Compressive Sensing based Intercell Interference ... - IEEE Xplore

2014 IEEE 15th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

Compressive Sensing based Intercell Interference Channel Estimation for Heterogeneous Network Lingchen Zhu, James H. McClellan Georgia Institute of Technology, 75 Fifth Street NW, Atlanta, GA 30308 Email: {lczhu, jim.mcclellan}@gatech.edu

Abstract—Heterogeneous network (HetNet) uses two-tier network architecture in which an unplanned femtocell layer is randomly deployed with no coordination between the coexisting macrocell layer. Both layers share the same spectrum so that spontaneous intercell interference is unavoidable and needs to be identified and canceled afterwards. In order to address the subsequent interference management, this paper proposes an intercell interference channel estimation scheme for HetNets using compressive sensing (CS). Applying CS to analog orthogonal frequency division multiple access (OFDMA) signals not only achieves lower cost sub-Nyquist sampling but also reduces interference by spreading out the energy of reference symbols of interference link that overlaps data symbols of desired link. Our scheme enhances both desired data symbols and interference channel by canceling the estimations of each other from the received signal in turn iteratively. Simulation results show that our scheme obtains accurate estimation of interference channel and is robust to variations in multipath number and signal-tointerference ratio (SIR). Index Terms—heterogeneous network, OFDMA, channel estimation, compressive sensing, intercell interference cancellation

I. I NTRODUCTION Recent survey[1] indicates that increasing demands on cellular networks are driven by indoor traffic. However, signal attenuation makes high-throughput, seamless coverage inside buildings difficult to achieve using only external macrocells. In order to provide better indoor coverage and offload data traffic from macrocells, Long Term Evolution-Advanced (LTE-A) supports femtocells [2] which are low-power, short-range base stations with about ten-meter coverage radius and broadband backhauling. Such two-tier networks consisting of femtocells underneath macrocells are called HetNets. Femtocells are randomly deployed in an ad-hoc manner and transmit much lower power than the overlaid macrocells. Allowing femtocells to share the same spectrum with macrocells introduces intercell interference, which has a negative influence on the performance of HetNets. For a synchronized OFDMA system, intercell interference occurs when both femtocell and macrocell users are allocated to the same subcarrier at the same time slot, especially under two scenarios: 1) a macrocell user equipment (MUE) happens to roam into the coverage area of a femtocell operating in close access mode but is not its subscriber; 2) a femtocell user equipment (FUE) tends to connect with an open-access macrocell providing stronger downlink (DL) signal strength than the desired femtocell. We generalize these two scenarios as a DL interfer-

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ence cancellation problem comprising two uncoordinated cochannel links: desired link and interference link. Previous literature in interference cancellation for OFDMAbased HetNets assumes perfect knowledge of interference channel at the desired receiver[3, 4]. Channel estimation takes advantages of reference symbols that are known by both the transmitter and receiver[5, 6] and often works well in a coordinated network. However, in uncoordinated HetNets, data symbols of desired link may overlap with reference symbols of interference link, which deteriorates its channel estimation accuracy. This problem has not been solved efficiently to the best of our knowledge. Meanwhile, higher-speed LTE-A systems come along with larger signal bandwidth, making Nyquist sampling more challenging as building high-rate analog-todigital converter (ADC) is costly. Furthermore, multipath channel is dominated by a small number of significant paths due to fast power decay, its estimation can benefit from CS with a much lower sampling rate[7, 8]. Recently, random and incoherent CS samplers convert analog signal to far fewer discrete measurements[9, 10], from which the original signal can be efficiently recovered. This paper proposes a CS-based channel estimation scheme for HetNets when intercell interference exists. CS sampling not only reduces the sampling rate but also spreads out energy of the overlapping subcarriers to alleviate their impacts on receiver performance. In particular, our proposed receiver estimates the interference channel and desired data symbols by canceling the estimations of each other from the received signal in turn iteratively. Iterations start with a minimum meansquare error (MMSE) estimation of the desired data symbols from the received CS measurements as if no interference exists. The estimation is then regenerated into CS measurements and canceled from the received measurements, so that sparse channel can be estimated by applying CS reconstruction on the residual. In future iterations, the regenerated measurements of previously estimated desired data symbols, or interference channel, is removed from the received measurements to obtain a better estimation of interference channel, or desired data symbols, respectively. Simulations show that this iterative method can separate out desired and interference links and obtain faithful estimations in a few iterations. II. S YSTEM M ODEL Without loss of generality, we consider an OFDMA-based HetNet including one femtocell and one macrocell coupled

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Fig. 2: Block diagram of the desired receiver

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Fig. 1: A HetNet with desired and interference transmitters

y(t) · p(t)dt.

