Organization of the Thesis

The organization of this thesis is as follow. In Chapter 2, some comb-type pilot interpolation methods will be introduced, after that, we will discuss our interpolation method in detail. In Chapter 3, the architecture of MIMO-OFDM that adopts the interpolation method we bring out will be introduced. In Chapter 4, we will show the simulations results and analysis of these methods. At last, the conclusion and future works will be discussed in Chapter 5.

Chapter 2

Comb-Type Pilot Aided Channel Estimation

2.1 The Signal Model of Comb-Type Pilot Channel Estimation in OFDM System

OFDM is becoming widely applied in digital wireless communication systems due to its high data rate and high bandwidth efficiency. For wideband mobile communication system, the channel between transmitter and receiver is usually frequency-selective and time-variant. However, most channel estimation methods for OFDM system are developed under the assumption that the channel is stationary within one packet, and those methods are not suitable under mobile communication channel. Therefore, a dynamic channel estimation method for frequency-selective and time variant channel is needed. The pilot-based channel estimation has been proven suitable for OFDM systems in mobile communication channels [7].

In comb-type pilot arrangement, as shown in Fig. 2.1, for each transmitted symbol, N_p pilots signals are uniformly inserted into N subcarriers in Xn. FS is the distance of two pilots subcarriers, where F_S=N/ N_p. In Fig. 2.2, a more detail com-type pilot arrangement in single OFDM symbol is illustrated.

Fig. 2.1 Comb-Type Pilot Arrangement

Fig. 2.2 Comb-type Pilot Arrangement in Single OFDM Symbol

Denote that the subcarrier address of pilots in each OFDM symbol is the same, and can be expressed as follow

( ) ( )

^{0 , 1 , ,}

(

^{p 1}

)

^{T, ( )}

⁽

¹

⁾

^{s k 1 p}

p p p N p k k F

⎡ ⎤

=⎣ − ⎦ = −

p = ∼ N

⎤ ⎦

(2.1) and the pilot values are

p T

P(N -1)

P P(0) P(1)

, , ,

X X X

= ⎣ ⎡

X

. (2.2) At the receiver, if pilot values and subcarrier address are known, then CFR at the pilot subcarrier can be calculated by some method, for example, LS estimator, equation 1.10, or

some other algorithms. Denote that the estimated CFR at the pilot subcarriers of n-th OFDM symbol as follow

p T

P(N -1)

P P(0) P(1)

ˆ

_n

= ⎣ ⎡ H ˆ

_n

, H ˆ

_n

, , H ˆ

_n

H ⎤ ⎦

]

(2.3) where . Since estimated CFR at the pilot subcarrier are acquired, there are some interpolation methods to interpolate the CFR between two pilot subcarriers. The channel estimation block diagram is shown as Fig. 2.3.

P(0) [

,P(0)

ˆ ˆ

n n

H = H

Fig. 2.3 The Channel Estimation Block Diagram

2.2 Prior Arts in One-Dimensional(1-D) Interpolation Methods for Comb-type Channel Estimation

One-dimensional interpolation is used to estimate the CFR at the data subcarriers, with reference to the estimated CFR at the pilot subcarriers, and some methods are summarized in the following section [7].

2.2.1 Piecewise-Constant Interpolation

The piecewise interpolation is the simplest interpolation method. With mathematical expression, the data subcarrier CFR between two pilot subcarrier and are estimated as follow, and illustrated in Fig. 2.4.

ˆP( )^k

Hn ˆ^{P(k 1)} Hn ⁺

P( ) P( )

ˆ_n ^{k i} ˆ_n ^k ,where i=1 FS 1

H ⁺ =H ∼ − (2.4)

Fig. 2.4 Piecewise-Constant Interpolation

2.2.2 Linear Interpolation (LI)

The LI interpolation method performs better than piecewise-constant interpolation, and the data subcarrier CFR between two pilot subcarrier and are estimated as follow equation, and illustrated in Fig. 2.5.

ˆP( )^k

Hn ˆ^{P(k 1)} Hn ⁺

[ ]

Fig. 2.5 Linear Interpolation

2.2.3 Low-pass Interpolation(LPI)

The LPI method is performed by inserting zeros in to , and then applying a low-pass finite-length impulse response (FIR) filter. The data in pilot subcarriers is not changed, and the data in the data subcarriers will be interpolated.

