Journal of Communications and Information Networks

Figure 1 Illustration of a multi-user MIMO scenario: a BS is equipped with an array of M antennas and simultaneously serves N randomly distributed single-antenna user terminals in the same time-frequency resource

Single-carrier, narrowband transmission is considered, and thus the channel matrix between BS and users are represented by an M×N matrix $G = H D^{1 / 2}$ , where $H$ is the M×N matrix modeling small-scale fading coefficients between the users and BS, and $D$ is the N×N diagonal matrix representing the large-scale fading (both path loss and shadow fading)coefficients. The(m, n)element of G can be modeled as $g_{m m} = h_{m n} \sqrt{β_{n}}$ p , with h_mn describing the instantaneous propagation channel from the nth user to the mth antenna of BS and β_n being the large-scale fading component. We assume a Rayleigh small-scale fading distribution such that h_mn is zero mean, circularly symmetric, complex Gaussian random variable with variance 1. Besides, β_n is assumed to be constant across the antenna array^[13].

Linear processing is used at a BS to perform uplink data detection. Under imperfect CSI assumption, the UEs send orthogonal pilot sequences of length τ symbols to the BS for channel estimation, which indicates that(T−τ)symbols are left to enable uplink data transmission. The entire uplink transmission process consists of the following three phases.

• First, the UEs send pilot sequences which are spatially orthogonal and τ symbols long. As seen from Fig.2, uplink pilot transmission takes place first and occupies τ symbols.

• Second, based on the received pilot sequences at the BS, channel estimation is performed for linear receiver processing. In this paper, we ignore the processing time needed for this phase.

• Third, all terminals send their uplink data to the BS using the remaining(T−τ)symbols, which is shown in Fig.2.

Figure 2

Figure 2 Illustration of the uplink transmission: uplink pilot signaling takes place first and occupies τ symbols; the subsequent(T−τ)symbols are used for uplink data transmission

A. Uplink Training and Data Transmission

In the uplink training phase, the UEs transmit a sequence of pilot signals which can be grouped into a matrix $ψ = [ψ_{1}, \dots ψ_{i}, \dots, ψ_{N}] \in ℂ^{τ \times N}$ with each element $ψ_{n} = [ψ_{n}] \in ℂ^{τ \times 1}$ being the pilot vector for the nth user and τ representing the number of training symbols. Assuming that each user terminal transmits the pilot symbol with equal transmit power p_t, the signal of ith symbol at the receive antennas $z_{i} \in ℂ^{M \times 1}$ is written as

z_{i} = \sqrt{τ p t} \sum_{n = 1}^{N} Ψ n i g n + w_{i}, (1)

where g_n denotes the nth column of the channel matrix $G$ , $ψ_{n i}$ is the ith element of $ψ_{n}$ , and wi represents the additive white Gaussian noise(AWGN)vector with zero mean and unit element-wise variance written as $w_{i} ~ L N (0, I_{M})$ .

In practical systems, an automatic gain control(AGC)variable gain amplifier is usually used before the ADC to adjust the dynamic range of the received signal $z_{i}$ . We assume the gain of AGC is set appropriately, and by adopting the additive quantization noise model, the output of the quantizer can be formulated as

{\hat{z}}_{i} = α z_{i} + {\hat{w}}_{i} = a \sqrt{τ p t} \sum_{n = 1}^{N} ψ_{n i} g n + a w_{i} + {\hat{w}}_{i}, (2)

where α=1−ρ with ρ being the inverse of the signal-toquantization-noise ratio, and ${\hat{w}}_{i}$ is the additive Gaussian quan tization noise vector that is uncorrelated with $z_{i}$ . The covariance matrix of ${\hat{w}}_{i}$ , which depends on the number of quantization bits b, is given approximately by

R_{{\hat{w}}_{i}} = a (1 - a) d i a g (R_{z_{i}}), (3)

where $R_{z_{i}} = E {z_{i} z_{i}^{*}}$ . Note that the values of ρ can be ap-proximated by $ρ = \sqrt{3 π} \cdot 2^{- 2 b - 1}$ when b> 5, and for b 6 5, the values are associated with the number of quantization bits, which can be found in Ref. [22]. Following Eq. (3), we further express $R_{{\hat{w}}_{i}}$ as

R_{{\hat{w}}_{i}} = α (1 - α) (1 + τ p t \sum_{n = 1}^{N} {| ψ_{n i} |}^{2} β_{n}) I_{M} . (4)

It can be seen from Eq. (4)that the covariance of ${\hat{w}}_{i}$ is dependant on the pilot symbols ψ_ni. In this paper we assume the pilot sequence ψ_n is the nth row of a discrete Fourier transform (DFT)matrix, which means that|ψ_ni|²=1/τ. Consequently, we have $R_{{\hat{w}}_{i}} = σ_{t}^{2} I_{M}$ , with