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(a) Regular RD framework t = mKTc

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p0 (t) 2 f§1g t = mKTc

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z[mK + k]

pk (t) 2 f§1g

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y = Hdd Tcp FH xd + Xi hdi + n (2) where Hdd is an (N + Ng ) × (N + Ng ) Toeplitz matrix with [hdd [0], hdd [1], . . . , hdd [L − 1], 0, . . . , 0]T as its first column and [hdd [0], 0, . . . , 0] as its first row, hdi = ˜ T , 0, · · · , 0]T ∈ [hdi [0], hdi [1], . . . , hdi [L − 1], 0, . . . , 0]T = [h di N ˜ di ∈ CL is C is the interference channel to be recovered. h (r) the first L elements of hdi that can take nonzero values. Xi is an (N +Ng )×N Toeplitz matrix containing reference symbols

mTc

z[m] = Φ(y) =

…

with a FUE and a MUE, respectively. If either pair has established a desired link, the transmitter in the other pair represents a synchronized interferer operating in the same time-frequency grid and its transmission to the desired receiver becomes an interference link, as depicted in Fig. 1. In the (d) (d) desired transmitter, data symbols sd ∈ CNd are multiplexed (r) with reference symbols sd ∈ CNr , resulting a frequencydomain symbol vector xd ∈ CN . Since control channels of the desired link have been decoded, we denote subcarriers ˜ ˜ ˜ d ∈ CN ( N within xd occupied by users as x ≤ N ). After N -point IFFT transforms xd into the time-domain signal Xd = FH xd , cyclic prefix (CP) of length Ng is added to the √ beginning of Xd yielding Xcp d = Tcp Xd where F = [(1/ N ) exp(−j(2π/N )pq)]p,q=0,1,...,N −1 is the FFT matrix and Tcp ∈ {0, 1}(N +Ng )×N is the CP insertion matrix[11]. Similarly, the CP-inserted time-domain signal of the interference transmitter is Xcp = Tcp FH xi where xi ∈ CN i (d) (d) is multiplexed by data symbols si ∈ CNi and reference (r) symbols si ∈ CNr . The desired symbol energy is σs2 and the interference symbol energy is σi2 , so that SIR = σs2 /σi2 . cp Then digital-to-analog converter (DAC) turns Xcp d and Xi cp cp into analog baseband signals Xd (t) and Xi (t), respectively. The proposed desired receiver whose block diagram is shown in Fig. 2 obtains the signal y(t) = hdd (t) ∗ Xdcp (t) + hdi (t) ∗ Xicp (t) + n(t) (1) where hdd (t) and hdi (t) are the channel response from the desired and interference transmitter to the desired receiver with a maximum order of L ≤ Ng , respectively. n(t) is additive white Gaussian noise (AWGN) with variance σn2 and SNR = σs2 /σn2 . Since interference occurs after the desired link has been established, we assume hdd (t) is already estimated to the desired receiver but hdi (t) isn’t. As y(t) is bandlimited with maximum frequency W , for future reference, its Nyquist-rate (fs = 1/Ts ≥ 2W ) samples can be expressed in a vector form

with [Xicp [−Ng ], Xicp [−Ng + 1], . . . , Xicp [N − 1]]T as its first column and [Xicp [−Ng ], 0, . . . , 0] as its first row. A CS sampler acquires discrete measurements from analog signal in the form of z = Φ(y) where Φ(·) represents a CS sampling framework. For random demodulator (RD)[9], Φ(·) multiplies the input with a chipping sequence p(t) that switches between {±1} randomly and integrates the product, which is then sampled at a sub-Nyquist rate fc = 1/Tc ≤ fs to obtain Z

t = mKTc R mKTc

(m¡1)KTc (²)dt

zK¡1 [m]

pK¡1 (t) 2 f§1g

(b) Parallel segmented RD framework

Fig. 3: Block diagram of RD framework Should a Nyquist-rate ADC provide N samples for each OFDMA symbol, the total sensing time is T = N Ts while a serial RD produces M = T /Tc = N Ts /Tc measurements after CP removal. Hence the sampling rate of the ADC on RD is fc = (M/N )fs where ρ , M/N ≤ 1 is compression ratio. A serial RD is shown in Fig. 3(a) while Fig. 3(b) shows a parallel RD structure[12] which processes input in a sliding window manner. Both are designed such that the discrete equivalence of Φ(y) is a matrix multiplication as z = Φy where Φ ∈ R(M +ρNg )×(N +Ng ) is an (M + ρNg )-block diagonal matrix that each block is a random chipping row vector p[n] ∈ {±1} of length 1/ρ. After that, matrix RΦ cp ∈ M ×(M +ρNg ) {0, 1} removes the measurements of length ρNg corresponding to the CP, yielding an M × 1 vector v = RΦ cp z