ˆ

P

H

n

This method minimized the mean-square error between real value and interpolated value [7]. The idea of this method is also the inspiration of our algorithms. To explain this method, we need to figure out what padding zeros and low-pass filtering do in both time domain and frequency domain. Assumed that the estimated CFR on pilot subcarrier are equal to the real CFR, that is

assume that the channel path is L, and then do N-points DFT to transform to frequency domain. After padding F -1 zeros into Hˆ ^P, it becomes

It is assumed that N/FS>L, which means no aliasing at time domain. And denote The relationship between and in frequency domain and time domain is shown in Fig. 2.6, and the elements of and can be expressed as following equation

S times repetitions of h_n and 1/F_S of its amplitude. Therefore, a low pass filter is used to pass through is needed to get one copy of the repetition CIR. Theoretically, the width of this filter is 2*N/FS, which is shown in Fig. 2.7. The time domain low-pass filter can be implemented by a finite length impulse (FIR) response filter, and applied after

, Fig. 2.8 shows the block diagram of LPI..

ˆ 0

_n

h

ˆ 0

_n

H

Fig. 2.6 The Relationships Between and in Frequency Domain and Time Domain

ˆ 0_n

H H_n

Fig. 2.7 The Low-Pass Filter that Gets The Real CIR

Fig. 2.8 The Block Diagram of LPI

2.2.4 Time Domain Interpolation (TDI)

The TDI method is a high-resolution interpolation based on DFT/IDFT. It first converts to time domain by IDFT, and then interpolate the time domain data to N-points by piecewise-constant interpolation or linear interpolation method. The block diagram of TDI is shown in Fig. 2.9.

ˆ

P

H

n

Fig. 2.9 TDI Block Diagram

2.2.5 Other Interpolation Methods

Besides the methods we introduced in 2.2.1 to 2.2.4, there are other 1-D interpolation methods, for example, Second-Order Interpolation (SOI) and Spline Cubic Interpolation (SCI) which exploit three adjacent pilot data and continuous polynomial fitting to estimate CFR respectively. The complexity analysis and simulation performance are shown in Table1 and Fig. 2.10 [7].

Estimation Scheme

Complexity Comments PCI Lowest

LI Lowest SOI Low

Simple estimation and interpolation methods

SCI LPI TDI

Moderated Interpolation methods are relatively complex, with fitted polynomial, low-pass convolution, and DFT/IDFT calculation, respectively

Table 1: Computational Complexity Analysis of Comb-Type Pilot 1-D Interpolation Methods

Fig. 2.10 SER Performance versus SNR for the Channel Estimators Based on LS with Comb-Type Pilot Arrangements

2.3 Proposed Moments-Assisted Low Pass Interpolation (MA-LPI)

In those 1-D interpolation methods, with mathematical analysis and simulation results, LPI method is the best performance. In LPI method, the width of low-pass FIR filter is decided by N-points FFT and the distance between two pilot subcarriers. However, there are still some places that can be improved. For example, in 2K-points FFT system, the distance of two pilot subcarriers is 4 (FS=4), and the channel length L is 40. In LPI method, the width of low-pass filter will be 2K/4, which is much longer than the channel length L, and a lot noise (AWGN noise, the red part) will be filtered in. This example is shown in Fig.

2.11; we can see that, a lot of undesired noises (the red part) are passed by the wide width low-pass filter. If the filter width can be tuned to get lesser noise, the signal to noise ratio (SNR) will be raised, and the SER/BER will be lowered. In Fig. 2.12, a more ideal filter in assumption is shown. Therefore, the idea is that if the range can be tuned with respect to the CFR or CIR, the new filter will be more ideal [11].

Fig. 2.11 The Example of LPI

Fig. 2.12 A more Ideal Filter in Assumption

A bigger question is how to exploit the CIR/CFR information to design the filter.

There are two parameters we eager to know, the first one is where the center of the CIR is, and the second one, what’s the length of the CIR is. If the CIR is acquired, these two parameters can be accessed easily. The center of the CIR can be accessed by setting the threshold of the amplitude of CIR, and then find out the center and the width of this band-pass filter. Of course, it needs DFT/IDFT blocks to transform to time domain.

