Joint Channel Estimation and Decoding in SISO-OFDM Systems

Since the communication devices become smaller, the devices will be moved more easily and the channel response will change through time more quickly. The receivers

8 10 12 14 16 1E-4

1E-3 0.01

BER

avg{ E_b/N₀}

Perfect CSI MB(7,56) MB(3,14) MB(3,16,6)

Figure 3.11: The BER Performance for the M B(3, 14), M B(3, 16, 6), and M B(7, 56) Channel Estimates.

8 10 12 14 16

1E-3 0.01 0.1

FER

avg{ E_b/N₀}

Perfect CSI MB(7,56) MB(3,14) MB(3,16,6)

Figure 3.12: The FER Performance for the M B(3, 14), M B(3, 16, 6), and M B(7, 56) Channel Estimates.

0 10 20 30 40 50 60 -0.2

0.0 0.2 0.4 0.6

RealPartoftheChannelResponse

sub-carrier index

True Channel Response Step1: Regression Function Step2: Hermite Curve

Figure 3.13: An example of the channel estimates using the two-stage M B(3, 16, 6) estimate when the average SNR is high.

20 30 40

0.3 0.4

RealPartoftheChannelResponse

sub-carrier index

True Channel Response Regression Function Regional Curve Fitting

Figure 3.14: The detail of the regional curve fitting.

need to use the information from not only the training signals but all the transmitted signals in order to estimate the channel response at present. The optimal receiver per-forms to detect the transmitted data given the all received signals (based on whatever information of the channel statistics that can be considered known). If the data bits are coded, after decoding the data, we can get the data part information, and this informa-tion can help estimating the channel response at different symbols, so the receiver can operate iteratively by joint channel estimation and data decoding.

Channel Equalizer

Channel Es tim ation

LDP C Dec oder

Data Sym bol Inform ation P ream ble S ym bols

ˆ

m

X

ˆ

m

ˆ X H

m

Y

m

Dec oded Data Outer loop

Inner loop

Figure 3.15: The joint channel estimation and decoding receiver in SISO-OFDM systems.

The receiver operation structure is shown as Fig. 3.15, and the corresponding factor graph is shown as Fig. 3.16. After the channel estimation at the preamble, we can get the channel estimate ˆH, and use this channel estimate to detect the data symbols X, and after several times of the LDPC decoding (here we use 25 iterations) the moreˆ reliable data symbols can be seen as the pilot symbols, the channel response at each symbol can be re-estimated by these pilot symbols. The factor graph representation of the joint channel estimation and decoding is shown in Fig. 3.16.

In Figs. 3.17, 3.18, 3.19, and 3.20, we examine the effect of k_out, the updating

LD PC Bit N ode

LD PC C heck N ode LS C hannel Estimator

X0 Y₀ M odel Based C hannel Estimation

with model order M_o

C ompute the soft infromation for the LD PC decoder

codeword length

Figure 3.16: The factor graph representation of joint channel estimation and decoding in SISO-OFDM systems.

8 10 12 14 16 1E-4

1E-3 0.01

BER

avg{ E_b/N₀}

Perfect CSI LS

MB(3,16,6) k_out=0 MB(3,16,6) k_out=1 MB(3,16,6) k_out=3

Figure 3.17: The BER performance of joint the M B(3, 16, 6) estimation and decoding with k_out= 0, 1, 3.

8 10 12 14 16

1E-3 0.01 0.1

FER

avg{ E_b/N₀}

Perfect CSI LS

MB(3,16,6) k_out=0 MB(3,16,6) k_out=1 MB(3,16,6) k_out=3

Figure 3.18: The FER performance of joint the M B(3, 16, 6) estimation and decoding with k_out= 0, 1, 3.

8 10 12 14 16 1E-4

1E-3 0.01

BER

avg{ E_b/N₀}

Perfect CSI LS

MB(7,56) k_out=0 MB(7,56) k_out=1 MB(7,56) k_out=3

Figure 3.19: The BER performance of joint the M B(7, 56) estimation and decoding with k_out = 0, 1, 3.