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H = RΦ cp Φ(Hdd Tcp F xd + Xi hdi + n)

= Ad xd + Ai hdi + nΦ

(4)


(r)

H Φ where Ad = RΦ and cp ΦHdd Tcp F , Ai = Rcp ΦXi Φ nΦ = Rcp Φn. A K-branch RD, as Fig. 3(b) suggests, further reduces the sampling rate of each branch to fc /K. In this case, Φ ∈ R(M +ρNg )×(N +Ng ) is a (M + ρNg )/K-block matrix where each block is a row-wise stack of K random chipping row vectors {pk [n]}K−1 k=0 for all branches, of which each length is K/ρ.

III. CS- BASED I TERATIVE I NTERFERENCE C HANNEL E STIMATION As we have mentioned in Section I, interference channel estimation is required when the reference symbol of interference link uncoordinatedly overlaps the ongoing transmission of desired link. Rather than seek help from symbol muting schemes such as almost blank subframes (ABS)[13] to avoid interference, we provide an iterative algorithm to estimate the interference channel and then use it as a basis for interference cancellation so that both links can be kept active.

Then the recovery can be accomplished by solving an `1 -norm minimization problem ˆ di = argmin khdi k1 s.t. kvi − Ai hdi k2 ≤ (9) h which is also named as basis pursuit denoising (BPDN)[15]. Random sensing matrices like those with independently and identically Gaussian distributed entries satisfies (8) with small δP only if M ≥ O(P log(N/P )), which is also known to be optimal[16]. However, the time of multiplying by Gaussian random matrices is always the bottleneck. Practical applications usually involve fast matrix-vector multiplications such as FFT or convolution, some previous literatures construct RIP matrices by setting up an appropriately structured matrix (e.g., a FFT matrix or a circulant matrix) and leaving only a random subset of its rows[17]. Though measurements of more polylogarithmic orders are needed, matrix-vector multiplication time can be greatly reduced, as is shown in Table I. TABLE I: Performance of different sensing matrices Matrix Random Gaussian Random FFT Random circulant

A. Desired symbol estimator based on MMSE Eq. (4) indicates that we need to recover hdi in the presence of desired symbols xd and noise nΦ . Since xd is hardly sparse neither known, its removal from v is desired so that the sparsity of hdi is more likely to be exploited via CS reconstruction. For such a scenario linear MMSE estimation is preferred because it provides an optimal estimation of xd that has a minimum mean-square error. However, neither locations nor magnitudes of the nonzero components of hdi are known at this time, we have to omit its covariance matrix and estimate ˆ d as follows, assuming no correlations between the desired x symbols and AWGN, −1 eH A e d Cx˜ A e H + Cn (5) ˆ d = Cx˜ d A x v d d Φ d e d ∈ CN ×N˜ is formed by selecting N ˜ columns where A ˜ d is formed from Ad with the same indices used when x from xd , covariance matrices Cx˜ d = σs2 IN˜ and CnΦ = Φ H σn2 (RΦ cp Φ)(Rcp Φ) . Here In denotes an n×n identity matrix. ˆ d from v results in a residual Removing x e dx e d (˜ (6) ˆ d = Ai hdi + A ˆ d ) + nΦ vi = v − A xd − x from which hdi can be better recovered. B. Reference symbol design for stable CS reconstruction Let the interference channel response hdi [n] have P significant paths satisfying P L for channel sparsity, it can be modeled as P X (di) hdi [n] = αp(di) δ[n − τp(di) ], τP ≤ L − 1 ≤ Ng (7) p=1 (di) αp

(di)

where is a complex gain for pth path and τp is its path delay. Stable reconstruction of P -sparse hdi requires (r) its sensing matrix Ai = RΦ ∈ CM ×N to satisfy cp ΦXi the restricted isometry property (RIP) with an RIP constant √ δP < 2 − 1 as[14] (1 − δP )khdi k22 ≤ kAi hdi k22 ≤ (1 + δP )khdi k22 . (8)

