In this paper, given a degree of non-orthogonality of the pilot sequences, we first derive the optimal pilot length that satisfies an upper bound on the channel estimation error, and we demonstrate the validity of using a non-orthogonal pilot sequence of set in the pilot assignment method to maximize the entire network capacity. To accurately communicate ideas, we define the following terms: a set of pilot sequences means a collection of pilot sequences, which are complex vectors of the same length. If all vectors in a set of pilot sequences are orthogonal, the set of pilot sequences is called an orthogonal pilot sequence set.
The size of the pilot sequence set means the number of pilot sequences determined by the sequence type, such as orthogonal sequence, Zadoff-Chu (ZC) sequence, or PN sequence. Thus, a pilot reuse scheme is proposed in [5], which shares the pilot sequence set with other cells. However, this approach suffers from a limitation on the number of pilot reuse patterns with a rectangular set of pilot sequences that can be generated up to 𝑁𝑝𝑖𝑙𝑜𝑡.
Contributions
The studies in [7][8] have an excellent view of a large-scale optimization rule for reusing a limited number of pilot sequences in a cellular network. The authors proposed an optimization method for a hierarchical pilot reuse strategy to reduce pilot contamination by sharing the pilot sequence among users across cells. The analysis of pilot reuse patterns provides the optimal pilot allocation strategy for the entire cells given the number of users 𝐾, the normalized coherence time 𝑁𝑐𝑜ℎ and the length of pilot sequences 𝑁𝑝𝑖𝑙𝑜𝑡.
As a result, they formulated the optimal pilot length and optimal pilot assignment status [7][8] for the given parameters. The other contributions to improving network capacity have led to studies on generating pilot sequences that themselves alleviate pilot pollution. One of these studies is to design complex non-orthogonal pilot sequences by solving the Grassmannian manifold line packing problem [1].
This approach quantifies the pilot contamination as an inner product between two complex number series and provides insight into the characteristics of non-orthogonal pilot series. According to the performance comparisons, we show that a non-orthogonal pilot set can support network users with higher net amounts than an orthogonal pilot set when the network has a short normalized coherence time, and the net amounts for a non-orthogonal pilot set orthogonal pilot set are higher than that for orthogonal pilot set, depending on the increase in the number of users. In Section II, we explain the wireless communication model based on a time-division duplex (TDD) mode system with the uplink pilot transmission.
In Section IV, we extend the pilot allocation strategy in [7][8] to the non-orthogonal cases.
System Model
In addition, 𝜌0 is the signal strength, 𝜏 is the length of the nonorthogonal pilot sequence, ϕ 𝑖𝑘 ∈ ℂ1×𝜏 is the nonorthogonal pilot sequence vector for the 𝑘-th user in the 𝑖-th cell, and 𝑊𝑙𝑡 ∈ 𝑀ℂ×𝜏 is the additive white Gaussian noise matrix for the 𝑡- of this user in the 𝑙th cell. According to [10], if the network uses an orthogonal pilot array, the signal-to-interference ratio (SIR) at the 𝑡-th user in the 𝑙-th cell during downlink when 𝑀 becomes infinite is given by .
Analysis Tools
- Pilot Sequence Set
- Pilot Assignment Vector
- MSE Channel Estimation
- Net-Sum Rate
It is figured out that non-orthogonal pilot sequences can be packed more than orthogonal pilot sequences as much as the empty space in the sphere. This advantage can be advantageously applied in pilot reuse schemes rather than orthogonal pilot series set. Let 𝑁𝑜𝑝(𝒑) represent the length of orthogonal pilot sequences with respect to the valid pilot assignment vector 𝒑 axis.
Note that 𝑁𝑜𝑝(𝒑) can also be interpreted as the size of the orthogonal pilot series set. In a similar way, we separately define 𝑁𝑛𝑝(𝒑𝜏, 𝜏) as the size of a non-orthogonal pilot sequence set consisting of non-orthogonal sequences of length 𝜏, where 𝒑𝜏 is pilot assignment vector by non-orthogonal number pilot sequences of non-orthogonal pilot sequences can vary by the number of orthogonal pilot sequences in a given length as we mentioned. In Section IV, we will explain how 𝑁𝑛𝑝(𝒑𝜏, 𝜏) affects the non-orthogonal pilot assignment vector 𝒑𝜏, and compare corresponding performances with the conventional method with orthogonal pilot sequence sets.
If non-orthogonal pilot sequences are used, interference increases with its non-orthogonality. By expressing the channel estimation error as a function of 𝐿, 𝐾, 𝜌0 and 𝑐𝑜𝑠𝜃, it is possible to analyze how non-orthogonal pilot sequences contribute to the estimation of channels and to choose the appropriate non-orthogonality which makes the allowed channel estimation . For the orthogonal pilot case, i.e. cos𝜃 = 0, the result of (8) is zero and there is no distortion for channel estimation by non-orthogonality.
On the other hand, 𝐶𝑠𝑢𝑚𝑛𝑝(𝒑𝜏) and 𝐶𝑛𝑒𝑡𝑛𝑝(𝒑𝜏) can be expressed by 𝜏 as. However, the feature of non-orthogonal pilot sequence set can be used as the key to achieving the further optimized net sum rate when there are not enough time resources for communication.
