A Post-Detection SNR-Aided Timing Recovery Loop ...

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

A Post-Detection SNR-Aided Timing Recovery Loop for MIMO-OFDM Receivers Wenzhen Li, Masayuki Tomisawa OKI Techno Centre (Singapore) Pte Ltd, 20 Science Park Road #02-06/10, Teletech Park, Singapore Science Park II, Singapore 117674, Fax: +(065) 6779-2382 Abstract --- In this paper, a novel third-order Phase Lock Loop (PLL) is proposed for the timing recovery in MIMO-OFDM systems. It differentiates from conventional timing recovery algorithms for OFDM systems in two aspects. Firstly a new Timing-clock-offset (TO) estimator is proposed, which utilizes the post MIMO detection SNRs to weight and combine TO estimates in frequency and spatial domains. This TO estimator efficiently handles the MIMO fading and the colored noise caused by MIMO detection to improve the estimation accuracy. Secondly, a novel third-order PLL structure is developed and its stochastic difference equation (SDE) that represents the dynamic behavior of the tracking loop is derived and used to evaluate the loop performance. The proposed PLL has an good loop noise performance for fast acquisition and stable tracking. The analysis and simulation show that the proposed timing recovery scheme outperforms the conventional timing recovery schemes in MIMO fading channel. Keywords --- MIMO-OFDM, timing clock offset, timing recovery, third-order PLL

I. INTRODUCTION Nowadays MIMO-OFDM technology is extensively employed in high data-rate wireless communications because the combination of MIMO and OFDM is able to take advantage of multipath propagation to increase throughput, range/coverage and reliability, and at the same time provide relatively low-cost implementation. Similar as OFDM systems, MIMO-OFDM is also vulnerable to synchronization errors [1][2], which induce phase rotation and inter-carrier interference (ICI), and degrade system performance. In a MIMO-OFDM transceiver, the synchronization includes carrier recovery, phase noise suppression and clock timing recovery. Generally the processing of the preamble takes care of the initial synchronization in the MIMO-OFDM receiver [1][2]. However, initial synchronization solely on its own is insufficient due to the residual carrier frequency offset (CFO) and sampling clock offset (here it is also briefly denoted as timing offset, i.e. TO), which introduce accumulated phase rotation and timing drift to deteriorate the receiver performance. Therefore we need an efficient CFO/TO tracking scheme to dynamically compensate for such phase/timing drift. Seldom works in the literature dedicatedly contributes to timing recovery for MIMO-OFDM systems except in [3]-[6], where TO estimation is roughly treated as a simple extension of that in OFDM systems. In OFDM systems, a conventional method is to use pilot subcarriers, i.e., the known symbols inserted into data symbols, to track the time drift. However,

most of the works for timing recovery in OFDM systems only addresses TO estimation instead of recovery [7]. An exception is the noncoherent delay-locked loop based sampling clock recovery scheme proposed in [8]. This method has some flavor of tracking the timing drift caused by the TO. However, its acquisition is slow and the tracking performance is poor particularly in the low SNR and multipath fading channels because of its poor TO estimation accuracy. A jointly weighted least squares CFO/TO estimation is proposed for OFDM systems in [9] to improve the TO estimation accuracy. However, its closed-form expressions of the optimal weighting factors are difficult to derive for MIMO-OFDM receivers due to the complicated MIMO fading channel and its spatial correlation. Moreover, TO estimation in [9] is not integrated with an efficient PLL implementation. Thus its acquisition and tracking performance is deteriorated by the serious loop noise. In this paper, a new TO estimator for the MIMO-OFDM receiver is proposed. This TO estimator is closely combined with the MIMO detection. A two-tier correlation is performed on the outputs of the MIMO detector to extract the TO estimates in spatial and frequency domains, and the twodimension (spatial and frequency) post-detection SNRs are utilized to weight and combine these TO estimates. This postdetection SNR-aided estimator efficiently combats the adversities caused by MIMO fading and colored noise due to the MIMO detection. Another important feature of the proposed TO estimator is that it is CFO independent. Because the effects of CFO on the TO estimates are eliminated, the estimation accuracy is improved, at the same time the complexity of TO estimator is much reduced when compared with the conventional joint CFO/TO estimation [9]. Moreover, a novel third-order PLL is developed in this paper, and its stochastic difference equation (SDE) that represents the dynamic behavior of the tracking loop is derived and used to evaluate the loop performance. The proposed PLL has an good loop noise performance for fast acquisition and stable tracking. The rest of the paper is organized as follows. In Section II, a MIMO-OFDM system model considering the impairments of CFO/TO is presented. Then a high efficient timing recovery scheme with a post-detection SNR-aided estimator and a third-order PLL is elaborated in Section III. Finally, we present the simulation results and conclusions in Section IV and Section V respectively.