σ_{t}^{2} = α (1 - α) (1 + p_{t} \sum_{n = 1}^{N} β_{n}) . (5)

With ${\hat{z}}_{i}$ representing the quantizer outputs of the ith symbol, when a block of pilot symbols Ψ is transmitted, the quantized signal of τ symbols can be grouped into an M×τ matrix $\hat{Z} = [{\hat{z}}_{1}, \dots {\hat{z}}_{τ}]$ . In addition, we let $\hat{W} = [{\hat{w}}_{1}, \dots {\hat{w}}_{τ}] \in ℂ^{M \times τ}$ and $W = [w_{1}, \dots, w_{τ}] \in ℂ^{M \times τ}$ such that the signal model after the ADC in matrix form becomes

\hat{Z} = a \sqrt{p_{t}} G Ψ + α W + \hat{W} (6)

The MMSE estimate of the channel G, from the received and quantized signal $\hat{Z}$ , between all UEs and all BS antennas is^[32]

\hat{G} = \frac{1}{α \sqrt{τ p_{t}}} \hat{Z} Ψ^{H} \hat{D}, (7)

where $\hat{D} = {[I_{N} + (1 / τ p_{t} + σ_{t}^{2} / (α^{2} τ p_{t})) D^{- 1}]}^{- 1}$ . $L e t Ξ = G - \hat{G} = [e_{m n}]$ denote the channel estimation error, and after some simple calculation, the variance of elements of $Ξ$ is

σ_{e_{m n}}^{2} = E {{| e_{m n} - E {e_{m n}} |}^{2}} = \frac{(α^{2} + σ_{t}^{2}) β_{n}}{τ α^{2} p_{t} β_{n} + α^{2} + σ_{t}^{2}} . (8)

One important observation from Eq. (8)is that the variance of the channel estimation error is independent of the BS antenna index m. This means that for one specific UE, the variance of channel estimation error between this user and all the antennas at BS remains the same. Besides, it can be found that $σ_{e_{m n}}^{2}$ is monotonically decreasing with α and touches the minimum at α=1, which indicates that the accuracy of channel estimate can be enhanced by increasing the precision of the ADCs.

After the training phase comes the data transmission process, in which all N terminals send their data to the BS using the remaining(T−τ)symbols. Uplink data symbols transmitted by all users are collected in vector $\sqrt{p d} x \in ℂ^{N \times 1}$ , where $E {x x^{H}} = I_{N}$ ? ? and p_d denotes the average transmitted power of each terminal. The M×1 received signal at the BS is represented as

y = \sqrt{p d} G x + n, (9)

where $n \in ℂ^{M \times 1}$ is the AWGN vector with zero mean and unit element-wise variance^{[22,33,34,35]}. Following Eq. (2), the received signals are quantized by ADCs and the quantization output is represented as

\hat{y} = α \sqrt{p d} G x + α n + \hat{n}, (10)

where $\hat{n}$ denotes the additive Gaussian quantization noise vector.

As we have obtained the channel estimates, we are able to combine the received vectors to express separate transmitted symbols of each UE. Here we consider linear detection techniques to minimizing the computational complexity. Linear processing matrices for simple MRC are formed with the channel estimates as $A_{M R C} = \hat{G}$ . The massive MIMO receiver applies the processing matrix to detect the transmitted signals as

\hat{x} = {\hat{G}}^{H} \hat{y} = α \sqrt{p d} {\hat{G}}^{H} (\hat{G} + E) x + α {\hat{G}}^{H} n + {\hat{G}}^{H} \tilde{n} . (11)

According to Eq. (11), the estimated symbols sent by the nth user are calculated as

{\hat{x}}_{n} = α \sqrt{p d} {\hat{g}}_{n}^{H} {\hat{g}}_{n} x_{n} + α \sqrt{p d} \sum_{i = 1, i \neq n}^{N} {\hat{g}}_{n}^{H} {\hat{g}}_{i} x_{n} +

α \sqrt{p d} \sum_{i = 1}^{N} {\hat{g}}_{n}^{H} e_{i} x_{i} + α {\hat{g}}_{n}^{H} n + {\hat{g}}_{n}^{H} \hat{n}, (12)

where ${\hat{g}}_{i}$ and ei are the ith columns of $\hat{G}$ and E, respectively. The last four terms in Eq. (12)account for different interference and noise sources, namely: intra-cell interference, channel estimation error, receiver noise, and quantizer noise, respectively.

