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Window Function and Interpolation Algorithm for OFDM Frequency-Offset Correction R. Zhang, T. T. Tjhung, Senior Member, IEEE, H. J. Hu, and P. He
Abstract—In orthogonal frequency-division multiplexing (OFDM) digital communication systems, an uncorrected frequency offset can lead to a severe degradation of the system performance. This paper describes a technique to mitigate the influence of frequency offset on system performance by applying a weighted discrete Fourier transform (DFT) to a novel OFDM frequency-assignment scheme in which null carriers are regularly inserted among modulated data carriers. This frequency-assignment scheme makes possible the application of a window function at the demodulator to broaden the signal spectrum and renders the demodulated signal more immune to the influence of frequency offset. In addition, a variant of this frequency-assignment scheme also leads to a DFT-based measurement technique for easy and accurate determination of the frequency offset. Interpolation algorithms and frequency-assignment schemes suitable for both initial frequency acquisition and subsequent frequency tracking are described in detail. The Rife and Vincent (Class-I) windows are examined and their estimation results are obtained for an additive white Gaussian noise (AWGN) and a multipath fading channel. Index Terms—Discrete Fourier transform; frequency offset correction and tracking; interpolation algorithm; orthogonal frequency division multiplexing; window function.
I. INTRODUCTION
O
RTHOGONAL frequency-division multiplexing (OFDM) is a bandwidth-efficient digital communications scheme that was first proposed by Chang [1]. Compared with the conventional time-domain single-carrier modulation (SCM), OFDM can be viewed as a frequency domain multicarrier modulation (MCM) technique. As a result, OFDM is robust to time-domain impulsive interference and multipath propagation effects that are the main obstructions to high data rate transmission for the SCM scheme. However, as a parallel transmission technique, OFDM is very sensitive to frequency-domain interferences such as frequency-offset and tone interference. Many factors can lead to frequency-offset interference; the most evident are the movement-induced Doppler shift and the frequency difference between local oscillators in the transmitter and the receiver. The frequency offset, after being normalized to the actual frequency spacing between transmitted carriers, Manuscript received March 24, 2000; revised March 26, 2001 and October 24, 2002. R. Zhang is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA (e-mail:
[email protected]). T. T. Tjhung and P. He are with the Institute for Communications Research, Singapore 117674, Singapore (e-mail:
[email protected]). H. J. Hu is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singapore. Digital Object Identifier 10.1109/TVT.2003.811333
can be divided into an integer-offset and a residual part with an absolute value of less than 0.5. The integer offset manifests itself as a frequency-index mismatch between the transmitted and received carriers while a nonzero residual offset can disturb the orthogonality between the discrete Fourier-transform (DFT) demodulated carriers. As a result, the signal amplitude of each demodulated carrier is reduced and in the demodulated signal of one carrier, there is interference coming from other carriers, so-called intercarrier interference (ICI). These effects can reduce the signal-to-interference ratio (SIR) in the demodulated signal and can degrade the system performance. In Section II of this paper, we will describe a novel OFDM frequency-assignment scheme in which null carriers are inserted among the modulated data carriers. We will also show that an OFDM symbol shaping with a proper window function on this novel frequency-assignment scheme leads to a SIR in the demodulated signal, which is more immune to the influence of the ICI. In Section III, we will introduce an accurate DFT-based frequency-estimation technique using this new OFDM scheme and so-called Rife and Vincent (Class-I) window functions. For a DFT-based OFDM demodulation, the implementation of our estimation algorithm is very simple indeed. The use of DFT in frequency estimation for single and multiple tones from a finite number of noisy discrete-time observations is first discussed in [2] and [3]. In [3], a window-functionbased algorithm is also used to suppress multifrequency tone interference. Some similar works on window-function-based estimation have also been reported in [4] and [5]. Rife and Vincent [6] provided a detailed discussion on estimation algorithms implemented for a class of window functions that later carried their names. In [7], Nogami and Nagashima introduced a DFTbased frequency-offset estimation algorithm using the Hanning window for OFDM systems. In Section III of this paper, we will generalize the estimation algorithm in [7] with the Rife and Vincent (Class-I) window functions [6]. We will also modify this algorithm to make it more suitable for real-time frequency estimation and tracking. In addition, we will provide a detailed analysis of the estimation accuracy and bit-error rate (BER) performance of the OFDM system with our proposed frequency-offset estimation algorithm and assuming quaternary phase-shift keying (QPSK) modulation. Many algorithms have been proposed for OFDM frequencyoffset estimation [7]–[13]. For most of these algorithms, frequency-offset estimation is implemented in two steps; the integer offset is first estimated and then the residual offset is determined. In general, these algorithms can be classified into the following two categories. The first is “blind” algorithm, i.e., no special training signals are required. This is usually achieved
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ZHANG et al.: WINDOW FUNCTION AND INTERPOLATION ALGORITHM FOR OFDM FREQUENCY-OFFSET CORRECTION
by exploiting some time- or frequency-domain redundancy in the OFDM signal structure. One typical example is the joint OFDM symbol-timing and frequency-synchronization scheme described in [8]–[10], which makes use of the cyclically extended guard interval inserted in the OFDM symbol frame. Another estimation algorithm using the signal-subspace resolution method [11] exploits the frequency-domain redundancy, i.e., the unused anti-aliasing carriers at each side of the signal spectrum, usually referred to as virtual carriers. In this algorithm, both integer and residual components of the frequency offset can be estimated simultaneously. In general, blind algorithms are particularly suitable for frequency tracking because they require no special signal-structure arrangements. However, they are not suitable for initial frequency acquisition because an accurate estimation usually needs an averaging over a large number of OFDM symbols. And in high-rate package-data transmission, the time for initial frequency synchronization needs to be as short as possible, preferably only one or two symbol intervals. In such cases, the second class of estimation algorithm that uses special training signals to provide fast synchronization may be more suitable. One example is the algorithm proposed in [12] and [13], which computes the differential phase between one OFDM block and its cyclic repetition. This algorithm can be implemented in either the time [12] or frequency domain [13]. Note that the algorithm mentioned earlier that uses the cyclically extended guard interval could be considered as a special case of that in [12] in which only a small fraction of signal block is cyclically repeated as the guard interval. However, the maximum number of carries that can be transmitted for the algorithm in [12] and [13] is only half of the total number of OFDM carriers, which may be too low for this algorithm to be suitable for frequency tracking. In Section III of this paper, we will describe OFDM symbol structures that can be efficiently implemented for initial frequency acquisition and subsequent frequency tracking. II. A NOVEL OFDM CARRIER- ASSIGNMENT SCHEME AND FREQUENCY-OFFSET CONTROL In this section, we will first describe a novel OFDM carrier-assignment scheme in which null carriers are inserted among modulated carriers. This frequency-assignment scheme is a generalized version of the one proposed by Nogami and Nagashima in [7]. Then we will analyze the effect of the frequency offset on the performance of this novel OFDM carrier-assignment scheme.
