Data

The DATA field contains the SERVICE field, PSDU, TAIL bits, and PAD

bits.

Figure 3.3 SIGNAL field bits

3.3.1.3.1 Service, Tail, and Pad

The SERVICE field has the length of 16 bits. The bits 0-6 are used to synchronize the descrambler in the receiver and bits 7-15 shall be “Reserved” for future use.

The TAIL is a 6-bits field. All of the bits shall be set to zeros to ensure that the convolutional encoder is back to zero state.

Pad bits shall be inserted after the convolutional encoder and puncturer to ensure that the encoded data stream is a multiple of the number of coded bits in an OFDM symbol (N_CBPS). The number of OFDM symbols (N_SYM), the number of data bits per OFDM symbol (NDBPS), the number of bits in the DATA field (NDATA) and the number of pad bits (N_PAD), are computed from the length of the PSDU (LENGTH) as follows:

NSYM = Ceiling [(16 + 8 LENGTH + 6)] / NDBPS] NDATA = NSYM NDBPS

NPAD = NDATA –(16 + 8 LENGTH + 6)

The function Ceiling ( ) is a function that returns the smallest integer value greater than or equal to its argument value. The appended bits (pad bits) are set to zeros and are subsequently scrambled with the rest of the bits in the MAC frame payload.

3.3.1.3.2 Data Scrambler

The DATA shall be scrambled with a length-127 scrambler. The generator polynomial of the scrambler is S(x) = x⁷ + x⁴ +1, which is operated with all ‘1’ for initial state to generate a 127 bit sequence. The sequence is: 00001110 11110010 11001001 00000010 00100110 00101110 10110110 00001100 11010100 11100111 10110100 00101010 11111010 01010001 10111000 11111111. The data scrambler adopts the 127 bits sequence generator to scramble all bits in the data field to randomize the bit patterns avoiding long streams of 1s and 0s.The scrambler is described in Fig. 3.4.

Figure 3.4 Data scrambler

3.3.1.3.3 Convolutional Encoder

Data are coded with a convolutional encoder of coding rate R = 1/2, 2/3, or 3/4. Fig. 3.5 illustrates the convolutional encoder of R = 1/2 using the industry-standard generator polynomials, g0 = 1338 and g1 = 1718. The bit denoted as

“A” should be the first bit generated by the encoder, followed by the bit denoted as

“B”. Another coding rates are derived from the basic rate of 1/2 using a technique called “puncturing”. Puncturing is “stealing” some of the encoded bits in the transmitter (thus reducing the number of transmitted bits and increasing the coding

rate) and inserting a dummy “zero” into the decoder bits in place of the stolen bits in the receiver (Fig. 3.6 & Fig. 3.7). Decoding the received signal is adopting the Viterbi algorithm to track the transmitted data.

Figure 3.5 Convolutional encoder

Figure 3.6 Puncturing (R=2/3)

Figure 3.7 Puncturing (R=3/4)

3.3.1.3.4 Data Interleaving

In IEEE 802.11a OFDM, the main purpose of the interleaving is used to avoid burst errors due to channel fading in the frequency domain. The data bits are interleaved by a block interleaver (the data bits is written in columns and read in rows), and the block size is correspondent to the number of bits in one OFDM symbol (NCBPS). Table 3.4 is an example of 6×8 block interleaver.

0 6 12 18 24 30 36 42 1 7 13 19 25 31 37 43 2 8 14 20 26 32 38 44 3 9 15 21 27 33 39 45 4 10 16 22 28 34 40 46 5 11 17 23 29 35 41 47

Table 3.4 Block interleaving

3.3.1.3.5 Mapping

OFDM modulation is employing BPSK, QPSK, 16-QAM, or 64-QAM. The binary data sequence is divided into group symbols of 1, 2, 4,or 6 bits, depending on the chosen data rate, and then the group symbols are converted into complex number points of the BPSK, QPSK, 16-QAM, or 64-QAM constellations. The group symbols of the constellation points are coded by Gray code illustrated in Fig. 3.8-3.10. The converted complex values are the form of ”I+jQ” multiplied by a normalization factor K_MOD listed in Table 3.5.