This idea, is rather similar to the combination of LPI and TDI, however, this is the first version of this interpolation method. The only drawback of this idea is the high cost of IDFT/DFT block. The block diagram of the current idea, called Parameters-Assisted Low Pass Interpolation in Time Domain (PA-LPITD), is shown in Fig. 2.13.

ˆ P

Hn

access the filter parameters in frequency domain. In this way, all the computation is completed in frequency domain, and the IDFT/DFT blocks are negligible. However, the center and the length of the CIR are not accessible in frequency domain, but the centroid (the first moment) and the second moment of CIR are accessible in frequency domain.

Therefore, in the second version algorithm, the center of the CIR is replaced by the first moment of CIR, the length of CIR is replaced by the second centre moment of CIR, and the IDFT/DFT blocks are removed. This method, which is called Moments-Assisted Low Pass Interpolation in Frequency Domain (MA-LPIFD), accesses these parameters in frequency domain, and it is discussed in Chapter 2.4. In Fig. 2.14, the block diagram of MA-LPIFD is shown.

Fig. 2.14 The Block Diagram of MA-LPIFD

2.4 The Channel Moment Estimation

In this section, we will focus on the estimation of the 1^st moment and 2^nd centre moments of CIR. In section 2.4.1, it will be discussed in continuous time, to figure out the method to access moments in frequency domain. In section 2.4.2, the discrete form of the moment estimators will be shown. In section 2.4.3, the down sampling effects will be discussed. In section 2.4.4, the relationships between moments and moments estimators are discussed, and some improvements of the estimators are also introduced in section 2.4.5.

Some simulation of this improvement is shown in section 2.4.6.

2.4.1 How to Access the Moments of CIR

Denote that the continuous time CIR is ^{h t , and}

( )

h t0

( )

= h t

( )

² is the square of absolute of ^{h t}

( )

in continuous time.

Because the CIR is complex number, the square of absolute value of CIR is used in following calculation. Of course, there are some differences between ^{h t}

( )

and ^{h t}

( )

^2,

but not much differences in the moments calculations. To design the filter, the following information of ^{h t}

( )

² is required: the 1^st moment m1, the 2^nd centre moment m2,centre and

and (2.9), (2.10), (2.11) are functions of continuous time. These functions need a bridge to connect frequency domain and time domain, and the Fourier transform is the one to

connect two different domains. The Fourier transforms of ^{h t}

( )

^{2 and} h t1

( )

² are shown in (2.12a) and (2.12b) respectively. (2.13) is the differential of (2.12) to frequency; (2.14) is the second differential of (2.12) to frequency.

{ }

And if f=0, (2.12a), (2.13), (2.14) can be written as follows:

(0) ( )2 (2.16a), the first moment can be expressed in frequency domain:

1 And the differential functions can express as follows:

1 And the approximate of (2.19) is

1

where F is division between two subcarrier frequencies. Since are complex conjugate, (2.20) can rewrite as

( ) and ( ) From (2.15) and (2.17b), the 2^nd centre moment is written as

'' And the approximation of (2.22) is

2

And the upper part of (2.24) can express as follows:

2

(2.24) can rewrite as follows:

{ }

And the approximation of (2.23) is similar to the 2^nd moment, the only difference is the m₁ phase shift, which is R_HH1( )f =R_HH( )f ×e^j2πfm1. And the approximation of (2.23) is

2.4.2 The Discrete Form of the Moments

Because all simulations are executed in discrete, it’s necessary to transform m1, m2, and m2,centre into discrete form. We can assume that sampling frequency

F = 1/ N

, where N is the FFT size, and the 1^st moment in discrete and the 1^st moment estimator in discrete is show as following equations:

[ ] equations of the 2^nd moment and the 2^nd moment estimator in discrete are as follow

[ ]

And the equations of the 2^nd centre moment and the 2^nd centre moment estimator in discrete are as follow

2.4.3 The Effect of Down-Sampling to the Moments Estimators

In chapter 2.4.2, the F is defined as 1/N, which means all subcarrier information is obtained. In fact, a receiver gets information only on pilot subcarriers. The CFR on pilot

downsampling

In (s1), the pilot signals are acquired at receiver, and then we use those pilots to access the autocorrelations. But all the autocorrelations we mentioned before are defined as steps (s2). It’s important to know what’s different in those two steps. With reference to Fig.

