ICI on OFDM System

corresponding to the ith transmit antenna and the jth receive antenna. A constant channel is assumed over the time interval

s s

n T t n T t=0

(

with indicating the start of the data part of the symbol.

)

[ ]

, j i

hl n for − ≤ ≤ −G n 1 and 0≤ ≤ −n N 1 represents the lth channel tap in the guard interval and data interval respectively.

Then the received signal r^{( )j}

at the jth receive antenna is the superposition of all distorted transmitted signals, which can be expressed as follows:

(2.5) uncorrelated additive white Gaussian noise (AWGN) with zero-mean and variance σ on the jth receive antenna.

2.4 ICI on OFDM System

After the OFDM demodulator in Fig. 1, the received OFDM data symbols at the jth receive antenna in frequency domain are given by

( )

[ ] {

^{( )}

[ ] }

^{1 ( )}

^{[ ]}

²

Substitute (2.1) (2.3) (2.5) into (2.6), we will find the relationship between and

)

[ ]

hand side of (2.8) represents ICI due to the Doppler spread caused by high mobility and cannot be neglected as the maximum Doppler frequency increases. Define Fl ^{j i,)}

( )

k

( )

( )

^{( )}

[ ]

as the DFT of the lth channel tap with respect to time-variation:

1 2

m which denotes the channel frequency response from the mth subcarrier on

the kth one can be defined as

( )^,

[ ]

^{( ), 1 ( )},0^{, 2}

is the average of the lth channel tap over the time duration of

[ ]

t N Ts

≤ ≤ × . Therefore, H^{(j i,)} k k,

[

represents the DFT of this average.

As the channel is time-invariant or we assume that the channel is quasi-static within one

OFDM data symbol, h nl

]

_l orthogonal property of the subcarriers, the second term of (2.8) becomes 0, which means no ICI, and (2.8) can be easily reduced as follows:

k H k k X k Z k

And the received pilot preamble at the jth receive antenna can be described by

[ ]

^,

[ ] [ ]

Chapter 3 V-BLAST   Detection    

In this chapter, the conventional V-BLAST detection method with MMSE-SIC algorithm is introduced to perform data detection in MIMO OFDM system. We first consider a MIMO system with transmit and receive antennas, which can achieve high data rates by transmitting simultaneously different data on the different transmit antennas, and

NT N_R

T R

N ≤N . A single data stream is split into substreams, each of which is transmitted using one of the transmit antennas. The transmit diversity introduces spatial interference. The signals transmitted from various antennas propagate over independently scattered paths and interfere with each other upon reception at the receiver. After passing through the channel (assuming quasi-stationary), each receive antenna receives the signals radiated from all transmit antennas. Let

denote the vector of transmit symbols, then the corresponding received signal vector can be written as

(3.1) where is an -component column vector of the received signals across the receive antenna, is a matrix with representing the channel frequency response between transmit antenna and receive antenna j , and is the AWGN noise

vector with zero mean and variance

Z

σ2.

In the MMSE detection algorithm with successive interference cancellation (MMSE-SIC) [8], the expected value of the mean square error between transmitted signal X and a linear

combination of the received vector w R^H is minimized as follows where is an matrix of linear combination coefficients given by

(

² T

)

¹

, the decision statistics for the symbol sent from antenna is obtained as

consisting of components. The estimate of the

symbol sent from antenna , denoted by X , is obtained by making a hard decision on Y_i

( )

In an algorithm with interference suppression only, the detector calculates the hard decisions

estimates by using (3.4) and (3.5) for all transmit antennas.

In a combined interference suppression and interference cancellation algorithm, the interference contribution from already-detected components of is subtracted from the received signal vector, resulting in a modified received vector in which fewer interferers are present. As mentioned in [2], when interference cancellation is used, the order in which the components of are detected becomes important to the overall performance of the system.

Let the ordered set S ≡

{

k k^{1, 2, ,}kNT

}

be a permutation of the integers 1

specifying the order in which components of the transmitted symbol vector are extracted.

, 2, ,N_T X

The optimal detection order is determined to maximize the minimum post-detection SNR of all data streams. A result is that simply choosing the best post-detection SNR at each stage in the detection process leads to the globally optimum ordering, S_opt.