Modulation KMOD

BPSK 1 QPSK 1/√2 16 QAM 1/√10 64 QAM 1/√42

Table 3.5 Mapping normalization factor

Figure 3.8 BPSK & QPSK constellation

Figure 3.9 16 QAM constellation

3.3.1.3.6 Pilot

In the IEEE 802.11a standard, pilot symbols are put in subcarrier –21, -7, 7, and 21 (Fig. 3.11). The pilot subcarriers for the nth OFDM symbol are produced by the sequence Pn after Fourier transformation, given by

P_{n –26,26} ={ 0,0,0,0,0,p_n,0,0,0,0,0,0,0,0,0,0,0,0,0,p_n,0,0,0,0,0,0,0, 0,0,0,0,0,0,pn,0,0,0,0,0,0,0,0,0,0,0,0,0,-pn0,0,0,0,0 }

The sequence p_n can be generated by the scrambler with the “all ones” initial state, and replacing “1” with “-1” and “0” with “1”, given by

p0-126= {1,1,1,1, -1,-1,-1,1, -1,-1,-1,-1, 1,1,-1,1, -1,-1,1,1, -1,1,1,-1, 1,1,1,1, 1,1,-1,1, 1,1,-1,1, 1,-1,-1,1, 1,1,-1,1, -1,-1,-1,1, -1,1,-1,-1, 1,-1,-1,1, 1,1,1,1, -1,-1,1,1, -1,-1,1,-1, 1,-1,1,1, -1,-1,-1,1, 1,-1,-1,-1, -1,1,-1,-1, 1,-1,1,1, 1,1,-1,1,-1,1,-1,1, -1,-1,-1,-1, -1,1,-1,1, 1,-1,1,-1, 1,1,1,-1, -1,1,-1,-1, -1,1,1,1, -1,-1,-1,-1, -1-1-1 }

Figure 3.11 Pilot subcarriers

3.4 Operating Frequencies

The frequency band of 5 GHz U-NII is specified in the IEEE 802.11a standard. In the United States, the FCC (Federal Communication Commission) is the

agency responsible for the allocation of the 5 GHz U-NII bands. The 5 GHz U-NII frequency band is segmented into three 100 MHz bands. The lower band range is 5.15-5.25 GHz, the middle band range is 5.25-5.35 GHz, and the upper band range is 5.725-5.825 GHz. The lower and middle bands provide 8 carriers in a total bandwidth of 200 MHz and the upper band provides 4 carriers in a 100MHz bandwidth. The frequency channel center frequencies are spaced by a 20 MHz frequency band. The two outermost carrier centers of the lower and middle bands are at a distance of 30 MHz from the band edges and the upper bands are 20 MHz from the band edges (Fig.

3.12). Transmitting power levels are specified as: 40 mW (lower band), 200 mW (middle band), and 800 mW (upper band) (Table 3.6).

Band (GHz) Channel Number Center Freq. (MHz) Max Output Power (mW) 36 5180

40 5200 44 5220 Lower

Band (5.12-5.25)

48 5240

40

(2.5mW/MHz)

52 5260 56 5280 60 5300 Middle

Band (5.25-5.35)

64 5320

200

(12.5mW/MHz)

149 5745 153 5765 157 5785 Upper

Band (5.725-5.825)

161 5805

800

(50mW/MHz)

Table 3.6 The 802.11a operating bands

Figure 3.12 The 802.11a frequency channel

Chapter 4

Interpolation

4.1 Polynomial-Based Interpolation Filter

The interpolator, which is used to recover the originally transmitted data in the presence of STO can be regarded as a finite impulse response (FIR) filter [31]. For the purpose of simplifying the filter coefficients, a polynomial-based filter can be

employed [32]-[35]. The continuous-time output of the filter is shown as )

( ) ( )

( _{I s}

m

s h t mT

T m x t

y , (4.1)

where {x(mTs)} is a sequence of signal samples taken at intervals Ts , and h_I(t) is the finite impulse response of a time-continuous analog interpolation filter. Sampling y(t) at time instants t= kTi , the sampled data would become as

) (

) ( )

( _{I i s}

m

s

i x mT h kT mT

T k

y . (4.2)

Refer to Fig. 1.1 and 1.2, the interpolated data after the interpolation filter can be expressed as

] ) (

[ )

(kT_i y m T_s

y

] ) [(

] ) [(

2

1

s I

s I

I i

T i h T i m

x . (4.3)

In (4.3), m is the largest positive integer which is less than or equal to T_i/T_s, is

the estimated STO, I₁ and I₂ are fixed numbers (I₁=-M/2 and I₂=M/2-1, M is the length of the filter impulse response). Let I = I2－I1＋1. Then, I is the number of branches used for the interpolation. Moreover, N=I－1 is the degree of the interpolator. If N =1 and 3, the filters are called “linear interpolation filter” and “cubic interpolation filter”, respectively.

