System model

Consider an OFDM system with N subcarriers signaling through a time-varying frequency-selective Rayleigh fading channel. The whole system in base band model is illustrated in Fig.1.

The data are modulated in blocks by means of a discrete Fourier transform (DFT),given by

[ ]

¹

[ ]

where the subscript t represents the OFDM frame. A cyclic prefix (CP) is inserted into the transmitted signal to prevent the intersymbol interference (ISI) between successive OFDM frames. After parallel to serial conversion, the signals are transmitted trough a frequency selective time-varying fading channel. At the receiver side, the received time-domain signal can be expressed as [15]

tth

[ ]

¹ ,

[ ]

where represents the channel impulse response (CIR) of the path during OFDM frame, L represents the length of the frequency-selective fading channel,

, and i represents the modulo-N operation. The fading channel coefficients N are modeled as zero mean complex Gaussian random variables. According to Wide Sense Stationary Uncorrelated Scattering (WSSUS) assumption, the fading coefficients in different delay paths are statistically independent. In the time domain, the fading coefficients are correlated and have a Doppler power spectrum density modeled as in Jakes [16], given by duration.. Then substitute equation (2.1) into (2.2):

[ ]

^{1 1}

[ ]

,

^{( )} [ ]

At the output of the DFT demodulator, the sequence can be expressed as

[ ] { } [ ]

w n is i.i.d complex AWGN noise due to the orthonormal transformation of the t

original noise n nt

[ ]

. It also can be written in a concise matrix form as

2.2 Inter-carrier interference in OFDM systems induced by the Doppler effect

OFDM is robust against frequency selective fading due to the increase of the symbol duration. With the introduction of the CP, the problem of the ISI in single-carrier systems can be greatly reduced. However, in mobile radio environment, multipath channels are usually time-varying. Channel variations within an OFDM block destroy the orthogonality among the subcarriers, resulting in ICI and performance degradation. Since the transmitted symbol duration is N times longer than that in a single-carrier system, the increase in the symbol duration makes the system more sensitive to the time variation of the channel. If ICI is modeled as an additive white Gaussian process and not adequately compensated, the ICI will lead to an error floor. Doppler frequency is used to indicate the rate of the channel variation, which is proportional to vehicle velocity and carrier frequency

Fd

v f , _c

given by

_d v f^c (2.9)

F c

= ⋅

Therefore, the ICI induced by time-varying channel is determined by Doppler frequency and the OFDM symbol duration . In order to analyze the symbol energy distribution and ICI on one subcarrier, equation (2.6) can be modified as following form

Fd T

The first term in (2.10) is the desired signal, the second term represents the ICI from the other subcarriers, and finally the third term is the additive noise. Hence, the energy of

[ ]

Fig. 2 Normalized symbol energy distribution

The energy of X n distributed to subcarrier t

[ ]

n v− to n v+ can be expressed as

The normalized symbol energy distributionΦ_v E_s is depicted in Fig. 2. It indicates that most of the subcarrier’s energy is spread over itself and its nearby subcarriers when f T_{d s} < , 1 and more than 99% of the subcarrier’s energy is distributed on itself and its two nearby subcarriers when . Therefore, the ICI on one subcarrier mostly comes from only few neighborhood subcarriers.

d s 0.1 f T =

Chapter 3

ICI Mitigation in Doppler Spread Channel

3.1 Pilot-based channel estimation

Pilot-based approaches are widely used to estimate channel properties and correct the received signal [14]. Usually, a comb-type pilot subcarriers arrangement is adopted, as depicted in Fig. 3. For each transmitted symbol, pilot signals are uniformly distributed within an OFDM symbol, while null and data tones are assigned to other subcarriers.

Fig. 3 Comb-type pilot arrangement

Without using any knowledge of statistics of the channels, the estimate of the channel at pilot subcarriers based on least square (LS) estimation is given by

[ ] [ ]

pilot data. Thus, channel estimation with comb-type pilot-aided symbol requires an

interpolation technique in order to obtain the channel information at null and data subcarriers. Before further discussion, assume that the number of pilots is large enough such that aliasing of channel impulse response will not occur. The channel estimation at the

p

N m l

N + -th subcarrier using linear interpolation is given by [17]

( ) ( ( ) )

The second-order interpolation performs better than the linear interpolation method, where channel estimates at the data subcarriers are obtained by weighted linear combination of the three adjacent pilot estimates, given by [7]

( ) ( )

The low-pass interpolation method is performed by inserting zeros into the LS estimates at pilot subcarriers and then applying a low-pass finite-length impulse response (FIR) filter, which allows the original data to pass through unchanged and interpolates such that the mean-square error (MSE) between the interpolated points and their ideal values is minimized. The time domain interpolation [4] is a high-resolution interpolation based on zero-padding and DFT/IDFT. First, it converts H^ˆp

[ ]

m to time domain by IDFT :

Then, based on the basic multi-rate signal processing properties, the -sample time domain sequence

Np

[ ]

G n is extended to an N-sample sequence p GN

[ ]

q by padding with

N−Np zeros samples at the “high frequency” region around N_p 2, given by

Finally, the estimate of the channel at all frequencies is obtained by:

[ ]

¹

[ ]

The performance among the comb-type estimation techniques usually ranks from the best to the worst as follows: low-pass, time-domain, second-order, and linear.

