Notation of MIMO-OFDM Systems

: the interleaved bit at the transmitter antenna

p th th

c

n

n p

i

.

: the modulated signal at the subcarrier and the OFDM symbol at the transmitter antenna

: the transmitted signal at time at the

^th

transmitter antenna

s (t)

′

p t p

: the index of modulated symbol sequence i

: time index

t

i

: the number of transmitter antennas

N

T

i

: the number of information bits

b n

{ } ^b _{n n} ^L

^b

₌₀ ^-1

1 ( ) s t ′

{ } ^{c n L -1} _n

^c

_{= 0}

{ } ^{c n 1 L -1} _n=0

^c

p ( ) s t ′

{ } ^{s 1 , k}

{ } ^{c n p L -1} _n=0

^c

{ } ^{c n N}

^T

_n=0 ^{L -1}

^c

{ } ^{s p , k =} _{= 0 k=0} ^{L -1 k=} _,

^s

^{, K}

{ } ^{s N , k}

^T

^{s ′ N}

^T

^{( ) t}

Fig. 2-14: Notations of a MIMO-OFDM transmitter

The notations of MIMO-OFDM receiver is shown in.Fig. 2-15

Fig. 2-15: Notations of a MIMO-OFDM receiver

where

: the received signal at time at the

^th

transmitter antenna

r (t)q

′

t q

i

.

: the modulated signal at the subcarrier and the OFDM symbol at the transmitter antenna

.

: the detected signal at the subcarrier and the OFDM symbol from the transmitter antenna

( ) c

_n : the a posteriori log likelihood ratio of deinterleaved bit

c

_n Λ

: the number of transmitter antennas

N

R

DeMUD Deinterleaver Viterbi

Decoder

{ } ^r

^q,k Calculation^MetricsBit

{ ^Λ ( ) ^c

^np

} { } ^H

^{q p,,k}

{ } ^y

^p,k

^{{ Λ ( ) c}

ⁿ

^} { } ^{b ˆ}

^{n L}_n^b₌₀^-1

Chapter 3:

Linear Multi-Stage Detection

This chapter considers MIMO-OFDM systems with bit-interleaved coded modulation (BICM) [7]. The maximum likelihood (ML) receiver has higher computation complexity. Therefore, we design a low-complexity receiver with a linear detector based on zero-forcing (ZF) and minimum mean squared errors (MMSE) algorithms. Here, we propose how to calculate the bit metrics for BICM on a ZF receiver and an MMSE receiver.

Equation Section 3

3.1 Bit Metrics for BICM

BICM technique is suited to multipath fast-fading channels, then the sub-channels of OFDM systems with bit-interleaver can be approximated as independently fast-fading. For better performance, the decoder is implemented by soft Viterbi decoding. Because bit interleaving is applied to the encoded bit before the M-QAM modulator, maximum likelihood decoding of BICM signals would require joint decoding and demodulation. According to the MAP criterion, estimate the coded bit sequence

{ }

^cnp _n=0^{L -1c at the}

p sub-stream by

^th

{ }

^{ˆ c =arg max p}

_{{ }}

_n^{p L -1c}

{ ^{{ } { }}

^{c , s}

}

n=0

L -1 L -1 =L -1,k=K -1

p p p

n n=0 c n n=0 k =0,k=0

c

⎡

c y

⎤

⎢ ⎥

⎣ ⎦ (3.1)

Thus, all possible coded and interleaved bit sequences would be calculate in (3.1).

Zehavi proposed a decoding scheme in [23] to compute sub-optimal simplified bit metrics to be used inside a Viterbi decoder for path metric computation.

Define the bit metrics of coded and interleaved bit

c

^p_,k,m for BICM system by ignoring the noise color, i.e., assuming the real part and the image part of noise are independent.

Interleaved coded bit sequences

Every log M coded bits to map a modulated symbol₂

where = M

^#

log M

₂

Fig. 3-1: To group M interleaved-coded bits to map a modulated symbol for MIMO-OFDM systems

where

k subcarrier, at the

th ^thOFDM symbol, and at the

p sub-stream. Because the

^th

computation of bit metrics of coded bit

c

^p_,k,mdepends only on detected signal

y

^p_,kat

the

k subcarrier, at the

^{th th}OFDM symbol, and at the

p sub-stream, we can ignore

^th the subcarrier indexk and the OFDM symbol index .