B. Uplink SE with Imperfect CSI

Under the classical assumption of worst-case uncorrelated Gaussian noise^[14], along with the fact that the variance of elements of receiver noise is 1 and the variance of elements of estimation error is $α_{e_{m n}}^{2}$ , the uplink SE(in bits · s⁻¹ · Hz⁻¹)of the nth UE is obtained by

R_{n} = \frac{T - τ}{T} E {1 b (1 + \frac{α^{2} p d {‖ {\hat{g}}_{n} ‖}^{4}}{})} (13)

where

J_{\hat{G}} = α^{2} p d \sum_{i \neq n} {| {\hat{g}}_{n}^{H} {\hat{g}}_{i} |}^{2} + α^{2} p d {‖ \hat{g} n ‖}^{2} \sum_{i} σ_{e_{m n}}^{2} + α^{2} {‖ \hat{g} n ‖}^{2} +

α (1 - α) {\hat{g}}_{n}^{H} d i a g (p d G G^{H} + I) \hat{g} n .

$T h e o r e m 1$ If MRC detection is applied with MMSE channel estimate in the quantized MIMO systems, we can approximate the uplink SE of the nth user as

R_{n} = \frac{T - τ}{T} 1 b (1 + \frac{τ α^{3} p d p t (M + 1) β_{n}^{2}}{τ Λ + (α^{2} + σ_{t}^{2}) (p d \sum_{i = 1}^{N} β_{i} + 1)}),

where $Λ = α^{2} p d p t β_{n} \sum_{i}^{N} β_{i} + (1 - 2 α) α^{2} p d p t β_{n}^{2} + α^{2} p t β n .$ .

$P r o o f$ See appendix.

The SE expression of Eq. (14) in Theorem 1 is different from Ref. [22]due to the imperfect CSI which causes unavoidable interference between the users. In particular, Theorem 1 reveals the effect of the number of antennas M, number of quantization bits b, pilot length τ and transmit power p_t, p_d on the SE performance. To have a full and profound understanding of Theorem 1, we give some asymptotic results as follows.

$C o r o l l a r y 1$ For the special case of b→∞with fixed user transmitter power and number of BS antennas, the uplink SE of the nth user $R_{n}$ in Eq. (14)can be simplified as

\lim_{b \to \infty} R_{n} = \frac{T - τ}{T} 1 b [1 +

\frac{p r p d (M + 1) β_{n}^{2} τ}{(p d p t β_{n} \sum_{i = 1, i \neq n}^{N} β_{i} + p t β_{n}) τ + p d \sum_{i = 1}^{N} β_{i} + 1}] . (15)

$P r o o f$ As b grows without bound, we have α→1. The result is directly obtained by setting α=1 in Eq. (14).

Corollary 1 shows a precise agreement with the result derived in Ref. [32], where the achievable SE for the infinite precision case is given. The result also validates the accuracy of our derived approximation in Eq. (14), which embraces the results in Ref. [32]as a special case. We will discuss how different pilot lengths influence the system performance and the optimal resource allocation scheme in the following sections. $C o r o l l a r y 2$ With fixed ADC quantization bits b and the number of BS antennas M, when the training signal power equals the data transmit power and tends to be unlimited, i. e. , p_d=p_t→∞, the approximation in Eq. (14)reduces to

\lim_{p d = p t \to \infty} R_{n}

\to 1 b [1 + \frac{α (M + 1) β_{n}}{\sum_{i = 1, i \neq n}^{N} + 2 (1 - α) β_{n} + \frac{(1 - α)}{β_{n} α τ} {(\sum_{i = 1}^{N} β_{i})}^{2}}] . (16)

$P r o o f$ To better observe the relation between $R_{n}$ and p_d, p_t, we rewrite Eq. (14)as

R_{n} = \frac{T - π}{T} 1 b (1 + S I N R), (17)

where

SINR

= \frac{α^{2} (M + 1) β_{n}^{2} τ p_{d} p_{t}}{Δ p d p t + ((1 - α) \sum_{n = 1}^{N} β_{n} + α β_{n} τ) p t + \sum_{n = 1}^{N} β_{n} p d + 1} (18)

with

Δ = β_{n} α τ \sum_{i = 1}^{N} β_{i} + (1 - 2 α) τ α β_{n}^{2} + (1 - α) {(\sum_{i = 1}^{N} β_{i})}^{2} . (19)

We further calculate Eq. (18)by dividing p_t p_d from both the numerator and denominator to get

SINR

= \frac{α^{2} (M + 1) β_{n}^{2} τ}{Δ + \frac{1}{p d} ((1 - α) \sum_{n = 1}^{N} β_{n} + α β_{n} τ) + \frac{1}{p t} \sum_{n = 1}^{N} β_{n} + \frac{1}{p d p t}} . (20)

Obviously, when p_d=p_t→∞, we have

S I N R = \frac{α (M + 1) β_{n}^{2} τ}{Δ} . (21)

Substituting Eq. (21)into Eq. (17), we finally obtain the result in Eq. (16).

