Shannon Capacity

SURVEY OF NON-LINEAR MIMO RECEIVERS AND THEIR IMPACT ON SPATIAL MULTIPLEXING PERFORMANCE, APRIL 2009

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Survey of Non-Linear MIMO Receivers and Their Impact on Spatial Multiplexing Performance John W. Thomas, Student, University of Texas at Dallas

Abstract—This paper is a short survey of a few available receivers and their impact on a Spatial Multiplexing (SM) scheme in a multiple-input, multiple-output (MIMO) environment. The ultimate advantage of SM is to increase capacity (b/s/Hz), but another aspect of performance is the rate of errors over a channel. Each used receiver has a different impact on the overall performance. This paper attempts to summarize some of those differences.

B. Multiple Transmit Antenna System - MISO

I. I NTRODUCTION

C. Multiple Receive Antenna System - SIMO

ρ 2 χ ) (2) M 2M Where ρ is equal to the average received SNR and χ22M is a chi-squared random variable with 2M degrees of freedom, while M is the number of transmit antennas. Capacity = log2 (1 +

T

HE choice of decoders at the receiver have an immense impact on performance. An even greater impact is felt by a spatial multiplexing MIMO system due to the multitude of receivers and independent demultiplexing of bits/symbols over multiple antennas at the transmitter. This creates a phenomenon where the different transmitted streams from each antenna interfere at the receiver. This is known as multi-stream interference (MSI). This paper attempts to summarize how a select choice of a decoders at a receiver impacts performance. A few assumptions made in this paper are that the channel experiences flat Raleigh fading, channel coding is absent, antennas are decorrelated or antennas spacing is at least half the wavelength of the signal, perfect channel knowledge is only available at the receiver, and their is additive white gaussian noise (AWGN) at each receiver. A. Organization of Paper This paper is outlined as follows: Section 2 will describe the capacity performance metric, Section 3 will summarize both linear and non-linear decoders used at the receiver, Section 4 will describe how the choice of receivers impact the error rate performance metric, and Section 5 will conclude the paper.

Capacity = log2 (1 + ρχ22N )

(3) χ22N

Where ρ is equal to the average received SNR and is a chi-squared random variable with 2N degrees of freedom, while N is the number of receive antennas. D. Multiple Transmit and Receive Antenna System - MIMO M X

Capacity >

log2 (1 +

k=M −N −1

ρ 2 χ ) M 2k

(4)

Where ρ is equal to the average received SNR and χ22k is a chi-squared random variable with 2k degrees of freedom. One can observe from Figures 1 and 2 that MIMO provides the greatest amount of capacity out of the displayed antenna systems, followed by SIMO. Another observation is that MISO and SISO are very close in terms of capacity regardless of how many transmit antennas are used. This shows that the amount of receive antennas, used only in MIMO and SIMO, has the greatest impact on capacity. Capacity at 10 dB AVG. SNR at RX 600 SISO SIMO MISO MIMO lower bound

II. C APACITY 500

Capacity describes the amount of bits that can be sent over the channel in one cycle or second per Hertz. The generalized Shannon Capacity describes the upper bound of error-free capacity and is shown below for different antenna configuration schemes. Figures 1 and 2 compare the capacity of multiple antenna systems with respect to the number of antennas used for transmission, reception, or both.

Capacity (b/s/Hz)

400

300

200

100

A. Single Antennas System - SISO 0

Capacity = log2 (1 +

ρχ22 )

1

2

3

4

5 6 # of antennas

7

8

9

10

(1) χ22

Where ρ is equal to the average received SNR and is a chi-squared random variable with 2 degrees of freedom.

Fig. 1.

