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

i

With the MIMO channel model, the received signal can expressed as follow

,1 2 0 2 1 1 ,1 Exploiting eq. 3.1, the channel information is accessible if the transmitted signal is known, which is as follow

( )

In Alamouti scheme, the CIR is assumed not changed during time t to time t+T, which

means the first and the second symbol suffer the same channel effects. Because the pilot information is inserted in frequency domain, the Alamouti encoder and decoder are implemented in frequency domain. Fig. 3.3 is the block diagram of MIMO-OFDM with Alamouti scheme in detail.

Fig. 3.3 The Block Diagram of MIMO-OFDM with Alamouti Scheme

3.4 How to Apply 1

^st

and 2

^nd

Moments in MIMO

With this MIMO-OFDM architecture, the complicated relationships of multi-antenna are solved by Alamouti CODEC and Alamouti equalizer blocks. After passing through Alamouti Decoder, each output stream can be regarded as a SISO-OFDM system. Therefore, all the channel estimation methods based on comb-type pilot interpolation in SISO-OFDM system are adaptable in MIMO-OFDM system without changing the algorithms. Since each output stream of Alamouti decoder represents each Tx-Rx path, the CFR information on pilot subcarrier is able to calculate the 1^st and 2^nd moments just as Chapter2 does.

Chapter 4

Simulation Results

4.1 Some Simulation Result of the Moment Estimators

The first series simulation results are the improvements of the 1^st and 2^nd moment estimator with factors x and y added. There are three types of power delay profiles (PDP) in this simulation, which are step function, exponential function, and normal function.

Denote that the total powers of them are equal. In Fig. 4.1, there are three types of PDPs.

Fig. 4.1 Three Types of PDPs

In the simulation, the channel length N is 1000, pilot subcarrier number is 333, and all

PDP has the same power, which is equal to one. In Fig. 4.2 and Fig. 4.3, the x-axis of uniform distribution is the width of step function PDP, the x-axis of exponential distribution is the decay parameter alpha, where

2 alpha alpha

The x-axis of normal distribution is the mean value mu, mu=1:200 p=50, and the x-axis of normal distribution1 is the variance value p, p=1:200 mu=50.

The normal distribution is

( )²

2

mu 2

normalizer 2 p2 l

The y-axis in Fig. 4.2 and Fig. 4.3 are the normalized MSE of the estimators in db, which are

⎟ respectively.

Fig. 4.2 The Normalized MSE of m1 Estimator

Fig. 4.3 The Normalized MSE of m2 Estimator

In Fig 4.2, the Normal Distribution1, the blue line with factor x get raised after mu

>160. The reason is when aliasing occurred, the estimated m1 is negative, which is wrong.

However, the x factor is negative too, and that is the reason the blue line get raised when mu>160. With respect to the summation paths of 1^st moment, we can see that when PDP is shorter than N/(4*F_S), the m1 estimator is reliable. In the simulation N/(4*F_S) = 83. And with respect to summation path of 2^nd moment, the estimator is reliable when PDP is shorter than N/(3*F_S), which is 111 in this simulation.

In Fig. 4.4 and Fig. 4.5 are the simulations of MSE of m1 and m2 respectively, which

Fig. 4.4 The m1 Estimator MSE of Theoretical Value and Monte Carlo Method

4.2 Some Simulation Results of the MIMO-OFDM with the Interpolation methods

MIMO-OFDM system parameters used in the simulations are illustrated in Table 3.

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.

Table 3 Simulation Parameters

Parameters Specifications

FFT Size 1024

Pilot Ratio 1/11

Guard Interval 64

Antenna 2*2 Signal Constellation QPSK

Channel Model Rayleigh fading Channel Length 20

Power Delay Profile Exponential

The simulation shows the BER of different interpolation methods used in MIMO-OFDM in different SNR. The simulation result is shown in Fig.4.6. The X-axis represent SNR in db, and the noise is added after the signal convolution to CIR, and the noise power is expressed as

SNR10 _ _

noise power = 10

_

FFT length CP length FFT length

− × + .

Fig. 4.6 BER Performance versus SNR for the Equalizer Based On Alamouti with Comb-Type Pilot Arrangements

Chapter 5 Conclusion

There are two contributions achieved in this thesis. First, a comb-type pilot aided channel estimation method (MA-LPIFD) is proposed. Second, the architecture of MIMO-OFDM that is able to apply to MA-LPIFD is designed. From the simulation result, we can see that the BER performance is better than traditional LPI, and the costs are N_P+3 multiplications and 2*N_P additions to access filter design indexes.

Denote that NP is the number of pilot subcarrier.

Besides the contributions, there are still some future works and some survey I haven’t completed. First, in chapter2, if the centroid (first moment) of CIR can be replaced by the center of CIR, the center of the filter can be designed more precisely, and the performance of BER must be improved. I think the key point is to figure out

Besides the contributions, there are still some future works and some survey I haven’t completed. First, in chapter2, if the centroid (first moment) of CIR can be replaced by the center of CIR, the center of the filter can be designed more precisely, and the performance of BER must be improved. I think the key point is to figure out

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