Corollary 2 indicates that when the transmit power of pilot and data signal tends to infinity, the approximate SE approaches a fixed value which is dependent of the pilot length τ and the quantization bits b. This observation reveals that increasing the transmit power in quantized massive MIMO systems will not eliminate the effects produced by the MMSE channel estimation.

Based on the expression in Eq. (14), we can investigate the optimization of pilot length τ and the joint optimization of quantization bits b and the number of antennas M to maximize the uplink SE, which will be carried out in the following sections.

Ⅲ. PILOT LENGTH

Under training-based schemes, the receiver usually requires the transmitter to send known training symbols to learn the channel. How training affects the capacity of a fading channel in MIMO systems has been studied in a number of works^[36,37,38], since less training leads to improperly learned channel and too much training results in less time for data transmission. In particularly, Ref. [38]investigated the problem of given a multi-antenna wireless link with fixed transmit antennas, receive antennas, coherence interval and signal-tonoise ratio(SNR), how much of the coherence interval should be spent training. However, it did not take the quantization effect into consideration.

In this section, we consider the quantized MIMO systems and aim at finding the pilot length τ to maximize the uplink achievable SE. Although Ref. [39]showed that allocating different powers for the uplink training and data transmission phases can improve the system performance, in this paper we follow the methodology in Ref. [13]that the training and data powers are set to be equal, i. e. , p_t = p_d = p. Furthermore, we assume that all users suffer the same large-scale fading, i. e. , β_n=β, ∀n. Under these assumptions, we define the optimization problem as follows

\max_{τ} J (τ) = \sum_{n = 1}^{N} R_{n}, (22)

s . t . N \leq τ \leq T,

where the optimization variable τ is an integer variable, and $S (τ)$ is the sum SE with $R_{n}$ given in Eq. (14). The uplink SE of the nth user $R_{n}$ can be rewritten as

R_{1} = \frac{T - τ}{T} 1b (1 + \frac{a τ}{b τ + c}), (23)

where

σ_{t}^{2} = a (1 - a) (1 + N p β),

a = (M + 1) α^{3} p^{2} β^{2},

b = (N + 1) α^{2} p^{2} β^{2} - 2 α^{3} p^{2} β^{2} + α^{2} p β,

c = (α^{2} + σ_{t}^{2}) (N p β + 1) .

Thus the sum SE $S (τ)$ in Eq. (22)becomes

S_{1} (τ) = (1 - \frac{τ}{T}) N 1 b (1 + \frac{a τ}{b τ + c}), (24)

and the pilot sequence length τ satisfies

τ^{*} = \arg \max_{N \leq τ \leq T} S_{1} (τ) . (25)

To find the optimal solution in Eq. (25), we first investigate the properties of $S_{1} (τ)$ . Through some basic calculations, the first and second derivatives of $S_{1} (τ)$ are given by

{S^{'}}_{1} (τ)

= - \frac{((b τ + c) (τ (a + b) + c) \ln (\frac{a τ}{b τ + c} + 1) + a c (τ - T))}{\frac{T}{N} \ln (2) (b τ + c) (τ (a + b) + c)}, (26)

and

{S^{″}}_{1} (τ) = - \frac{a c N (c (a + 2 b) (τ + T) + 2 b τ T (a + b) + 2 c^{2})}{T \ln (2) {(b τ + c)}^{2} {(τ (a + b) + c)}^{2}} (27)

Since a, b and c are all positive, we have ${S^{″}}_{1} (τ) ＜ 0$ for τ> 0.Moreover, it is easily seen from Eq. (26)that ${S^{″}}_{1} (τ)$ is monotonically decreasing function of τ, with ${S^{″}}_{1} (0) = a N / (c \ln 2) ＞ 0$ and ${S^{'}}_{1} (τ) \to - N 1 b (1 + a / b) / T ＜ 0$ . Therefore we draw the conclusion that $S_{1} (τ)$ is strictly concave in τ for τ> 0.In particular, $S_{1} (τ)$ first increases and then decreases in the interval(0, ∞).

Note that above we treat τ as a continuous variable in (0, ∞), while the truth is that the pilot length must be a positive integers in the interval[N, T]. Considering the constraint of both $τ \in ℤ$ and N 6 τ 6 T, we have the following proposition.

Proposition 1 The pilot length τ^∗that maximizes the uplink sum SE in Eq. (25)obeys the following properties.

• If ${S^{″}}_{1} (N) ＞ 0$ , then τ^∗ can be obtained by Newton’s method and then rounded to the nearest integer.

• If ${S^{″}}_{1} (N) \leq 0$ , the pilot length satisfies τ^∗=N.