8 10 12 14 16

1E-3 0.01 0.1

FER

avg{ E_b/N₀}

Perfect CSI LS

MB(7,56) k_out=0 MB(7,56) k_out=1 MB(7,56) k_out=3

Figure 3.20: The FER performance of joint the M B(7, 56) estimation and decoding with k_out = 0, 1, 3.

times of the outer loop, and at each time of the outer loop, there are 25 inner loop iterations. The channel estimate used by the outer loop are M B(3, 16, 6) and M B(7, 56), respectively, show that at each iteration of the channel re-estimation will bring some performance improvement. k_out = 0 means the channel response only estimated at the preamble but not re-estimated after decoding. k_out= 1, 3 denotes the iterative number of channel re-estimating, and the total iterations of the LDPC decoding is equal to 50, 100, respectively.

Finally, we observe that re-estimating channel once after 25 decoding iterations can bring about 1.5dB gain in BER and FER performance. If there are 3 times of channel re-estimation, as the performance curves labelled by M B(3, 16, 6) 3 and M B(7, 56) 3 have indicated, we then come very close to the theoretical lower bound–that achieved by a receiver with perfect channel estimate.

Chapter 4

Joint Channel Estimation and

Decoding in MIMO-OFDM Systems

The combination of MIMO signal processing with OFDM waveform offers a practical solution to achieve very high spectral efficiencies and unprecedented data rates. With a rich scattered multi-path propagation environment and appropriate signal processing at the receiver side, the received streams can be separated so that a MIMO wireless channel can be viewed as virtual parallel independent channels.

LDPC

Figure 4.1: A LDPC-coded MIMO-OFDM system model.

A MIMO-OFDM system model is depicted in Fig. 4.1 in which the data stream is first encoded by a single LDPC encoder, then serial-to-parallel converted into N_{T x}

parallel data sub-streams, where N_{T x} is the number of transmit antennas. Each data sub-stream is mapped onto a stream of symbols and an IFFT is performed on each sub-stream of symbols and a cyclic prefix is inserted in front of each OFDM symbol as the GI. Denoting the signal vector transmitted from N_{T x} transmit antennas on the mth sub-carrier by

Xm = [X_m¹, X_m², ..., X_m^{NT x}]^T, (4.1)

and assuming the channel delay spread is less than the GI so that the ISI can be fully removed, we can express the received signal vector Ym at the mth sub-carrier as

Ym =HmXm+Nm, (4.2)

where Ym ∈ C^NRx×1, Hm ∈ C^{NRx×NT x}, Xm ∈ C^{NT x×1}, and Nm ∈ C^NRx×1, and H_m^ij denotes the channel response from the ith transmit antenna to the jth receive antenna at the mth sub-carrier, and N_{T x},N_Rx is the number of transmit and receive antennas, respectively.

Separating the desired signal X_mⁱ from the interference from other transmit antennas Xⁱm
, we have

Ym =Hⁱm· Xmⁱ +Hⁱm
Xⁱm
+Nm,

where

Hⁱm = [H_mⁱ¹, H_mⁱ², ..., H_m^iNRx]^T

Hⁱm
= [H¹m,H²k, ...,Hⁱm⁻¹,Hⁱm⁺¹, ...,H^Nm^{T x}]^T Xⁱm
= [X_m¹, X_m², ..., X_mⁱ⁻¹, X_mⁱ⁺¹, ..., X_m^{NT x}]^T,

and Hⁱm
Xⁱm
is called the co-antenna interference (CAI).

Because of the presence of CAI in a MIMO-OFDM system, to estimate the channel response between each transmit antenna and each receive antenna based on Y_m^j, j = 1, 2,· · · , NRx is a significant challenge.

In the remaining part of this chapter, we first deal with the (initial) channel estima-tion problem using the special preamble symbol structure of the 802.11n specificaestima-tion.