4.2 Filter Structure

According to equation (4.3), we find that the continuous-time impulse response )h_I(t is a function of variable, .In other word, the filter coefficients of the interpolation filter are changeful because of different value of . In order to simplify the structure, the interpolation filter coefficient might be evaluated by a suitable approach devised by Farrow [36]. The interpolation filter impulse response would become as

] ) ( [ )

( _{I s}

I t h i T

h ^l

N

l

l i

b

0

)

( , (4.4)

where N is the degree of interpolation filter, and b_l(i) are filter coefficients for h_I(t). Note that all the filter coefficients are fixed. In practical implementation, the number of multipliers can be reduced if the number of calculations required for the determination of filter coefficients can be decreased. In this case, the interpolated data can be written as

2

1

) (

) ( ) (

I

I i

k

i x m i

T k y k

y ^l

N

l

l i

b

0

) (

N

l l 0

2

1

) ( ) (

I

I i

l i x m i

b . (4.5)

Obviously, the recovered data depends highly on . Consequently, effective interpolation necessitates accurate STO estimation. An example of the cubic

interpolation filter with the Farrow structure is shown in Fig. 4.1, and the filter coefficients are listed in Table 4.1. In this figure, the degree of filter is N=3 and the length of the filter impulse response is I=4. Thus, the number of filter branch is 4 and each filter branch has four taps. We will apply this cubic interpolation filter for our simulation.

Figure 4.1 Farrow structure of cubic interpolation filter

i l=0 l=1 l=2 l=3 -2 0 -1/6 0 1/6 -1 0 1 1/2 -1/2 0 1 -1/2 -1 1/2 1 0 -1/3 1/2 -1/6

Table 4.1 Filter coefficients bl(i) for cubic interpolator

4.3 Another Interpolation Filter

4.3.1 Lower Degree Interpolator

In the previous section, the cubic interpolation filter (N=3) was discussed, and a good 2-degree interpolation filter is introduced in this section. In [37], the authors provide a fixed coefficient of the 2-degree filter, and the filter coefficients are shown in Table 4.2. Simpler structure of this 2-degree interpolation filter is shown in Fig. 4.2 [38].

i l=0 l=0 l=0

-2 0 -0.6741 0.6741 -1 0 1.4542 -0.4542 0 1 -0.5458 -0.4542 1 0 -0.6741 0.6741

Table 4.2 The coefficients (bl(i)) of 2-degree interpolation filter

4.3.2 Efficient Implementation of Interpolator

There is a new method to make Farrow structure more efficient by symmetrizing the fixed coefficients. Table 4.3 describes two different length and degree of interpolation filters called Interpolator I and Interpolator II. The coefficients of each interpolation filter are shown in Table 4.4 and Table 4.5. [39]

Degree Length Interpolator I 2 4 Interpolator II 3 8

Table 4.3 Degree and length of two interpolator

i=0 i=1

b

0

(i) 0.58071 -0.09914 b

1

(i) 0.5 0

b

2

(i) -0.008071 0.09914

Table 4.4 The filter coefficient of interpolator I

i=0 i=1 i=2 i=3

b

0

(i) 0.61442 -0.15277 0.04856 -0.01132 b

1

(i) -0.60188 0.04089 -0.00402 -0.00076 b

₂

(i) -0.11442 0.15277 -0.04856 0.01132 b

3

(i) 0.10188 -0.04089 0.00402 0.00076

Table 4.5 The filter coefficient of interpolator II

The Farrow structure of Interpolator I is shown in Fig. 4.3. Obviously, we can find that the coefficients of the structure are fixed and symmetrical. The structure consists three FIR filter branches multiplying by (2 －1)ⁿ and then summing them.

