A Theoretical Answer to "Does the IRC-SINR of an Interference Rejection Combiner always Increase with an Increase in Number of Receive Antennas?"
aa r X i v : . [ c s . I T ] F e b A Theoretical Answer to “Does the IRC-SINRof an Interference Rejection Combiner alwaysIncrease with an Increase in Number ofReceive Antennas?”
Karthik Muralidhar
Abstract
Interference rejection combiners (IRCs) are very popular in 4G/5G systems. In particular, they areoften times used in co-ordinated multi-point (CoMP) networks where the antennas of a neighboringcell’s base station (BS) are used in an IRC receiver, in conjunction with the antennas of the BS of acell-edge UE’s own cell, to improve the IRC-SINR of a cell-edge user. But does the IRC-SINR alwaysincrease with an increase in the number of antennas? In this paper, we attempt to answer the questiontheoretically. We give a theoretical derivation that quantifies the improvement in the IRC-SINR whenthe number of antennas increases by unity. We show that this improvement in IRC-SINR is alwaysgreater than or equal to zero. Thus we prove that increasing the number of antennas even by unity willalways improve the IRC-SINR. Selecting the extra antennas of the neighbouring cell can be viewed asa special case of antenna selection described in [1]. We also present the IRC-SINR improvement inan uplink CoMP scenario by simulations and verify that it indeed matches with the theoretical gainsderived in this paper.
Index Terms - Antenna selection, CoMP and IRC.
I. I
NTRODUCTION
IRCs are gaining popularity in 4G, 5G systems [2], [3] and [4]. They are used for cell-edgeUEs in uplink co-ordinated multi-point (CoMP) framework, where the received signal at theantennas of the BS of two or more neighbouring cells is processed at a central processing unit(CPU). In [1], the authors dealt with IRCs in the context of antenna selection. In this paper, weshow a theoretical derivation that as the number of antennas increase by unity, the IRC-SINRincrease is always greater than or equal to zero. We show simulation results for IRC receivers
February 9, 2021 DRAFT in the context of uplink CoMP and show that the increase in IRC-SINR due to adding antennasof neighbouring cells to the IRC receiver completely matches the theoretical derivation done inthis paper. While there could be many ways of doing the theoretical derivation, we start withand analyze in detail Equation (8) of [1], to derive the theoretical derivation.
Notation : We follow Matlab notation to access parts of a vector or a matrix. Conjugate of x is denoted by x ∗ . Real part of x is denoted by R { x } . Identity matrix of appropriate dimensionis denoted by I .II. IRC-SINR GAIN DUE TO INCREASE IN NUMBER OF ANTENNAS
In this section, let us assume that h i , i = 1 , . . . , Z are all channel vectors of Z single-antennaUEs to ( N R + 1) receive antennas. The channel vectors are of dimension ( N R + 1) × . Wewill derive the IRC-SINR gain of the first UE as the number of antennas increase from N R to N R + 1 . Let h = ¯h ˜ h , P = ¯P ρ = (cid:20) h h · · · h Z (cid:21) (1)where ¯h , ¯P are the channel vector of the first UE and channel matrix of the next ( Z − UEsto the first N R antennas. The channel of the first UE to the ( N R + 1) st antenna is ˜ h . Likewisethe channel vector of the next ( Z − UEs to the ( N R + 1) st antenna is ρ = h ˜ h , . . . , ˜ h Z − i .Let σ be the variance of AWGN per antenna. Let us define A = (cid:16) σ I + ¯P H ¯P (cid:17) − , ¯h H ¯PA ρ H = y, ρ A ρ H = t · (2)From Equation (8) in [1], the IRC-SINRs of an IRC receiver using N R and N R +1 antennas aregiven by IRC-SINR ( N R ) = ¯h H ¯h σ − σ ¯h H ¯P (cid:16) σ I + ¯P H ¯P (cid:17) − ¯P H ¯h , IRC-SINR ( N R +1) = h H h σ − σ h H P (cid:16) σ I + P H P (cid:17) − P H h · (3)We also define A = (cid:16) σ I + P H P (cid:17) − · (4)Using Woodbury’s identity [5] and (1), we can write the recursive relation for A as A = A − A ρ H ρ A ρ A ρ H · (5) February 9, 2021 DRAFT