Let

c

_m^p

= c

^p_,k,m,

y

^p

= y

^p_,k and the bit metrics of

c

_m^p is

A posteriori probability log likelihood ratio (LLR) can be shown as

(1)

By the Bayes rules,

(1) (1)

Because all symbols on the constellation are transmitted with equal probability, then equation (3.6) can be modified to

By equations (3.5) and (3.7), the bit metrics is equal to

( )

⁽¹⁾

Sub-optimal simplified LLR can be reduced by the log-sum approximation

( ( ) )

i max i

i i

ln

⎛⎜

x

⎞⎟≈

ln x

⎝

∑

⎠ ^(3.9)

The log-sum approximation is a good approximation if the summation in the left-hand side of equation (3.9) is dominated by the largest term. Then, at high signal-to-noise ratio (SNR), the bit metrics can be approximated by the log-sum approximation, see

In this section, use ZF approach to detect signal. Because the MIMO-OFDM systems is used in the indoor WLAN scenario, we can assume the MIMO channel is multipath quasi-static Rayleigh fading channel. The frequency response

H

_k of MIMO channel in the

k

^thsubchannel of OFDM systems is defined as.

=

The transmitted signal vector before IFFT/GI is defined as

s

_.k

T ._k

= ⎡ ⎣ s

¹._k

, , s

^N._k^T

⎤ ⎦

s

_(3.12)

The received signal vector after FFT/remove-GI is defined as

r

_.k

T ._k

= ⎡ ⎣ r

¹._k

, , r

.^N_k^R

⎤ ⎦

r

_(3.13)

And the received signal vector after FFT/remove-GI can be represented as

._k

=

_k ._k

+

._k

r H s n

(3.14)

where

n

_.k

=[ n

¹_.k

, , n

^N_.k^R

]

is the received noise vector.

Define the coefficient of the linear detector in the

k

^thsubchannel is

ZF H

To detect signal at the

p sub-stream based on the zero-forcing criterion.

^th

ZF

ˆ

Then, the output signal of the ZF receiver is

ZF ZF ZF

._k

=

_k ._k

=

_{k k} ._k

+

_k ._k

y G r G H s G n

(3.18)

Assume there is perfect channel estimation. Then the detected signal vector is

ZF

3.2.1 Approximation of Bit Metrics

Observe the

p sub-stream detected signal

^th

y

^p_.k ,

( )

^H

.

=

.

+

.

p p p

k k k k

y s g n

_(3.20)

The received noises

n

¹_.k

, , n

^N_.k^R are statistically independent and identically distributed complex Gaussian random variables with zero mean and variance

σ

_n². Define over all noise term in the equation (3.20) is

( )

^H

.

=

.

=

.

+ +

^{p,NR N}.^R

p p p,1 1

k k k k k k k

z g n g n g n

(3.21)

Then

z

^p_.k is still a complex Gaussian random variable with zero mean and variance

Then the detect signal is shown as

.

=

.

+

.

p p p

k k k

y s z

(3.23)

The conditional pdf of

y

^p_.k is a complex Gaussian distribution,

( )

²

( )

Then, define the coefficients of bit metricsc^p_{. ,k m} for BICM

So the coefficient of bit metricsc^p_{. ,k m} for BICM is directly proportional to the signal-to-noise ratio of detected signaly^p_.k.

3.2.2 Simulation Results

Our simulation platform is based on the proposal of TGn Sync. The signal bandwidth (BW) is 20MHz. The transmitter and receiver use 128-points IFFT and FFT, respectively. The antenna spacing in the transmitter and receiver are equal to 0.5 wavelength. The decoder uses soft Viterbi algorithm to decide information bits with trace back length of 128. Assume there are perfect synchronization in the receiver, i.e. without frequency offset, clock offset, and phase rotation. The channel is well-kwon in the receiver. There are 8000 information bits per packet. There are at least 500 packet errors down to 1% packet error rate (PER) or a total of 10,000 packets in our simulation. The detector design in this section is based on the ZF criterion. Compare the equal weight

W

_.^p_k=1and weighted

.

Case1: Channel B of IEEE802.11n, 2x2

From the simulation results Fig. 3-2 and Fig. 3-3, we can discover that the performance of weighted coefficients for bit metrics computation is much better than those of equal gain. There are about 4~5 dB improvement under the PER=0.1.