(

^{i 1}

)

The receiver starts from first iteration = whose detection order is and computes

its signal estimate by using (3.4) and (3.5). The received signal in this iteration is denoted by . For calculation of the next iteration

k1

X is subtracted from the received signal and this modified received signal denoted by is used in computing the decision statistics for iteration 2 from (3.4) and its hard estimate in Eq. (3.5). This process continues for all other iterations.

R1

R2

i ˆ

ki

After detection of iteration i whose detection order is k , the hard estimate X is subtracted from the received signal to remove its interference contribution, resulting in modified received signal :

Ri the interference contribution caused by X in the received vector. is the received vector which is free from interference coming from

i+1

. For estimation of the nest stage, this signal is used in (3.4) instead of . Finally, a deflated version of the channel matrix is calculated, denoted by

ki

H , by zeroing column k k₁, ₂, ,k_i of H . The

deflation is needed as the interference associated with the current symbol has been estimated

and cancelled.

The full V-BLAST with MMSE-SIC detection algorithm is described as a recursive procedure, including determination of the optimal ordering, as follows:

( )

Since the post-detection SNR for the th detected component of is

2

j corresponds to the strongest detected component of . Similarly, corresponds to the strongest detected component among the remaining − +

components. Thus, it determines the elements of Sopt, the optimal ordering.

The V-BLAST architecture is essentially a single carrier signal processing algorithm.

Therefore, to combine it with OFDM, the V-BLAST detection process has to be performed on

every subcarrier at the receiver to achieve high data rate transmission in frequency selective fading channels. As OFDM effectively divides the frequency selective channel into a number of flat fading subchannels, the MIMO OFDM system comprises a number of narrow band MIMO systems on different subcarriers. As the same detection algorithm is used on each subcarrier, MIMO OFDM system is basically a per-subcarrier MIMO structure, which performs V-BLAST detection on each subcarrier.

Chapter 4 Iterative Receiver  

The iterative receiver based on joint channel estimation and data detection mainly consists of three-stage processing. In this section, we first present the MPIC-based decorrelation method for the initialization stage. In the following tracking stage, we adopt a

β

-tracker followed by DF-DFT based channel estimation method to estimate the frequency response of average channel variations of each path. With the estimated channel frequency response, the detection of transmitted signals can be performed by applying V-BLAST with MMSE-SIC algorithm described in chapter 3. In the final stage, we approximate channel variations of each path with a linear model. Then, a two-dimensional V-BLAST method is utilized to perform ICI cancellation and data detection. The block diagram of the channel estimation in the tracking stage and the ICI cancellation in the final stage is shown in Fig. 3.

β

Fig. 3 Block diagram of the channel estimation in the tracking stage and the ICI cancellation in the final stage.

4.1 Initialization Stage: The MPIC-Based Decorrelation Method

In this stage, we roughly estimate multipath delays and multipath complex gains through the preamble placed at the beginning of each OFDM frame. We all know that CIR can be estimated by using the preamble placed at the beginning of each OFDM frame, while the difficulty is that for most wireless standards, the preamble does not have ideal auto-correlation due to the use of either guard band or non-equally spaced pilot tones. Fig. 4 outlines the MPIC-based decorrelation method to estimate CIR path-by-path by canceling out already known multipath interference. Since the preambles transmitted from different antennas do not interfere with each other at the receiver side, channel estimation can be independently performed for each transceiver antenna pair, and therefore the antenna indices j and i are omitted in the following. In step 1, we first define two parameters and which represent a multipath observation window and a presumed number of paths in a mobile radio channel, respectively. Next, we calculate the cyclic cross-correlation

Wb N_p

[ ]

CRP τ between the received and the transmitted preamble by

{

[ ] [ ] [ ]

^*

}

^{, 0}

RP P

C τ =IDFT R k ⋅P k τ = …, ,N−1 (4.1)

[ ]

CPP

The normalized cyclic auto-correlation τ of the transmitted preamble can also be calculated by

{

[ ] [ ] [ ]

^*

}

^{, 0, , 1}

CPP τ =IDFT P k ⋅P k τ = … N− (4.2)

ρ and κ, which stand for a path counting variable and the number of legal paths Both

found by the MPIC-based decorrelation method, respectively, are initialized to zero. In step 2, we start by increasing the value of the path counting variable ρ by one, and picking only one path whose time delay τ_ρ yields the largest value in CRP