Figure 4.2 The simpler structure of 2-degree interpolation filter

Figure 4.3 Farrow structure of Interpolator I

Chapter 5

STO Estimation

From Fig. 1.1 we can see that STO is the only parameter which will influence the calculation of the interpolator. For effective operation of the interpolator, the STO must be estimated accurately. In the specification of IEEE 802.11a WLAN standard, four pilots and short preamble are used for time offset estimation. Using the four pilots for estimation is so-called feedback estimation, and applying short preamble to estimate is called feedforward estimation.

5.1 Feedback Estimation

In feedback timing recovery loop, the STOis estimated by calculating the difference of the phase rotation between two of the four-pilots in the TED, shown in Fig. 1.1. The estimated STO of the (n-2)th symbol using the pilots of (n-1)th and nth symbols is given by

μ

(n-2)=average (

2

2 p1 p

N (

1 2

,p p pilot p

( ˆ , 1

p

Xn

-, 2

ˆnp

X ) - (

, 1

ˆ 1 p

Xn

-, 2

ˆ 1 p

Xn ) ) )

(5.1) where p1, p2 = -21, -7, 7, and 21, N is the length of OFDM symbol , and

, 1

ˆnp

X means the phase of p₁ pilot in the nth symbol. With those four pilot subcarries, there

exists a total of six terms in the summation. [40] [41]

5.2 Feedforward Estimation

Feedback estimation has the drawback of long acquisition time. In the burst-mode transmissions, the receiver needs to achieve synchronization rapidly, and the feedforward estimation can accomplish it. So we adopt feedforward method to get the only changeable parameter ( ) of interpolator. In this section, we introduce two feedforward estimating methods and modify one of them to suit the OFDM system.

5.2.1 Maximum Likelihood Estimation

Synchronization algorithms of estimating the STO are based on the maximum likelihood estimation (MLE) theory, and these algorithms are used to find the STO which maximizes the log-likelihood function (Λ(τ)). [8]

Λ(τ) =

N

n 1

a^*(n) m(n,τ) (5.2)

where a^*(n) are the conjugate of the deterministic known preamble sequence, m(n,τ) are the received short preamble sequence, andτ(0≦τ＜T_s)is the timing error. N is the number of used samples and Ts is the sampling interval.

In this system, the received signals are sampled at the double sampling rate and a cubic interpolation filter is used for an interpolator. Because of the double sampling, we can use two different polynomial approximations for the log-likelihood functions as

Λ₁(2τ) =

N

n 1

a^*(n) m(2n-1,2τ) , for 0≦τ＜0.5

Λ₂(2τ-1) =

N

n 1

a^*(n) m(2n,2τ-1), for 0.5≦τ＜1 (5.3)

The range of 2τand 2τ-1 are in the interval [0,1), and therefore, 2τand 2τ-1 can be replaced by μ. The values of m(n,μ) are produced by the cubic interpolator in Farrow structure using the received short preamble as input.

m(n,μ) =

3

0 i

μⁱ f i(n) (5.4)

where f i(n) are the branch filter outputs of cubic interpolation filter. Consequently, the log-likelihood function become:

Λ1(μ) =μ³ X ^{3 1} +μ² X ^{2 1} +μX ^{1 1} + X ^{0 1}

Λ₂(μ) =μ³ X 3 2 +μ² X 2 2 +μX 1 2 + X 0 2 (5.5) where

Xi 1 =

N

n 1

a^*(n) f i(2n-1) for i = 0,….,3

X i 2 =

N

n 1

a^*(n) f i(2n) for i = 0,….,3 (5.6)

The block diagram of maximum likelihood estimation is shown in Fig 5.1 that achieves the following:

1. When the short preamble is received,

A . computes the outputs of the interpolator branches f i(n) B. divides f _i(n) into two parts (odd and even).

C. computes X i 1 and X i 2

2. obtain the timing offset , μ

A .Λ1(μ) orΛ2(μ) which has the larger maximum value that is the log-likelihood function we select.