In the ensuing section we discuss the MIMO signal detection issue, i.e., the problem of extracting soft information for the LDPC decoder in the presence of CAI. We close this chapter with a presentation of an iterative joint channel estimation/tracking and decoding algorithm and its performance.

4.1 Channel Estimation in MIMO-OFDM Systems

As shown in Fig. 2.1, the coded bit stream is de-multiplexed (spatially parsed) into substreams that are to be transmitted via individual antennas. Each frequency domain data stream is interleaved before being modulated by the associated constellation mapper. As mentioned in Chapter 2, the packet format of the IEEE 802.11n standard, PPDU, is divided into two concatenated units: the preamble unit and the data unit.

Such a signal structure is a common practice in almost all wireless protocols. In other words, it calls for the transmission of a preamble prior to the payload that contains the information block.

The preamble structure and the contents of the HT-LTF for the different number of transmit antennas can be found at A.1. An example of tone interleaving across 4 transmit antennas is shown in Fig. 4.2. The HT-LTFs set 0, set 1, set 2, and set 3, as given in Table A.1, form a partition of the 56 tones in the HT-LTF sequence 4. At the OFDM symbol interval, each set of tones maps to single transmit antenna. Only after the overall preamble is transmitted, all sets get mapped to each transmit antenna.

By using the preamble unit in which each tone is assigned to a single antenna, one can easily separate the signals transmitted from different antenna, channel estimation can thus be carried out by using the same algorithm as that for the SISO-OFDM systems.

HT-LTF 0

HT-LTF 1

HT-LTF 2

HT-LTF 3

Tx 0 Set 0 Set 3 Set 2 Set 1

Tx 1 Set 1 Set 0 Set 3 Set 2

Tx 2 Set 2 Set 1 Set 0 Set 3

Tx 3 Set 3 Set 2 Set 1 Set 0

time

0 1 2

-Transmit

antenna

Figure 4.2: HT-LTF tone interleaving across 4 transmit antennas.

4.2 MIMO Signal Detection

Unlike the received signals in SISO-OFDM systems, the baseband samples received by the jth receive antenna at the mth sub-carrier Y_m^j is a combination of the transmitted signals (X_m¹, X_m², ..., X_m^{NT x}) from all transmit antennas. Before decoding, one needs to suppress CAI and estimate the transmitted data Xm from different transmit antennas based on Ym and the channel estimate obtained during the preamble period.

4.2.1 MIMO MMSE detector

The MIMO MMSE detection scheme is proposed by Haykin [10] for Turbo-BLAST.

The estimator for X_mⁱ is given by

Xˆ_mⁱ =w^Hi Ym− ui,

wherewi represents tap weights of a linear filter, and u_i is the corresponding weighted combination of the estimated CAIs.

Invoking the MMSE principle to minimize the cost function

( ˆwi,uˆ_i) = arg min

(wi,u_i)E[|Xmⁱ − ˆX_mⁱ |²], (4.3) we obtain

wˆi = (P + Q + ΣN_Rx)⁻¹Hⁱm (4.4) ˆ

u_i = wˆiz (4.5)

where

P = HⁱmHⁱm

H ∈ C^NRx

Q = Hⁱm
[IN_{T x}−1− Diag(E[Xⁱm
]E[Xⁱm
]^H)]Hⁱm

H ∈ C^NRx Σ_NRx = σ_N²IN_Rx ∈ C^NRx

z = Hⁱm
E[Xⁱm
]∈ C^NRx×1

Note that the expectations E[Xⁱm
] are the LDPC decoder soft outputs and z repre-sents our estimate of CAI. A simplified sub-optimum solution is obtained by assuming E[Xⁱm]E[Xⁱm]^H = 1 ∀i, m so that Q = 0N_Rx×NRx and the desired signal becomes

