Channel Equalizer Design Spec - 用於 UWB 及 WIMAX之雙模通道等化器設計

Chapter 1. Introduction

1.5. Channel Equalizer Design Spec

In OFDM system, the multi-path channel, AWGN, Carrier frequency Offset (CFO) between transmitter and receiver, and Sampling Clock Offset (SCO) between DAC and ADC, are the four main data distortion issues. In order to solve the multi-path channel effect, preambles are used to estimate the fading channel. Then, receiver uses the estimated channel to do the equalization. Because of the existence of AWGN, it will cause channel estimation (CE) error. This CE error will be propagated to the equalizer and leads to performance degradation. The way to lower down the CE error will be introduced in chapter 3.

Residual CFO and SCO will cause phase rotation in frequency domain, and result in ICI. Phase Error Tracker (PET) is applied in OFDM systems for phase rotation tracking and compensation. The detail algorithm for PET will also be introduced in chapter 3.

Chapter 2. Design Specification of Dual-Mode Channel Equalizer

2.1. Wireless Channel Model

In order to simulate the practical data transmission, the wireless channel model must be established. The channel model comprises multi-path fading channel, CFO, SCO, AWGN. The details are introduced individually below.

2.2. Intel Proposed UWB Multi-paths Channel Model

The Intel proposed channel model [4] is based on the observation that usually multi-path contributions generated by the same pulse arrive at the receiver grouped into clusters. The time of arrival of clusters is modeled as a Poisson arrival process with rate Λ:

( 1)

( |_n _n 1) ^{T T}ⁿ ⁿ

p T T

_  

e

^ ^ ^ (2.3)

Where

T

n and

T

n-1 are the times of arrival of the n-th and the (n-1)-th clusters.

Within each cluster, subsequent multi-path contributions also arrive according to a Poisson process with rate λ:

( ( 1) )

(

_nk

|

( 1)_n _k

)

^nk ⁿ ^k

p  

_

  e

^^{ } ^^ ^ (2.4) Where

τ

_nk and

τ

_(n-1)k are the time of arrival of the n-th and the (n-1)-th contributions

within cluster k.

The channel impulse response of the IEEE model can be expressed as follow:

( )

The channel coefficient

α

_nk can be defined as follows:

p

nk nk







(2.2)

where

p

nk is discrete random variable assuming values ±1 with equal probability and β_nk is the log-normal distributed channel coefficient of multi-path contribution

k belonging to cluster n. Theβ

_nkterm can be expressed as follows:

10^x20^nk

 

nk (2.3) Where

x

nkis assumed to be a Guassian random variable with mean

μ

nk and the

standard deviation

σ

_nk .

Variable

x

nk in particular, can be future decomposed as follows:

nk nk n nk

x





 

 

(2.4)

where

ζ

_n and

ξ

_nk are two Gaussian random variables that represent the fluctations of the channel coefficient on each cluster and on each contribution. We indicate the variance of

ζ

_n and

ξ

_nk by



_²and



_². The

μ

_nk value is determined to reproduce the exponential power decay for the amplitude of the multi-path contribution with in each cluster. One can write:

The amplitude gain X in eq. 2.1 is assumed to be a log-normal random variable:

10

20^g

X 

(2.7)

Where

g

is Gaussian random variable with mean

g

0 and variance



_g² . The

g

0 value depends on the average total multi-path gain G.

0 10ln( ) ln(10)

ln(10)

G

^g 20

g 

  (2.8)

The value can be determined as indicated below:

0 suggested in (Ghassemzadeh and Tarokh, 2003) for different propagation environments: A0=47dB and γ=1.7 for a LOS environment, and A0= 51dB and γ=3.5 for a NLOS environment.

According to the above definitions, the channel model represented by the impulse response is fully characterized when the following parameters are defined:

l Λ : The cluster average arrival rate l λ : The ray average arrival rate

l Γ : The power decay factor for clusters

l γ : The power decay factor for rays in a cluster

s : The standard deviation of the fluctuations of the channel coefficients

for clusters

s : The standard deviation of the fluctuations of the channel coefficients

for rays in each cluster

s : The standard deviation of the channel amplitude gain

The IEEE suggested an initial set of values for the above parameters. These values will shows in Table 2.1.

Table 2.1 –Parameter Settings for the IEEE UWB Channel Model

Scenario

Λ λ Γ γ

s

The channel impulse response (CIR) and the corresponding channel frequency response (CFR) of CM2, CM4 are shown in Fig. 2.1.