B. find the timing offset μ which maximizes the log-likelihood function

3. using μ to track in the interpolator

Figure 5.1 Block diagram of the ML estimation

5.2.2 Fourier Transform Estimation

In [9], authors proposed a feedforward approach to estimate the STO using correlations of the known preamble sequences preceding a data packet is proposed.

Let a be the deterministic known preamble sequence, and let N be the length of the _m sequence used in the correlation. Furthermore, let x(t) be the received sequence and let )r_xx(i denote the correlation. Then, the correlation can be described as

) ( r_xx i =

1

0 N

m

a^*_m x( iT_s + 2mT_s), . (5.7)

in which ^* denotes complex conjugate. Assume that each symbol consists of two samples. That is, the symbol period is 2T . After some derivations, we find that the _s STO, , can be represented as

)]

R(e [ 2 arg j

, (5.8)

where R(e^j ) is the Fourier transform of rxx(i), shown as:

R(e^j ) =

i

r_xx(i) e^{j i}. (5.9)

Based on (5), enumeration of R(e^j ) requires infinite number of terms. To make the Fourier transform implementation realistic, the authors truncate the r_xx(i sequence ) before taking Fourier transform. Specifically, only four samples of r_xx(i sequence ) are used in the original work, as shown in (5.10).

RT( e ^jπ/2) = [rxx(0) - rxx(2)] + j[rxx(-1) - rxx(1)]. (5.10)

Thus the estimating equation of STO would become as (5.11), and the estimation structure is described as Fig. 5.2.

μ= 2

arg( RT( e ^jπ/2) ) (5.11)

Figure 5.2 The stricture of STOestimation in [9]

Obviously, the structure of the Fourier transform becomes much simpler.

This, however, calls for an extra hardware circuit for the receiver. Recall that in an OFDM receiver, there is an FFT processor used for demodulation. For the 802.11a

OFDM system, operation of the FFT processor is not necessary during the STS period.

This motivates us to use the FFT processor for the computation of the Fourier transform during its idle period. To probe further, we take correlation of the STSs to get the r_xx(i sequences and then use the existent FFT processor to enumerate the ) Fourier transform, depicted in Fig. 5.3. In this way, the performances shown in Fig 5.4 are very close, but the additional hardware requirement described in [39] can be eliminated.

Figure 5.3 Structure of using the FFT for the sampling time offset estimation

It is necessary to address the feasibility of the new scheme. Specifically, we need to justify that the FFT processor in the standardized 802.11a receiver can achieve our mission. According to the IEEE 802.11a specifications, the received signal should be detected within the STS t to ₁ t (see Fig. 5.5). This implies that the symbol ₇ time offset can be estimated during the interval t to ₁ t and the FFT window must ₇ be adjusted accordingly, except the possible STO. For our scheme, the symbol time can be detected by observing the peaks of the r_xx(i sequence. This information is ) used to control the FFT window start position to within an STO interval. Thus, misalignment of the FFT window start position will be only STO, if ant, after the t ₇ STS.

Figure 5.4 The result of using Dengwei’s estimation and FFT estimation

Another issue that should be considered regarding to the feasibility of our scheme is that whether there is sufficient time for the FFT processor to carry out the task without obstructing its duty work. Recall that the FFT processor is always functioning during the LTS interval for channel estimation or residual frequency offset correction. This means that the FFT processor can be employed to do the computation only before the start of the LTS. From the above description, we can see that the STS from t₈ to t₁₀ and the guard interval G12 prior to the LTS are available. Thus, there are 4μs (=2.4μs+1.6μs) in total available for our need. This is greater than the FFT window size 3.2μs specified in the standard. Thus, there is enough duration for the FFT processor to accomplish the STO calculation. In the following, we present detailed operation of STO estimation using the FFT processor.