2.15, the down-sampling effects to the CIR in time domain are shown clearly.

P

Hn

Fig. 2.15 Down Sampling Effects

And the differences in time domain of these two steps are shown in (2.30).

[ ] [ ] [ ] and the moments estimator is usable. In other words, if the channel length is longer than N/F

[ ]^,

3_{n l}

h h1_{[ ]n l,}

S, there are aliasing in , and its autocorrelation will not equal to , and the estimator is not work. With sapling theory, if the number of pilot subcarriers is not enough, the obtained information is unable to recover the original data, which means the N/FS will be shorter than the channel length.

[ ]^,

0_{n l}

h h1_{[ ]n l,} h3_{[ ]n l,}

2.4.4 The Relationships between Moments and Moments Estimator in Time Domain:

To evaluate the performance of an estimator, there is some background knowledge that needs to know. The relationship of the moments and the moment estimators in time domain can help with the analysis of the estimator performance. Therefore, we have to transform the frequency domain estimator to time domain to compare with the original moments.

The frequency response of channel is (1.3a):

1 L−

∑

[ ] [ ] reason to use N instead of L is that it’s convenient to combine the effect of down sampling to the estimators. Use (2.31) to the estimators, the autocorrelation, (2.27b) and (2.28b), can be expressed in time domain as follow:

[ ]^{n l,}

summation path of m1 is linear and the summation path of m₁ is a sine wave. Fig 2.16 shows the summation paths of the 1^st moment estimator and the1st moment. The left one is the summation path of the estimator, the right one is the summation path of the original moment, and the blue one is the CIR. Denote that the FS equal to F, which means all subcarrier information is obtained, and there is no down sampling effects in Fig. 2.16:

Fig. 2.16 The Summation Paths of the 1^st Moment and its Estimator From Fig. 2.16, (2.27a), and (2.33), we can know that the slope of summation paths is almost equivalent at the beginning paths, which is about N/4 paths. It tells that if the CIR is centered on first few paths, the estimator will be very accurate.

When it comes to the performance of the 2^nd moment and the 2^nd centre moment, there is a simplification has to do to the estimator, thus we can see the relations more clear.

The following equation is the relation between m2 and m2,centre in statistic.

{

^2,centre

} { ( ( ) )

²

} ^{{ }}

²

^{( { } )}

²

E m =E x−E x =E x − E x ,

where x is the CIR. The simplification is: Assume that m₁ is perfectly estimated, which E{x}= m1, and then E{x²}=m2 is the only parameter that affects the performance of the 2^nd centre moment estimator. After this simplification, the performance evaluation of the 2^nd

original moment in time domain, and their summation paths are shown in Fig. 2.17. The cosine line is the summation path of the estimator, and the red line is the summation path of the original moment. Denote that FS=F , which is same as the figure of 1^st moments.

Fig. 2.17 The Summation Paths of the 2^nd Moment and its Estimator

From the upper figure, we can see that the summation path of the estimator is almost equivalent to the one of the original moment before N/3 paths. It means that if the CIR is longer than N/3 paths, then the 2^nd moment estimator will be failed.

It is assumed that all subcarrier information is obtained in upper two figures, however, often the FS does not equal to F, and the summation paths also have some changes. These changes are effect from down sampling, which is discussed in Chapter 2.4.3. The

summation path of first moments with down sampling effects is show in Fig. 2.18, and the red line is the summation path of m₁,and the black sin waves is the summation path of , we may consider to adjust the amplitude of sine wave to gain a more accurate estimator with comparison to m

m1

1. With down sampling effects, the m1 estimator equation has a little change, which is as follow

[ ]

The summation length is reduced from N-1 to (N-1)/FS because of the FS repetitions in time domain, and the F_S in denominator is used to reduce the amplitude of sine wave.

Fig. 2.18 The Summation Paths of m1 with Down Sampling Effects

On the other hand,m2 estimator has the same down sampling effects as m1 estimator does.