[ ]

τ , for τ∈ W . If the time _b delay τ_ρ

RP 0

C ⎡ ⎤ =⎣ ⎦τ_ρ

κ th

is larger than the length of the CP, this path is treated as an illegal path, and we discard it by setting . Otherwise, we increase the number of legal paths found,

, by one, and then reserve this path as the κ legal path with time delay τˆ_κ =τ_ρ

[

and complex path gain μ^ˆ_κ =CRP τ^ˆ_κ

]

. The replica of the interference associated with this legal path is regenerated and subtracted from CRP

[ ]

τ to obtain a refined cross-correlation function:

[ ] [ ]

^{ˆ ˆ , \}

{

^{: 1, , 1}

}

RP RP PP b i

C τ ←C τ −μ_κC ⎡⎣τ τ− _κ ⎤⎦ τ∈W τ i= … ρ− (4.3) where “←” is the assignment operation. We continue the iterative process of the step 2 until

ρ reaches the presumed value of N_p.

[ ] [ ] [ ] { [ ] [ ] }

[ ] { [ ] [ ] }

*

*

Step1: Set preassumed number of paths & observation window Calculate & by

Step2 : Estimate multipath delays and comples gains coarsely:

while

Fig. 4 The MPIC-based decorrelation method in the initialization stage

4.2 Tracking Stage: β -tracker

Through the initialization stage, we are able to obtain information on the number of paths

( )^{j i,}

(

^Np

)

κ ≤ , the multipath delays τˆ_l^{( )j i,} , and the multipath complex gains μˆ_l^{( )j i,} , for

{

^{1, ,} ( )^{j i,}

}

l∈ … κ . Without loss of generality, we assume that the multipath delays do not vary over the duration of each OFDM frame. In this stage, we also suppose that the channel is quasi-static within one OFDM data symbol, which means the multipath complex gains of channel model are unchanged, so as to estimate the average channel variations of each path.

[ ]

Accordingly, the corresponding channel estimator Hˆ( )^{j i,} k k,

(

for the frequency response

[ ]

^{( )}

In this stage, a

β

-tracker followed by DF DFT-based channel estimation is applied to track channel. We start by utilizing the decision feedback (DF) DFT- based channel estimation

method, using the ML criterion [9] [10][11][12], which is provided in the following:

( )^, ( )^,

(

( )^,, ( )^,,

)

^{1 ( ),,}

( )

^{( ) 1 ( ),}

the estimated channel frequency response between the ith transmit antenna and the jth receive antenna at the vth iteration, ^{Xˆ i =diag X}

{

^{ˆ i[Θ1],...,Xˆ i[ΘΘ]}

}

Θ Q

consists of the decision data symbols in which is a subset of used to track channel variations and Xˆ [^{( )i} k can be ] obtained by applying the previously estimated channel frequency response to the V-BLAST data detection with respect to the received signals. F_{d DFT}^{( )}_,^{j i,} represents the Θ×κ^{( )j i,}

truncated IDFT matrix which can be equivalently expressed as

j i j i H

d IDFT = d DFT

F F , and F_DFT^{(j i,)} is the N×κ^{( )j i,} truncated DFT matrix. For simplification, the

subscripts “DFT” and “IDFT” are omitted, then F_{d DFT}^{( )}_,^{j i,} , F_{d IDFT}^{( )}_,^{j i,} and F_DFT^{(j i,)}

(

are replaced by

)

, j i

Fd ^,

( )

^{Fd( )j i, H, and F(j ,i)} respectively. We assume that the data interference signals from other transmit antennas can be reconstructed and cancelled perfectly from the received signals

( )

^,

( )^j

R . Thus, the refined signals corresponding to the j i

( ) ( )

[ ]

^{( )}

[ ]

^{( )}

[ ]

^{( )}

[ ]

^{( )}

[ ]

th antenna pair can be represented as Moreover, since the refined signals from different antenna pair do not interfere with each other, we can perform the channel estimation for each transceiver antenna pair and, for simplification, the superscript “

( )

^{j, i ＂ and “}

( )

ⁱ ＂are dropped hereafter. Therefore, the DF-DFT based channel estimation method can be expressed as follows:

( )