Fig. 2.1 CIR, CFR of the CM2 and CM4

2.3. WIMAX SUI Channel Model

In 802.16d system, a series of Stanford University Interim (SUI) [5] channel with three terrain types are selected for application on fixed broadband wireless.

According to the different terrain and tree density, the six SUI channels can be classified into three categories as showed in Table 2.2

Table 2.2 –SUI Channel

Model Category

SUI1

Flat terrain with light tree densities SUI2

SUI3

Hilly terrain with light tree densities SUI4

SUI5

Hilly terrain with moderate-to-heavy tree densities SUI6 standard deviation was found to be approximately 8 dB. K-factor model is presented as follows:

0 S h b

K F F F K d u

 ^ (2.10)

u

is a lognormal variable which has zero dB mean and standard deviation of 8.0 dB We use the method of filtered noise to generate channel coefficients with the specified distribution and spectral power density. For each tap a set of complex zero-mean Gaussian distributed numbers is generated with a variance of 0.5 for the real and imaginary part. Total average power of this distribution is 1. This yields a normalized Rayleigh distribution for the magnitude of the complex coefficients. For the Ricean distribution (K>0), a constant path component m has to be added to the Rayleigh set of coefficients. The ratio of powers between this constant part and the Rayleigh (variable) part is specified by the K-factor. The following equation shows the distribution of the power by stating the total power

p

of each tap.

2 2

Where m is the complex constant and



² is the variance of the complex Gaussian set. The generated coefficients of channel taps have a white spectrum since they are independent of each other. The SUI channel model defines a specific power spectral

density (PSD) function for these scatter component channel coefficients is given as:

ì - + £

To arrive at a set of channel coefficients with this PSD function, we correlate the original coefficients with a filter which amplitude frequency response is derived from eq. 2.12 as

( ) = ( )

H f S f

(2.13)

Without changing the total power of transmitted signal, the total power of the Doppler filter has to be normalized to one. Consequently, after passing through the Doppler filter, normalized factor is applied to normalize these multi-path fading taps.

Table 2.3 shows the parameters of six SUI channels. Table 2.4 shows the normalization factor for each SUI channel.

SUI-4 1.257 Delay(us) 0 1.5 4 [0 0 0]

Power(dB) 0 -4 -8

Doppler Frequency (Hz) 0.2 0.15 0.25

SUI-5 2.842 Delay(us) 0 4 10 [0 0 0]

Power(dB) 0 -5 -10

Doppler Frequency (Hz) 2 1.5 2.5

SUI-6 5.240 Delay(us) 0 14 20 [0 0 0]

Power(dB) 0 -10 -14

Doppler Frequency (Hz) 0.4 0.3 0.5

Table 2.4 -Normalization Factors

SUI Channel Models Normalization Factor (dB)

SUI-1 -0.1771

SUI-2 -0.3930

SUI-3 -1.5113

SUI-4 -1.9218

SUI-5 -1.5113

SUI-6 -0.5683

For system bandwidth equal to 20MHz, the examples of CIR and CFR for SUI-1 to SUI-3 are shown in Fig. 2.2 to Fig. 2.4.

Fig. 2.2 CIR and CFR for SUI-1

Fig. 2.3 CIR and CFR for SUI-2

Fig. 2.4 CIR and CFR for SUI-3

2.4. Carrier Frequency Offset Model

Carrier Frequency Offset (CFO) [1] is caused by the mismatch in RF local frequency between transmitter and receiver. When CFO is existed in transmission path, the received sub-carrier will be influence by the other sub-carriers, which is

the frequency offset. The received signal after down conversion is given as:

2 2 ( ) 2

( ) ( ) e^{j f t}^c e ^j ^f^c ^{f t} ( ) e ^j ^{f t}

R t



S t

 ^  ^ ^ ^ 

S t

 ^ ^^{ } (2.12) Finally, CFO effect is simulated by multiplying the original signal with a phase rotation. This phase rotation is increased by the time as it can be seen from eq. 2.12.

2.5. Sampling Clock Offset Model

SCO is caused by ADC/DAC sampling frequency mismatch. Because of SCO, the sampling points will slowly shift with time. This shift in time domain will result in phase rotation in frequency domain and ICI. Fig. 2.6 shows the concept of SCO, where

T

_s is the sampling time and

μ

is the value of timing offset.

m

( )

T

Fig. 2.6 SCO effect

The SCO model is established by the concept of interpolation with 20 taps raised cosine filter [6]. The FIR filter structure of the interpolator is shown in Fig.