Xˆ_mⁱ = (Hⁱm

HHⁱm+ σ²_N)⁻¹Hⁱm

H(Ym− Hⁱm
E[Xⁱm
]), (4.6)

with the initialization E[Xmⁱ ] = 0. This data estimator will be referred to as MMSE detector henceforth. Given the channel estimate obtained by applying the M B(3, 16, 6) estimate at the preamble, one can detect the data by first use the MIMO MMSE detec-tion algorithm and then proceed to perform the LDPC decoding. An iterative MMSE detection and LDPC decoding process that feedback the soft LDPC decoder output for MMSE detection is expected to enhance the BER (or FER) performance. Since the LDPC decoder converges rather slow, to feedback reliable and substantially improved soft values to the MMSE detector, many LDPC decoding iterations need to be carried out. We compare the FER and BER performance of the above three scenarios in Figs.

8 9 10 11 1E-5

1E-4 1E-3 0.01

BER

avg{ E_b/N₀}

MMSE Detector Perfect CSI, k_in=5 MB(3,16,6), k

in=0 MB(3,16,6), k_in=1 MB(3,16,6), k_in=5

Figure 4.3: Influence of the inner loop iteration number k_in on the BER performance of 2× 2 MIMO-OFDM systems with MMSE detector.

8 9 10 11

1E-3 0.01

FER

avg{ E_b/N₀}

MMSE Detector Perfect CSI, k_in=5 MB(3,16,6), k_in=0 MB(3,16,6), k_in=1 MB(3,16,6), k_in=5

Figure 4.4: Influence of k_in on the FER performance of 2× 2 MIMO-OFDM systems with MMSE detector.

4.4 and 4.3, respectively. In these figures, the curve labelled by M B(3, 16, 6), k_in rep-resents the scenario that soft output is obtained after k_in LDPC decoding iterations, k_in = 0 represents the no-feedback case. We notice that the system performance im-proves as the number of LDPC decoding iteration increase. For k_in >5, the performance gain becomes negligible and, for the sake of brevity, the corresponding curves are not shown here.

4.2.2 MIMO MLD detector

Another MIMO detector called the soft-input/soft-output maximum likelihood de-coder (MLD) is used in the Zelst’s Turbo-BLAST [11] as an ideal BLAST de-mapper.

To produce soft-decision outputs with the MLD, the log-likelihood ratio is used as an in-dication for the reliability of a bit. If X_mⁱ is the transmitted signal form the ith transmit antenna at the m sub-carrier and with QPSK mapping, then we have to derive estimates (soft values) of the corresponding bits, bⁱ_m,0 and bⁱ_m,1, from the received samplesYm. For example, the a posteriori log value of the estimated bit b¹_m,0 can be expressed as

L_{M LD,p}(bⁱ_m,0|Ym) = lnP(b¹_m,0 = 0|Ym) P(b¹_m,0 = 1|Ym)

= lnP(b = [0, 0, .., 0]|Ym) + ... + P (b = [0, 1, .., 1]|Ym) P(b = [1, 0, .., 0]|Ym) + ... + P (b = [1, 1, .., 1]|Ym) (4.7) whereb represents the transmitted coded bits [b¹m,0, b¹_m,1, b²_m,0, ..., b^N_m,^{T x}₁].

Applying the Bayes rule, the above expression render the decomposition

L_{M LD,p}(bⁱ_m,0|Ym)

= lnP(Ym|b = [0, .., 0])P (b = [0, .., 0]) + ... + P (Ym|b = [0, .., 1])P (b = [0, .., 1]) P(Ym|b = [1, .., 0])P (b = [1, .., 0]) + ... + P (Ym|b = [1, .., 1])P (b = [1, .., 1])

= lnP(Ym|b = [0, .., 0])P (b¹m,0 = 0)P (b¹m,1 = 0)· · · P (b^Nm,^{T x}1 = 0) + ...

P(Ym|b = [1, .., 0])P (b¹m,0 = 1)P (b¹m,1 = 0)· · · P (b^Nm,^{T x}1 = 0) + ...

= lnP(b¹m,0 = 0)

P(b¹m,0 = 1) + lnP(Ym|b = [0, .., 0])P (b¹m,1 = 0)· · · P (b^Nm,^{T x}1 = 0) + ...