Fig. 3-2: PER of bit metrics calculation with equal and weighted coefficients by ZF detector for BPSK and QPSK in channel B, 2x2

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 10^-2

10^-1

10⁰ P E R vs. S N R _d B C h -B 2x2 (p erfect C S I) Z F D etecto r

S N R _ d B

PER

B P S K ,R_c=1/2 QP S K ,R_c=1/2 QP S K ,R_c=3/4

W=SINR W=1

Fig. 3-3: PER of bit metrics calculation with equal and weighted coefficients by ZF detector for 16-QAM and 64-QAM in channel B, 2x2

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 10^-3

10^-2 10^-1

10⁰ BER vs. SNR_dB Ch-B 2x2 (Perfect CSI) ZF detector

SNR_dB

BER

BPSK,R_c=1/2 QPSK,R_c=1/2 QPSK,R_c=3/4

Fig. 3-4: BER of bit metrics calculation with equal and weighted coefficients by ZF detector for BPSK and QPSK in channel B, 2x2

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 10^-2

10^-1

10⁰ P E R vs. S N R _d B C h -B 2x2 (p erfect C S I) Z F D etecto r

S N R _d B

PER

16-QA M ,R_c=1/2 16-QA M ,R_c=3/4 64-QA M ,R_c=2/3 64-QA M ,R_c=3/4

W=SINR W=1

W=SINR W=1

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 10^-4

10^-3 10^-2 10^-1

10⁰ BER vs. SNR_dB Ch-B 2x2 (Perfect CSI) ZF detector

SNR_dB

Fig. 3-5: BER of bit metrics calculation with equal and weighted coefficients by ZF detector for 16-QAM and 64-QAM in channel B, 2x2

3.3 MMSE Criterion

In this section, use MMSE approach to detect signal. It is similar to a ZF receiver. Assume the MIMO channel is multipath quasi-static Rayleigh fading channel. The received signal vector after FFT/remove-GI is defined in (3.14) and the output signal vector

y of MMSE detector is defined as

_.k

MMSE ._k

=

_k ._k

y G r

(3.28)

Now, base on the MMSE criterion to minimize the error of the detected signal vector

y and a transmitter signal vector

_.k

s

_.k

{ } { }

See Appendix A, assume the energy of signal is equal to 1.

Then, the coefficients of an MMSE detector is

( )

^H

( )

^{H -1}

3.3.1 Approximation of Bit Metrics

Observe the

p sub-stream detected signal

^th

y

^p_.k ,

( )

^H

( )

^H

( )

^H

In [21], H.V. Poor and S.Verdu show that the MMSE estimate approximates a Gaussian distribution. Hence, the co-antenna interference and noise are considered together as complex Gaussian noise

z

^p_.k with Gaussian approximation.

( )

^H

( )

^H

.

( )

Then the detect signal is shown as

( )

^H

.

=

.

+

.

p p p p p

k k k k k

y g h s z

_(3.37)

The conditional pdf of

y

^p_.k is a complex Gaussian distribution,

( )

²

( ( ) )

By the way, the signal-to-interference-and-noise ratio of

y

^p_.k is

{ } { } _{( )} ^{( )}

So the coefficient of bit metrics

c

^p_{. ,k m} for BICM is directly proportional to the signal-to-interference-and-noise ratio of detected signaly^p_.k.

3.3.2 Simulation Results

Our simulation platform is based on the proposal of TGn Sync. The signal bandwidth (BW) is 20MHz. The transmitter and receiver use 128-points IFFT and FFT, respectively. The antenna spacing in the transmitter and receiver are equal to 0.5 wavelength. The decoder uses soft Viterbi algorithm to decide information bits with trace back length of 128. Assume there are perfect synchronization in the receiver, i.e. without frequency offset, clock offset, and phase rotation. The channel is well-kwon in the receiver. There are 8000 information bits per packet. There are at least 500 packet errors down to 1% packet error rate (PER) or a total of 10,000 packets in our simulation. The detector design in this section is based on the MMSE criterion. Compare the performance of equal and weighted coefficients of bit metrics calculation. The SNR is defined in chapter 2.