2.7. This filter performs a linear combination of (

I

2+1=20) signal samples

x(nT

)

taken around the basepoint

m

k , and the operation can be shown in eq. 2.13. In eq.

2.13,

h 

( )

is the coefficient of FIR filter tap, the sampled value of the raised cosine function. The corresponding sampled value is shown in eq. 2.14 and Fig. 2.8.

Fig. 2.8 Sampled value of the raised cosine function

2.6. Additive White Gaussian Noise

The AWGN is established by the random generator (

randn). The output

random signal is normally distributed with zero mean and variance equal to 1. The AWGN noise can be modeled as

10 10

[ (1, ) (1, )] 2

P SNRS

w randn n j randn n



    (2.15)

Where ^P_S is signal power, SNR is signal-to-noise ratio (dB), and n is length of data signal.

Chapter 3. Design of Dual-Mode Channel Equalizer

In this chapter, a low complexity and high performance dual-mode channel equalizer is proposed. It includes channel estimator, equalizer, phase error tracker and modified normalized least-mean-square (NLMS) channel tracker. Data distortion caused by multi-path fading, AWGN, residual CFO and SCO can be eliminated by these four parts. Following is the block diagram of the proposed channel equalizer. After passing through the FFT, channel estimator uses received preambles to estimate the channel frequency response (CFR). The estimated CFR will be sent to the equalizer. Then, equalizer uses the estimated CFR to equalize the distorted data sub-carriers. After doing the equalization, equalized data sub-carriers will be sent to the phase error tracker and compensator. The tracked phase error combined with unequalized data, equalized data and estimated CFR are then sent to modified NLMS channel tracker to calculate the updated CFR. Finally, compensated data sub-carriers are sent to De-Mapper.

Fig. 3.1 Block Diagram of Proposed Channel Equalizer

3.1. Dual-Mode Design of Channel Estimation

The signal is transmitted over frequency-selective fading channel. After removing the cyclic prefix and guard interval, the post-FFT signal of the l-th OFDM symbol can be expressed by: post-FFT signal can be simplified as an equivalent model [7]:

e p f caused by AWGN, symbol timing offset, ICI, and other non-ideal parameters. With the assumption that the timing synchronization is perfect, there is no timing offset.

The equivalent model can be simplified as:

a f

p f ⁺

=

^{[ 2} ⁽ ^)/ ^]

+

( ) ( ) ( )

^{j l} ^k ^{T T}^u ^g ^T^u

l k l k l k

Y k X k H e W

(3.3)

Where the attenuation factor ( )

a f is very close to 1 and therefore can be neglected.

3.1.1. Channel Estimation for UWB

In UWB system, use zero-forcing (ZF) methodology with received two preambles to estimate the fading channel. Using the average value, the estimation error can be reduced and the performance will be improved. Simulation result shows that it will contribute 1~2dB gain in SNR. Considering that the pre-defined signal

L( )

X k in preamble, the estimated of the k-th sub-channel is given by:



1 2



( ) ( ) ( ) / 2 ( )

E L L L

H k



Y k



Y k X k

(3.4)

Where

Y

_L₁( )

k

Y

_L₂

( ) k

are the first and second received preamble.

In channel estimation, these two received preambles are also used to estimate the noise power which can be described as follows:

1 2 1 2 2

Based on the correlative property between adjacent sub-channels, the estimation of sub-channels can be further improved by delivering them into the 3-taps smoothing filter. The smoothing filter is a finite impulse response filter [8], which can be described as:

After doing channel estimation, the estimated sub-channel

H k will be

_E( ) convoluted by the smoothing filter. The equation is given by:

=-= å

- ×

( )

( ) ( )

S E

H k

H k m S m

(3.7)

After smoothing, each sub-channel will be a weighted summation of itself and nearby sub-channels. Simulation reveals that the system performance on PER will improve about 1dB for different transmission modes.

3.1.2. Channel Estimation for WIMAX

In WIMAX system, long preamble is used to estimate the fading channel. In standard of 802.16-2004, only even sub-carriers are utilized in long preamble. DC sub-carrier and odd sub-carriers are transmitted with zero value. Consequently, the ZF methodology is used to estimate the CFR of even sub-carriers at first, and then use 6-tap raised-cosine interpolator to get CFR of DC sub-carrier and odd sub-carriers. Following is the description of channel estimation for WIMAX system.

