Typical sphere decoders for MIMO channels can only handle the case where
R T
N ≥ N [1]. These sphere decoders fail when NT > NR because H does not have full column rank and therefore cannot be QR-factorized. Here, a
Hs1
b1
b2
c1
c2
modification is proposed to deal with the case NT > NR.
The idea is to augment H into a matrix with full column rank. Let the augmented matrix be
T R T R R
in which the bottom NR rows comprise the original channel matrix, I is the identity matrix, and a is either a small real or complex number depending on the modulation scheme. The pseudo received vector is defined as
1
and the noise vector is augmented as
1
to make the final augmented received vector to be
( ) 1
By this augmentation,
H
has full column rank and can be decomposed via standard QR factorization algorithms. The SD algorithm can now be applied with similar effectiveness for the case NT > NR. This method is similar but more straightforward than the method in [12] in which an augmented diagonal matrixα I
is added to the matrixH H
H to make it full-rank. More comparisons will be made when the effect of a is analyzed.The concept of pseudo-antenna augmentation is shown in Fig. 4.2 where a simple 2 1× MIMO channel is augmented to a 2 2× MIMO channel. Fig.
4.3(a) shows the space diagram of the transmitted symbol vector, fig. 4.3(b) shows the pseudo received signals space, and the augmented received signals space was shown in fig 4.3(c). From (3.6) and (3.8), the smaller the value of a is ,
is also shown in Fig. 4.3(a)-(c).
Figure 4.2 The diagram of an augmented 2 2× MIMO system.
s
1s
2Figure 4.3(a) The space diagram of the transmitted symbol vectors.
Figure 4.3(b) The pseudo received signal vectors. Assume NT = 2, NR = 1, BPSK modulation and h1 >h2 >0 for simplicity. Define b1 = +h1 h2,
2 1 2
b = −h h , b3 = − +h1 h2, and b4 = − −h1 h2 for convenience.
s1
s2
h1
h2
a x’
0 n’
y n
x
as1
h1s1+h2s2
(b2, a) (b1, a)
(b3, a) (b4, a)
Figure 4.3(c) The augmented received signal vectors.
The effect of the value taken by a can be further analyzed as follows. The set of constellation points resulting in received signals inside the hypersphere D is found as
{
2 2}
.sD = x d ≥ y−Hx (4.9)
The inequality in (3.9) can be expanded to
( )
2The lower bound of the radius d with which the correct symbol s lies in the hypersphere, i.e.,
x = ∈ s s
D , depends on the noise condition and a. AssumeThe expected lower bound is thus
{ }
LB22 (
T R)
2 R 2.
E d ≥ N − N a + N σ
(4.12)As can be seen clearly in (4.11), if a is small, the lower bound on the radius with which the correct symbol vector can be included is essentially independent of a.
But if a is large, the radius needs to be large.
h1s1+h2s2
b2 b1
b3
b4
0
Figure 4.4 The space diagram of the hypersphere D when a is very large.
Assume BPSK and a 2 1× MIMO channel for simplicity.
Fig. 4.4 shows the diagram of a simple example with a 2 1× MIMO channel, BPSK, and a large a. Let point p1 be the augmented received signal and z1 the pseudo received signal. The total number of possible received points is 4.
As is said before, the radius of the sphere needs to be large. However, when setting the radius, it is extremely difficult for the decoder to find a radius barely large enough to include the lattice point corresponding to the correct symbol while avoid including wrong lattice points in the sphere simultaneously. In Fig.
4.4, the sphere not only contains the correct point z1 but also z2. If a more sophisticated modulation such as 64-QAM is used, and the number of transmit antenna is larger, much more lattice points will inevitably be included in the large hypershpere, and the efficiency of SD will be greatly diminished.
Therefore, a should be as small as possible, as long as the numerical stability is maintained in the computing process. With a small a, the complexity of SD is essentially independent of a and the same as that of usual SD algorithms, i.e.,
h1s1+h2s2
z1
as1
p1
z2
roughly O N
( )
T3 when SNR is high [1]. The efficiency of the method in [12], on the contrary, depends on the choice ofα
, and the optimal choice ofα
depends on noise condition and is not easy to find.
After the set of all candidate points is generated, the final step of the modified SD algorithm for MIMO channels works the same as the ML detector does. The estimated transmitted symbol vector
ˆs
is obtained by exhaustive search and equals toˆ arg min .
∈
= −
Hs
s y Hs
D (4.13)
Chapter 5
Simulation Results
Figure 5.1 The BER curves of SD and brute-force ML detector. Assume NT=6, NR=3, QPSK, spatial multiplexing, and a = 0.1 + 0.1 j .
Fig. 5.1 shows the performance of SD compared to that of ML receiver. The value of a is set to be very small and the BER performance is equal to that of a brute-force ML receiver.
Fig. 5.2 shows the average number of candidates found in D when
different values of a and
0
Eb
N and the proposed initial radius are used. Notice that when a is getting smaller, say, less than 0.1 0.1 j+ , the number of candidates found in D is essentially independent of a and is only function of
SNR. Also notice that when SNR is moderately large, e.g., in the applications of spatial multiplexing, the number of candidates is close to 1. This means the proposed SD algorithm is operating in a very efficient manner.
Figure 5.2 The average number of candidates inside sphere D with different values of a and
0
Eb
N . Assume NT = 6, NR = 3 and QPSK modulation.