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conventional OFDM carrier-assignment scheme, all except anti-aliasing carriers at each side of the allocated bandwidth active carriers. Fig. 1(a) are transmitted, resulting in shows a novel carrier-assignment scheme in which only out carriers with frequency are modulated with of , . Note that and . Let , be the frequency spacing between two adjacent transmitted carriers. B. Frequency Offset and SNIR In a channel with frequency offset, the received complex , corresponding to the Nyquist baseband equivalent signal sampling rate of , is
(1) is the channel-induced frequency offset, is the where transfer function of the channel at frequency and is assumed to be unchanged during one OFDM symbol period (this correare sponds to a slow-fading radio frequency channel), and the equivalent lowpass additive white Gaussian noise (AWGN) samples and are assumed to be independent complex Gaussian random variables, each with zero-mean and variance, . Let the frequency offset be written as (2) where is an integer and denotes the residual offset . Because we have assumed that the OFDM lowpass-equivis asalent signal is band-limited, the maximum value of sumed to be less than the number of unused anti-aliasing carriers at each edge . The conventional demodulator for the received OFDM disignal is implemented by feeding the sequence rectly into an -point DFT processor. However, with the new carrier-assignment scheme described in Fig. 1(a), we present here an alternative demodulation procedure by first with a real-valued window sequence , multiplying before the DFT demodulation. is assumed to For convenience, the window sequence . The sequence satisfy the condition is then fed into the DFT demodulator to obtain the -point , i.e., weighted DFT sequence
A. Novel OFDM Carrier-Assignment Scheme Let the OFDM symbol period excluding the cyclically (seconds) and the lowpass extended guard interval be equivalent bandwidth (two-sided) of the OFDM signal be (Hz). Assume that the time-frequency product is equal and, therefore, in one OFDM symbol, to an even integer orthogonal carriers with frequencies , a total number of can be transmitted. The frequency in our analysis is normalized to the actual intercarrier frequency . In addition, we assume that is large 16 , spacing which is usually true for most practical OFDM systems. In the
(3) Here we define the discrete-time Fourier transform (DTFT) of as (4)
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(a)
(b) Fig. 1.
(a) Novel OFDM carrier assignment scheme and (b) spectral sequence of the demodulated OFDM signal.
It can be seen from (3) and (4) that the sequence is actuat a set of integer frequencies ally obtained by sampling , . correIn Fig. 1(b), we show the spectral sequence , sponding to the set of transmitted carriers at frequencies, in Fig. 1(a). It can be seen that there spectrum clusters in , corresponding to the origare carriers with modulation signals , inally transmitted . In each cluster, more than one nonzero DFT elements is observed due to the fact that the window func, tion tends to broaden the signal spectrum. Let denote the element in for and denote its adjacent elements. Then
(5) denotes the largest integer that is less than its where , and represent, respectively, argument. . the signal, the ICI, and the noise components in and denote, respectively, the DTFT of Let and the squared window sethe window sequence , i.e., and quence . Then
condition that and where denotes the delta function. That is, the modulation points are . independent with zero-mean and constant average power In addition, the average channel gain is assumed to be constant, . With these assumptions and a large , i.e., is a Gaussian random variable with we can regard that zero mean and variance
(8) We define the average OFDM symbol energy per car. It can be easily shown that rier as where is the one-sided power spectral density of the AWGN in the radio fredenote the average quency (RF) channel. Let signal-to-noise-plus-interference ratio (SNIR) in the demodu, then lated signal
(9)
where
(6)
(10)
(7)
(11)
, is the DFT of the and and can be shown to be a set weighted noise sequence of complex Gaussian random variables with zero mean and cowhere denotes variance the complex conjugate. We now describe the statistical properties of the ICI compo. First, we assume that the signal satisfies the nent
(12)
(13)
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TABLE I ENBW AND SL(0:5) FOR CLASS-I RIFE AND VINCENT WINDOWS OF ORDER Z
, it can be shown that the DTFT of the Class-I Rife and Vincent windows of order is Fig. 2. The normalized amplitude spectrum for Rife and Vincent (Class-I) 1, 4. The corresponding time response is given window function of order Z in the box.
(16)
In (9), the processing loss [4] is a window function-performance index to reflect the reduction of the SNIR after the . As given in DFT demodulation of the weighted signal and the (10), it is the ratio of the squared scalloping loss equivalent noise bandwidth (ENBW). The scalloping loss reflects the amplitude reduction of the demodulated DFT signal due to the residual frequency offset . The equivalent noise bandwidth ENBW is defined as the ratio of the weighted to the nonweighted (assuming a rectangular window) noise powers. It is a measure of the noise-power enhancement after windowing. is a measure of the amount of residual The normalized ICI ICI in the demodulated signal after the side-lobe spectrum suppression of the window function. It is dependant on , , and the frequency spacing between carriers. In summary, the application of the window function has three main effects, as follows: broadening the spectrum of the demodulated signal, suppressing the ICI component, and amplifying the noise component in the demodulated signal.
