# Co-ordinate interleaved spatial multiplexing with channel state information

## Summary (2 min read)

### Introduction

- Diagonalizing the MIMO channel through singular value decomposition (SVD) and transmitting multiple data streams over the resulting parallel eigen sub-channels or eigenmodes, is a well known and simple linear diagonal MIMO transceiver.
- Analytical results presented in the paper show that co-ordinate interleaving across eigenmodes k and l, having diversity gains gd(k) and gd(l), respectively, results in a diversity gain of max{gd(k), gd(l)} on both the channels.

### II. SYSTEM MODEL

- The discrete time baseband input-output relation of the MIMO channel is y =.
- The channel gains {hij} are assumed to be independent, frequency-flat Rayleigh fading and hence, {hij} are i.i.d with hij ∼ CN (0, 1).
- For uncorrelated Rayleigh MIMO channels, rank(H) = n = min{Nt, Nr}.
- K denotes the number of active eigenmodes or eigen subchannels and the transmission scheme discussed above is referred to as SVD transceiver in rest of the paper.

### A. Performance Measures: Diversity gain and DiversityMultiplexing Tradeoff

- Diversity gains of most of the advanced diagonal transceivers are same as that of SVDTR [10].
- The diversity-multiplexing tradeoff (DMT) [2] is a more fundamental performance measure of a MIMO transceiver in slow-fading scenario as it captures diversity gain and multiplexing gain.
- Note that gd is the diversity gain for fixed input data rate (i.e., gd = d(0)), and to distiguish from d(r), it is referred to as classical diversity gain [14] wherever necessary.
- The DMT for SVDTR has been evaluated in [12].

### III. CO-ORDINATE INTERLEAVED SPATIAL MULTIPLEXING

- Co-ordinate interleaved spatial multiplexing (CISM) is described.
- The proposed scheme is based on coordinate interleaving (CI), a technique which was originally proposed to exploit the co-ordinate, or, component level diversity for single antenna transmission over Rayleigh fading channels [15], [16].
- The idea is to interleave the real and imaginary parts of the complex symbols at the transmitter such that they go through independently fading channels.
- The effect of rotation angle and the optimal angle of rotation are discussed in subsequent sections.

### A. CISM Transceiver

- As can be seen from (7), the received signals gets decoupled and hence can be decoded by single-symbol maximum likelihood (ML) decoding.
- Among the K eigenmodes used for transmission, the two symbols transmitted on strongest and weakest eigenmodes are interleaved and the two symbols transmitted on second strongest and second weakest eigenmodes are interleaved and so on.

### IV. DIVERSITY GAIN OF CISM

- The error rate performance of CISM, in particular, the diversity gain, is analyzed in this section.
- For illustrative purpose, the authors consider 4-QAM signaling and analysis for any M -QAM can be done in a similar way.
- 1 is lower bounded by the PEP corresponding to confusing xA with its nearest neighbor.
- As both the upper bound and lower bound on P 4-QAMs,1 have the same SNR exponent, it follows that data transmitted on first eigenmode has a diversity gain of 2m.
- SEP analysis for any M - QAM can be carried out in a similar way to show that CISM always achieves the maximum diversity gain 2m offered by the channel when rank(H) = Downloaded on July 17, 2009 at 02:07 from IEEE Xplore.

### V. DIVERSITY-MULTIPLEXING TRADEOFF OF CISM

- In the following, the authors compute the SEP Ps,k when Rk, data rate on kth eigenmode, scales with SNR as Rk = rk log SNR b/s/Hz, rk ∈ [0, 1], and obtain the DMT of CISM.
- Thus, CISM improves the diversity gains of the weaker eigenmodes resulting in a significant gain in the overall diversity, as shown by the following remark.
- Dividing the total input data rate R = r log SNR, r ∈ [0, 2], equally between the two eigenmodes results in the following.
- The tradeoff for SVDTR (or, multiple beamforming), and MEBF are derived in [12].

### A. CISM with multi-dimensional signaling

- Intrigued by (22), the authors investigate the DMT of CISM over rank n MIMO channels when the input symbols are from rotated n-dimensional QAM constellations.
- An n-dimensional QAM signal set is obtained as the Cartesian product of n/2 two-dimensional QAM signal sets [16].
- To send b bits in n-dimensions, the n-dimensional constellation will have (at least) 2b points.
- DMT of CISM with 2-dimensional signaling is plotted according to (26) and (27) and it outperforms SVDTR and MEBF.

### VI. SIMULATION RESULTS

- This section reports SEP of the proposed CISM transceiver, evaluated through Monte Carlo simulations.
- Consider CISM over 2 × 2 MIMO channel with data symbols from 4-QAM constellation rotated by θopt = 27.9o, also known as Example 2.
- Fig. 4 compares the SEP obtained through simulations with the union bound given by (9).
- As the data stream transmitted on second eigenmode does not get effected by interleaving, it has same SEP as SVDTR Eigch2.