Case1: Orthogonal AWGN channel, 2x2

The signal is transmitted through the AWGN channel with orthogonal MIMO channel. equal and weighted coefficients are almost the same. That’s because the frequency response of all subchannel are equal.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

10^-2 10^-1

10⁰ PER vs. SNR_dB PerfectCSI (AWGN)

SNR_dB

Fig. 3-6: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector in AWGN channel, 2x2

W=SINR W=1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 10^-6

10^-5 10^-4 10^-3

10^-2 BER vs. SNR_dB PerfectCSI (AWGN)

SNR_dB

BER

BPSK,R_c=1/2 QPSK,R_c=1/2 QPSK,R_c=3/4 16-QAM,R_c=1/2 16-QAM,R_c=3/4 64-QAM,R_c=2/3 64-QAM,R_c=3/4

Fig. 3-7: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector in AWGN channel, 2x2

Case2: Channel B of IEEE802.11n, 2x2

From Fig. 3-8 and Fig. 3-9, we can find that the performance of weighted gain for bit metrics computation is better than those of equal gain. There are about 1dB improvement for BPSK and QPSK, about 3dB improvement for 16-QAM and about 4dB improvement for 64-QAM under the PER=0.1. Compare to ZF detectors, the improvement of MMSE detects is smaller than those of ZF detector, especially for lower modulation. That is because we use the Gaussian approximation in MMSE detector. Then in the low modulation scheme and fewer sub-streams, the Gaussian approximation of interference is loose.

W=SINR W=1

Fig. 3-8: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 2x2

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 10^-2

10^-1

10⁰ PER vs. SNR_dB Ch-B2x2 (Perfect CSI) MMSE Detector

SNR_dB

PER

16-QAM,R_c=1/2 16-QAM,R_c=3/4 64-QAM,R_c=2/3 64-QAM,R_c=3/4

Fig. 3-9: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 2x2

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 10^-2

10^-1

10⁰ P E R vs. S N R _d B C h -B 2x2 (P erfect C S I) M MS E D etecto r

S N R _d B

PER

B P S K ,R_c=1/2 QP S K ,R_c=1/2 QP S K ,R_c=3/4

W=SINR W=1

W=SINR W=1

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

10^-1 BER vs. SNR_dB Ch-B 2x2 (Perfect CSI) MMSE detector

SNR_dB

BER

BPSK,R_c=1/2 QPSK,R_c=1/2 QPSK,R_c=3/4

Fig. 3-10: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 2x2

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 10^-4

10^-3 10^-2 10^-1

10⁰ BER vs. SNR_dB Ch-B 2x2 (Perfect CSI) MMSE detector

SNR_dB

Fig. 3-11: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 2x2

W=SINR W=1

W=SINR W=1

Case3: Channel B of IEEE802.11n, 2x3

In this case, the receiver uses three antennas to receive signal. From Fig. 3-12 and Fig. 3-13, we can find that the performance of weighted gain for bit metrics computation is better than those of equal gain. There are smaller than 0.5dB improvement for BPSK and QPSK, about 1dB improvement for 16-QAM and about 1.5dB improvement for 64-QAM under the PER=0.1. Compare to case2, the receiver in the case3 uses more receiver antenna than those in case2, and then the receiver has more diversity gain. Therefore, the weight for bit metrics is close to equal.

Fig. 3-12: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 2x3

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

10^-2 10^-1

10⁰ P E R vs. S N R _dB C h-B 2x3 (P erfect C S I) MMS E D etector

S N R _dB

PER

B P S K ,R_c=1/2 QP S K ,R_c=1/2 QP S K ,R_c=3/4 W=SINR W=1

Fig. 3-13: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 2x3

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

10⁰ BER vs. SNR_dB Ch-B 2x3 (Perfect CSI) MMSE detector

SNR_dB

BER

BPSK,R_c=1/2 QPSK,R_c=1/2 QPSK,R_c=3/4

Fig. 3-14: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 2x3

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 10^-5

10^-4 10^-3 10^-2 10^-1

10⁰ BER vs. SNR_dB Ch-B 2x3 (Perfect CSI) MMSE detector

SNR_dB

BER

16-QAM,R_c=1/2 16-QAM,R_c=3/4 64-QAM,R_c=2/3 64-QAM,R_c=3/4

Fig. 3-15: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 2x3

Case4: Channel B of IEEE802.11n, 3x3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 10^-2