In Fig. 2, we show the normalized amplitude spectrum for the Class-I windows of order , 4, and their corresponding time responses. From the spectrum plots, it can be seen that a higher order Class-I window suppresses the side-lobes more substantially, but broadens the main-lobe width by a larger amount. This implies that a higher order window is capable of suppressing the amount of ICI in (9) more substantially. However, this and, thus, the only happens if the spacing between the transmitted carriers is kept at least at twice of the normalized one-sided main-lobe for the Class-I window of bandwidth (BW), which is order . It should be noted that the Class-I windows satisfy the for integer and . This condition that implies that the orthogonality between the transmitted carriers is is maintained, provided that the intercarrier spacing kept to be at least BW. at for the Class-I Table I shows the ENBW windows of order from 0 to 4. Note that although a higher order window tends to have less scalloping loss due to its larger BW, it suffers from a greater ENBW due to the weighted noise. , which is a measure of Fig. 3 shows the processing loss and ENBW for the Class-I winthe combined effect of dows of order from 0 to 4. From the above results, the choice of the Class-I window for in (9) can be made for some special cases. a minimized , 1) For a channel without frequency offset, i.e., in (9) becomes . Hence, the the rectangular window should be chosen because it gives the minimum processing loss at zero frequency offset, i.e., , compared with all the other Class-I windows becomes equal as is shown in Fig. 3. In this case, . to 2) For a noisy channel with frequency offset, i.e., when is quite low such that , the can then be approximated as . In this case, the window function should be chosen to ; that is, to minimize the processing maximize
=
C. Class-I Rife and Vincent Window Function As an illustration of the above parameter definitions relating to the window function, we describe here a family of Class-I Rife and Vincent window functions [6] with weights
(14) where denotes the order of the window function and the coin (14) are efficients (15) Note that the Class-I windows of order 0 and 1 are the wellknown rectangular window and the Hanning window, respecis large, i.e., and tively. Under the condition that
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Fig. 3.
The processing loss versus the residual-frequency offset for Rife and Vincent (Class-I) window function of order Z from 0 to 4.
loss. Hence, from Fig. 3, a lower order Class-I window should be chosen. 3) For a noiseless channel with frequency offset, i.e., when is high enough to make , is then approximated as . In this case, the window function chosen should be able to minimize and, as well, have a moderate value of the processing loss.
D. SNIR for Rectangular and Hanning Window Functions It can be easily seen from (9) that only the residual offset can affect . component in the frequency offset is an even function of . In order to study In addition, the effect of on the demodulated carrier SNIR, the in (9) is plotted versus in Fig. 4 for the rectangular and the dB. It is assumed that Hanning windows and for the OFDM carrier-assignment scheme in Fig. 1(a), the intercarrier spacing between the transmitted carries is equal, i.e., , . In addition, is chosen to be at least for each plot in order to maintain the orthogonality between transmitted carries at zero . We also assume that the spectrum utilization efficiency , defined as the ratio of to , in the coris maximized. This is achieved when responding carrier-assignment scheme for each plot. It is observed that, compared with the case without windowing (i.e., with the rectangular window), the OFDM signal with the Hanning window is more immune to the residual-frequency offset and, hence, leads to a less SNIR degradation as increases. However, at zero , the OFDM signal with the Hanning window has a 1.76-dB loss in the SNIR, compared with the case without windowing, due to the fact that the Hanning windowing amplifies the noise component. Another price that has to be paid for
the Hanning windowing is that the maximum spectrum utilization efficiency is now only half that for the case without windowing. This is because the main-lobe bandwidth of the Hanning window is twice that of the rectangular window. Therefore, a higher level constellation for data modulation must be used with the Hanning window to compensate for the loss in spectrum-utilization efficiency. Because a significant improvement of the SNIR can be achieved with a higher order Class-I window when the residual frequency offset is present, the carrier-assignment scheme in Fig. 1(a), together with a higher level constellation of carrier modulation, is a better solution than the conventional OFDM carrier assignment scheme. III. FREQUENCY-OFFSET ESTIMATION AND TRACKING In Section II, we have described and analyzed a novel OFDM carrier-assignment scheme in which frequency nulls are assigned between modulated carries. This scheme makes possible the use of window function to broaden the received signal spectrum and by so doing renders the demodulated signal more immune to the effect of residual frequency offset in the received OFDM signal. As is shown in [7], this new OFDM scheme, together with the Hanning window, can also provide a frequency-domain estimation algorithm for the frequency offset. In this section, we will spend our effort to make a more thorough analysis of this algorithm. In addition, we will generalize this algorithm to make it applicable to other window functions and suitable for both initial frequency acquisition and subsequent frequency tracking. Lastly, we will discuss some issues on the algorithm implementation. A. Initial Frequency Acquisition 1) Carrier-Assignment Scheme: In Fig. 5(a), we illustrate a typical carrier-assignment scheme suitable for initial frequency
ZHANG et al.: WINDOW FUNCTION AND INTERPOLATION ALGORITHM FOR OFDM FREQUENCY-OFFSET CORRECTION
Fig. 4. The SNIR versus the residual frequency offset for E =N
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= 30 dB and Rife and Vincent (Class-I) window function of order Z = 0, 1.
(a)
(b) Fig. 5.