P(Ym|b = [1, .., 0])P (b¹m,1 = 0)· · · P (b^Nm,^{T x}1 = 0) + ...

= L_a(bⁱ_m,0) + L_{M LD,e}(bⁱ_m,0|Ym) (4.8)

where we have assumed that the coded bits are independent so that the joint probabil-ity is equal to the product of the corresponding marginal probabilities, e.g., P (bⁱ_m,0 = 0, bⁱ_m,1 = 0) = P (bⁱ_m,0 = 0)· P (bⁱm,1 = 0).

The above equation says that L_{M LD,e}(bⁱ_m,0|Ym) is the difference between the a priori and the a posteriori log value at the MIMO MLD detector and consists of channel information and extrinsic information. The L_{M LD,e} are taken as the a priori values to the LDPC decoder for further iterative decoding steps. And after several iterations of LDPC decoding, a posteriori log values output from the LDPC decoding can feedback to the MIMO MLD detector as a priori log values. After estimating the channel response at the preamble by applying the M B(3, 16, 6) estimator, the MIMO MLD detector can detect the data and then proceed to perform the LDPC decoding.

An iterative MLD detection and LDPC decoding process that feedback the soft LDPC decoder output for MLD detection can also be expected to enhance the BER (or FER) performance. Since as the result as the previous subsection, after 5 LDPC decoding iterations, to feedback reliable and substantially will improve soft values to the MLD detector. We compares the BER and FER performance of the above three scenarios in Figs. 4.5 and 4.6, respectively.

4.3 Joint Channel Estimation and Decoding in MIMO-OFDM Systems

We have shown that excellent performance is attainable with joint channel estimation and decoding in SISO-OFDM systems. It is thus natural apply the same approach to MIMO-OFDM systems. The proposed receiver structure is shown in Fig.4.7. The initial channel estimate ˆH derived from the preamble is sent to the MIMO detector to help estimating data symbols ˆX from different transmit antennas. The output from the MIMO detector is forwarded to the LDPC decoder whose soft output is feedback to both the MIMO detector and the channel estimator.

8 9 10 11 1E-5

1E-4 1E-3 0.01

BER

avg{ E_b/N₀}

MLD Detector

Perfect CSI, k_in=5 MB(3,16,6), k_in=0 MB(3,16,6), k_in=1 MB(3,16,6), k_in=5

Figure 4.5: Influence of k_in on the BER performance of 2× 2 MIMO-OFDM systems with MLD detector.

8 9 10 11

1E-3 0.01 0.1

FER

avg{ E_b/N₀}

MLD Detector

Perfect CSI, k_in=5 MB(3,16,6), k_in=0 MB(3,16,6), k_in=1 MB(3,16,6), k_in=5

Figure 4.6: Influence of k_in on the FER performance of 2× 2 MIMO-OFDM systems with MLD detector.

Y

1

T entative Data S ym bols Dec is ions from LDPC Dec oder Output

X ˆ

b ˆ

Figure 4.7: An iterative receiver structure for LDPC-coded MIMO signals.

Grant [12] proposed a simple method for joint decoding and channel estimation for MIMO channels. The decoder obtains a tentative data sequence estimate by using the current channel estimate. The initial channel estimation can be derived from the preamble. Let ˆX_mⁱ be the decoder’s tentative output for the data from the ith transmit antenna and denote by ˆHⁱm the ith column of ˆH_m. One then original compute an equivalent SISO received vector by subtracting from the original MIMO received vector

the tentative data vectors associated with other transmit antennas

Hˆⁱm[t] =

Ym−

k=i

Xˆ_m^kHˆ^km[t− 1]

Xˆ_mⁱ . (4.9)

Removing the estimated CAI, one then go on to compute a new channel estimate by our SISO-OFDM solution proposed before.