3.2 The proposed method

First we will recapitulate why the LS estimator presented in the previous section is not adequate at combating ICI. The LS estimator with 1-D interpolation compensates for the frequency-selectivity fading channel, assuming that the channel is stationary during one symbol interval. Usually, if f T is les than 0.01, the channel can be assumed constant _d during one symbol interval. But the equalizer considers the ICI as an additive Gaussian random process, the performance of equalizer degrades significantly due to ICI for larger channel variation, as for f T_d ≥0.01.

To fully count the ICI effect, (2.7) should be used to solve , and it is necessary to estimate the channel matrix then calculate its matrix inverse. Nevertheless, accurate estimation of the transfer function requires complete knowledge of the time-variation of the CIR for each OFDM symbol, which is not usually available. When the OFDM symbol duration is smaller than 10% of the channel coherence time, the variation of the channel during a block period can be assumed in a linear model [10]. By utilizing the above

Xt

Ht

assumption, the estimation problem of the channel matrix can be greatly simplified, since the value of the slope of the linear model uniquely determines the ICI. The component of the channel matrix can be expressed as

Ht

It is shown in (3.7) that the main diagonal of the estimated channel matrix depends on the average of the channel frequency response during one block period and the ICI term is only determined by the channel variation and OFDM parameters. Then substitute (3.7) and (3.8) into (2.7),we get

Since only depends on OFDM parameters, it is possible to precalculate it once at initialization. There are several schemes proposed in the following. The procedure for the estimation of the channel matrix is shown in Fig. 4 and described as follows.

Φq

Fig. 4 Block diagram of the estimation of the channel matrix

First, a time domain pilot signal is inserted at the end of every symbol, as shown in Fig. 5.

A similar method can be found in [10].

0 1 0 cp1 Data1 0 1 0 cp2 Data2

L_P 1 L_P L L_P 1 L_P L

Fig. 5 Transmitted data format 1

The pilot symbol is composed of 2L_P+ samples. The first 1 L samples are used to _P avoid ISI, while remaining L_P+ samples are inserted for CIR estimation. Then, by 1 comparing the CIR changes between the received signals corresponding to the (i−1)th pilot symbol and th pilot symbol for each path, the CIR variation during the block period is estimated using linear interpolation. Although using pseudo-delta function to do the channel estimation is straightforward, it may have some drawbacks. In order to coincide with the signal power spectrum density, the pseudo-noise can be used as time domain pilot in place of the delta function, as shown in Fig.6.

i

Fig. 6 Transmitted data format 2

Before further discussion, we assume that the length of time domain pilot is M- sample.

Assuming the channel variation during the M samples can be negligible. The time domain convolution can be expressed as a matrix vector multiplication. The linear

convolution matrix is formed from the time domain pilot.The least-square channel estimate, assuming P P^H has full rank, is given by

( )

h = P Pˆ ^{H -1}P r^H (3.10)

and the corresponding MSE is given by^σn2tr

{ (

^{P PH}

)

⁻¹

}

. Based on minimizing the channel estimation MSE, it can be achieved if and only if has equal eigenvalues. This is achieved when

P PH

P P =^H E_PI (3.11)

where denotes the linear convolution matrix. It can be observed that the time domain pilot should be a shift-orthogonal sequence. The corresponding minimum MSE is

P

2

n P

Lσ E . Then, the CIR variation for each path during the OFDM symbol can be estimated with linear interpolation, given the LS estimates at the inserted pilot symbols. Finally, the proposed method reconstructs the channel matrix and calculates its matrix inverse. Since the channel matrix can have a large size, it is difficult to process in real time. Based on the structure of the channel matrix , whose energy is concentrated on the neighborhood of the main diagonal, the computation complexity can be reduced by considering only the elements nearby the main diagonal and ignoring remaining elements, as shown in Fig. 7.