Carrier-assignment scheme for (a) initial frequency acquisition and (b) frequency tracking.
acquisition. Here, we assign maximum number of carriers number of transmitted pilot that can be transmitted 256, 7, frequency of the first pilot carrier 96, carriers spacing between adjacent pilot carriers , , where is a pseudorandom sequence known to the receiver. In this case, . This pilot OFDM symbol is transmitted before data communication is commenced. It can be used to implement many tasks such as frequency and timing synchronization as well as channel estimation. In general, the initial frequency offset is a random variable and may be of a large value in the order of sev. Therefore, the initial frequencyeral intercarrier spacing acquisition range needs to be sufficiently large in order to accommodate anypossible frequency-offset values. This end is achieved by assigning unequal and pseudorandom spacing between transmitted carriers, as shown in Fig. 5(a) and described in [7]. 2) Maximum-Likelihood Estimation of Frequency Offset: As shown in Appendix A, a maximum-likelihood for the carrier (ML) estimation of the frequency offset assignment scheme in Fig. 5(a) can be obtained to be (17)
where is the DTFT of the and corresponds to the maximum received signal is the defined in (4) for acquisition range. Note that the rectangular window function. For an ML estimator, the estimation variance tends to be equal to its unbiased Cramer–Rao (CR) lower bound. In Appendix A, the conditional estimation variance of the ML estimator in (17) is derived as (18) is the average symbol energy over all transwhere mitted carriers. It is to be noted that the conditional estimation variance in (18) for this ML estimator is in fact not dependent . on the actual frequency offset for The ML estimation by (17) is actually a search of . In all the values of within the range of practice, this search routine is difficult to implement if a fast algorithm for the frequency acquisition is needed. Another difficulty for implementing this ML algorithm is that at the initial frequency-acquisition stage, the channel estimation for in (17) may not be available before the frequency synchronization is successfully achieved. Hence, we have to consider suboptimal estimation algorithms to implement this search routine.
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(a)
(b) Fig. 6.
The discrete-time cross-correlation function g (d).
As we have mentioned earlier, the frequency offset in the initial acquisition is usually random and of a large value. Therefore, as is suggested in [2], the search routine for can be efficiently implemented in a two-step operation, i.e., a coarse-search step followed by a fine-search step. If we re-exas press (19) , is a nonnegative integer and dewhere notes the integer closest to its argument. It can be easily verified . It should be noted that when , and that in (19) become and defined in (2). In the coarse search, we will estimate the value of and locate the unknown frequency in a small range of . In the subsequent fine offset search, we will then proceed to estimate the residual frequency offset . 3) The Coarse Search: In this search, we first evaluate at a set of discrete frequencies defined as
(20) . The practical value of can be taken to where or , with the corresponding value of to be 0 or 1. be , the sequence becomes the -point Note that if , . For , the DFT of can be obtained by constructing a new sequence sequence , for which for and otherwise and then feeding
into a
(21) -point DFT processor.
Let denote the discrete-time function of two sequences, , i.e.,
cross-correlation and for
(22) is related to the set of frequencies assigned for where in Fig. 5(a), i.e., for transmitted carriers and is zero otherwise. Then, the output of the coarse search, which is a suboptimal estimate of in (19), becomes (23) , corresponding to the In Fig. 6(a) and (b), the function carrier assignment scheme in Fig. 5(a), is plotted for two cases, and , respectively. The value of in each . To illusplot is shown in Table II. For both cases, , it is assumed that there is no noise trate the mean value of and fading in the channel. In Fig. 6(a) and (b), one global peak . and several local peak regions can be seen in each plot of It can be shown that the number of local peak regions and their amplitudes relative to the global peak are actually determined by and the maxthe autocorrelation function of the sequence imum frequency-acquisition range. It is obvious that the global peak region corresponds to the actual frequency offset and that satisfies the condition the global peak position (see Table II). It can be seen in Fig. 6(a) that for , the global peak tends to become a single peak when the residual offset approaches 0, but splits into two equal peaks when approaches 0.5. A similar observation can be made for the in Fig. 6(b). In summary, when the frecase with approaches even-integer multiples of , quency offset
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TABLE II FREQUENCY-OFFSET VALUES IN FIG. 6
i.e., approaches zero, a single peak can be detected in the approaches odd-integer mulglobal peak region while when , i.e., approaches , two nearly equal tiples of amplitude peaks are present in the global peak region. becomes The fact that the global peak position when the residual frequency ambiguous from approaches its limiting values of should not hinder the detection of in the coarse search. This is because, as long as the global peak is detected to be either one of these two peaks, we can correctly determine that the actual frequency will be between and . However, offset as will be explained in the following fine search, the correct choice of the actual peak in the global peak region from its adjacent elements is critical for the accurate estimation of the residual frequency offset . a) Peak-detection probability: In order to quantify the performance of the residual frequency-offset estimation algorithm to be formulated later in the fine search, we define for the correct detection here the peak-detection probability . In Appendix B, of the actual peak as approaches , which is an even function we derive an expression of of . Here, we only show in Fig. 7(a) and (b) the plots of versus for values of at 30, 35, and 40 dB and , respectively. It can be seen and for , say 10 , the threshold that for a specific value of beyond which the correct peak cannot be detected is value . This indicates a monotonically increasing function of can lead to that given a specific value of , increasing and, thus reduce, the range of within a higher value of which the peak cannot be correctly detected. : When and, b) Peak detection when may become indistinguishdue to the noise corruption, as approaches or from able from as approaches . Hence, we have to find an alternative solution for the correct detection of the actual global peak . One practical way is to set a threshold value position defined (with reference to Appendix B) as
(24) and compare it with the ratio . If either one of these two ratios is greater than and will then double the DFT size , i.e.,
, , we ,
(a)
(b) Fig. 7. 1
0P
versus .