$P r o o f$ Based on the properties of $S_{1} (τ)$ , we know that $S_{1} (τ)$ is strictly concave in τ for τ > 0.Fig.3 shows the trend of the sum SE $S_{1} (τ)$ as a function of the pilot length τ. By using simple calculus we can verify that ${S^{'}}_{1} (T) = - (N / T) 1 b (a T / (b T + c) + 1)$ is less than 0, and if ${S^{'}}_{1} (N) ＞ 0$ , we obtain that $S_{1} (τ)$ first increases and then decreases over the range N 6 τ 6 T. Under this condition, $S_{1} (τ)$ reaches a maximum value at τ^∗ which satisfies ${S^{'}}_{1} (τ^{*}) ＞ 0$ , just as shown in the left part of Fig.3. Therefore the value of τ^∗can be obtained by Newton’s method and then rounded to the nearest integer as described in Proposition 1. On the other side, when ${S^{'}}_{1} (N) \leq 0$ , ${S^{'}}_{1} (τ)$ is a monotonically decreasing function of τ in the interval[N, T], as shown in the right part of Fig.3. As a result, the pilot length τ^∗equals N.

Figure 3

Figure 3 The different trends of sum SE $S_{1} (τ)$ over pilot lengths:(a) ${S^{'}}_{1} (N) \geq 0$ ;(b) ${S^{'}}_{1} (N) ＜ 0$

Proposition 1 shows that the pilot length τ^∗is dependent on the value of ${S^{'}}_{1} (τ)$ . According to Eq. (26), ${S^{'}}_{1} (τ)$ relates to several system parameters, such as the number of BS antennas, the number of users, the transmit power, the number of quantization bits and so on. In other words, the pilot length τ^∗ is varied along with the system parameters which will be validated by the simulation results.

Ⅳ. ADC RESOLUTIONS

Although the system SE performance loss due to equipping low resolution ADCs can be compensated by increasing the number of receiving antennas^[22], the large number of BS antennas also leads to higher total power consumption. This has stimulated the design of energy efficient massive MIMO systems where a trade-off between ADC resolutions and a number of BS antennas must be found. In this section, we will investigate the quantization distortions of ADCs and number of BS antennas to maximize the system total SE while keeping the power consumption below a certain threshold.

A. Power Consumption Model

An accurate modeling of the system power consumption (including signal processing and antenna hardware)is of sig-nificant importance when optimizing b and M under a total power consumption constraint. Based on Refs. [30,31], we present a general power consumption model taking the reso-lution of ADCs into account. Total power consumption of the BS in the uplink can be expressed as

P = P_{T X} + P_{C P}, (28)

where P_TX accounts for the average uplink power amplifier (PA)power(in Watt)defined as the power consumed by the PAs, and P_CP is the circuit power consumption.

In many related works, P_CP is defined as a constant value including the fixed power consumption needed for baseband processors and backhaul infrastructure^[6]. In fact, this model is not accurate enough to be as close as hardware reality, for it neglects the fact that each antenna at the BS requires dedi-cated circuits with non-zero power consumption, and the sig-nal processing tends to be increasingly complex when more and more antennas at BS are added. Hence, the circuit power consumption can not be simply modeled as a constant quan-tity. A more accurate and functional modeling of P_CP is of primary importance when an energy-efficient massive MIMO system is designed by optimizing the number of antennas(M) and number of quantization bits(b). In the following, we aim at presenting an appropriate model of P_CP(b, M)as a function of the two principal design parameters which are also our op-timization objective.

The circuit power consumption P_CP consists of the power consumed by different analog components and digital sig-nal processing^[6]. For a multi-user MIMO system, the circuit power consumption model is defined as^[30]

P_{C P} = P_{F L X} + P_{T C} + P_{C E} + P_{L P}, (29)

where P_FIX accounts for the fixed power as mentioned before, P_TC represents the power consumption of the transceiver chains, P_CE is the power consumed by channel estimation, and P_LP is the power consumed by linear processing at the BS. Next we will give each term in Eq. (29)a realistic and accurate analysis of their relation (linearly or non-linearly) with the main system parameters(b, M).

1) Transceiver Chains: The power consumption of the transceiver chains P_TC comprises of a set of typical transmitters and receivers, and it can be defined as

P_{T C} = M P_{B S} + P_{S Y N} + N P_{U E}, (30)

where P_BS denotes the power consumption needed to run the circuit components attached to each antenna at the BS, such as analog-to-digital/digital-to-analog converters, mixers, filters and so on. P_SYN accounts for the power required by the local oscillator, and P_UE represents the power consumed by all circuit components of each single-antenna UE, such as amplifiers, oscillator, mixer, and filters.