Multiple Antenna System w/ avg. receive SNR of 10 dB

SURVEY OF NON-LINEAR MIMO RECEIVERS AND THEIR IMPACT ON SPATIAL MULTIPLEXING PERFORMANCE, APRIL 2009

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5) Moore-Penrose psuedoinverse: In this paper, we will assume that H is at full rank and if H is not a square matrix then the Moore-Penrose psuedo-inverse matrix will be applied ˜ −1 . This is described in the following equations: to find H

Capacity at 10 dB AVG. SNR at RX 600 SISO SIMO MISO MIMO lower bound 500

Capacity (b/s/Hz)

400

(8)

H = HH˜−1 H

(9)

˜ −1 H = (H ˜ −1 H)H H

(10)

˜ −1 = (HH˜−1 )H HH

(11)

300

200

100

0

Fig. 2.

˜ −1 = H ˜ −1 HH ˜ −1 H

1

2

3

4

5 6 # of antennas

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Multiple Antenna System w/ avg. receive SNR of -10 dB

C. Z inf ty Receiver D. Minimum Mean-Squared Estimate Receiver

III. L INEAR R ECEIVERS One aspect that makes linear receivers distinct is the methodology of using a linear filter to separate each of the distinct transmitted streams. This filter is represented by the weighted matrix G. The symbol estimate, sˆ is found by selecting the symbol, s, multiplied by the weighted matrix, G, and the known channel matrix H, that is closest in Euclidean distance to the received symbol, y. This is displayed in the below equation. ˆs = mins kGy − GHsk2

(5)

The Minimum Mean-Squared Estimate Receiver (MMSE) also minimizes MSI, but also minimizes noise to lower errors. The linear filter matrix, G, is found by choosing the G that minimizes the expected Euclidean distance between the product of the linear filter matrix and the received symbol and the transmitted symbol. This is represented by the equation below: 2

G = minG E[kGy − sk ]

(12)

E. Joint-Channel Diagonalization F. Sphere Detector Receiver IV. N ON -L INEAR

A. Maximum Likelihood Receiver A Maximum Likelihood (ML) Receiver provides the optimal performance. In this case the linear filter matrix, G, is equal to the identity matrix. G=I

A. Successive Cancellation Receiver B. Ordered Successive Cancellation Receiver VBLAST is an example of an OSUC receiver.

(6)

C. Sphere Detection Receiver

This inhibits an exhaustive search through all available symbols, given the used modulation technique, to find each estimated symbol, ˆs. Decoding complexity in an ML receiver is considered to be high, but there are algorithms, such as sphere decoding, that have been proposed that combat this.

D. Lattice Reduction Receiver V. P ERFORMANCE C OMPARISON One can observe from figure 3, 4, and 5 that the SER/BER performance is much better when using a combination of an ML detector with an OSUC receiver, VBLAST. An ML estimator should provide you with optimal detector.

B. Zero-Forcing or Inverse Channel Receiver Zero-Forcing (ZF) receivers look to eliminate interference between each independent stream originating from one of the multiple transmit antennas. This is known as Multiple Stream Interference (MSI). This is done through inverting the known channel matrix, H, and setting it equal to G. G = H −1 1) 2) 3) 4)

Simple Analytical Solution: Block-wise Analytical Solution: Gaussian Elimination: LU decomposition:

(7)

VI. C ONCLUSION R EFERENCES [1] On Limits of Wireless Communications in a Fading Environment when Using Multiple Antennas .

SURVEY OF NON-LINEAR MIMO RECEIVERS AND THEIR IMPACT ON SPATIAL MULTIPLEXING PERFORMANCE, APRIL 2009

BER/SER Plots

0

10

BER SER

−1

BER/SER

10

−2

10

−3

10

−4

10 −10

Fig. 3.

−5

0

5

10 SNR [dB]

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Performance of QPSK using ML/VBLAST receivers in 2x2 MIMO

BER/SER Plots

0

10

BER SER

−1

BER/SER

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−2

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−3

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−4

10 −10

Fig. 4. MIMO

−5

0

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10 SNR [dB]

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Performance of QPSK using MMSE/VBLAST receivers in 2x2

BER/SER Plots

0

10

BER SER

−1

BER/SER

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−2

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−3

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−4

10 −10

Fig. 5.

−5

0

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10 SNR [dB]

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Performance of QPSK using ZF/VBLAST receivers in 2x2 MIMO

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