and reset if or if . To illustrate how this algorithm works, we and . It can look at Fig. 6(a) for the case of be seen that there are two peaks in the global peak region with 4 or 5 due to . Hence, we can conclude that or that of with a high probability, the actual value of will be greater than . However, if we double the but with , DFT size for the same value of becomes 0 and it can be seen that in Fig. 6(b) there is only one . single peak present and 4) The Fine Search: After the coarse search and the correct determination of , the residual frequency offset within the will be accurately estimated in this section. range of , and In the coarse search, peaks
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are found in . It is, thus, conjectured and its adjacent that can be estimated by interpolating in each peak region, because these elements elements bear the information on and, as well, have a high SNIR. How. In this case, as ever, this may not be true when approaches zero, it can be easily seen from Fig. 6(a) that the and decreases to zero and, hence, SNIR of both cannot be used for estimating . This is because the normalized one-sided main-lobe BW for the rectangular window is only one. Therefore, it is reasonable here to apply a window function can be in order to broaden the signal spectrum and then in used to estimate . We will consider the DTFT of (4) at a set of discrete frequencies, i.e.,
The final estimation of the frequency offset and fine searches becomes
after the coarse (31)
or in (28). It should where can be chosen to be either be noted that for a high SNIR, can be assumed to be correctly determined in the course search. Therefore, the estimation acin (31) is in fact mainly determined by . curacy of c) Estimation variance of : We now consider the estimation accuracy of . First, for a high SNIR, the estimator of in (28) can be assumed to be unbiased. Hence, the estima, can be taken as a good accuracy tion variance of index, which is derived in Appendix C as
(25) as in the coarse search. Similar to , peaks in , , and . With the same approach as in Section II, we and its adjacent elements in terms can express each of the signal, ICI, and noise components. For a high and negligible ICI, we can assume that and for each peak region and
where there are also
(26) We note that because the coefficients of the window function is symmetric are real values, the amplitude spectrum . is then estimated and, hence, from (26) from each peak region to be
(32) is dependent on the actual From (32), it can be seen that residual frequency offset and is inversely proportional to both and . We will next look at the effect of Class-I Rife and Vincent . In Fig. 8, we show given window function on dB, , and various by (32) versus for and DFT size . combinations of window function order Note that the acquisition range of in each plot with DFT size is . The corresponding plot of is with respect to the vertical just the reflection of that of axis and, thus, is not shown in Fig. 8. We also include, in the for the ML method given same figure, the curve of for the differential phase in (18) and the curve of (DP) method given by (33) [12] and [13], for comparison (33)
(27) denotes the inverse function of where output of the fine search becomes
. The final
(28) where is the combining weight that minimizes the estimation (see Appendix C) and is equal to variance of (29)
is not available at the freIf the channel estimation for quency-synchronization stage, the combining weight in (29), under the condition of a high SNIR, can be approximated as (30)
Several observations can be obtained from Fig. 8. First, is higher than for all combinations of window-function order and DFT-size . This is due to the fact that our algorithm, rather than using the whole spectrum, and , from each uses only two DFT elements, i.e., . This obviously leads to a loss of peak region to determine information. Second, the higher order Class-I window results , compared with the lower order in a larger value of in (32) is inversely Class-I window. This is because and from Fig. 3 a higher order window proportional to in the given range of . gives a smaller value of is of the same order as . This is not Finally, surprising because the DP method in [12] and [13] is actually a ML method as well. It can also be seen that the acquisition is fixed to be . range of and : The estimation acd) Choosing between can be further improved if we assign a rule for curacy of and as the final estimate of in (31). choosing between increases significantly From Fig. 8, it can be seen that , the estias becomes less than zero. However, for , by symmetry, should be much smaller mation variance for
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Fig. 8.
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var[^ ] versus for E =N = 30 dB, N = 256, and Rife and Vincent (Class-I) window function of order Z = 0, 1, 2.
Fig. 9. var[^ ] versus for E =N = 20, N = 256 and Rife and Vincent (Class-I) window function of order Z = 0, 1.
than tween
. This motivates us to set the rule of selection beand to be if if
(34)
From (35),
can be reexpressed as (36)
can be shown, with a similar approach as in Apwhere pendix B, to be
which can be implemented as (37)
if (35) if
is defined in (A2-9) and [see (38) at the bottom of where the next page]. In Fig. 9, we show the new estimation variance
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given in (36) for the rectangular window with and the Hanning window with , dB, and . Compared with the corresponding plots of and in the same figure, it can be seen that a significant reduction of the estimation variance is achieved by applying the selection rule given by (35). In addition, it can be seen that is an even function of . 5) Summary of the Initial Frequency Acquisition: Finally, we summarize the algorithm for the initial frequency acquisition as follows: Step 1) Choose a proper DFT size and obtain , . in (22) and deStep 2) Calculate . Compare the ratio , for termine with . If both of the ratios , go to Step 4; are smaller or equal to , and go otherwise, denote Step 3. and obtain Step 3) Double the DFT size . Reset . the new in (23) and set Step 4) Determine . , Step 5) Obtain , for a chosen window function. Calculate . If the difference the difference is greater than or equal in (28) to determine ; to zero, use . otherwise, use Step 6) Estimate the actual frequency in (31). Control the local offset as oscillators in the receiver to correct this frequency-offset error. The frequency-synchronization system will then switch from the initial acquisition mode to the tracking mode. 6) BER Performance Comparison: In this section, we will compare the performance of different methods presented earlier for OFDM frequency-offset estimation based on an analysis of the BER in the AWGN channel. For simplicity, we assume that the carrier modulation for OFDM data symbols is QPSK. The BER corresponding to a normalized frequency offset will then become (39) is the complementary error function as defined in where is defined in (9). It is assumed here that (A2-9) and the rectangular window is applied to the OFDM data symbol.
The initial residual-frequency offset before correction , defined in (19), can be assumed to be uniformly distributed bewhere is the acquisition range in the fine tween and for the case search. For example, for the DP method , of our proposed method . Let be the resultant frequency offset after the correction, i.e., where is the estimated frequency offset in the fine search and is its probability density function (pdf) conditional on . By numerical simulation, we have verified that for both the DP and our proposed method, under the assumption of high in the pilot symbol, is approximately normal with zero demean and variance equal to the estimation variance fined in (33) and (36), respectively. The average BER for the OFDM data symbol after the offset correction can be then expressed as (40) where (41) In Fig. 10, we show the data symbol BER after the offset corin (41) versus the initial frequency offset for the rection OFDM data symbols. For comparison, the BER before the offset for correction is also shown in the same figure. The SNR the pilot symbols for offset estimation is set to be 20 and 30 dB, respectively, indicated by the dash and solid lines in the figure. for the data symbols is set to be The SNR per carrier 13 dB. It is noted that the BER of the data symbol improves significantly after the frequency-offset correction. In addition, the BER improvement after the correction is not dependent on due to the fact that the estimation variance of the frequency in (32) and (33) does not change significantly with offset . It is also noted that with the pilot symbol SNR increasing from 20 to 30 dB, the BER improves for all methods is inversely proportional to . due to the fact that becomes equal to 30 dB, the BER conHowever, when verges to that of the single-carrier QPSK in the AWGN channel for all methods. in The average BER over all initial frequency offsets (40) is plotted in Fig. 11 versus different pilot symbol SNR . The data symbol SNR per carrier is still 13 is inversely proportional to , the dB. Because BER performance also increases proportionately. However, is greater then 32 dB, all methods provide when approximately the same BER performance as the single-carrier QPSK. It is inferred from this that the BER performance is linearly related to the estimation variance up to a threshold . Beyond this threshold value of the pilot symbol SNR
(38)
ZHANG et al.: WINDOW FUNCTION AND INTERPOLATION ALGORITHM FOR OFDM FREQUENCY-OFFSET CORRECTION
Fig. 10.