Ht

Fig. 7 Banded channel matrix with color region corresponding to ICI concentration

Another way to avoid straight matrix inversion is to divide the task into ICI reduction and a simple one-tap equalizer. In the following, the ICI reduction procedure will be described in detail. According to equation (2.7), it can be found that ICI component is determined by not only the variation of the channel but also the transmitted data. The decision-feedback techniques can be utilized to acquire the estimates of the transmitted data, then the estimated channel matrix in (3.9) is used to subtract the ICI components from the received signal. The resulting ICI reduction method is as follows

[ ] [ ] [ ]

where Dec denotes a slicer in the demapper. Assuming that the number of data carriers is large, the effect of incorrect hard decisions will not have a significant impact. Finally, the ICI-cancelled Y m can be used in a conventional one-tap equalizer. For severe Doppler t

[ ]

effects, it also can be combined with an iterative method to enhance the ICI estimation accuracy, as shown in Fig. 8. In addition, the hard decision in (4.1) can be replaced by a MMSE equalization, given by

only consider the main diagonal of the estimated channel matrix and

solve the linear equation

Fig. 8 Block diagram of the proposed ICI-reduction method

In the same manner, the computation complexity of the ICI reduction can be reduced by considering only the ICI component due to the nearby subcarriers without degrading much system performance, given by

q

Even, it can be ignored the off-diagonal elements of the estimated channel matrix and use one-tap equalizer without ICI reduction. It can be observed in (3.7) that the main diagonal elements are the average of the channel frequency response during one block period. Based the linear property, it can be obtained by

( ) 1 (3.16)

where the subscript denotes the OFDM frame and denotes the main diagonal of the matrix. Then, utilize one-tap equalizer without ICI reduction.

t t^th diag()

3.3 Computation complexity analysis

Utilizing the time domain pilot inserted at the end of every OFDM symbol to acquire the time domain channel estimates in (3.10) requires multiplications. The matrix can be pre-calculated at the initialization. Based on the linear property of the channel variation, the computation of the channel matrix in (3.8) and (3.9) approximately requires

L2 Φ

2

( )

2Nlog N

)

N multiplications. The method that directly uses the inversion of the estimated channel matrix requires computational complexity. The method using one-tap equalizer with ICI reduction approximately requires

multiplications. The simplified method in (3.14) and (3.15) using one-tap equalizer with ICI reduction which only consider the elements of the estimated channel matrix in the adjacent of the main diagonal requires

( 3

multiplication. As the size of the channel matrix increases, the simplified method conserves more computation complexity than that of which utilizes the whole estimated channel matrix to do ICI-reduction. The method using one-tap equalizer without ICI reduction approximately requires only multiplication. The complexity of the proposed methods are summarized in Table 1

2

( )

log

N N +

Table 1 Computational complexity analysis

Estimation Scheme Computational complexity Comments

One-tap equalizer

Chapter4

Simulations and Discussions

4.1 Simulation result

1) System parameters:

OFDM system parameters used in the simulations are illustrated in Table 2. Since the aim is to observe channel estimation performance, it assumed to be perfect synchronization in the simulations. Moreover, the guard interval is assumed to be longer than the maximum delay spread of the channel. Simulations are carried out for various signal-to-noise ratio (SNR) and Doppler spreads.

Table 2 Simulation parameters

Parameters Specifications

FFT Size 128

Pilot Ratio 1/16 Guard Interval 16 Carrier frequency 5M

Bandwidth 500K Signal Constellation QPSK,16-QAM Channel Model Jakes

2) Channel model

Assume a symbol spaced, tap-delay-line channel model of 4 paths, where each channel tap is generated with the Doppler spectrum based on the Jakes’ model. The

generation of the tap gains is illustrated in Fig. 9.

Fig. 9 Generation of the tap weight processes

It start with a set of independent, zero-mean complex Gaussian white noise processes, which are filtered to produce the appropriate Doppler spectrum, as depicted by the Jakes model. These are then scaled to produce the desired power profile. In the simulations, the path gains follow the exponentially-decayed power profile as depicted by

( )

² constant. We choose the power delay time constant such that the last path power is 20dB below the first path. Before further discussion, we assume the overhead of the proposed method and DFT-based method are the same. Moreover, the suboptimal time domain pilot we used is the product of the shift-orthogonal sequence and a finite-duration window function, such as Hanning window.

Fig. 10 shows the BER performance for the case f T_{d s} =0.04. The modulation scheme used for this simulation is 16-QAM. Assume the pilot subcarriers are equispaced along the frequency domain such that DFT-based channel estimator can be used. The DFT-based channel estimation is also shown as reference, which has comparable performance to that of MMSE channel estimation method. In addition, the overhead of the proposed method are

the same as that of the DFT-based method in the simulation. From Fig. 13, it can be observed that the ICI-reduction algorithm effectively reduces the error floor, as is more evident especially in high SNR than low SNR. The method using the one-tap equalizer with ICI reduction has comparable performance to that of directly using the inversion of the estimated channel matrix, while costs much less computation.