¹

This is, however, not a good solution in fast time-varying channels because decision data symbols easily induce the error propagation effect. Therefore, pilot tones play an important role in fast fading channels. While in the slowly fading channels, decision data symbols are more reliable than pilot tones due to the number of data subcarriers are much more than pilot tones. Hence, we can modify the first iteration by adopting pilot tone as well as decision data symbols simultaneously to perform channel estimation at the first iteration. This, we modify the first iteration with v=1 by the

β

-tracker in which the current channel can be weighted with the channel estimated over data symbols in the previous time slot and the channel estimated over pilot signals in the current time slot due to that the current channel is related to the previous channel, which depends on how fast the channel varies. Therefore, The

β

-tracker for the first iteration (v=1) can be described as follows:

where represents the channel estimated over data symbols in previous time slot, denotes the channel estimated by DF DFT-based channel estimation over pilot subcarriers in which is the diagonal matrix of pilots, is the corresponding received pilot signals, and is the J×κ^{( )j ,i} truncated DFT matrix over pilot subcarriers. Likewise, since the received pilot signals from different antenna pair do not interfere with each other, we can also perform the channel estimation o for each transceiver antenna pair. In order to initialize the channel estimator of (4.8), the CSI estimated in the last iteration of previous time slot has to be taken as the initial value of the CSI for the current time slot.

Then, we can construct an MMSE cost function over all subcarrier indices, and derive the minimum β as follows:

where is the true value of current channel frequency response. Each component of β can be derived separately in the following. From (4.8) and (4.9), we have

( )

Let the estimation of be the summation of the true previous value plus a noise term, and the estimation of

v−1

H

H also be the summation of the true current value plus a noise term P

as follows:

( )

Furthermore, the autocorrelation function of the channel frequency response is as follows:

(4.14)

where denotes the zero-order Bessel function of the first kind, and f is Doppler frequency in hertz, σ_h² is the channel power, and is the time duration of one OFDM symbol after adding the guard interval.

T

Hence, from (4.12) ~(4.14) , (4.10) is equivalent to

⎡ ⎤ ⎡ ⎤

Similarly, we can obtain

(4.16)

Therefore, apply (4.15)~(4.17) into (4.9) then we can get the following equation:

(4.18)

We can observe from the above equation that as the mobile speed is slow,

and then

It is noted that, after the first iteration, we execute the channel tracking process of (4.7) for the second and subsequent iterations until a stopping criterion holds. The stopping criterion is to check whether the iteration number reaches the maximum value of . The channel

tracking process for the current time slot will be stopped when the above condition holds.

β

Fig.4 outlines the -tracker followed by DF DFT-based channel estimation method to estimate the average channel variations in the tracking stage. We perform the channel estimator described above for each transceiver antenna pair respectively, and then after each iteration we have ^{Hˆ( )j i, = ˆ( )j i,}

[ ]

^{k∈Ω= 0,}

{

^…,N−1

}

i= …^{1, ,}NT detect data by performing V-BLAST with MMSE-SIC algorithm on each subcarrier as mentioned in chapter 3.

( ) ( )

[ ] { }

β

-tracker followed by DF DFT-based channel estimation method in the tracking stage Fig. 5

4.3 ICI Estimation

In the presence of Doppler caused by high mobility, due to the ICI term of (2.8), using the estimate of average channel for data detection results in poor performance. This motivates the need to perform ICI estimation.

In this stage, we adopt a linear model with a constant slope over the time duration of one OFDM symbol to approximate time variations of each path. When the linear model is applied, we will derive the frequency domain relationship, similar to (2.8). We first consider the SISO case with one transmit and one receive antenna by omitting the superscript “

( )

^{j i,}

)

The received signal of SISO case is as follows:

X k H k k X m H k m Z k

the channel frequency response from the mth subcarrier on the kth one which can be expressed as:

Furthermore, we assume

β

-tracker and can

be expressed as

[ ]

¹ ,0 ²

where μ_l,0 is the average channel variation of the lth path. Through an IDFT, we have

[ ]

²

linearization, knowledge of the channel at one time instant in the symbol is necessary. As mentioned in [7], let

Consider linearization around 1

l 2

h ⎡_⎢N ⁻ ⎤_⎥^{. Then,}

[ ]

⎣ ⎦ ^{h nl}

[ ]

can be approximated as follows:

,1 1 ,0 0 1

l l 2 l

h n =μ ^⎛⎜⎝n−N + +^⎞⎟⎠ μ ≤ ≤ −n N

[ ]