10^-1

10⁰ PER vs. SNR_dB Ch-B 3x3 (PerfectCSI) MMSE Detector

SNR_dB

PER

BPSK,R_c=1/2 QPSK,R_c=1/2 QPSK,R_c=3/4

Fig. 3-16: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 3x3

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 10^-2

10^-1

10⁰ PER vs. SNR_dB Ch-B 3x3 (PerfectCSI) MMSE Detector

SNR_dB

PER

16-QAM,R_c=1/2 16-QAM,R_c=3/4 64-QAM,R_c=2/3 64-QAM,R_c=3/4

Fig. 3-17: PER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 3x3

W=SINR W=1

W=SINR W=1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

10⁰ BER vs. SNR_dB Ch-B 3x3 (PerfectCSI) MMSE Detector

SNR_dB

BER

BPSK,R_c=1/2 QPSK,R_c=1/2 QPSK,R_c=3/4

Fig. 3-18: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for BPSK and QPSK in channel B, 3x3

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 10^-4

10^-3 10^-2 10^-1

10⁰ BER vs. SNR_dB Ch-B 3x3 (PerfectCSI) MMSE Detector

SNR_dB

Fig. 3-19: BER of bit metrics calculation with equal and weighted coefficients by MMSE detector for 16-QAM and 64-QAM in channel B, 3x3

W=SINR W=1

W=SINR W=1

Case5: Compare MMSE and ZF detector in channel B of IEEE802.11n, 2x2

Fig. 3-20: PER of bit metrics calculation with weighted coefficients by MMSE detector and ZF detector for BPSK and QPSK in channel B, 2x2

Fig. 3-21: PER of bit metrics calculation with weighted coefficients by MMSE detector and ZF detector for 16-QAM and 64-QAM in channel B, 2x2

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 10^-2

10^-1

10⁰ P E R vs. S N R _dB C h -B 2x2 (perfect C S I) W =S IN R

S N R _d B

PER

B P S K ,R_c=1/2 QP S K ,R_c=1/2 QP S K ,R_c=3/4

MMSE Detector ZF Detector

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 10^-2

10^-1

10⁰ P E R vs. S N R _d B (perfect C S I) W =S IN R

S N R _dB

PER

16-QA M,R_c=1/2 16-QA M,R_c=3/4 64-QA M,R_c=2/3 64-QA M,R_c=3/4

MMSE Detector ZF Detector

3.4 Conclusions

In this chapter, we derived the approximation of bit metric for MMSE detector and ZF detector, respectively. We analyze the performance of bit metric calculation with equal and weighted coefficients for the MMSE detector and the ZF detector There are about 3~4dB improvement by using weighted coefficients compared to equal coefficients. But in the lower modulation scheme, the Gaussian approximation of the interference would be loose. Hence, the improvement for BPSK and QPSK is only about 1dB in the MMSE detector. By the way, the ZF detector has noise enhancement so the performance of MMSE detector is better than those of ZF detector about 1~4dB, especially at lower SNR. At high SNR, the performance of the ZF detector is close to those of the MMSE detector.

Chapter 4:

Low-Complexity Iterative Detection

Under the condition that the transmitter architecture is of no change and the receiver only uses available received signals, this chapter utilizes an iterative method to improve the performance of MIMO BICM systems. The receiver joints signal detection and soft decoding with turbo principles to suppress the strong co-antenna interference in MIMO systems. The receiver returns soft information of the MAP decoder back to the multistage detector to enhance the ability of detecting signals. The subchannel, i.e. subcarrier, of MIMO-OFDM system has constant channel gain on the multipath Rayleigh fading channel. The MIMO-OFDM receiver detects signals per subcarrier. It is similar to the receiver of MIMO Single-Carrier system on the flat fading channel. Here, our proposed algorithm can be used for general MIMO systems.

It is more convenient to me to depict our proposed algorithm for MIMO BICM systems. The block diagram of MIMO transmitter structure is shown in Fig. 4-1.

Fig. 4-1: A MIMO transmitter

{ } b

n n^Lb₌₀^-1

{ } ^c

^{n -1L}_n^c₌₀

{ } ^c

^np _n^Lc₌^-1₀

^{{ } s}

^{tp tL=s0-1}

{ } ^c

^{n1 L}_n^c₌^-1₀

{ } ^c

^{nNT L}_n^c₌^-1₀

{ } ^s

^１t _t^L₌^s₀^-1

{ } ^s

^tNT _t^L₌^s₀^-1

where

: the number of transmitter antennas

N

T

i

: the number of information bits

b n

The MIMO channel is shown in Fig. 4-2.