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

II. SYSTEM MODEL In a MIMO-OFDM system, the transceiver signal can be expressed in the frequency domain using relatively straightforward matrix algebra. Considering a Nt*Nr MIMOOFDM system, here Nt and Nr are the numbers of transmit and receiver antennas respectively, the Nr *1 received signal vector on the k-th subcarrier is written as (1) y (k ) = H ( k )x ( k ) + n (k ) T Here y ( k ) = (y1 ( k ), y2 (k ),…, y N r ( k ) ) , ( )T denotes the operation of the transpose, and x (k ) is the Nt*1 signal vector transmitted from Nt transmit antennas on the k-th subcarrier, which is written as

(

x(k ) = x1 (k ), x 2 (k ), … , x N t (k )

)T

The channel frequency response of the k-th subcarrier from the j-th transmit antenna to the i-th receiver antenna is denoted by H i , j ( k ) , and H1, 2 (k )  H1,1 ( k )  H k H ( )  2,1 2, 2 ( k ) H( k ) =    H N ,1 ( k ) H N , 2 ( k ) r r 

(

H1, N t ( k )   … H 2, N t ( k )   …  … H N r , N t ( k )  …

n(k ) = n1 (k ), n2 ( k ), …, n N r (k )

)

T

ICI l ( k ) =

N −1

 2π k (lN s + N g )φ pk   H ( p )x l ( p ) s (πφ pk )  N  

∑ exp  j

p = 0, p≠k

Here the definitions of s(πφ pk ) and s (πφ kk ) are the same as those in [9], i.e. s (πφ pk ) =

sin( πφ pk ) N sin(

πφ

pk

N

, )

and s(πφ kk ) = s (πφ pk ) if p = k Here p and k are subcarrier indexes and , . In the case of small η and φ pk = (1 + η )(ε + p ) − k ε = ∆f ⋅ T

ε , s (πφ kk ) is close to 1 and s(πφ pk ) is near zero. This means

is the spatially independent

Gaussian noise vector for multiple receiver chains. In the above system model, an ideal synchronization in the receiver has been assumed. Actually there is CFO/TO in the receiver due to the mismatch between the transmitter and receiver oscillators or the channel Doppler frequency shift. The OFDM system model involving the synchronization error has been investigated extensively in the literature [1][7]-[9]. The same analyzing method can be extended to MIMOOFDM systems. In the following we use this method to derive the MIMO-OFDM system model suffering from the synchronization error. In this paper, we assume that the CFO and the normalized TO are ∆f andη respectively. It should also be noted that there are multiple radio chains in the MIMO-OFDM transmitter and receiver sides respectively. These multiple radio chains share the same frequency synthesizer, i.e., these multiple radio chains in the transmitter or receiver side are synchronized. In the transmitter, the information signals are encoded and modulated to form an OFDM symbol in the frequency domain. After Inverse-Fast-Fourier-Transform (IFFT) and pass-band modulation, the time-domain transmitted signals pass through a MIMO fading channel. In the receiver, the signals firstly pass through the analog demodulation chain and the initial frequency and timing synchronization modules. Then the guard interval is removed and the l-th received baseband OFDM symbol is represented by N samples. The demodulation of these received samples via an N-point FFT yields the received OFDM symbols in the frequency domain y l (k ) = exp( j 2π∆f (lNs + N g )(1 + η)Tl ) ⋅  2πk (lNs + N g )η  H(k )xl (k )s(πφkk ) + ICIl (k ) + nl (k ) exp j  N  

Where T is the sample duration, Ng and Ns are the numbers of samples in the guard interval and the whole OFDM symbol including guard interval respectively. The distortion on the kth subcarrier data caused by synchronization error includes the phase rotation and the magnitude scaling, s (πφ kk ) . Moreover, the synchronization error destroys the orthogonality between subcarriers and results in ICI, which is the function of s(πφ pk ) and can be expressed as