Using the M B(3, 16, 6) channel estimate obtained during the preamble period, we perform iterative joint MMSE detection (or MLD detection) and decoding to obtain a tentative data vector estimate. Regarding this tentative decision as pilots, we can then re-estimate the channel response via the M B(2, 11, 6) estimate. We use a lower-order channel model for fear of severe CAI-induced estimation error. Our choice of M_O = 2 means a straight line is used as a local approximation. Additional Hermite curve fitting will smooth the piece-wise linear estimate and result in a much better approximation of the true channel response. Such an iterative detection scheme consists of two iteration loops: the inner loop performs iterative MMSE detection (or MLD detection) and LDPC decoding while the outer loop represents the iterative process between the channel estimator and the inner loop.

Figs. 4.8 and 4.9 are the BER and FER performance of the above double-loop iterative receiver by using MMSE detector, where Figs. 4.10 and 4.11 are using MLD detector. The curves labelled by M B(3, 16, 6) k_in, M B(2, 11, 6) k_outrepresent the perfor-mance of the detection strategy that performs k_out outer loop iterations where an outer loop iteration mean a renewed channel estimation after a M B(3, 16, 6) k_in round. The performance of the M B(3, 16, 6) 5 iterative scheme, which re-estimate the received data for every 5 LDPC decoding iterations without channel re-estimation, is also included for convenience of comparison.

8 9 10 11 1E-5

1E-4 1E-3 0.01

BER

avg{ E_b/N₀} MMSE Detector

Perfect CSI, k_in=5 MB(3,16,6), k_in=5

MB(3,16,6), k_in=5; MB(2,11,6), k_out=1 MB(3,16,6), k_in=5; MB(2,11,6), k_out=3

Figure 4.8: Influence of k_in and the outer loop iteration number k_out and with the M B(3, 13, 6) estimation at the preamble and the M B(2, 11, 6) estimation at the data symbols on the BER performance of 2× 2 MIMO-OFDM systems with MMSE detector.

8 9 10 11

1E-3 0.01

FER

avg{ E_b/N₀} MMSE Detector

Perfect CSI, k_in=5 MB(3,16,6), k_in=5

MB(3,16,6), k_in=5; MB(2,11,6), k_out=1 MB(3,16,6), k_in=5; MB(2,11,6), k_out=3

Figure 4.9: Influence of k_in and k_out and with the M B(3, 13, 6) estimation at the pream-ble and the M B(2, 11, 6) estimation at the data symbols on the FER performance of 2× 2 MIMO-OFDM systems with MMSE detector.

8 9 10 11 1E-5

1E-4 1E-3 0.01

BER

avg{ E_b/N₀} MLD Detector

Perfect CSI, k_in=5 MB(3,16,6), k_in=5

MB(3,16,6), k_in=5; MB(2,11,6), k_out=1 MB(3,16,6), k_in=5; MB(2,11,6), k_out=3

Figure 4.10: Influence of k_in and k_out and with the M B(3, 13, 6) estimation at the preamble and the M B(2, 11, 6) estimation at the data symbols on the BER performance of 2× 2 MIMO-OFDM systems with MLD detector.

8 9 10 11

1E-3 0.01

FER

avg{ E_b/N₀} MLD Detector

Perfect CSI, k_in=5 MB(3,16,6), k_in=5

MB(3,16,6), k_in=5; MB(2,11,6), k_out=1 MB(3,16,6), k_in=5; MB(2,11,6), k_out=3

Figure 4.11: Influence of k_in and k_out and with the M B(3, 13, 6) estimation at the preamble and the M B(2, 11, 6) estimation at the data symbols on the FER performance of 2× 2 MIMO-OFDM systems with MLD detector.

Chapter 5 Conclusion

In this thesis, we have investigated and developed the structures and the associated signal processing algorithms for both SISO and MIMO-OFDM systems. Iterative joint channel estimation and decoding algorithms are proposed and used for detecting IEEE 802.11n signals. We present a new efficient channel estimate with modest complexity and enhanced performance, and faster turbo BLAST scheme with reduced complexity is developed. Numerical examples show that our solution provide BER performance and FER performance that is less than 0.2 (1) dB away from the theoretical BER and FER lower bounds for SISO (MIMO)-OFDM systems.