BER versus initial frequency offset.
Fig. 11.
BER versus pilots symbol SNR.
SNR, reducing the estimation variance further has negligible effects on improving the BER performance for both the DP and our proposed method. B. Frequency Tracking After the initial frequency acquisition, the frequency-offset error in the receiver oscillators will be corrected and data communication is initiated. However, due to the various random factors in the transmission channel, such as Doppler shift and frequency drift in the oscillators, the frequency offset may be consistently present in the demodulated signal and if not compensated accurately, may cause severe performance degradation.
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Hence, frequency tracking is still necessary to maintain reliable data communications. 1) Carrier-Assignment Scheme: In Fig. 5(b), we show a carrier-assignment scheme in which frequency tracking is incor, porated into data transmission. We assign number of unused anti-aliasing carriers (for each side) 32, number of transmitted data carriers 170, number spacing between adjaof transmitted pilot carriers 4, 1, spacing between adjacent data cent data carriers frequencies for pilot carriers and pilot carriers 4, and . Obviously, in this OFDM symbol, we have chosen four pilot carriers out of the total of 174 transmitted
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Fig. 12. channel.
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var[1f^] versus 1f for E =N = 30 dB, Rife and Vincent (Class-I) window function of order Z = 0, 1 in the AWGN channel and the multipath
carriers for frequency tracking. Compared with the carrier-assignment scheme for initial frequency acquisition in Fig. 5(a), the overhead in terms of the number of pilot and null carriers is now reduced significantly and the spectrum utilization effifor Fig. 5(b) is ciency
(42) is determined completely From (42), given , , and , . Obby and the spacing between pilot and data carriers are kept small. By adviously, is large when both and , a flexible value of can be obtained. With justing and is equal to 68% as comthe parameter values in Fig. 5(b), for the DP method in [12] and [13], pared to the maximum and . which is only 37.5% assuming that Unlike in the initial frequency acquisition, the possible values of frequency offset in the tracking mode are usually limited in a much smaller range. This is because the frequency offset in the OFDM system, like the Doppler shift or the frequency drift of oscillators, usually cannot change abruptly with time. Hence, the OFDM symbol with the carrier assignment shown in Fig. 5(b) can be inserted regularly into the normal OFDM data symbols to achieve frequency tracking. In addition, the fact that the value of the frequency offset in the tracking mode is limited in a smaller range leads to a simplified estimation algorithm here, compared to that at the initial frequency-acquisition stage.
This is so because the coarse search adopted at the initial acquisition stage for determining the large frequency deviation is actually unnecessary here in the tracking stage. Therefore, we can assume here that an algorithm similar to that described in the fine search with the frequency offset at the tracking stage assumed to be equal to . However, here, the ICI components in the demodulated pilot signals have to be taken into consideration because the number of null carriers in Fig. 5(b) has now been reduced significantly. 2) Simulation Results: In order to demonstrate the effect of the ICI on the estimation accuracy at the tracking stage, we implement the Monte-Carlo simulation in the AWGN channel and the multipath fading channel, respectively. In the simulation, we assume that the carrier modulation is taken to be QPSK. For the multipath fading channel, the channel is assumed to be slow paths with delays of and comfading and have , . For convenience, plex amplitude gains and the other , we assume that are independent random variables that satisfy the condition that and where denotes the . It is also inserted guard interval and is assumed to be , are independent complex assumed that Gaussian random variables with zero mean and equal variance, , for all . Then, the transfer function of the i.e., becomes channel at frequency , for (43)
ZHANG et al.: WINDOW FUNCTION AND INTERPOLATION ALGORITHM FOR OFDM FREQUENCY-OFFSET CORRECTION
Note that the defined in (1) is equal to at frequency . In Fig. 12, we show the simulation results for the estimaversus frequency offset for tion variance dB and two cases with the rectangular and Hanning window, for each plot shown is respectively. The negative half of only its symmetrical reflection with respect to the vertical axis and, hence, is not shown in the figure. Note that the frequency for each plot with offset-tracking range is . We ran the Monte-Carlo simulation 1000 DFT size times for each data point in the plots. Several new observations obcan be obtained here. First, compare the plot of shown in (36) tained for an AWGN channel with that of . It can be seen that for the rectangular window with the former is significantly greater than the latter due to the effect of ICI. However, with the Hanning window, their performances become indistinguishable and the effect of ICI has been reduced significantly by the spectrum side-lobe suppression of the Hanning window. Therefore, it can be easily seen that the Hanning window or other higher order Class-I windows should be chosen for frequency estimation at the tracking stage. Second, is also plotted for the Hanning window case in a multipath fading channel, for two values of the number of pilot carand , respectively. It is evident from (36) riers , that the estimation variance depends only on the received signal over all pilot carriers, rather than that for each indienergy the performance in a vidual pilot carrier. Therefore, for multipath fading channel is unaffected by the multipath, com, the espared to that in an AWGN channel. However, for timation variance degrades significantly because the single pilot carrier suffers from selective fading in the frequency domain. A rule of thumb here is that we should assign the number of pilot carriers (interspaced as distantly as possible) to be greater than to achieve the frequency-domain diversity of the reover all pilot carriers. This is so because the ceived energy is usually designed to be equal to the channel ratio of coherent bandwidth and, hence, the total OFDM signal bandwidth can be divided into an approximate number of independently faded regions. In the simulation, and ; hence, needs to be at least 4.