(2)

that the magnitude distortion and the ICI term can be ignored. Thus from Eq.(2), the phase rotation, θ l (k ) can be expressed as θ l (k) = 2π∆f (lNs + N g )(1 + η )T +

2πk (lNs + N g )η N

(3)

The first component is the inter-play of carrier and sampling frequency offset, and the third component is contributed by sampling frequency offset exclusively. When only TO exists, the phase rotation is proportional to the subcarrier index as well as the symbol index. III. Post-Detection SNR-Aided third-order Timing Recovery PLL When there is a TO in the receiver, the symbol timing will drift linearly forward or backward. It is well known that the feedback technique based PLL structure has a good tracking ability to automatically track slowly varying parameter changes, which is a good solution to track the timing drift due to TO. The proposed timing recovery scheme is shown in Fig.1. After the initial symbol synchronization is done in the time domain with the help of preambles, the symbol timing is assumed to be strictly synchronized for the first OFDM symbol. The proposed timing recovery algorithm starts its function in the frequency domain (after FFT). It has two stages: TO estimation and timing drifting adjustment. In the first stage, a two-tier correlation based TO estimator catering for the MIMO-OFDM receiver is developed. Then this TO estimate is used as the phase discrimination of a third-order PLL to track and compensate for the drifting timing of each OFDM symbol. The tracked timing output is divided into an integer part and a fractional part. The integer part is used to adjust the symbol timing, namely, the FFT window position. The FFT window controller moves the window position forward or backward on a sample basis. At the same time, the fractional

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

part is used to adjust the timing phase, which is done in the discrete-time domain by rotating the phase of each subcarrier in the OFDM symbol. Post-det. 2-d SNR

MIMO Chnn. est.

already know that in the case of small η and ε , the magnitude distortion and the ICI term can be ignored. In the light of this consideration, the detected signal in m-th pilot subcarrier of l-th OFDM symbol suffering from the synchronization error is represented as (6)

xˆ l ( m ) = exp( j θ l ( m )) x l ( m ) + G ( m ) n l ( m ) FFT FFT

FFT window controller

Phase rotation

MIMO Det.

Demap.

TO Est.

mod (ψˆ l ,2π ) Acc.

ηˆ

   ∑ηˆl  ⊗ f (l )  l 

Ts 1 − Z −1

wsft

The above equation can be rewritten as the following independent equations

CFO/PN tracking

 xˆl ,1 ( m ) = exp( jθ l ( m )) xl ,1 ( m ) + ξ l ,1 ( m )    xˆ ( m ) = exp( j θ ( m )) x ( m ) + ξ ( m ) l l, Nt l,Nt  l,N t

Acc Ts 1 − Z −1

m = −1,

∑ηˆ

l

l

Fig. 1, a third-order PLL for timing recovery in MIMOOFDM receivers A. Post-Detection SNR-Aided TO estimator The proposed TO estimator operates on the MIMO detection output and the detection output SNRs are utilized for the improved accuracy. In this paper, the linear MIMO detection is employed since it is the simplest and efficient spatial multiplexing receiver in practical MIMO systems, particularly those with a large number of transmit and receiver antennas. The linear receiver utilizes matrix multiplication to separate the data streams transmitted simultaneously. xˆ l (k ) = G (k )y l (k ) (4) = exp( jθ l (k ))G(k )H(k )xl (k )s (πφ kk ) + G (k )ICIl (k ) + G (k )nl (k ) In the above equation, xˆ l (k ) denotes the estimate of the

transmitted symbol vector, and the detection matrix G (k ) can be computed according to zero-forcing (ZF) or minimum mean square error (MMSE) criterion. The detection is performed assuming that G (k ) H (k ) = I , where I is an identity matrix. For ZF detection, this assumption is correct. For MMSE, it is an approximated assumption, remaining valid when SNR is high enough. According to the MMSE criterion, G (k ) is calculated as [13]

(

G (k ) = H H (k )H(k ) + N 0 × I

)

−1

H H (k )

here N 0 is noise spectral density, and and transpose of the matrix.

(7)

,J

Here ξ l ,i (m) is the colored noise due to MIMO detection, and it is a function of MIMO channel.