For the power consumption of circuit components P_BS, one important part is the power required by the converters, among which the ADCs are responsible for a major portion. Each antenna generally demands two separate ADCs(in-phase and quadrature branches)to digitalize the analog signal. We define the power consumption of each ADC as P_ADC, and the remaining portion of P_BS as ${\bar{P}}_{B S}$ , the total power required at BS is quantified as

P_{B S} = 2 P_{A D C} + {\bar{P}}_{B S}, (31)

where P_ADC can be further calculated as

P_{A D C} = 2^{φ D}, (32)

where $D_{1} \geq 0$ is constant coefficient with a typical value as $D_{1} \in [10^{- 8}, 10^{- 7}]$ [W/conversion step], andϕ> 0 is also set constant with a typical value asϕ∈[1, 2]. The choice of $D_{1}$ andϕ depends on the figure-of-merit(FoM)of ADCs where smaller $D_{1}$ is paired with largerϕand vice versa. This power consumption model of the ADC encompasses varying architecture and implementations of ADCs which are described in Refs. [40,41,42].

2)Channel Estimation: Channel estimation in the uplink is performed locally at the BS with computational efficiency of current microprocessors known as[bit operations/J](also in flops/Watt). The pilot-based channel estimation is carried out once per block. In the uplink of multi-user MIMO systems, the BS receives the pilot signal as an M×τ matrix $\hat{Z}$ and then estimate each user’s channel by multiplying with the corresponding pilots of all UEs Ψ. From numerical linear algebra computations, this is a standard operation and consume

P_{C E} = \frac{B_{W}}{L} \frac{τ}{T} 2 M N b, (33)

where L is the computational efficiency at the BS, and B_W is the bandwidth the system operates over.

3)Linear Processing: The received vector of information symbols at the BS are processed by maximal ratio combining as $\hat{x} = {\hat{G}}^{H} \hat{y}$ . The amount of power required for signal processing is also related to the number of quantization bits used by the ADCs. We calculate the number of bit operations required for linear processing, and divide it by the computational efficiency to get

P_{L P} = \frac{B_{W}}{L} (1 - \frac{τ}{T}) 2 M N b . (34)

To sum up, we combine Eqs. (30), (33) and (34) into Eq. (29)to obtain the circuit power consumption as

P_{C P} = P_{F I X} + M (P_{A D C} + {\bar{P}}_{B S}) + P_{S Y N} + N P_{U E} +

\frac{B_{W}}{L} \frac{τ}{T} 2 M N b + \frac{B_{W}}{L} (1 - \frac{τ}{T}) 2 M N b . (35)

Recall that the total power consumption of the BS in the uplink of a multi-user MIMO system is P=P_TX+P_CP, which can be rewritten in the more concrete form

P (b, M) = \frac{1}{η} \sum_{i = 1}^{N} p_{i} + C_{0} + C_{1} N + D_{0} M + 2 M D_{1} 2^{φ b} + \frac{2 M N b}{L}

= C_{0} + D_{0} M + M D_{1} 2^{φ b + 1} + \frac{2 M N b}{L}, (36)

where $C_{0} = P_{S Y N} + P_{F I X}$ , $C_{1} = P_{U E}$ , $D_{0} = {\bar{P}}_{B S}$ , and ${\tilde{C}}_{0} = \frac{1}{η} \sum_{i = 1}^{N} p_{i} + C_{0} + C_{1} N$ . The first term in ${\tilde{C}}_{0}$ accounts for the transmission power consumption with η being the efficiency of the power amplifier. $C_{0} ＞ 0$ represents the fixed power consumption required for site-cooling, control signaling, and so on. $C_{1} ＞ 0$ , $D_{0} ＞ 0$ denote the power consumption per antenna of the BS and UE respectively which are independent of the number of quantization bits b.

B. Problem Statement and Solving

Despite the fact that equipping highly economical ADCs with low resolution in practical massive MIMO systems is feasible, it is not sure whether such low ADC precision is acceptable in terms of SE and power consumption performance. This subsection aims at analyzing the number of BS antennas M and ADC resolution b that maximizes the uplink sum SE, defined as $R_{S U M} = \sum_{n = 1}^{N} R_{n}$ with $R_{n}$ given in Theorem 1, subject to a total power consumption constraint. An SE-optimal multi-user MIMO setup is achieved by solving the following optimization problem:

\max_{b, M \in ℤ_{+}} R_{S U M} (b, M),

s . t, P (b, M) \leq γ, (37)

where γ accounts for the maximum value of the system total power consumption, and the optimization variables b and M are non-negative integers whose set is denoted by $ℤ_{+}$ .