M needed for RIP O(P log(N/P )) O(P log N log3 P ) O(P log2 N log2 P )

Multiplication time O(M N ) O(N log N ) O(N log M )

In our channel estimation problem, Ai can be also regarded e ∈ RM ×N and a as the product of a RD sampling matrix Φ (r) N ×N circulant matrix Rcp Xi ∈ C , meaning that removing compressive measurements of CP after RD sampling in our scheme is equivalent to sampling with RD after removing all Nyquist-rate CP samples. Specifically, each row of Ai can be regarded as a linear combination of several rows (r) from Rcp Xi and the coefficients are random sign flips e both of which can be designed to achieve {±1} within Φ, RIP. Inspired by [17], we let the reference symbols xi be complex numbers with the same magnitude and uniformly distributed random phases on [0, 2π]. In this way, Toeplitz (r) matrix Xi is generated by the vectors following the same (r) random pattern. Multiplying such Rcp Xi with hdi is called random convolution, whose purpose is to evenly spread out the sparse energy of hdi across all of N slots. Suppose hdi is normalized to unit energy, then the expected energy for (r) each element of Rcp Xi hdi is 1/N . RD then takes a random e with sum over a block of N/M elements by multiplying Φ (r) Rcp Xi hdi , yielding that the expected energy of each block is 1/M , and the total energy of all measurements goes to 1. Hence RD assists to retain all of the signal energy on average. Recent work [18] suggests that such sensing matrix Ai has a stable `1 -norm recovery given M ≥ O(P log N log2 P ). C. Iterative algorithm By applying BPDN on the residual vi , we’ve obtained the ˆ di . initial rough estimation of the interference channel as h ˆ d would Should it be perfect the desired symbol estimation x ˆ di is removed from v be interference-free optimal as long as h ˆ di = A ˆ di ) + nΦ . (10) e dx ˜ d + Ai (hdi − h v d = v − Ai h ˆ di and hdi When AWGN exists, estimation error between h exists and thus vd is not interference-free. Nevertheless, the

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D. Update of the covariance matrices A necessary condition for BPDN to provide a good reconˆ di is the fidelity of which, in our case, refers to an struction h upper bound of both residual signal power and noise power in the measurements. Therefore, it needs to be updated in each iteration. By utilizing the invariability of trace under cyclic permutations, is bounded by the traces of some covariance matrices which are determined as follows e d (˜ ˆ d ) + nΦ k2 kvi − Ai hdi k2 = kA xd − x e d (˜ ˆ d )k2 + knΦ k2 ≤ kA xd − x q p e d Cx˜ −ˆx A e H ) + tr(Cn ) = tr(A Φ d d d q e dA e H ) − σ 2 tr(A e dA e H C−H e eH = σs tr(A vd Ad Ad ) s d d q Φ H + σn tr(RΦ cp Φ(Rcp Φ) ) ≤ .

In simulations for simplicity we consider desired and interference links which are both LTE systems with 20MHz transmission bandwidth, i.e., N = 2048 subcarriers shared by all active users within a single OFDMA symbol. In order to estimate the channel of the interference link, reference symbols with the same magnitude and uniformly random phases are inserted in its OFDMA symbol with a frequency domain spacing of 6 subcarrier slots, regardless of their occupancy in the desired link. The sampling frequency at the transmitter DACs is fs = 30.72MHz while both receivers employ parallel RDs with K = 4 branches and each ADC runs at fc /K = 1.92MHz. Therefore, receivers acquire M = 512 measurements which gives an overall sampling ratio of ρ = M/N = 1/4. SPGL1[20], a highly efficient solver for sparse reconstruction, is employed to solve the BPDN problem.

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Fig. 4: (a) Reconstructed interference channel and (b) MSE of interference channel versus number of iterations

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On the other hand, as we can see from Algorithm 1, covariance matrix Cvd also needs to be updated in each iteration. The update relies on the stability of prior BPDN reconstruction, which is a consequence of the RIP[19]. Theorem: Let x ∈ CN be a compressible signal and xS its best S-term `1 -norm approximation that only retains the locations and values of its S-largest elements. Suppose √ the sensing matrix A satisfies the RIP with constant δS < 2 − 1 and the CS measurements y = Ax + n where ≥ knk2 . The ˆ of BPDN satisfies solution x kx − xS k1 √ kˆ x − xk2 ≤ C0 + C1 (11) S√ √ 2)δS 4 1+δ √ √S . where C0 = 2−2(1− , C1 = 1−(1+ 1−(1+ 2)δ 2)δ