Figure 5.5 Preamble of the IEEE 802.11a WLAN

Equation (5.10) can be obtained by dividing the four truncated samples, rxx(-1)~ rxx(2), into odd part (multiplied by “j”) and even part. Then, a 4-points FFT can be employed to do the transformation. An alternative is to use a 32-points FFT, which is a half of the 64-points FFT specified in the IEEE 802.11a, to perform the computation. We take correlations of the STSs to provide correlation sequences based on (3). Then, choose 32 terms of the correlation sequence, rxx(-16) ~ rxx(15), and divide these 32 terms into an odd part and an even part. The odd part consists of the number of the FFT input from 0 to 15 (after being multiplied by j), and the even part is comprised of the input from 16 to 31. All the remainder inputs of the FFT are set to zeroes. Based on this input arrangement, we can find that the 64-points FFT is equivalent to two 32-points FFT, and we choose the upper 32-points FFT for our purpose. Since the number of the adopted correlation sequence terms is eight times of the number of terms dictated by (5.10), we can use eight terms of the FFT outputs for estimation. If the output pin number (0), (8), (16), (24), (32), (40), (48), and (56) of the FFT (these terms are the upper eight output pins of a 64-points FFT with butterfly structure depicted in Fig. 5.6 [15]) are selected, only four inputs, rxx(-16), rxx(-15), r_xx(0), and r_xx(1) are needed. It is noted that these four inputs represent two peak positions of estimation equation (5.7) which we need. Alternatively, if we only select

the first seven terms of the above eight outputs, the value of (5.7) will depend heavily on the four “peak” input terms and little on the other 28 input terms. Gathering the minor contribution resulted from these 28 inputs can reduce the effect of channel noise. This can improve the STO estimation accuracy. In this work, an angle is computed based on the average of the selected seven output terms. Multiplying the angle by 4/π yields the vale of as illustrated in Fig. 5.7

Figure 5.6 The butterfly structure of 64-points FFT

Figure 5.7 Structure of using the FFT for the sampling time offset estimation

5.3 Simulation and Some Results

We consider burst-mode transmissions of the OFDM-based IEEE 802.11a WLAN for our simulations. The number of burst block is 250, and each burst block contains 25 OFDM symbols. Performance of the system that uses the FFT processor to calculate the STO will be compared to that which can be achieved by using a Maximum-Likelihood (ML) estimation method. In the sequel, the notation “FT” and

“ML” will be used to denote a system that employs the FFT calculation and the ML algorithm, respectively.

For AWGN channels and QPSK modulation, Fig. 5.8 and 5.9 display the bit error rate (BER) of a system that can be achieved by using FT and ML method, as a function of signal to noise ratio (SNR). We have simulated for two values of N, which is the number of STS samples used in the calculation, N=96 and 32. From these figures we can see that for any value of N, the FT method outperforms the ML method. In addition, we can also find that larger N will result in better performance.

This is the direct consequence that large number of samples will average out the variance of AWGN noise. Using N=96 and in AWGN channels, we compare the BER performance of FT and ML with QPSK and 16-QAM in Fig. 5.10. It can be observed that the FT method can perform better than the ML method regardless of the modulation modes. Similar characteristics can be found for the BER performance under a Rayleigh multipath fading channel [26] shown in Fig. 5.11 and 5.12. But we can see from Fig. 5.11 and 5.12 that the BER performance will level off when the SNR is larger than some specific values. This is because that when SNR is high (AWGN noise is low), the interference resulted from the multipath effect will dominate the performance. Thus, further increase of SNR will not significantly

improve the performance.

For 16-QAM mode in AWGN channels with N=96 and SNR=12dB, the simulated jitter of STO estimation that can be achieved by using the FT method and the ML method is depicted in Fig. 5.13. From this figure we can see the FT method can result in smaller jitter than that which can be achieved by the ML method. This consequence can be realized from the description mentioned in the following. For the ML method, the can be obtained by differentiating (5.5) with respect to and setting the derivative to zero. These are expressed by the following two equations:

i i

i X X

X _{3 2 1}

2 2

3 (5.12)

i

i i i

i

X

X X X

X

3

1 3 2

2 2

6

12 4

2

. (5.13)

Figure 5.8 BER of FT and ML methods for QPSK in AWGN channels with N=96

Figure 5.9 BER of FT and ML methods for QPSK in AWGN channels with N=32

Figure 5.10 BER of FT and ML for QPSK and 16-QAM in AWGN channels

Figure 5.11 BER of FT and ML methods for 16-QAM in Fading channels

Figure 5.13 Jitter of sampling time offset estimation.

whereX_ni will contain channel noises. Based on (5.13), calculation of the STO necessitates multiplication of X_ni. This will magnify the effect of channel noise on the STO estimation. In contrast, calculation of STO based on (5.8) for the FT method involves only r_xx (i , which has similar noise property to ) X_ni. Thus, the FT method can result in better jitter performance. This can also be interpreted as that the FT method is more robust to channel noises.