To find the optimal solution of Eq. (37), we first provide the properties of the sum SE R_SUM(b, M)and the total power consumption P(b, M)in the following lemma. $L e m m a 1$ Both P(b, M)and R_SUM(b, M)are monotonically increasing functions of b. In addition, P(b, M)increases linearly with M while R_SUM(b, M)increases nonlinearly with M. $P r o o f$ By using some basic calculations, we get that

\frac{\partial P (b, M)}{\partial b} = φ M D_{1} 2^{φ b + 1} \ln 2 + \frac{2 M N}{L}, (38)

and we have∂P(b, M)/∂b> 0, which indicates that P(b, M) increases monotonically with b. Moreover, we rewrite the total power consumption as $P (b, M) = (D_{0} + D_{1} 2^{φ b + 1} + 2 N b / L) M + {\tilde{C}}_{0}, w i t h D_{0} + D_{1} 2^{φ b + 1} + 2 N b / L ＞ 0$ , thus it can be deduced that P(b, M)is linearly increasing functions of M. Note that here we treat b and M as continuous variables, but the conclusion applied to the situation when b and M are nonnegative integers. With similar procedures, the properties of R_SUM(b, M)are easy to be found.

Assuming that{b^∗, M^∗}is the optimal solution to Eq. (37), and it satisfies P(b^∗, M^∗)< γ. From Lemma 1, we know that P(b, M)increases monotonically with b, therefore there exists a value b₁> b^∗ such that P(b₁, M^∗)=γ. Recall that R_SUM(b, M)is also a monotonically increasing function of b which means R_SUM(b₁, M^∗)> R_SUM(b^∗, M^∗). This is a contradiction to the presumption of P(b^∗, M^∗)< γ. Thus if the optimization problem is feasible, the maximum R_SUM can only be found when P(b, M)=γ. As a result, the optimal values of b and M satisfy

M (b) = \frac{L (γ - {\tilde{C}}_{0})}{L (D_{0} + D_{1} 2^{φ b + 1}) + 2 N b} . (39)

To solve the optimization problem in Eq. (37), we can conduct an exhaustive search algorithm where b and M are both non-negative integers. Notice that the optimal solution is found for M being a decreasing function of b (according to Eq. (39)), which indicates that the largest possible M is obtained when b=1. This gives us a finite set of M as M∈{1, 2, …, bM(1)c}to conduct the exhaustive search over, and for each M we can solve P(b, M)=γ to get the corresponding value of b. To this end we present the following definition and proposition.

$D e f i n i t i o n 1$ The Lambert-W function (also called the omega function or product logarithm)is the branches of the inverse relation of the function f(w)=w · e^w where e^w is the exponential function and w is any complex number. For any complex value z, there is

z = W (z) e^{W (z)} . (40)

$P r o p o s i t i o n 2$ When the number of BS antennas M is fixed, the corresponding b, which satisfies P(b, M)=γ, is calculated as

b (M) = ⌊ b_{R} (M) ⌋, (41)

where

b_{R} (M) = - \frac{W (\frac{L D_{1} φ \ln 2}{N} 2^{\frac{L φ (γ - {\tilde{C}}_{0} - D_{0} M)}{2 M N}})}{φ \ln 2} + \frac{L (γ - {\tilde{C}}_{0} - D_{0} M)}{2 M N} . (42)

$P r o o f$ Consider the equation with x being the variable p^x=ax+c, where p> 0 and a=0.To solve x from the equation, we first substitute−t=x+c/a, and the equation can be transformed into

t p^{t} = R = - \frac{1}{a} p^{- \frac{c}{a}}, (43)

given

t = \frac{W (- \frac{1}{a} p^{- \frac{c}{a}} \ln p)}{\ln p}, (44)

which yields the final solution

x = - \frac{W (- \frac{\ln p}{a} p^{- \frac{c}{a}})}{\ln p} - \frac{c}{a} . (45)

Rewrite P(b, M)=γ as

{\tilde{C}}_{0} + D_{0} M + M D_{1} 2^{φ b + 1} + \frac{2 M N b}{L} = γ, (46)

and after some basic simplification, the above equation can be converted to

{(2^{φ})}^{b} = - \frac{N}{L D_{1}} b + \frac{γ - {\tilde{C}}_{0} - D_{0} M}{2 M D_{1}} . (47)

Let p=2^ϕ, $a = - N / (L D_{1})$ , $c = (γ - {\tilde{C}}_{0} - D_{0} M) / (2 M D_{1})$ , and we obtain

b_{R} (M) = - \frac{W (\frac{L D_{1} φ \ln 2}{N} 2 \frac{L φ (γ - {\tilde{C}}_{0} - D_{0} M)}{φ M N})}{φ \ln 2} + \frac{L (γ - {\tilde{C}}_{0} - D_{0} M)}{2 M N} (48)

As b is constrained to be non-negative integer, we finally have $b (M) = ⌊ b_{R} (M) ⌋$ .

