Improving the Performance of the Zero-Forcing Multiuser MISO Downlink Precoder Through User Grouping
Summary (4 min read)
I. INTRODUCTION
- Multiple-Input Multiple-Output (MIMO) technology holds the key to very high throughput downlink communication in fading wireless channels by exploiting the spatial dimension [1] .
- To keep the low-complexity benefit of the ZF precoder and yet improve the overall sum rate (specially when the channel is ill-conditioned), the authors propose a user grouping based precoder.
- Another special case of the proposed precoder is when there are N u groups with each user being a separate group.
- The following notations have been used in this paper.
USERS
- Hence, for any given user, its effective channel gain is proportional to the Euclidean length of the projection of its channel vector onto the space orthogonal to the space spanned by the channel vectors of remaining users.
- In case of ill-conditioned channels, since the channel vectors of all the users are "nearly" linearly dependent, the effective channel gain of each user would be small, implying low achievable rates.
- By grouping users into groups of size larger than one, beamforming can be done to nullify only inter-group interference.
- Since the projection of a vector onto a subspace of some space G is of lesser Euclidean length than its projection onto the space G, it follows that the effective channel gain for U ij is higher with the proposed user grouping based precoder as compared to that with the ZF precoder.
- For a given user grouping the sum rate is maximized by the waterfilling power allocation across all the users (the effective channel gain of each user is considered).
IV. PROPOSED USER GROUPING BASED PRECODER
- This section is organized into several subsections.
- With the proposed beamforming the original N u -user Gaussian broadcast channel is transformed into N g parallel g-user Gaussian broadcast channels.
- Subsequently in Section IV-B, using the fact that the effective channel is lower triangular the authors use Dirty Paper Coding to cancel interference between the users within a group.
- The authors also show that for a fixed user grouping, the information sum rate is maximized by the waterfilling power allocation.
- The authors also present expressions for the sum rate achieved by the ZF precoder.
A. Beamforming to cancel inter-group interference
- The information symbols are assumed to be i.i.d.
- This interference can be nullified by choosing the precoding matrix D[k] for the k-th group in such a way that its columns are orthogonal to the channel vectors of all the users in the other groups.
- From (11) it is clear that each group of users does not have any interference from the other groups.
- C g×g is an upper triangular matrix with positive diagonal entries (since F[i] is full rank), and Q[i] ∈ C Nt×g is a matrix with orthonormal columns.
- Using (10) along with the fact that the columns of Q[k] are orthonormal, the sum power constraint in ( 8) is given by EQUATION where the authors have used the fact that Q[i] has orthonormal columns and Tr denotes the trace operation for matrices.
B. Dirty Paper Coding to cancel intra-group interference
- Since the BS has perfect CSI and it knows the transmitted information symbol for the first user (i.e., u i1 ), it knows the interference term for the second user, and can therefore perform known interference pre-cancellation using the Dirty Paper Coding scheme [19] , [20] , [21] .
- This has been discussed in detail in [20] as the ZF-DP precoder.
D. The proposed precoder achieves a higher information rate than the ZF precoder
- The following theorem shows that irrespective of the channel realization H and P T , the sum rate achieved by the proposed precoder with any arbitrary user grouping having g ≥ 2 is greater than that achieved by the ZF precoder (i.e., proposed precoder with g = 1).
- Rayleigh faded with each entry distributed as a circular symmetric complex Gaussian random variable having zero mean and unit variance.
- The above analysis shows that, even with random user grouping, the proposed grouping based precoder is more power efficient than the ZF precoder.
E. Motivating the need for "optimal" user grouping
- So far the authors have not bothered much about the choice of user grouping.
- In Fig. 1 , the authors plot the sum rate (vertical axis) versus the ordered user grouping number (horizontal axis).
- Note that the sum rate of the proposed user grouping scheme is significantly higher than that of the ZF precoder even for small g =.
V. PARTITIONING USERS INTO GROUPS
- 13 Therefore for large N u the authors propose an iterative "Joint Power Allocation and User Grouping Algorithm" , which solves (32) 11.
- The sum capacity of the broadcast channel is computed using the sum power iterative waterfilling method proposed in [4] .
- Further, the complexity of performing practical DPC is also implementation dependent (e.g., LDPC, convolutional codes).
- Let P (q) be the user grouping after the q-th iteration of JPAUGA.
- The JPAUGA algorithm is summarized in the table above.
A. Generalized User Grouping Algorithm -GUGA
- Therefore in the following the authors propose a low complexity approximate solution to (35), called "GUGA".
- The authors remind the reader that R[k] is implicitly dependent on the chosen grouping.
- The proposed user grouping algorithm (GUGA) needs to initially compute the rate of all possible subsets of S of size g.