Considering that the pre-defined signal

X k in preamble, the estimated CFR

_L( ) of the k-th even sub-carrier is given by:

( ) ( ) / ( )

Where D is the spacing of even sub-carriers, and β is the roll-off factor. And, let k equal to -5, -3, -1, 1, 3, 5, the coefficient of six taps can be generated.

But, using 6-tap raised-cosine interpolator to get CFR of sub-carrier number -99, -97, 97, 99 will result in large error [9]. For example, to get CFR of sub-carrier number -99, it needs CFR of sub-carrier number -104, -102, -100, -98, -96, -94. But, sub-carriers are transmitted with zero value at number -104, -102. Receiver doesn’t have any information at these two places. If zero value is applied to substitute the CFR of these two places, it will result in large error. So, the linear interpolation is used to get the CFR of sub-carrier number -99, -97, 97, 99.

Otherwise, DC sub-carrier (number 0) is transmitted with zero value. In order to get CFR of sub-carrier number -5, -3, -1, 1, 3, 5, CFR of DC sub-carrier is needed.

To interpolate the DC value, CFR of sub-carrier number -10, -6, -2, 2, 6, 10 are used to get the CFR of DC sub-carrier [9]. Next, DC value is used to get the CFR of sub-carrier number -5, -3, -1, 1, 3, 5. For example, to interpolate the CFR of sub-carrier number 5, CFR of sub-carrier number DC, 2, 4, 6, 8, 10 are used as inputs to the interpolator. Following the same rule, CFR of sub-carrier number -5, -3, -1, 1, 3 can be generated.

According to the above statement, 6-tap raised-cosine interpolator is needed.

Using 6-tap raised-cosine interpolator to get CFR of odd sub-carrier can be described as follows:

33 (MMSE) methodology is very suitable to equalize the distorted signal. Traditional zero-forcing equalization will increase the noise and degrade the system’s performance. Consequently, MMSE equalization is used to suppress the interference and noise effects in UWB system. The compensating value of the MMSE

channel estimation part, X k_L( )²is transmitted preamble power, which can be fixed as 1 in UWB system. Finally, the signal is equalized as:

ˆ ( )

_l _l

( )

_MMSE

( )

X k = Y k C × k

(3.12)

For WIMAX system, there are four different modulation types, such as BPSK, QPSK, 16-QAM, 64-QAM. For 16-QAM, 64-QAM conditions, using MMSE

methodology to do equalization will suppress the signal and noise simultaneously.

The attenuation signal will cross the decision boundary, and result in decision error.

Consequently, MMSE methodology is not suitable for WIMAX system. In this place, ZF methodology is used to equalize the distorted signal. Following is the compensating value of the ZF equalization:

( ) 1/ ( )

ZF E

C k  H k

(3.13)

Next, the signal is equalized as:

ˆ ( )

_l _l

( )

_ZF

( )

X k = Y k C × k

(3.14)

3.3. Dual-Mode Design of Phase Error Tracking and compensation

CFO and SCO are the other two data distortion sources for OFDM systems, which are caused mainly by a crystal oscillator frequency mismatch between transceiver and DAC/ADC mismatch. Generally, timing synchronization is applied to compensate these two effects. Nevertheless, other impairments will make the synchronization imperfect. The residual CFO and SCO will cause the data phase rotation in frequency domain and result in inter-carrier interference (ICI). In case of a coherent demodulation scheme, the error due to the rotation of a signal constellation will be fatal for transmission bursts, because the accumulated phase rotation will shift the constellation point to cross the decision boundary. It will cause high error rates and therefore the phase error tracking (PET) is applied in OFDM systems for phase rotation tracking and compensation.

Following the eq. 3.3, after doing the equalization, the equalized data can be

35 effect. The phase rotation of the l-th symbol and k-th sub-carrier can be derived as:

p pz

æ + ö

The first term is caused by CFO effect. This phase rotation is constant for all data sub-carriers in an OFDM symbol and increases with symbol index. The second term is caused by SCO effect, which is a linear phase rotation with sub-carrier index.

This also increases with symbol index. So, the PET is used to track the mean phase error caused and phase error slope caused by these two effects.