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Hence, in order to reduce the complexity of the estimation algorithm, we may apply the Class-I Rife and Vincent windows with the DFT size . Obviously, the Hanning window is a good choice for many circumstances in our earlier discussion because it can provide an accurate estimation at both initial frequency acquisition and frequency tracking. However, for the Class-I Rife and Vincent window with other cannot be obtained. DFT sizes, a simple expression of Here, we suggest that a -point linear piecewise interpolator at values of other than may be used to approximate the selected points. We have verified numerically that for the Class-I Rife and Vincent windows of order 0 to 4 and , the maximum square error between the output of a 21-point (equal-spaced) linear interpolator and the actual value of the , is less than 10 . This is much corresponding function smaller compared to the required estimation variance in most practical applications. From Nonweighted DFT 2) Obtaining Weighted DFT : Next, we will introduce a very efficient frequency-domain implementation method for obtaining the weighted DFT from the nonweighted DFT sequence , sequence with the same DFT size . This implementation is required at both initial frequency acquisition (the fine search) and frequency-tracking stages. For the Class-I Rife and Vincent windows given in (14), it can be shown that (45) From (45) for
i.e., (46)
is the sequence obtained by shifting (cycliwhere by . Equation (46) implies a cally) the sequence with the Class-I much simpler method of obtaining , window function from the nonweighted DFT sequence as compared to direct DFT processing of the product sequence .
C. Algorithm Implementation In this section, we will briefly discuss some issues on the implementation of the estimation algorithm with the Class-I Rife and Vincent window functions. in 1) Implementation of Inverse Function (27): First, we will look at the practical methods to implement the algorithm given in (27) for the residual frequency-offset in estimation. We note here that the inverse function (27) may not be of a simple form for most practical window functions. Therefore, numerical techniques are necessary to approximately. However, if the Class-I Rife determine and Vincent window function is used, it can be shown from a simple inverse function can (16) that when be obtained as (44)
IV. CONCLUSION In this paper, we introduce a novel OFDM carrier-assignment scheme in which null carriers are inserted among transmitted carriers. At the receiver, we apply window function on the received signal samples before DFT demodulation. Because the weighted signal spectrum has a broad main-lobe bandwidth and substantially suppressed side-lobe amplitude, the influence of frequency offset on the SIR of the demodulated signal can be significantly reduced. We show that the orthogonality among transmitted carriers is maintained if a Class-I Rife and Vincent window function is applied. The trade off for the use of window function is shown to be the enhanced influence of the additive noise and reduced spectrum utilization efficiency. A DFT-based algorithm suitable for both initial frequency acquisition and subsequent frequency tracking has also been presented in this paper.
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APPENDIX I MAXIMUM-LIKELIHOOD ESTIMATE OF FREQUENCY OFFSET Let and denote, respectively, length- row vectors repcomplex values of and given in resenting the . In addi(1) in the main text denote diagonal matrix with its diagtion, let , onal elements corresponding to the sequence of . Under the assumption that the in are independent complex Gaussian noise elements random variables with zero mean and equal variance , beand comes a random vector with the mean vector of the conditional joint pdf with respect to the unknown frequency given by offset
the correct detection of the global peak , which given in (22) of the main text, is achieved when maximizes is greater than all the other values of for and . As illustrated in Fig. 6 of the main text, , , which may most likely exceed the values of , are those corresponding to all the local peaks , and those corresponding to the adjacent values of , i.e., , . Hence, without loss of accuracy, as we can make an approximation of for all and
(A1-1) where mate of
denotes the Hermitian transpose. Then the ML estican be shown to be
(A1-2) From (A1-2), (17) in the main text can be derived. If the ML estimation in (A1-2) is unbiased, i.e., , the unbiased CR lower bound for the conditional estimation is given by the following expression: variance of (A1-3)
Note that that
. It can be easily shown from (A1-3)
(A1-4) Hence, (A1-5)
If we assume that the average signal power and , are both large from (A1-5), (18) in the main text can be derived.
(A2-1)
For the carrier-assignment scheme in Fig. 5(a) of the main text, we have assigned a sufficiently large number of null carriers betweenpilotcarriers.Therefore,wecanassumethattheICIcompois negligible and, hence, nent in the demodulated sequence can be neglected in the analysis. In addition, if we assume that the is sufficiently high and the sequence posses value of of a good cross-correlation behavior (i.e., high correlation peak at zero delay and low correlation values at nonzero delays) in the fre, the probability quency-acquisition range of in (A2-1) can be assumed to be one. Furthermore, from Fig. 6 , the probability of the main text, as approaches in (A2-1) can be approximated as if and if . With the above provisions, we can assume that [see equation (A2-2) at the bottom of the page]. To evaluate (A2-2), we need to describe . For the -point DFT sethe statistical prosperities of , given in (20) quence and denote, respectively, of the main text, let the peak element and its right element where . Then, and are noncentral chi-square distributed random variables and under the condition of a high and , it can be shown that [see (1) in the main text]
APPENDIX II PEAK DETECTION PROBABILITY IN THE COARSE SEARCH (A2-3)
In this appendix, we will derive an expression for the peak dedefined in Section III-A3a as the residual tection probability . In the coarse search, frequency offset approaches
if if
(A2-4)
and and
(A2-2)
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(A2-10)
(A2-5)
The estimation variance of can be approximated to be
given in (27) of the main text
(A3-5)
(A2-6) in the above equations correspond to the rectNote that angular window. For the rectangular window, it can be easily verified from (16) of the main text that (A2-7) Next, we will consider the sequence of random variables , . According to the central-limit theorem, we can assume that the pdf for large is approxiof in (A2-2) for and mately normal and, hence, the can be derived to be
with If we expand the function and into the Taylor series with respect to the point where and and then truncate the series beyond the second-order terms, we have
(A3-6) By applying (A3-1)–(A3-4) and (A3-6), it leads to (A3-7)
(A2-8)
where
is the complementary error function defined as (A2-9)
and [see (A2-10) at the top of the page]. It can be shown that and . (A2-8) also holds for APPENDIX III RESIDUAL FREQUENCY-OFFSET ESTIMATION VARIANCE In this appendix, we will derive (32) in the main -point weighted DFT sequence , text. For the given in (25) of the main text, and denote, respectively, the peak element let . Similar and its adjacent elements where and as in Appendix B, under the condition of a high and , it can be shown that negligible ICI in
(A3-8) or If we assume that the estimates obtained from for are independent, by (A3-5) and (A3-8), in (28) of the main text becomes the estimation variance of
(A3-1) (A3-2) (A3-3) (A3-4)
(A3-9) From (A3-9) and assuming be derived.