In addition to the performance superiority as described above, the FT method possesses another advantage, the flexibility. It is noted that whereas a fixed cubic interpolation filter must be used for the ML method, cubic interpolation filter for the FT method can be replaced with a simple and efficient interpolator filter. For instances, a 2-degree interpolation filter or a symmetrical Farrow structure is proposed in section 4.4 [33]-[35]. By using these interpolation filters, the number of filter

branches can be reduced without sacrificing significant performance, as compared to the ML estimation with cubic interpolation filter. Fig. 5.14 and Fig. 5.15 show the BER of 16-QAM and QPSK under AWGN channels with N=96 employing the 2-degree interpolation filter. From the two figures we can see that performances of FT method using the lower degree interpolator can be similar with ML method. Fig. 5.16 and Fig. 5.17 illustrate the BER of 16-QAM and QPSK under AWGN channels with N=96 adopting the symmetrical-coefficient interpolator. Although the degree of interpolator is lower in Fig. 5.16 to Fig. 5.17, the performances are better than ML estimation with cubic interpolator.

Figure 5.14 The BER comparison of the 2-degree interpolator with 16-QAM

Figure 5.15 The BER comparison of the 2-degree interpolator with QPSK

Figure 5.16 The BER comparison of the symmetrical interpolator with 16-QAM

Figure 5.17 The BER comparison of the symmetrical interpolator with QPSK

5.4 Use of Curve Fitting Functions

STO estimation using the FT method described above is not very accurate, as can be seen from Fig. 5.18, which compares the actual and the ideal estimated value under noise free environments. This figure shows that there exist errors between the estimated values and the actual STOs. To improve the estimation accuracy, we can employ a curve-fitting function to the proposed scheme to modify the estimation error.

The commonly adopted curve-fitting function can be classified into linear, quadratic, and cubic, as represented in (5.14), (5.15), and (5.16), respectively.

1 1 1

ˆ 0

1 C

c

c . (5.14)

2 2 2 1 0

ˆ2

1 ˆ C

c c c

. (5.15)

3 3

3 2 1 0 3

2 ˆ

ˆ

1 ˆ C

c c c c

. (5.16)

In (10)-(12), is the actual STO, ˆ is the estimated STO, andc_i' are the s coefficients to be determined for which the estimation error is minimized in the sense of Minimum Mean Square Error (MMSE). From equation (5.14) to (5.16), C_i^'s can be expressed as

3

~ 1 for , )

( ¹ i

C_{i i i i} , (5.17)

where ˆ is the estimated STO under noise free conditions. Substituting the determinedC into (5.14)~(5.16) and replacing ˆ with estimated STO, we can _i compute the STO with curve-filtering function amendment in noisy channels.

For noise free channels with N=96, Fig. 5.19 illustrates the performance improvements resulted from the three curve-fitting functions. It can be observed that generally a higher order curve-fitting function can achieve better performance. If the cubic curve-fitting function is adopted, the estimated STO is very close to the real STO.

Figure 5.18 The actual estimation value of FT method and ideal estimation value

Figure 5.19 Estimation error of FT estimation with curve-fitting function amendment

Moreover, by using curve-fitting functions, only two terms of (5.7) are needed for the evaluation of (5.8). Thus,

)]

1 ( r j (0) [r 4 arg

XX

XX . (5.18)

This significantly reduces the implementation complexity. For the three curve-fitting functions, estimation errors that can be achieved by using this simple algorithm are depicted in Fig. 5.20.

From this figure it is seen that the estimation error can be suppressed considerably.

Based on (5.18) with cubic curve-fitting function, the BER of 16-QAM under AWGN channel with N=96 is depicted in Fig. 5.21 From this figure we can see that by employing the cubic curve-fitting function, difference in BER which can be achieved by the simple method and the FT estimation is insignificant.

Figure 5.20 Estimation error of the simple estimation with curve-fitting function amendment

Figure 5.21 The BER of simplified estimation with cubic curve-fitting function

在文檔中 中 華 大 學 (頁 37-79)