(4.25) Inserting (4.25) into (4.21), we will have

2

Subtract the non- ICI term from the received signals and we will have k

[ ] [ ] [ ]

¹

[

[ ] [ ] [ ]

For simplification, let

[ ]

and we can modify the cost function as

[ ] [ ] [ ]

Direct calculation yields that (4.31) is equivalent to

[ ] [ ]

^'

Now we define Equation (4.35) can be written in matrix form as

(4.36)

Hence, the slope of each path can be estimated by

(4.37) and with the knowledge of the slope of each path, we can reconstruct the channel frequency

response ^{H k m,}

]

as shown in (4.26).

Now, with the ICI estimation for SISO case, we can apply it to MIMO case by modifying the received signal ^{Y k}

[ ]

in (4.20) and perform the ICI estimation for each antenna pair. In MIMO environment, received signal at each receive antenna suffers from inter-antenna interference including intercarrier interference (ICI) as follows:

(4.38)

To let the received signal in MIMO case corresponding to the i th antenna pair, named it as refined signal, be the same as ^{Y k}

[ ]

in SISO case, we need to subtract the inter-antenna interference including ICI from the received signal ^{R( )j}

[ ]

k as follows:

( )^{j i}

[ ]

antenna pair can be treated as ^{Y k}

[ ]

in SISO case. However, in the beginning of this stage,

( )^{j i,}

[ ]

R k

( )^{j n,}

[

^,

]

H k m is not available, so we cannot obtain . Therefore, with the simulation in the following, we use another refined signal ^{R( )j i,}

[ ]

^k

(

which is obtained by subtracting inter-antenna interference excluding ICI term from the received signal ^{R j)}

[ ]

k as follows to estimate the slope of each path:

In the following simulation, we assume that average channel variations μl are known and inter-antenna interference excluding ICI can be perfectly subtracted from received signal

[ ] [ ]

( )^j ( )^{j i,}

R k . We adopt the refined signal R k , and perform ICI estimation to estimate the

slope of each path for each antenna pair, and then we have Fd*T=0.02 for 6 path Fd*T=0.02 for 2 path

Fig. 6 NMSE of the channel estimation with the assumption that the average channel variations μ_l^{( )}_,0^{j i,} 0,for l= …,L^{( )j i,} are known versus E_b/N_o

Fig. 6 shows the normalized mean square error (NMSE) of channel estimation with the assumption that the average channel variations μ_l^{( )}_,0^{j i,} 0,for l= … L, ^{( )j i,}

equals to 0.02 and 0.1, respectively. As can be observed in Fig. 6, the MSE of channel variations are all quite small even in low region. Hence, to perform ICI estimation for MIMO case, we just need to modify (4.20) as follows:

[ ]

^{( )j i,}

[ ]

^{( )j}

[ ]

^{( )j n,}

[ ]

^{, ( )n}

[ ]

In the above derivation, we have assumed that the transmitted signals for each antenna are known. However, in reality, the transmitted signals are obtained by applying the V-BLAST detection as shown in chapter 3 to received signals.

4.4 Two Dimensional V-BLAST Detection

In section 2.4, we have known that the ICI effect on OFDM systems and its mathematic representation. Now, we change the mathematic representation in section 2.4 into matrix form which is convenient to be used in this section.

[ ]

^{( )}

[ ]

Then the received signals suffer not only from inter-antenna interference but also from intercarrier interference and (2.8) can be expressed as follows:

(4.43) In this section a two-dimensional V-BLAST method is utilized to perform ICI cancellation

and data detection. From [13], we know that most of the ICI effect on a subcarrier comes from neighboring subcarriers, so we assume the group size of the ICI effect is A=2a+1

which means that ^{X( )i}

[ ]

n causes interference to ^{R( )j ⎡}_⎣

( (

^{n a−}

) )

_N^⎤_⎦^{~R( )j ⎡}_⎣

( (

^n+a

) )

_N^⎤_⎦ Fig. 7 ICI effect from neighboring subcarriers with group size

( )_m¹ ~ ( )_m^NT

S S

The basic idea here is to cancel the component of , n≠m in Y_n^{( )j} . And treat

The basic idea here is to cancel the component of , n≠m in Y_n^{( )j} . And treat

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