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Algorithm 1: CS-based iterative interference channel estimation

IV. S IMULATION RESULTS

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IterCancel_CS (5−tap, ρ = 1/4, SIR=0dB) IterCancel_CS (5−tap, ρ = 1/4, SIR=−10dB) IterCancel_CS (5−tap, ρ = 1/4, SIR=−20dB) LS(SIR=0dB) LS(SIR=−10dB) LS(SIR=−20dB)

IterCancel_CS (5−tap, ρ = 1/4, SIR=0dB) IterCancel_CS (10−tap, ρ = 1/4, SIR=0dB) IterCancel_CS (15−tap, ρ = 1/4, SIR=0dB) LS −1

MSE

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e d Cx˜ A eH Initialization : Cv = A d + CnΦ d H e d C−1 ˆ d = Cx˜ d A x v v ˆ d does not converge do while x e dx e d (˜ ˆ d = Ai hdi + A ˆ d ) + nΦ vi = v − A xd − x ˆ di = argmin khdi k1 s.t. kvi − Ai hdi k2 ≤ h ˆ di = A ˆ di ) + nΦ e dx ˜ d + Ai (hdi − h vd = v − Ai h H e d Cx˜ A eH Cvd = A ˆ di Ai + CnΦ d + Ai Chdi −h d −1 eH ˆ d = Cx˜ d A x d Cvd vd end

In our case hdi is exactly P -sparse so that once we let S = P the first term in (11) becomes 0 and therefore khdi − ˆ di k2 ≤ C1 . Assuming that each element in hdi − h ˆ di is h independent and identically distributed, its covariance matrix ˆ di k2 = tr(C keeps diagonal. Since khdi − h ˆ di ) and let S 2 hdi −h ˆ di , we can approximate C be the support set of h ˆ di as hdi −h   1 (C1 )2 , i ∈ S, j ∈ S Chdi −hˆ di (i, j) = |S|  0, otherwise

Magnitude

ˆ di progressively reduces signal-to-interferenceremoval of h ˆ d estimated from vd plus-noise ratio (SINR) and so that x ˜ d . When subcarriers is expected to be a cleaner version of x of desired data symbols and interference reference symbols ˆ di from v would overlap, people might argue that removing h ˜ d so that its following estimation result in some loss of x would be impaired. However, it is not necessarily the case. Though two kinds of symbols from interlayer cells overlap in the frequency domain, random convolution spreads out the energy of hdi and averages it over all N subcarriers, letting the overlapping energy be insignificant. As a summary, the algorithmic framework of CS-based iterative interference channel estimation is presented in Algorithm 1.

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Fig. 5: MSE for various (a) multipath numbers (b) SIRs Fig. 4(a) demonstrates one example of the reconstructed interference channel with 10 multipaths under SNR = 30dB. Not only does our method detect the path delays but also their

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magnitudes. Its MSE performance versus number of iterations is shown in Fig. 4(b). We can see that for sparse channels, our method converges in at most 4 iterations. Fig. 5(a) depicts the MSE performance of the reconstructed interference channel for different multipath number P ranging from 5 to 15, SIR = 0dB and input SNR between 15dB and 35dB. Fig. 5(b) plots the MSE-SNR curves for different SIRs with P = 5. When P grows large, MSE gets worse because RIP requires M to be increased as well so as to keep the same performance. When SIR decreases, the increased reference symbol energy from the interference link makes the CS reconstruction more accurate, thus better MSE performance is achieved. For comparison’s sake, the least squares (LS) ˆ di = (X(r)H X(r) )−1 X(r)H y channel estimate given by h i i i produces an error floor because it cannot eliminate the overlapping subcarriers from the desired link. Thanks to random convolution, which spreads the energy of reference symbols into all subcarriers, mutual influences between the desired and interference links can be minimized.

[7]

[8]

[9]

[10]

[11]

V. C ONCLUSION Intercell interference happens in OFDMA-based HetNets due to the incoordination between the femtocell and macrocell. In this paper, we adopted CS-based samplers at the frontend of our proposed receiver that directly convert the received signal into low-rate measurements. Such means not only reduces the sampling rate but also suppresses interference energy using random convolution. As an important step for interference cancellation, we also presented an iterative algorithm to estimate the channel of the interference link. Our approach prevails regular reference-assisted channel estimation methods, especially when reference symbols of the interference link happen to overlap with data symbols of the desired link.

[12]

[13]

[14]

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