is large, (32) in the main text can
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REFERENCES [1] R. W. Chang, “Synthesis of band-limited orthogonal signals for multichannel data transmission,” Bell Syst. Tech. J., vol. 45, pp. 1775–1796, Dec. 1966. [2] D. C. Rife and R. R. Boorstyn, “Single-tone parameter estimation from discrete-time observations,” IEEE Trans. Inform. Theory, vol. 20, pp. 591–598, Sept. 1974. , “Multiple tone parameter estimation form discrete-time observa[3] tions,” Bell Syst. Tech. J., vol. 55, pp. 1389–1410, Nov. 1976. [4] F. J. Harris, “On the use of windows for harmonic analysis with the DFT,” Proc. IEEE, vol. 66, pp. 51–83, Jan. 1978. [5] C. Offelli and D. Petri, “Weighting effect on the discrete time Fourier transform of noisy signals,” IEEE Trans. Instrum. Meas., vol. 40, pp. 972–981, Dec. 1991. [6] D. C. Rife and G. A. Vincent, “Use of the discrete Fourier transform in the measurement of frequencies and levels of tones,” Bell Syst. Tech. J., vol. 49, pp. 197–228, Feb. 1976. [7] H. Nogami and T. Nagashima, “A frequency and timing period acquisition technique for OFDM systems,” Pers., Indoor and Mobile Radio Commun. (PIMRC), pp. 1010–1015, Sept. 1995. [8] J. van de Beek, M. Sandel, and P. O. Borjesson, “ML estimate of timing and frequency offset in OFDM systems,” IEEE Trans. Signal Processing, vol. 45, pp. 1800–1805, July 1997. [9] M.-H. Heieh and C.-H. Wei, “A low-complexity frame synchronization and frequency offset compensation scheme for OFDM systems over fading channels,” IEEE Trans. Veh. Technol., vol. 48, pp. 1596–1609, Sept. 1999. [10] N. Lashkarian and S. Kiaei, “Class of cyclic-based estimation for frequency-offset estimation of OFDM systems,” IEEE Trans. Commun., vol. 48, pp. 2139–2149, Dec. 2000. [11] U. Tureli, H. Liu, and M. D. Zoltowski, “OFDM blind carrier offset estimation: ESPRIT,” IEEE Trans. Commun., vol. 48, pp. 1459–1461, Sept. 2000. [12] T. M. Schmidl and D. C. Cox, “Robust frequency and timing syncrounization for OFDM,” IEEE Trans. Commun., vol. 45, pp. 1613–1621, Dec. 1997. [13] P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun., vol. 42, pp. 2908–2914, Oct. 1994.
R. Zhang received the B.Eng and M.Eng degrees in electrical engineering from the National University of Singapore in 1999 and 2001, respectively. He is currently working toward the Ph.D degree in electrical engineering at Stanford University, Stanford, CA. His current research interests include wireless communications. Mr. Zhang is a recipient of the Oversees Ph.D. Scholarship from the Agency for Science, Technology & Research (A-STAR) of Singapore.
T. T. Tjhung (SM’84) received the B.Eng. and M.Eng. degrees in electrical engineering from Carleton University, Ottawa, ON, Canada, in 1963 and 1965, respectively, and the Ph.D. degree from Queen’s University, Kingston, ON, Canada, in 1969. From 1963 to 1968, he worked for Acres-Inter-Tel Ltd., Ottawa, ON, Canada, as a Consultant involved with the design of frequency-shift keying (FSK) systems for secure radio communication. In 1969, he joined the Department of Electrical Engineering, National University of Singapore, where he was appointed Professor in 1985. He retired from the University in March 2001 and joined the Institute for Communications Research, Singapore, in April 2001 as Distinguished Member of Technical Staff. His present research interests include multicarrier and code-division–multiple-access (CDMA) techniques, channel estimation in multiple-input–multiple-output (MIMO) communication systems. Dr. Tjhung is a Fellow of IES Singapore. He is also a Member of the Association of Professional Engineers of Singapore.
H. J. Hu received the B. Eng. degree from Northwestern Polytechnical University, Xi’an, China, in 2000. He is currently working toward the M. Eng. degree at the National University of Singapore, Singapore. From September 2000 to August 2001, he was with Huawei Technologies, Shanghai, China, where he was involved in the design and development of demodulation algorithms for WCDMA base station. His research interest is in digital wireless communications.
P. He received the B. Sc. and M. Eng. degrees from the University of Electronic and Science Technology of China, Chendu, China, in 1983 and 1986, respectively, and the Ph. D. degree from Xidian University, Xian, China, in 1994. From 1986 to 1996, he was with the Department of Electrical and Communications Engineering, Xidian University, where he his work was concerned with HF MODEM and its key techniques, such as adaptive equalization, coded modulation/demodulation, and encoding/decoding. Since 1996, he has been with the Centre for Wireless Communications, National University of Singapore, Singapore, where he is currently a Senior Member of Technical Staff of Communication System and Signal Processing Group. His current research interests include mobile radio digital communications, such as multiuser detection of code division multiple access (CDMA), OFDM systems in wireless local area networks (LAN) or Hiperlan II, and synchronization and equalization of communications systems.