- For a given group, its rate is a function of the corresponding upper triangular matrix representing the effective channel for that group.
B. Complexity of the proposed precoder based on JPAUGA
- The whole precoding operation can be broadly divided into two phases.
- Through numerical simulations the authors have observed that JPAUGA converges very fast, and few iterations (less than five) are required irrespective of (N u , N t ).
- From the expressions for the complexity of the two phases as discussed above, it then follows that the complexity of computing the JPAUGA user grouping and power allocation is completely dominant over the complexity of computing the beamforming matrices for each group and beamforming information to users every channel use.
- With g = 2, due to the lower triangular nature of the effective channel matrix, the first user in each group gets its information symbol interference free, but the second user gets its information symbol along with some interference from the first user's information symbol.
- Near-optimal interference pre-subtraction can be performed at practical complexity as shown in [8] .
Downlink Ergodic Sum Rate (bpcu)
- The achievable sum rate for each precoder is random due to the random channel gains.
- It can be observed from the figure that the probability of the sum rate assuming small values (compared to the mean value, i.e., ergodic rate) is much higher for the ZF precoder than for the proposed user grouping based precoders.
- It can be seen that the performance improves with increasing number of iterations.
- 4 it is clear that the probability of the sum rate being small is larger when random user grouping is used.
- In the plot the realizations have been reordered so that channel realization number 1 is the realization for which the sum rate achieved by the ZF precoder is the least among the sum rates achieved by the ZF precoder for all the ten thousand realizations.
VII. FUTURE RESEARCH DIRECTIONS
- It is clear that the sum rate versus the ordered channel realization number curve is smooth and monotonically increasing for the ZF precoder, whereas that for the proposed precoder is fluctuating.the authors.
- 17 For channel realization indices between 1 and 400 the ZF precoder achieves a sum rate less than 3 bpcu.
- MIMO broadcast channels where users can have more than one receive antenna.
- The advantage of this extension is that the signal received at any antenna of a user is free of interference from information signals communicated to the other antennas of that user.
- Intergroup interference can be cancelled in exactly the same way as discussed in Section IV-A for MISO broadcast channels.
VIII. CONCLUSIONS
- Due to the lower triangular structure of the equivalent g × g broadcast channel for each group, successive DPC can be used to pre-cancel the intra-group interference within each group.
- This method of precoding is shown to achieve a significantly better performance than the ZF precoder, especially when the channel is ill-conditioned.
- The sum rate achieved by the proposed precoder is also shown to be sensitive towards the chosen user grouping, and therefore a novel low-complexity joint power allocation and user grouping algorithm is proposed.
- From the above arguments, EQUATION which proves (46) and subsequently (45).
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Citations
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Additional excerpts
...3) SVD operation [30], [31]: ψ {SVD (A)} ∼ O mn2 + O m2n + O m3....
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15 citations
Cites background from "Improving the Performance of the Ze..."
...For the parallel problem in RF, the traditional approach is from an information-theoretical perspective [22], [23]....
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9 citations
9 citations
Cites background or methods from "Improving the Performance of the Ze..."
...Traditional techniques such as ZF and MMSE typically experience inferior throughput performance for MU-MIMO and mmWave systems, particularly under illconditioned channels [11, 15, 16]....
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...In all four cases, throughput under conventional HB-MMSE and HB-ZF methods appears to be inferior to others, as MMSE and ZF are not designed for mmWave systems and the poorly conditioned channel greatly degrades MMSE/ZF’s performance [11, 15, 16]....
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...MMSE typically experience inferior throughput performance for MU-MIMO and mmWave systems, particularly under illconditioned channels [11, 15, 16]....
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8 citations
Cites background or result from "Improving the Performance of the Ze..."
...For comparisons, we also present the rates of the UG-DP precoder in [20] for sum-rate maximizations, which are inferior to the GZF-DP precoder with ν = Ng − 1 and similar DPC complexity....
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...Recently, Mohammed and Larsson [20] propose a user-group based DPC precoder (UG-DP), which splits the N users into g disjoint groups with each group containing Ng users....
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...To reduce the DPC complexity, Mohammed and Larsson [20] propose a lowcomplexity UG-DP precoder....
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References
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"Improving the Performance of the Ze..." refers methods in this paper
..., ui1 ), it knows the interference term for the second user, and can therefore perform known interference pre-cancellation using the Dirty Paper Coding scheme [19]–[21]....
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3,291 citations
"Improving the Performance of the Ze..." refers background or methods or result in this paper
...Hence, in the special case of MISO broadcast channel (which we consider in this paper), the block diagonalization precoder in [16] reduces to the ZF precoder....
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...This distinction is the same as that between the work in [16] and that in [13]....