LPF Ts / u1 u2 + 1− Z −1 u1

,− J ,1,2,

2 J ⋅ Nt

(5)

( )H denotes the conjugate

We assume that 2J pilots are inserted among N subcarriers and that the indexes of those pilot subcarriers are m = −1, , − J ,1, 2, , J . Here all positive and negative pilot pairs {-m,m} are assumed to have equal distance of K. We

We know the fact that a timing offset in the time domain results in a phase rotation in the frequency domain, and this phase rotation is proportional to both the sub-carrier index and the OFDM symbol index as shown in Eq.(3). Thus the phase rotation of the known pilot symbol, which is observed at the MIMO detector output, can be utilized to estimate the TO. In the proposed estimation method, two-tier correlations are performed on the pilot subcarrier output from the MIMO detection. The first tier cross-correlation is performed in the frequency domain over the pairs of pilot subcarriers with positive and negative indexes respectively. Then the second tier auto-correlation is performed in the time domain over the first-tier correlation outputs with one OFDM symbol delay. This two-tier correlation can be expressed as

[

][

zl ,i (m) = xˆl ,i ( m) xˆl*,i ( − m) ⋅ xˆl −1,i ( m) xˆl*−1,i (− m)

]

*

(8)  2πKN sη  = C exp j  + el ,i (m) N   where C = ±1 is the sign bit, whose value depends on the specific pilot pattern. e l ,i ( m) is the noisy item, which is independent of the useful component. We can see that the output of this two-tier correlation is the exponential function of the TO, which is disturbed by the colored noise. Utilizing this method, J ⋅ N t TO two-dimension (in spatial and frequency dimensions) estimates are obtained in each OFDM symbol. Since the estimation is performed after MIMO detection, the adversity of channel fading has been suppressed substantially in the TO estimation. However, some estimates may be seriously affected by the noises colored by MIMO detection. Thus the weighted combination instead of a simple average of all these estimates should be employed for the improved accuracy. Here a TO estimator based on the criteria of the Best Linear Unbiased Estimation (BLUE) is proposed as ηˆ l =

 ∑ ∑ z l ,i ( m) wi, m  N arg i m 2πKN s ∑ ∑ wi, m  i m 

    

(9)

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

here wi ,m is the weighting factor for the estimate zl ,i ( m) , which is a function of the covariance of el ,i (m) in Eq.(8). In most cases it is infeasible to derive the optimal weighting factors for this BLUE estimator. In particular in the considered MIMO–OFDM system, where a two-tier correlation is involved in the frequency and time domains, and the output zl ,i ( m) is a complicated interplay of the signals in the time, frequency, and spatial domains. However, we know that the noise is colored by MIMO detection matrix G (m ) , and the noise power varies as a function of channel. Therefore for the linear MIMO detection the post-detection SNR is the parameter of interest and this information can be exploited to get the potential weighting factors. When MMSE MIMO detection is employed, the post-detection SNR in the frequency and spatial domain for i-th sub-stream and m-th pilot subcarrier is given as [12] 1 (10) SNR = −1 i, m

[

]

−1

N 0 H H (m)H(m) + N 0 I i ,i

here, [⋅ ]i−, 1i denotes the reciprocal of the i-th element on the diagonal of the inverse matrix. Since the WLAN channel is considered to be time invariant, the same SNR is applied for the successive OFDM symbols in the time domain. There are 2J ⋅ Nt SNR parameters involved in TO estimation, SNRi ,m i = 1, , Nt , m = − J , , J , but there are only J ⋅ N t TO estimates in the proposed TO estimator (see Eq.(8)).

Each TO estimate is related to one pair of SNRs in the positive and negative pilot subcarriers. In the first-tier correlation, i.e., the cross-correlation of two pilot subcarriers, the output reliability is dominated by the less reliable pilot subcarrier. Therefore these SNRs are utilized as the weighting factor in Eq.(9) in the following way (11) wi ,m = min {SNRi , m , SNRi ,− m } B. Third-order PLL for timing recovery In the effort to obtain an efficient solution to timing recovery, we find the estimation accuracy is not the unique factor dominating the performance of the feedback tracking loop. In conventional timing tracking loop designs [8][9], the estimated TO is smoothed by a low pass loop filter (LPF) and then followed by an accumulator (NCO, numerical control oscillator). In this case, the high accurate TO estimate means a fast acquisition stage and a short time to enter into the locking and tracking stage. However, in the practical implementation, this kind of recovery loop generally takes a long time to enter into the stable state. In the following we analyze this phenomenon in detail. When a feedback PLL is employed, the estimator output is the residual TO. For example, assuming that the estimate deviation is −2% , the first two estimation outputs are 98%η, and 98%(η − 98 %η ) ≈ 2%η . This means the input of the loop