Fig. 10 BER performance of ICI reduction algorithm with f T_{d s} =0.04

Fig. 11 The comparison of partial ICI reduction method

Fig. 11 also shows the BER performance for the case with choosing different number of the nearby subcarriers for ICI reductions. From Fig. 11, it can be observed that the simplified scheme gives comparable results to the method using whole estimated channel matrix but has low computation complexity.

d s 0.04 f T = q

Fig. 12 and Fig.13 shows the BER performance respectively for the case f T_{d s} =0.1. and . From Fig.12, it can be observed that the proposed method From Fig.13, it can be observed that the proposed method gives rise to an error floor as the assumption of the linear property of the channel variation no longer holds.

d s 0.2 f T =

Fig. 14 and Fig. 15 shows the BER performance for the case f T_{d s} =0.04 respectively under various conditions of carrier frequency offset and timing offset . From Fig. 14, it can be observed that the proposed method is robust to the effect of carrier frequency offset. From Fig. 15, it can be shown that the proposed method is sensitive to the timing offset.

Fig. 12 BER performance of ICI reduction algorithm with f T_{d s} =0.1

Fig. 13 BER performance of ICI reduction algorithm with f T_{d s} =0.2

Fig. 14 BER performance of proposed method with f T_{d s} =0.04 and various CFO

Fig. 15 BER performance of proposed method with f T_{d s} =0.04 and various timing offset

Consider that the tolerance interval to symbol timing error is samples and the maximum channel length is

k

L. The comparison among the proposed methods and DFT-based method are summarized as in Table 3, including overhead, computational complexity and BER performance.

From Table 3, it can be concluded that the overhead of the proposed method is slightly more than that of the DFT-based method but the performance of the proposed method is much better than that of the DFT-base method. Moreover, it is worthwhile to increase the computational complexity for the sake of performance, as shown in Table 3.

Table 3 Overhead, complexity and performance of the various schemes

Estimation Scheme Overhead Complexity Performance

One-tap equalizer without ICI reduction

Moderate (2L+3K−1)

Low Moderate

One-tap equalizer with ICI reduction

Moderate (2L+3K−1)

High Very good

One-tap equalizer with partial ICI reduction

Moderate (2L+3K−1)

Moderate Very good

One-tap equalizer with DFT-based method

Low (2L+2K−1)

Low Moderate

4.2 Conclusion

Usually, DFT-based receivers considers ICI as an additive noise and it is not adequately compensated. In this thesis, we propose the time domain pilot-based estimation scheme combined an iterative method to suppress the ICI induced by time varying channels. It is shown through simulation that ICI can be compensated by the proposed method if the normalized Doppler frequency change is in the range of ( 0.01≤ f T_{d s} ≤0.1). The reason is that for the channel with a very high Doppler frequency, the assumption that the channel parameters vary in a linear fashion within a block period is no longer a good approximation and gives rise to an error floor. On the other hand, the channel can be assumed invariant during a block period if the normalized Doppler frequency less than 0.01.Under these circumstances, the proposed method only slightly outperforms the DFT-based method.

Furthermore, we also proposed a simplified method using a one-tap equalizer with partial ICI reduction; it has comparable performance to that using whole estimated channel matrix for ICI reduction.

Reference

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[2] J. Armstrong, P. M. Grant, and G. Povey, “Polynomial cancellation coding of OFDM to reduce intercarrier interference due to Doppler spread,” in Proc. IEEE Global Telecommunications Conf., vol. 5, pp. 2771-2776, 1998

[3] A. Seyedi, and G.J. Saulnier, “General self-cancellation scheme for mitigation of ICI in OFDM systems,” IEEE Communications Conf., VOL. 5, PP. 2653-2657, June 2004.

Communication Systems Based on Pilot Signals and Transform-Domain Processing,” Proc.

VTC’97, pp. 2089-2094

[5] Y.-S. Choi, P. J. Voltz, and F. A. Cassara, “On channel estimation and detection for multicarrier signal in fast and selective Rayleigh fading channels, ” IEEE Trans. Commun., vol. 49, pp. 1375-1387, Aug. 2001.

[6] O. Edfors, M. Sandell, J.-J. van de Beek, S. K. Wilson, and P. O. Brjesson, “OFDM channel estimation by singular value decomposition,” IEEE Trans. Commun., vol. 46, no. 7, pp. 931–939, Jul. 1998.

[7] Coleri, S. Ergen, M., Puri, a., and Bahai, A., “Channel Estimation Techniques Based on

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