### VII. CONCLUSIONS

- With perfect CSI at both ends of the link, a MIMO channel can be diagonalized and multiple data streams can be sent in parallel on the resulting eigenmodes.
- In most of the linear diagonal transceivers proposed to date, the weaker eigenmodes having low diversity gains drastically degrade the overall error rate performance.
- This paper proposed a novel transceiver, referred to as co-ordinate interleaved spatial multiplexing (CISM), that improves the diversity gains of the weaker eigenmodes.
- CISM diagonalizes the channel through SVD and interleaves the co-ordinates of the input symbols (from rotated QAM constellations) transmitted on strong and weak eigenmodes.
- By computing the upper bound and lower bound on symbol error probability of the eigen sub-channels, the diversity gains of CISM have been determined.

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##### Citations

165 citations

### Cites background from "Co-ordinate interleaved spatial mul..."

...originally proposed for communications over single-antenna channel [14], [15] and MIMO channels [16]–[18], to achieve signal-space modulation diversity....

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20 citations

### Additional excerpts

...A similar result was reported in [15] with a technique employing the rotated Quadrature Amplitude Modulation (QAM) constellation....

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18 citations

6 citations

### Cites background or methods from "Co-ordinate interleaved spatial mul..."

...1, the CISM scheme introduced in [40] is an uncoded case of the proposed...

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...In [40], the CISM scheme with SVD precoding is studied....

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...We proved that the optimum spatial Q-component interleaver is exactly the interleaver proposed in [40] and consummate the theoretical analysis of [40]....

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...As introduced in [13, 40, 45], for the M layer spatial multiplexing MIMO system, the precoding and detection process can be expressed as linear transformations z = Ū[M]HV̄[M]s+ Ū[M]n....

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...In [40], the optimal rotation angle obtained by SEP analysis is only applicable to the uncoded CISM scheme with SVD precoding....

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4 citations

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### "Co-ordinate interleaved spatial mul..." refers background in this paper

...Multiple paths can be exploited to obtain diversity gain by transmitting the same symbol over all the paths, and multiple degrees of freedom can be used to increase the data rate through spatial multiplexing [3]....

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##### Frequently Asked Questions (10)

###### Q2. What is the diversity gain of a SVDTR?

A diversity gain d(r) is achieved at a multiplexing gain r, if the data rate scales as R = r log SNR b/s/Hz and the SEP isPs(r log SNR) .= SNR−d(r) (4)where .= denotes exponential equality [2].

###### Q3. What is the symbol vector for the kth eigen sub-channel?

Let x ∈ CK×1, K ≤ n, be the symbol vector with xk ∈ Xk, 1 ≤ k ≤ K , where Xk is a unit energy QAM signal set employed on the kth eigen sub-channel, and E[xxH ] = IK .

###### Q4. What is the n-dimensional input data rate?

To achieve an overall input data rate of R = r log SNR b/s/Hz, r ∈ [0, n], the authors choose two n-dimensional symbols (say, x1 and x2) carrying R/2 bits each.

###### Q5. what is the coding gain of P 4-QAMs?

Pr{xA → xB} ≈ κCABSNR−2m (12) where κ = 82mm 4 ( 4m−1 4m 4m−3 4m−2 · · · 12 ) andCAB = 1(cos2 θ)m+1(sin2 θ)m−1 + 1 (cos2 θ)m−1(sin2 θ)m+1− 2(m− 1) m 1 (cos2 θ sin2 θ)m (13)Similarly,Pr{xA → xC} ≈ κCADSNR−2m (14) whereCAD = 1ψm+11 ψ m−1 2+ 1ψm−11 ψ m+1 2−2(m− 1) m 1 (ψ1ψ2)m(15)Hence, at high SNRs,P 4-QAMs ≤ κ(2CAB + CAD)SNR−2m (16) θopt, the optimal rotation angle, is computed by maximizing the coding gain (2CAB + CAD).

###### Q6. What is the diversity gain of CISM?

Remark 2: The overall (classical) diversity gain of CISM with K ≤ n is given bygCISMd = ( m− ⌊ K + 12⌋ + 1 )( n− ⌊ K + 12⌋ + 1 ) (21)SVDTR, as well as many advanced diagonal transceivers such as those proposed in [8], have an overall diversity gain of gd = (m−K+1)(n−K+1) [10].

###### Q7. What is the diversity gain of kth eigen subchannel?

The diversity gain gd(k) of kth eigen subchannel has been determined in [10] asgd(k) = (m− k + 1)(n− k + 1), k = 1, . . . , n (3)and was also shown that gd(k) does not improve with spatial power allocation.

###### Q8. What is the diversity gain of a CISM?

Remark 1 shows that, coordinate interleaving two data streams transmitted over two eigenmodes having different diversity gains would make the diversity gain of both the streams equal to the diversity gain of stronger eigenmode.

###### Q9. What is the optimal tradeoff curve for UCDVBLAST?

The UCD-VBLAST transceiver achieves the optimal open-loop tradeoff [9] and hence, the “Optimal tradeoff” curve also shows the DMT of UCDVBLAST.

###### Q10. What is the diversity gain for a fixed input data rate?

Note that gd is the diversity gain for fixed input data rate (i.e., gd = d(0)), and to distiguish from d(r), it is referred to as classical diversity gain [14] wherever necessary.