These effects are shown in Fig. 2.19. And the m2 estimator has the some changes too, which as follow:

[ ]

Fig. 2.19 The Summation Paths of m2 with Down Sampling Effects

And (2.35) and (2.36) are the equations used to evaluate the estimator performance in next section.

the receiver, we can see that the amplitude of sine wave and cosine wave is the key to the accuracy performance. If we can slightly adjust the amplitude of sine and cosine wave based on the channel power delay profile, it would be a good solution to improve the accuracy of these estimators. To adjust the amplitude, the factor x and y is added into the denominator of m1 and m2 estimators respectively, and the estimators can rewrite as:

[ ]

Use the method of MMSE to find out the optimal x, the procedures are as eq. 2.38, and it is assumed that the total power is constant, which _{[ ]}

/ 1 2 And the MSE and BIAS of the 1^st moment estimator are:

{

^{,1 ,12}

}

^{2 / 1 [ ], 22}

{ } { }

,1 ,1 ^{/ 1} _{[ ]}, ²

Assume that h[n,l]~complex normal distribution N(0,σ , then the probability density _l²) function (PDF) of h_{[ ]n l,} is Rayleigh distribution R(σ_l ^π₂, 2σ_l²), where R(a, b): a=E{x},

We can see that MSE function is also a function of factor x:

/ 1 2

Then the differential to x of f(x) is

/ 1 2

The f(x) will be minimized when x equals to

/ 1

And the MMSE and the BIAS with MMSE factor x are:

/ 1 2

It’s obviously that m1 is a biased estimator. After evaluated m1 estimator, we use the same method to find out the y factor of the 2^nd moment, the MSE and BIAS of m₂ estimator are: And MSE and BIAS become:

2

And let f(y) =MSE, and when df(y)/dy=0, and y is the MMSE factor, which is

/ 1 2

And mn,2,centre and mn,2 estimators with factor y can write as:

,1, ,1, ,1, ,1,

Chapter 3

Pilot-Aided Channel Estimation for MIMO-OFDM

3.1 Reasons to extend SISO-OFDM to MIMO-OFDM

Extending SISO-OFDM to MIMO-OFDM aims to provide spatial diversity gains by multi-antenna. The multi-antenna technique is a popular method, and there are many new algorithms designed to fully utilize its tremendous potential. Channel estimation for MIMO systems is important; the algorithm proposed in Chapter2 is designed based on SISO-OFDM, and if we want to use the MA-LPIFA method in MIMO-OFDM system without changing the algorithm, the architecture of

MIMO-OFDM must be considered. In this chapter the Alamouti scheme and MA-LPIFA are combined to provide an effective channel estimation method for MIMO-OFDM.

3.2 The Architecture of the MIMO-OFDM

To adapt those pilot interpolation methods in MIMO without changing the interpolation algorithms, the pilot interpolation must be completed in single channel information, which means there is a space decoder block needed to separate the multi-input for each receiver. Fig. 3.1 shows the idea of the architecture of MIMO-OFDM with comb-type pilot inserted, noted that the architecture after the spatial decoder is as same as the single channel estimation.

Fig. 3.1 The Architecture of the MIMO-OFDM Channel Estimation in Current Idea

3.3 The Space-Time CODEC for MIMO-OFDM

The Space-Time codes are exploited to extract the relations between different antennas; one of the famous S-T code schemes is Alamouti scheme [12]. In this these, the Alamouti scheme is the only method we use to extract the multi-antenna equations.

In Alamouti scheme, the antennas are design as 2 transmitters and M receiver, which is 2M diversity. The encoding and transmission sequence for the two-branch transmit diversity scheme is shown in table2.

Antenna0 Antenna1

Time t (n-th symbol) S0 S1

Time t+T (n+1-th symbol) -S1* S0*

Table 2 The Encoding and Transmission Sequence for the Two-Branch Transmit Diversity Scheme

In Table2, S0 and S1 are denoted as the n-th transmitted symbol by antenna0 and antenna1 respectively, and * is the complex conjugate notation. In Fig. 3.2 is the channel model with 2*M diversity.

Fig. 3.2 The 2*M MIMO Channel Model Denote that

2

2 1

The channel of Tx antenna0 and Rx antenna The channel of Tx antenna1 and Rx antenna

i The noise of Tx antenna0 and Rx antenna at time t+T (the second symbol)

i

The received signal at time t (the first symbol) of Rx antenna The received signal at time t+T (the second symbol) of Rx antenna

The received signal at time t (the first symbol) of Rx antenna The received signal at time t+T (the second symbol) of Rx antenna

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