Optimization Strategy
The Validity of Non-orthogonal Pilot Sequence Set for The Pilot Assignment Strategy
Equation (17) represents the difference between two pilot contamination values 𝑁𝑑+1 and 𝑁𝑑 for a non-orthogonal pilot sequence set which can be approximated to zero. 𝑂𝐼𝑑 ≫ 1 and 𝑁𝐼𝑑/𝑂𝐼𝑑 ≫ 1 when the network is dense and therefore a non-orthogonal pilot sequence set can be conveniently applied to (6). On the other hand, the value of 𝐶𝑑+1𝑛𝑝 − 𝐶𝑑𝑛𝑝 approaching zero means that there is no significant difference in the net throughput as the pilot recycling factor increases because interference from non-orthogonality is dominant in all pilot contamination, i.e. using non-orthogonal pilot sequences has less benefit in reducing pilot contamination by increasing the spacing of cells that use the same pilot sequences, so an increase in 𝜏 has less impact on net throughput growth.
In this view, it is shown that the set of non-orthogonal pilot sequences are applicable to the pilot assignment strategy, but also the application benefits can be questioned since the net throughput is less sensitive to the pilot reuse factor than cases orthogonal. However, the pilot assignment strategy clearly minimizes pilot contamination, although the net throughput improvement with respect to the pilot reuse factor is small even in the practical scenario where users are in cells.
The Formulation with Zadoff-Chu Sequence set for The Pilot Assignment Strategy
According to (22), an increase in 𝜏 leads to a decrease in the channel estimation error, that is, the longer it reaches 𝜏, the more accurate the channel estimation. However, randomly expanding 𝜏 is incorrect in terms of pilot overhead (the ratio of the time slot spent allocating data to the total time slot). Therefore, it is necessary to find 𝜏 to maximize the net sum rate within the allowable channel estimation error.
On the other hand, the random expansion of 𝜏 is not correct in terms of the pilot cost (the ratio of the time slot used for data allocation to the total time slot). Therefore, the choice of pilot sequence length has a trade-off between channel estimation error and net sum rate. From this point of view, we find the optimal 𝜏 to maximize the net sum rate within the allowable channel estimation error.
Through (12) and (23), the length of pilot sequences in terms of improving net sum rate can be simply determined as follows: the longer 𝜏 is, the worse net sum rate is in terms of the pilot overhead. Since intracellular interference does appear when 𝜏 = 𝜏∗< 𝐾, we can obtain the optimal 𝜏 as the closest and larger prime number than 𝐾 in a similar way to Case 1.
Numerical Results
Performance Comparison for The Number of Users
Net sum rate is supposed to be reduced as the number of users in cells increases to avoid interfering with each other in the limited transmission resources, but non-orthogonal pilot sequences can become a key rule to solve the problem by using numerous pilot sets. To confirm this assumption, the simulation comparing the net sum rates with an orthogonal pilot set and a non-orthogonal pilot set is performed to represent the effect of increasing users to net sum rate. This result indicates that using the orthogonal pilot sequence set will exacerbate pilot pollution by reusing the same pilot signals, while using the non-orthogonal pilot sequence set can mitigate pilot pollution of the pilot reuse scheme by assigning distinct pilot sequences to users.
Moreover, having a short pilot length with a large pilot set, which are the characteristics of a non-orthogonal pilot set, is advantageous to improve the net sum rate when a short coherence time is given by mobility. 5 is performed under the same conditions, except for the shorter normalized coherence time, 𝑁𝑐𝑜ℎ = 120, then 𝑁𝑐𝑜ℎ = 110 to confirm the advantage of a non-orthogonal pilot sequence set. The large-scale trend of net sum rates is similarly demonstrated in Figure 2.
Performance Comparison for Normalized Coherence Time
7 illustrates 𝐶𝑛𝑒𝑡𝑛𝑝(𝒑𝜏)/𝐾 and 𝐶𝑛𝑒𝑡𝑜𝑝(𝒑)/𝐾 under the same conditions as the simulation for Figure 7 except 𝐾 = 19 to compare the net amount as the number of users increases. As a result of performance improvements by a non-orthogonal pilot set and an orthogonal pilot set with respect to different 𝐾, increasing the number of users makes an orthogonal pilot set more disadvantageous than a non-orthogonal pilot set in terms of sharing a pilot set. 7, 𝐶𝑛𝑒𝑡𝑛𝑝(𝒑𝜏)/𝐾 becomes higher than 𝐶𝑛𝑒𝑡𝑜𝑝(𝒑)/𝐾 over the entire period of 𝑁𝑐𝑜ℎ, which specifically represents the advantage of a non-orthogonal pilot series in comparison with Fig.
Furthermore, the non-orthogonal pilot series set gives a lower variation of capacities over change of 𝑁𝑐𝑜ℎ, which implies that more stable capacity can be provided to user even when the velocity changes rapidly.
Performance Comparison for The Size of Pilot Sequence Set
Conclusion
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