Define ξ ( N R + 1 , N R ) = IRC-SINR gain = IRC-SINR ( N R +1) − IRC-SINR ( N R ) · (6)From (3), we have that ξ ( N R + 1 , N R ) = (cid:12)(cid:12)(cid:12) ˜ h (cid:12)(cid:12)(cid:12) σ − Ω σ (7)where Ω = (cid:26) h H P (cid:16) σ I + P H P (cid:17) − P H h (cid:27) − (cid:26) ¯h H ¯P (cid:16) σ I + ¯P H ¯P (cid:17) − ¯P H ¯h (cid:27) · (8)From (1), we have the relation h H P = ¯ h H ¯ P + ˜ h ∗ ρ . Using this relation and the definition of A , the first term in (8) can be written as Ω = h H P (cid:16) σ I + P H P (cid:17) − P H h = (cid:16) ¯ h H ¯ P + ˜ h ∗ ρ (cid:17) A (cid:16) ¯P H ¯h + ˜ h ρ H (cid:17) · (9)Using (5) we simplify Ω further as Ω = ¯h H ¯PA ¯P H ¯h − | y | t + 2 R n ˜ h ∗ ρ A ¯P H ¯h o + (cid:12)(cid:12)(cid:12) ˜ h (cid:12)(cid:12)(cid:12) ρ A ρ H · (10)Substituting the above expression for Ω in (8), we arrive at Ω = 2 R n ˜ h ∗ ρ A ¯P H ¯h o + (cid:12)(cid:12)(cid:12) ˜ h (cid:12)(cid:12)(cid:12) ρ A ρ H − | y | t · (11)We now substitute (5) in the above equation to simplify Ω further as Ω = 2 R n ˜ h ∗ y ∗ o + (cid:12)(cid:12)(cid:12) ˜ h (cid:12)(cid:12)(cid:12) t − | y | t − (cid:12)(cid:12)(cid:12) ˜ h (cid:12)(cid:12)(cid:12) t t − R n ˜ h ∗ y ∗ t o t · (12)Using the above relation in (7), we get ξ ( N R + 1 , N R ) = 1 σ (cid:12)(cid:12)(cid:12) ˜ h (cid:12)(cid:12)(cid:12) − R n ˜ h ∗ y ∗ o − (cid:12)(cid:12)(cid:12) ˜ h (cid:12)(cid:12)(cid:12) t + | y | t + (cid:12)(cid:12)(cid:12) ˜ h (cid:12)(cid:12)(cid:12) t t + 2 R n ˜ h ∗ y ∗ t o t (13)which gets simplified to ξ ( N R + 1 , N R ) = | y | + (cid:12)(cid:12)(cid:12) ˜ h (cid:12)(cid:12)(cid:12) − R n ˜ h ∗ y ∗ o σ (1 + t ) = (cid:12)(cid:12)(cid:12) y − ˜ h ∗ (cid:12)(cid:12)(cid:12) σ (1 + t ) (14)which is always greater than or equal to zero. Hence proved. The IRC-SINR gain as number ofantennas increase by a is ξ ( N R + a, N R ) = N R + a − X i = N R ξ ( i + 1 , i ) · (15) February 9, 2021 DRAFT
III. S
IMULATION RESULTS
We consider four cells. Each cell has two UEs each with a single antenna. The BS in eachcell has four antennas. The single-cell IRC-SINR of an UE is obtained by applying the IRCalgorithm to the signals received at the four antennas of the BS in the cell to which the UEbelongs. The multi-cell IRC-SINR of an UE is obtained by applying the IRC algorithm to all 16antennas of the combined BSs of all four cells (this happens in the CPU of cloud radio accessnetwork). Let h S be the × channel vector from an UE’s antenna to all antennas of the BSin the cell to which it belongs. Let h I be the × channel vector from an UE’s antenna toall antennas of any BS in the neighbouring cell. Signal to interference ratio (SIR) is defined as SIR = h HS h S h HI h I . The various channel vectors are all generated such that h HS h S = 1 and h HI h I issuch that the SIR is satisfied.We define the spectral mean of a set of SNRs as follows. Suppose there are A SNRs,
SNR , . . . , SNR A , the spectral mean (SM) SNR of these SNRs is given by SNR SM = A q (1 + SNR ) . . . (1 + SNR A ) − · (16)The SNR SM of a set of A SNRs is the SNR which will give the same spectral efficiency (as thesum of spectral efficiencies of corresponding to the A SNRs) if each of the A SNRs is replacedby
SNR SM . From Fig. 1, we can see the gain of IRC-SINR in the case of multi-cell scenariocompared to the single-cell scenario. The gain is significant at low SIRs where the interferenceis high. The “multi-cell(theoretical)” curve in Fig. 1 is obtained by starting with the “single-cell(simulations)” curve in Fig. 1 and applying (15) with the values N R = 4 , a = 12 . We see that“multi-cell(theoretical)” and “multi-cell(simulations)” curves overlap, thereby validating (14).Now we give a numerical example to verify (14). Let N R = 4 , σ = 0 . , h = . . i − . − . i − . . i . . i . − . i , P = . . i − . . i − . . i . − . i . − . i − . − . i − . . i . − . i . − . i . − . i − . − . i − . . i − . − . i − . . i . . i · (17) February 9, 2021 DRAFT
SIR (dB) S p ec t r a l m ea n o f I RC - S I NR ( d B ) Singfle-cell and Multi-cell IRC-SINRsIRC multi-cell (simulations)IRC single-cell (simulations)IRC multi-cell (theoretical)
Fig. 1. Comparisons between single-cell and multi-cell IRC-SINRs. σ = 0 . . Results correspond to 25 iterations. The values of IRC-SINR ( N R ) , IRC-SINR ( N R +1) in (3) are 5.8966 and 5.3994. respectively. Thegain is IRC-SINR ( N R +1) − IRC-SINR ( N R ) = 0 . . The theoretical gain as per (14) is indeed ξ ( N R + 1 , N R ) = 0 . . IV. C ONCLUSIONS
We derived and verified a theoretical analysis that gives the increase in IRC-SINR when thenumber of antennas increases by unity. We showed that this increase in IRC-SINR is alwaysgreater than or equal to zero. Simulation in the context of uplink CoMP was carried out andverified with the theoretical analysis derived in this paper.R
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February 9, 2021 DRAFT [5] https://en.wikipedia.org/wiki/Woodbury matrix identity.[5] https://en.wikipedia.org/wiki/Woodbury matrix identity.