Fig. 4-2: The MIMO channel

The received signal is

=0

The block diagram of MIMO iterative receiver architecture is shown in Fig. 4-3.

Fig. 4-3: A MIMO iterative receiver

where

The index

i and

o denote the log likelihood ratio (LLR) associated with the inner detector and outer decoder, respectively. And the superscripts a and e denote a priori (intrinsic) information and extrinsic information, respectively. ^π

( ) ^•

^{is an}

interleaver function.

This chapter is organized as follows: In the section 4.1, to describe the optimal detector based on MAP algorithm and MAP (BCJR) decoder. In the section 4.2, to provide the suboptimal low-complexity linear detector based on MMSE algorithm, and we propose four approximations to reduce the computation complexity of iterative MMSE receiver. Finally, in the section 4.3, the performances of various

Channel

iterative MMSE receiver schemes proposed in this chapter are examined.

Equation Section 4

4.1 Optimal Receiver Based on MAP Algorithm

Assume MIMO channel is an flat quasi-static Rayleigh fading channel matrix H.

The received signal

r

_t^q at the

q receiver antenna at time

^th

t is

Then the received signal vector

r is defined as

_t

NT 1

How to design an optimal receiver for MIMO system is to maximize a posteriori probability of information bit

b with all received signal vectors.

_n

{ { }

^=0-1

}

Define a posteriori log likelihood ratio of

b as

_n

( ) { }

Detect information bit

b ,

_n

( ) ( )

By the total probability theorem, the a posteriori probability of

b can be shown as

_n

{ } { } { }

Due to information bit

b depending on detected signal vector sequences

_n

{ } s

t t^L₌₀^s-1, then

The channel is a flat fading and discrete memoryless channel so the detected signal vector

s at time t only depends on the received signal vector

_t

r at time t. Then,

_t

Finally, the optimal receiver is able to calculate a posteriori LLR of information bit

b .

_n

( ) { }

But the computation complexity of the optimal receiver is too high. It is impossible to realize an optimal receiver. In order to reduce the computation complexity, we divide the receiver into two parts: inner detector and outer decoder, as Fig. 4-4 .

Fig. 4-4: A inner detector and a outer decoder

4.1.1 MAP Detector

The optimal detector for iterative receiver is an a posteriori probability (APP) detector.

{ }

At the first iteration, there is no soft information about transmitted signal vector

s

_t. It means thatp s

[ ]

t are equal. Then, the MAP detector is a maximum-likelihood (ML) detector.

{ } { }

=arg max =arg max

t t

t

p

t t

p

t t

∈Ψ ⎡⎣ ⎤⎦ ∈Ψ ⎡⎣ ⎤⎦

s s

s s r r s

(4.12)

The computation complexity of MAP detector (ML detector) is order of

M

^NT . MAP detector is not feasible for larger number of transmit antennas or higher modulation schemes. The suboptimal detector is a linear detector based on MMSE criterion.

4.1.2 MAP (BCJR) Decoder

In this section, we describe how to use a MAP decoder as an optimal decoder and how to calculate the soft information pass to inner detector. Because the transmitter uses a bit interleaver after a convolutional encoder to overcome Rayleigh fading channel, the receiver needs to calculate the bit metrics before a bit de-interleaver for soft Viterbi decoding or MAP decoding. The de-interleaved codeword is denoted by

c

_n. It is an encoder output tuple by encoding information bit

b

_n. Assume the code rate of a convolutional encoder is

R = 1/2 .

_c

The a posteriori log likelihood ratio of

c

_{n, j} for MAP decoder is denoted as

The a posteriori probability can be written as

{ ( ) } { ^{( )} }

By [19], the authors tell us,

By the equations(4.14) and(4.15), the a posteriori LLR is

( ) { ^{( )} }

( ) ( ) ( )

Finally, the a posteriori LLR can be shown as

( ) ( ) ( )

= extrinsic information

S' S S S' (S',S)

If want to reduce the computation complexity of a decoder, you can use a suboptimal decoder, SOVA decoder.