Table II. lists the probabilities of when the minimum column norm coincides the minimum distance under different settings. For most of the time, when NT = NR, they do coincide. When NT > NR, the probability is not high.
However, simulation (Fig. 5.2) shows that the minimum column norm is still an effective radius setter with moderate SNRs, judging from the low number of candidates found.
2 4 6 8
2 4 6 8
0.7770
N
RN
T0.9327 0.5650 0.3610 0.4537
0.8339 0.1804 0.0553
0.9490
0.9859 0.9170 0.8550 0.9010
0.9668 0.7970 0.6650
Table II. The probability of minimum column norm equal to minimum decision distance.
Chapter 6
Conclusion
SD algorithm can significantly lower the computational cost of ML detectors by reducing the number of possible candidates before executing the final step of exhaustive search. In this paper, two special features are introduced to enhance the capability of SD. First, a radius-setting method is used to keep the number of candidate lattice points consistently low. Second, a pseudo-antenna augmentation scheme is employed to cope with the situation where the number of transmit antennas is large than that of receive antennas, which happens often in real-world applications. In short, the modified SD algorithm constitutes an attractive option for practical MIMO receiver design.
Future work
Refer to [1], herein we will show a closed form of expected complexity of SD algorithm. From (1.1), 22 2 22 2
σ y−Hs =σ n is a
χ
2random variablewith 2
n degrees of freedom where n = 2NR due to complex Gaussian noise
vector. From (3.2), we may choose the radius d in such a way that with a high probability we find the transmitted vector inside the hypersphere D as
2 1 found, we can increase the probability
1 − ε
, adjust the radius, and search again. Apply to the radius setting method in (4.4), if the radius in (A.1) is large than that in (4.4), we may enlarge the radius used in SD algorithm.The complexity of SD algorithm is proportional to the number of nodes visited on the tree in searching tree as Fig. 3.1 and, consequently, to the number of points visited in the spheres of radius d and dimensions k = 1, 2, …, m. Hence the expected complexity is proportional to the number of points in such spheres that the algorithm visits on average. Thus the expected complexity of SD algorithm is given by
2
The coefficient f (k) = 2k + 17 is the number of elementary operations
(additions, subtractions, and multiplications) that the Fincke-Pohst algorithm performs per each visited point in dimension k.
Assume st is the transmitted vector, sa is an arbitrary lattice points, the probability that the k-dimensional lattice point
s
ka lies inside the hypersphere D aroundy = Hs
t+ n
with radius d can be expressed as the incompleteIn communication applications, the expected number of points in k-dimensional hypersphere depends on the modulation we use. Therefore the expected complexity C(m,d2,ε ) of SD algorithm to find the optimum solution is 1. for a 2-PAM constellation is
⎝ ⎠ is the number of k-dimensional lattice points with
2 a t
q = s −s , and
d
i is the radius used for i-th search. For QPSK modulation, it can be treated as two dimensional 2-PAM constellation and modify n = 2NR, m = 2NT.where gkl(q) is the coefficient of xq in the polynomial
∑
is the number of k-dimensional lattice points with2 a t
q = s −s . And 16-QAM modulation can also be treated as two dimensional 4-PAM constellation from.
3. for a 8-PAM constellation is
And 64-QAM modulation can also be treated as two dimensional 8-PAM constellation from. Similar expressions can be obtained for 16-PAM, etc., constellations.
When the Gram-Schmidt process is used to compute the QR factorization
=
H QR
, roundoff error can build up as the vectors (Q)i are calculated one by one on a computer. For large i, j , and i≠ j, the scalar products( ) ( )
Q iH Q jmay not be sufficiently close to zero. Interestingly, a rearrangement of the calculation, known as modified Gram-Schmidt (MGS), yields a much sounder computational procedure[13]. If orthonormality is critical, then MGS should be used to compute orthonormal bases only when the vectors to be orthogonalized are fairly independent, even though the computational complexity of MGS requires about twice as much arithmetic.
Figure 6.1 shows the flowchart of SD algorithm, m is the number of transmit antennas (m = NT). In this figure, we can make a roughly estimation of the computation complexity of SD algorithm. The complexity of a tree search in SD algorithm is 2(NT+3) flops, and the complexity of MGS is about 2NRNT2 flops. When NT = NR = 4, 64-QAM modulation, total number of source nodes of search trees is 643+642+64 = 266304. If we use the radius setting method in this paper and assume
0
Eb
N = 16dB, the expected number of source nodes of search trees is about 70, and the probability of the number of source nodes of search trees that less than 200 is about 95%. Therefore we can use 200 as a terminate condition of the number of source nodes of search trees.
Figure 6.1 The flowchart of decoding algorithm estimation of the complexity of SD algorithm for 802.11n standard in high data rate(40MHz) mode is as follows
108(2×4×42 + 2×(4 + 3)×200)/(3.6*10-6) = 8.784×1010 flops
Appendix
Proof of probability density function(pdf) of min ( )i
i H , where H is a m-by-n complex Gaussian matrix：
The i-th column norm of matrix H can be expressed as：
2
For convenience, let
r.v.Yi = ( ) , 1, H i i= … ,M ,
The cumulative distribution function (CDF) of random variable Z is derived as follow
( )
By differentiating the CDF of random variable Z we obtain the pdf
( )
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