Our numerical experiment follow the IEEE 802.11n standard. However, there is no knowing if the LDPC code used is the best choice in terms of performance and complexity. Reduction of the decoder convergence speed is highly desirable and further complexity reduction on the MIMO detector is also welcome.

Appendix A

802.11n Specification

A.1 Training symbols specification

The number of HT-LTFs is equal to N_{LT F}. For any PPDU, there must be at least as many HT-LTFs as spatial streams in the HT Data portion of the PPDU, i.e.N_{LT F} = N_SS (the number of spatial streams).

The first HT-LTF consists of two Long Training Symbols (LTS) as in 802.11a/g and a regular GI of 0.8µs, giving a total length of 7.2µs. When present, the second and all subsequent HT-LTFs each consist of a single HT-LTS with a regular GI of 0.8µs, giving a total length of 4µs.

In 20M Hz, for N_{T x} = 1, sequence 1 is:

HT L_−28:28 = [−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, 0, −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] (A.1)

In 20M Hz, for N_{T x} = 2, sequence 2 is:

HT L_−28:28 = [−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, 0, −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] (A.2)

In 20M Hz, for N_{T x} = 3, sequence 3 is:

HT L_−28:28 = [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, 0, 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] (A.3)

In 20M Hz, for N_{T x} = 4, sequence 4 is:

HT L_−28:28 = [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, 0, −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] (A.4)

Since the HT-LTF is tone interleaved across antennas, the 56 tones are partitioned across the antenna array during each OFDM symbol. Tone partitioning into sets for 20M Hz is shown in Table 26. At each OFDM symbol interval, each set of tones maps to one transmit antenna. And over time, all sets get mapped to an antenna.

N_SS set 0 set 1 set 2 set 3

1 [-28:1:-1] [1:1:28]

2 [-28:2:-2] [2:2:28] [-27:2:-1] [1:2:27]

3 [-28:3:-1] [2:3:26] [-27:3:-3] [3:3:27] [-26:3:-2] [1:3:28]

4 [-28:4:-4] [1:4:25] [-27:4:-3] [2:4:26] [-26:4:-2] [3:4:27] [-25:4:-2] [4:4:28]

Table A.1: Tone partitioning into sets for 20M Hz (56 tones)

A.2 Pilot symbols specification

In each OFDM symbol, four of the carriers in 20M Hz mode and six of the sub-carriers in 40M Hz mode are dedicated to pilot signals in order to make the coherent detection robust against frequency offsets and phase noise. These pilot signals shall be put in sub-carriers −21, −7, 7, and 21 in 20MHz mode, and in sub-carriers −53, −25,

−11, 11, 25 and 53 in 40MHz mode.

The contribution of pilot sub-carriers at the i_{T x}th transmission stream in 20M Hz mode is given by

P_−28..28,n^{iT x} ={0, 0, 0, 0, 0, 0, 0, P_−21,n^{iT x} ,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, P_−7,n^{iT x} ,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, P_7,n^{iT x},0, 0, 0, 0, 0,

0, 0, 0, 0, 0, 0, 0, 0,−P_21,n^{iT x},0, 0, 0, 0, 0, 0, 0}, (A.5)

where non-zero element of P_k,n^{iT x} is generated by the following rule

P_k,n^{iT x} = exp{jπ

2 (i_{T x}− 1)m}, m = (1 2(k

7 + 3) + n)mod 4, (A.6) where i_{T x} = 1, 2, ... is the index of transmit antenna, k is the location of pilot, n is the index of the OFDM symbol.

A.3 LDPC code specification

Parity check matrices H of the encoding procedure, are derived from one of the base

Parity check matrices H of the encoding procedure, are derived from one of the base

在文檔中 編碼多天線頻域正交多工信號在時變通道下之偵測 (頁 36-63)