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...We also clarify that, the proposed precoder is entirely different from the block diagonalization based precoder proposed in [16], which considers a MIMO broadcast channel, in which each user could have multiple receive antennas....
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...In [16] the authors propose a block diagonalization method for precoding of information to already selected users, whereas in [13] the authors propose a method to find the subset of users to be scheduled so that the information sum rate (using a block diagonalization precoder) is maximized....
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Frequently Asked Questions (16)
Q2. Why is the DPC used to cancel the inter-group interference?
Due to the lower triangular structure of the equivalent g × g broadcast channel for each group, successive DPC can be used to pre-cancel the intra-group interference within each group.
Q3. How many iterations are required to compute the user grouping?
Through numerical simulations the authors have observed that JPAUGA converges very fast, and few iterations (less than five) are required irrespective of (Nu, Nt).
Q4. What is the way to determine the optimal power allocation for a given grouping of users?
(21)Maximization of r(H, PT ,P ,p) over p yieldsr(H, PT ,P) ∆= max p ∣ ∣ Nu∑i=1pi=PT , pi≥0r(H, PT ,P ,p) (22)In (22), the optimal power allocation for a given grouping of users is given by the waterfilling scheme [22], i.e.pkj =[µ− 1 R[k]2(j,j)]+, (23)where k = 1, 2, . . . , Nu/g , j = 1, 2, . . . , g and µ > 0 is such thatNu/g∑k=1g ∑j=1pkj = PT . (24)The authors note that the ZF precoder is a special case of the proposed user grouping scheme with g = 1, i.e., Nu groups with one user per group.
Q5. How does the proposed precoder achieve a fixed sum rate?
The authors will also show that to achieve a given fixed sum rate, the ZF precoder asymptotically (i.e., as PT → ∞) requires about 2.17 dB more power than the proposed precoder (with g = 2 and random user grouping).
Q6. What is the way to extend the user grouping precoder to 16?
17In this section the authors briefly discuss two possible ways in which the proposed user grouping precoder could be extended to16Since the ordering is based on the sum rate achieved by the ZF precoder, it is clear that the sum rate versus the ordered channel realization number curve is smooth and monotonically increasing for the ZF precoder, whereas that for the proposed precoder is fluctuating.
Q7. What is the optimum sum rate for the user grouping?
The authors then order the user groupings in increasing order of the sum rates achieved by them, i.e., the ordered user grouping number 1 achieves the least sum rate and the ordered user grouping number 120 achieves the largest sum rate.
Q8. What is the effective channel gain for a given grouping?
For a given user grouping the sum rate is maximized by the waterfilling power allocation across all the users (the effective channel gain of each user is considered).
Q9. Why does the i-th group have a lower triangular structure?
Due to the lower triangular structure of the effective channel matrix for the i-th group, from (18), the authors observe that the j-th user in the i-th group (i.e., Uij ) has interference only from the symbols of the previous (j − 1) users in the same group (i.e., Ui1 , · · · Ui(j−1) ).
Q10. What is the way to compare the ZF-DP scheme to the proposed user grouping?
The authors observe that the ZF-DP scheme is near sum capacity achieving and has a better sum rate performance than the proposed user grouping precoder with g = 2 (optimal pairing).
Q11. What is the average information sum rate of the proposed precoder?
Also with random user grouping (curve marked with diamonds) the average information sum rate achieved by the proposed precoder is 3 bpcu at PT = 10 dB.
Q12. Why is the sum rate of the proposed precoder unable to be plotted?
The authors are unable to plot the p.d.f. of the sum rate achieved by the proposed precoder with optimal user grouping due to its prohibitive complexity (with g = 2 the number of possible groupings is only 120 when Nu = 6, but which increases to 665280 when Nu = 12).
Q13. What is the way to calculate the information sum rate of the ZF precoder?
As an example, at PT = 10 dB the information sum rate of the ZF precoder is only 0.31 bpcu when compared to 4.75 bpcu achieved by the proposed precoder with optimal user grouping.
Q14. What is the precoder for a given user grouping?
Motivated by the sensitivity of the proposed precoder w.r.t. user grouping the authors define the optimal user grouping as one which maximizes the sum rate.
Q15. What is the channel vector from the BS to the k-th user?
The channel vector from the BS to the k-th user is denoted by hHk ∈ C1×Nt , with its i-th entry h∗k,i representing the channel gain from the i-th transmit antenna to the receive antenna of the k-th user.
Q16. What is the definition of the g orthonormal columns of Q[i]?
From the definitions of P[i] and Q[i] in (5) and (13), it is clear that P[i] is the projection matrix for H⊥i which is also the space spanned by the columns of Q[i] and thereforeP[i]Q[i] = Q[i].