Dempster, A.P., Laird, N.M., and Rubin, D.B., Maximum Likelihood from Incomplete Data via the EM Algo rithm, J. Roy. Stat. Soc., Ser. B (Methodol.), 1977, vol. 39, no. 1, pp. 1–38. 24. Vahalin, A., and Holte, N., Optimal Finite Duration Pulses for OFDM, IEEE Trans. Commun. , 1996, vol. 44, no. 1, pp. 10–14. 25. Landström, D., Arenas, J.M., van de Beek, J.J., Börjesson, P.O., Boucheret, M.L., and Ödling, P., Time and Frequency Offset Estimation in OFDM Systems Employing Pulse Shaping, Proceedings of IEEE International Conference on Universal Personal Communication (ICUPC), San Diego, California, USA, 1997, pp. 279–283. 26. Yang, B., Letaief, K.B., Cheng, R.S., and Cao, Z., Burst Frame Synchronization for OFDM Transmission in Multipath Fading Links, Proceedings of IEEE Vehicular Technology Conference (VTC), Houston, Texas, USA, 1999, vol. 1, pp. 300–304. 27. Schmidl, T.M., and Cox, D.C., Robust Frequency and Timing Synchronization for OFDM, IEEE Trans. Com mun., 1997, vol. 45, no. 12, pp. 1613–1621. 28. Höher, P., TCM on FrequencySelective LandMobile Fading Channels, Proceedings of the Tirrenia Interna tional Workshop on Digital Communications, Tirrenia, Italy, 1991, pp. 317–328. 29. Edfors, O., Sandell, M., van de Beek, J.J., Wilson, S.K., and Börjesson, P.O., OFDM Channel Estimation by Singular Value Decomposition, Proceedings of IEEE Vehicular Technology Conference (VTC), Atlanta, Georgia, USA, 1996, pp. 923⎯927. 30. Chini, A., Multicarrier Modulation in Frequency Selective Fading Channels, PhD thesis, Ottawa: Carleton University, 1994. 31. Li, Y., Cimini, L.J., Jr., and Sollenberger, N.R., Robust Channel Estimation for OFDM Systems with Rapid Dispersive Fading Channels, IEEE Transactions on Communications, 1998, vol. 46, no. 7, pp. 902–915. 32. Negi, R., and Cioffi, J., Pilot Tone Selection for Channel Estimation in a Mobile OFDM System, IEEE Trans. Consum. Electron., 1998, vol. 44, no. 3, pp. 1122–1128. 33. Aboltins, A., and Klavins, D., Synchronization for OFDMBased Communication System: a Brief Overview, Riga Technical University 50th International Scientific Conference, Riga, Latvia, 2009.
AUTOMATIC CONTROL AND COMPUTER SCIENCES
Vol. 44
No. 3
2010