filter is highly dynamic in the initial stage. The adopted low pass loop filter should have a wide passband to adapt to this dynamics, which implies a poor loop noise performance and a long acquisition time. In this paper, a new third-order PLL for sampling timing recovery is implemented in Fig.1. In this feedback loop, the TO estimator output ηˆ passes through an accumulator first to get

∑ηˆl , then it is smoothed by a low pass filter to get l

   ∑ ηˆ l  l

   ⊗ f ( l )   

, and is further accumulated over the duration

of successive OFDM symbols to obtain the estimate of the accumulated phase rotation for the l + 1 -th OFDM symbol, ψˆ l +1 . Thus, the overall third-order PLL operation is described by,    = ∑  ∑ ηˆl  ⊗ f (l ) Ts  l  l 

ψˆ l +1

(12)

Where T s is one OFDM symbol duration with the guard interval, and ⊗ denotes convolution. f (l ) is the impulse response function of the LPF filter, and its system response function is, T /u u F (Z ) = s 1 + 2 . 1 − Z −1

u1

Here Z is unit delay. u1 and u2 are the constants of the LPF filter, which determine the pass bandwidth and stop bandwidth of the filter. Since the estimation of TO, ηˆ , is independent of the carrier frequency offset ∆f , we only consider the TO. In this case, the actual phase rotation for the timing phase adjustment at l-th OFDM symbol is, (13) ψ l = 2π (lNs + Ng )η −1

We define the estimation phase error as, eψ (l ) = ψˆ l − ψ l Combining Eq.(12) with Eq.(13) and eψ (l ) , and then differentiating Eq.(12), we can derive the stochastic difference equation (SDE) , which represents the dynamic behavior of the tracking loop η=

ψ l +1 − ψ l Ts

=−

∆eψ (l ) Ts

  +  ∑ ηˆ l  ⊗ f ( l )  l 

(14)

Where ∆eψ (l ) = eψ (l + 1) − eψ (l ) . Based on this SDE equation, the third-order PLL loop performance can be analytically evaluated. From the above analysis, we can see the difference of this PLL scheme from the conventional PLLs is that one more accumulator is used after the TO estimator to form a thirdorder PLL. In this feedback loop, the estimator output is the residual frequency offset ∆ηˆ ,thus its accumulation is the estimate of the initial frequency offset ηˆ , but with a better accuracy in the progress of accumulation. Since the proposed estimator has a good accuracy, its accumulator output can

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

quickly enter into the stable state and the corresponding dynamics is quite low. This implies a narrow passband LPF with good loop noise performance can be employed. This scheme provides a good solution to fast acquisition and stable tracking at the same time. Since only one more accumulator is needed, this superiority is achieved with a relative low complexity and cost.

to get the correct start-point of the FFT window, and the AutoFrequency-Control (AFC) to perform rough CFO correction. The residual CFO of 1000Hz and the TO of 40ppm is considered.

The loop is closed by using the accumulator output ψˆ l to adjust the FFT window start position and the timing phase. The window shift function is expressed as w sft

 1  = − 1  0 

ψˆ l > π ψˆ l < − π

(15)

otherwise

Here 1 means the window shift forward. For timing phase adjustment, we define ψˆ l − 2π  ˆ mod (ψ l , 2π ) = ψˆ l + 2 π  ψˆ l 

ψˆ l > π ψˆ l < − π

(16)

Fig.2, performance comparison of the proposed TO estimator and conventional estimator in Channel D

otherwise

The timing phase adjustment is performed on the received vector y l (k ) to obtain phase updated vector yˆ l (k ) for the MIMO detection.  k ⋅ mod (ψˆ l ,2π )  (17) yˆ l ( k ) = exp  j yl (k ) 

N



IV. SIMULATION RESULTS The proposed algorithm is verified in an AWGN channel and a 2*2 MIMO-OFDM based WLAN system platform respectively. For MIMO platform, the antenna configuration is linear array and the distance between two adjacent antennas is a half carrier wavelength. Two spatial streams are transmitted over a MIMO fading channel with 20MHz frequency band by utilizing spatial division multiplexing (SDM) technique and 64-point IFFT modulation. The bandwidth of the subcarrier is 4.125 MHz. The convolutional codes with different puncturing rates are adopted. In the considered WLAN platform, various modulation schemes including BPSK/QPSK/16QAM/64QAM are employed. The employed data frame structure and preamble structure follow the next generation WLAN specification, IEEE 802.11n, which is completely compatible with existing WLAN systems, IEEE 802.11a/g. The detailed system specification can be found in WLAN 802.11n standard draft [10].