4.2 Iterative MMSE Detector

The optimal detector of the iterative receiver, MAP detector, causes a large computational complexity. A suboptimal and low complex detector is using adaptive linear filter techniques. A linear minimum mean squared error (MMSE) detector is a simplified approach compared with an MAP detector. An MMSE detector has higher performance than other linear detector.

The received signal vector

r as (4.2), can be decomposed three part: desired

_t signal, co-antenna interferences and noise, see (4.30).

desired signal interference noise

= + =

^{p p}

+

^{p p}

+

First step, to estimate the co-antenna interference

μ based on soft information

_t^p

^λ

^ia

( ) ^c

^{nj ,}

see(4.31). Assume the channel estimation is perfect.

= (i) The modulator maps the coded bits to complex symbol

s .

_t^j

(

0 log2 -1

)

=map , ,

j j j

t t, t, M

s c c

(4.32)

Calculate

^s

^tj(i)=E

{ } ^s

^tj based on a priori information

{ ^λ

^ia

( ) ^c

^t,mj

}

^log_m=0^2M-1 from a MAP decoder.

Then, to remove the co-antenna interference

(

⁽ⁱ⁾

)

= = + +

p p p p p p p

t t t t

s

t t t t t

x r μ

－

h H s

－

s n

(4.33)

Output signal of adaptive linear detector

y is

_t^p

( )

^H

p= p p

t t t

y g x (4.34)

To calculate the coefficients of adaptive linear detector based on MMSE Criterion,

( )

( )

{ { } }

The coefficients of adaptive linear detector

( ) ^g

^{tp His}

( ) g

^{tp H=E}

{ ^s

^tp

( ) ^s

^{tp *}

} ^{( )} h

^{tp H⎡}⎢⎣

h

^tpE

{ ^s

^tp

^{( ) s}

^{tp *}

} ^{( )} h

^{tp H+}

H V H

^{tp tp}

^{( )}

^{tp H+}

^σ

ⁿ²

I

^NR⎤⎥⎦^-1(4.39) Before bit de-interleaving and MAP decoding, we need to calculate bit metrics with output signal of adaptive linear detector

y

_t^p.

We redefine coded and interleaved bit

c

_n^pto be

c

_t,m^p , as Fig. 4-5,

Interleaved coded bit sequences

Every log M coded bits to map a modulated symbol₂

where =

M^# log M₂

Fig. 4-5: To group log M₂ interleaved-coded bits to map a modulated symbol for MIMO systems

where symbol, and at the

p sub-stream. By the simplified computation of bit metrics of

^th coded bit

c

_t,m^p , it is can be presented as

(1)

The extrinsic information is defined as

( )

⁽¹⁾

And the a priori (intrinsic) information is defined as

( ) ⁼¹

By the turbo principle, the inner detector forwards the extrinsic information

^λ

^ie

( ) ^c

^{t,mp to}

the MAP decoder. We need to ensure that the equation (4.44) being PURE extrinsic information. It means that the conditional probability

p y s

^⎡_{⎣ t}^p _t^p=

ψ

^⎤⎦ should not depend

Fig. 4-6 : The block diagram of the proposed iterative MMSE receiver

The computation complexity of this iterative MMSE detector is proportional to

T s iteration 2

N

⋅ ⋅

L N

⋅

log M

, where

N

_iterativeis the number of iterations.

4.2.1 Approximation I of the proposed iterative MMSE detector

The computation complexity of the proposed iterative MMSE detector is very high. It needs to compute

N

_T⋅ ⋅

L N

_{s iteration}⋅

log M

₂ times the coefficients of iterative MMSE detector (pseudo inverse operations). In order to reduce the computation complexity, let ^E

{ } s

t^{p =0} and^E

{ ^s

^tp

( ) ^s

^{tp *}

}

⁼¹ when the receiver detects the

p

^th spatial stream signal at time t . Then, the coefficients of adaptive linear detector

( ) ^g

^{tp H}is simplified to proposed iterative MMSE detector.

r t

Adaptive MMSE

Fig. 4-7: The block diagram of the approximation I of the proposed iterative MMSE receiver

The computation of

( ) ^g

^{apIp His}

^N

^{T⋅ ⋅}

^{L N}

^{s iterative} pseudo inverse operations. It does not

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