The PET is designed based on the known pilot sub-carriers. In order to achieve correct tracking of phase error exceeding ±

p

, the pre-compensation [10] is added in the PET. Before PET, the pilot sub-carriers will first be compensated with the phase error tracked in the previous OFDM symbols. Therefore, only the difference of the phase error between previous and preset OFDM symbols needs to be tracked with pilot sub-carriers. This will enhance the PET accuracy. The PET algorithm with pre-compensation is given as:

K=K

WIMAX ,

l=1 means the first received OFDM symbol, 

,lK is the detected phase

error of l-th OFDM symbol, 



_,lK is defined as the pilot phase after pre-compensation with the previous tracked phase error _l_₁  K _l_₁.

Next, use the pre-compensation pilot phase to calculate the mean phase error

l, which can be shown in eq. 3.18 and phase error slope

g

l, which can be shown in eq. 3.19, and the related parameters for UWB and WIMAX systems are shown in eq. 3.20.

where

g

_l_-₁is the previous, Wg , C are constant terms, K⁺ is index list of positive

pilot sub-carriers , K^- is index list of negative pilot sub-carriers.

{ } { }

To consider the hardware complexity, the proposed phase error tracker has the advantage of reducing the multiplicative computation. In the pilot phase pre-compensation part, it only needs one multiplier. Using mean-average method, it only needs to do constant multiplication of two times in each OFDM symbol.

After PET, the data sub-carriers are compensated according to the tracked phase error, which can be indicated as:

( ) ˆ ( ) exp( ( ))

Consequently, interpolator is used to get the CFR of odd sub-carriers. Because the ideal interpolator is not practical, there is some aliasing effect occurred in interpolation. Aliasing effect will increase the channel estimation error, and degrade

the performance of equalizer. Otherwise, noise, interference and time-variant characteristic will also increase the channel estimation error. Consequently, channel tracker is applied to mitigate the above non-ideal effects and enhance the accuracy of channel estimation.

Standard LMS methodology suffers from a gradient noise amplification problem. To overcome this difficulty, NLMS methodology normalizes the adjustment of tap-weight vector. But, in OFDM system, residual CFO and SCO will result in phase rotation in frequency domain. The traditional NLMS methodology can not be directly used. So, the modified-NLMS methodology is proposed with consideration the phase rotation. Following is the detail algorithm of proposed modified-NLMS channel tracking.

The equalized signal after compensating with tracked phase error is sent into slicer (map the signal to nearest constellation point according to modulation type) to compute the decision signalX k_l( ). For each sub-carrier

k, a single tap frequency

domain NLMS filter is used. The modified-NLMS error

E k is as follows:

_l( )

( ) ( ) exp( ( ) ( )

_l 1

( )

l l l l l

E k  Y k   j   k   X k H 

_

k

(3.22)

Where Y k_l( ) is unequalized signal of

l-th OFDM symbol, 

 k 

_l is the compensated phase error com from PET, H_l_₁( )k is the previous estimated CFR.

The gradient at sub-carrier

k (

H k_l( )) of

l-th symbol is given by:

* 2

( ) ( ) ( )

( )

l l l

H k E k X k

  

X k

(3.23)

Finally, using the gradient and previous estimated CFR, the update eq is given as:

( )

( ) ( )

l l l

H k  H

_

k     H k

(3.24)

Where  is the constant gain.

Then, the updated CFR H k_l( )can be used for next OFDM symbol.

Chapter 4. Simulation Result and Performance Analysis

In order to verify the proposed design, complete system platforms are established according to the UWB standard (802.15.3a) and the fixed WIMAX standard (802.16d) on Matlab. The platform has been introduced in chapter 1.

Channel estimation accuracy, PET performance, and system performance will be simulated and compare to system constraint of two standards.

4.1. Performance Analysis of the Proposed Dual-Mode Channel Equalizer for UWB system

To analyze the performance of a proposed scheme, we simulate the PER (%) for different SNR and different CFO, SCO value with CFO and SCO variation being considered. The simulation is processed with multi-paths CM2 channel with data rate 480 (Mb/s). The simulation result is shown in Fig. 4.1. The cross-section view for PER (%) with different CFO, SCO values is shown in Fig. 4.2. From the simulation result, it shows that the proposed PET can track the phase error correctly and compensate the phase error under 60 ppm of CFO, SCO value. Above 60 ppm of CFO, SCO value, resulted ICI and phase error will increase grossly to degrade the system performance. Under IEEE 802.15.3a demanded [2], the CFO, SCO value should be less than total 40 ppm (±20 ppm), and the proposed PET can track the phase error accurately.

Fig. 4.1 PER (%) simulation for different SNR and CFO, SCO value under CM2, data rate

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