In this MIMO-OFDM platform, fading channel model D [11] for wireless indoor environment is employed in the system simulation. The root-mean-square (rms) delay spread of the multipath fading is 50ns (nano second), and its maximum multipath delays is around 400ns, which is less than the guard interval of the MIMO-OFDM symbol. In the MIMO-OFDM receiver, the Least Square (LS) MIMO channel estimation without smoothing and MMSE MIMO detection are employed. Moreover, the Viterbi decoding with trace-back length of 128 is adopted. The processing of the preamble takes care of the initial synchronization, such as frame/symbol synchronization

Fig.3, the acquisition and tracking behavior of the proposed third-order PLL

Two experiments are performed to verify the proposed TO estimator and third-order PLL timing recovery scheme. The concern of the first experiment is TO estimator. Here BPSK and 64QAM modulations are considered. The deviations of ψˆ l and the system BER performance for different TO estimators are shown in Fig.2. The proposed post-detection SNR-aided TO estimator and the conventional un-weighted estimator are compared. The compare results clearly indicate that the proposed TO estimator out-performs the conventional unweighted TO estimator. In the second experiment, the performance of the third-order PLL and the related system performance are verified. The acquisition and tracking behavior of the proposed third-order PLL in AWGN channel is shown in Fig.3, here the phase is scaled by the factor of 64, thus the phase discontinuities occur at ± π 64 instead of ±π . We can see the acquisition time is very short and is actually the estimate duration, i.e., only two

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

OFDM symbol time are used because of the novel third-order PLL structure as discussed in Section III. We also see that the acquisition and tracking start almost at the same time, and the transition time to stable tracking status is negligible. This substantially decreases the acquisition and tracking error. Thus the timing drifting can be tracked and compensated for with a high accuracy.

the second subfigure. These results also clearly indicate that the proposed third-order PLL scheme has better performance than the conventional second-order PLL. This advantage is more significant for 64QAM modulation with the same CPE compensation in an OFDM symbol. V.CONCLUSIONS In this paper, a post-detection SNR-aided TO estimator catering for MIMO-OFDM system is proposed. This CFO and phase noise independent TO estimator has good capability to combat the adversities of MIMO fading channel and colored noise by utilizing post-detection SNRs. Combined with this TO estimator, a new third-order PLL with fast acquisition and stable tracking is developed for the timing recovery in MIMOOFDM receivers. The model of the dynamic behavior of the timing recovery loop is derived and its superiorities are demonstrated by analysis and system simulation. REFERENCE

Fig.4, the performance comparison for BPSK in Channel D, packet length: 2048 bytes, 40ppm

Fig.5, the performance comparison for 64QAM in Channel D, packet length: 2048 bytes, 40ppm

The packet error rate (PERs) for BPSK and 64QAM modulation in this MIMO-OFDM platform are shown in Fig.4 and Fig.5 respectively. Since it’s difficult to regenerate the same platform and algorithms for conventional second-order PLL schemes [8][9], for simplicity, here the performances of the proposed third-order PLL scheme and the conventional second-order PLL scheme are compared on the same platform. Moreover, the CPE (common phase error caused by residual CFO) compensation scheme is combined with the proposed TO estimation and compensation. In the adopted CPE compensation scheme, one OFDM symbol is divided into several segments with the pilot subcarrier as the center point. The CPE is estimated and compensated for segment by segment. This CPE compensation method successfully alleviates the residual TO effect for better performance because the phase rotations for all subcarriers in one OFDM symbol are not the same once the residual TO exists. This can be easily seen from two subfigures in Fig.4 and Fig.5 respectively, where the performance in the condition of the same CPE for the whole OFDM symbol is also presented in

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