From Proposition 2, we know that when the total power consumption P(b, M)is fixed, the optimal solution is found for b being a decreasing function of M. Thus, the largest feasible b is found when the BS antennas M equals to 1, which means the possible interval for b is b∈{1, 2, …, b(1)}. This gives us a different way to conduct the exhaustive search. Moreover, Proposition 2 also indicates that the quantization bits which maximize the sum SE are in dependent with the number of users N. When N increases, the corresponding b gets smaller, which implies that the low-resolution quantizers can be adopted in massive MIMO systems where N takes relatively bigger values. This property is reasonable because the maximum sum SE can only be achieved when P(b, M)=γ as explained before. Recall the expression of P(b, M)in Eq. (36), and it is easily found that N only linearly affects the last term of Eq. (36), which means the increasing N would result in the decreasing b to maintain the constant value of γ.

Ⅴ. NUMERICAL RESULTS

This section conducts numerical simulations to validate the analytical results obtained in the previous sections. We consider a hexagonal cell with radius r_c, and the users are distributed randomly and uniformly over the cell, with the exclusion of a central disk radius r_h meters. The large-scale fading of the nth user consists of path loss and shadow fading which can be modeled as β_n=z_n/(r_n/r_h)^κ, where z_n is a log-normal variable with standard deviation σ_shadow, r_n is the distance between the nth user and BS, and κ is the path loss exponent. It should be pointed out that β_n is constant across the antenna array and the expectation is taken over the fastfading coefficients. In the following simulations, we set the number of single-antenna users N=10, the coherence time of the channel T =196 in symbols, the radius of the hexagonal cell r_c=1 000 m, and the exclusion of a central disk radius r_h=100 meters. In addition, we assume that the power of training and data transmission phases are fixed, i. e. , p_t=p_d. The rest of the selected values of the main parameters are listed in Tab.1.

Table 1 Simulation parameters

parameter	symbol	value
const.related to the ADC FoM	D1	10⁻⁷ W/conv-step
const.related to the ADC FoM	ϕ	1
power amplifier efficiency	η	0.39
static power consumption	C0	10 W
circuit power per active UE	C1	0.1W
circuit power per BS antenna	D0	0.1W
computational efficiency	L/B_W	20 480[bit op./W]
the pass loss exponent	κ	3.8
the shadow fading exponent	σ_shadow	8 dB

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A. SE

In this subsection, we evaluate the uplink sum SE of a quantized MIMO system with imperfect CSI and validate the accuracy of our proposed approximation in Theorem 1. Additionally, we analyze the pilot length which maximizes the uplink sum SE.

Fig.4 shows the sum SE R_SUM as a function of the number of BS antennas M, where we set p_t = p_d =1 dB, and τ=10.We compare the simulated SE in Eq. (13)with its corresponding analytical approximation in Eq. (14)for three different quantization bits b=1, 2, and∞, respectively. It can be observed from the figure that there is a precise agreement between the simulation results and our theoretical results, which validates our analysis in Theorem 1. For an MRC receiver, as M increases, the sum SE grows obvious for three kinds of ADC resolution. Furthermore, we can see that the gap between high and low quantization bits decreases with b, which implies that adopting low-resolution ADCs(e. g. , 1-2 bits)is a promising solution in massive MIMO systems and the system performance loss can be compensated by increasing the number of BS antennas. For example, in order to achieve the sum SE of 4. 143 bit·s⁻¹·Hz⁻¹, systems with perfect quantization (b=∞)require about 150 antennas at BS while the BS antennas in system with low-resolution ADCs(e. g. , 1 bit)need to reach 350.The trade-off between M and b has been studied in the previous section and will be discussed in later simulations.

Figure 4

Figure 4 Uplink sum SE[bit·s⁻¹·Hz⁻¹]versus the number of BS antennas M,for b=1,2,∞,τ=10 and p_t=p_d=1 dB

To better understand the effect of ADC resolution on the system SE performance, Fig.5 shows the sum SE with respect to the quantization bits b. We observe from the figure that the sum SE increases with the number of quantization bits and converges to a fixed value obtained in Eq. (15), which agrees with the result derived in Ref. [32]for the infinite precision case. This conclusion is reasonable because α →1, which means b→∞, indicates that the quantization error brought by ADC can be neglected. Moreover, the growth rate slows down with the increase of b so that when the quantization bits reach about 8 bits, the SE performance of the quantized MIMO system is close to the traditional MIMO systems with perfect quantization.

Figure 5

Figure 5 Uplink sum SE[bit·s⁻¹·Hz⁻¹]versus the number of quantization bits b,for M=200,400,τ=10 and p_t=p_d=1 dB

B. Pilot Length and ADC Resolution

First, we give the sum SE with respect to the pilot length τ under different transmit power and quantization bits in Fig.6, where we assume that all users have the same large-scale fading coefficient β =1. Considering the tightness between the analytical approximations and the simulation result, we use the approximation in Eq. (14) for the following numerical analysis. From Fig.6, we can observe that when the transmit power p=−10 dB, the sum SE increases first and then decreases for all three different values of quantization bits. Under this setting, ${S^{'}}_{1} (N) ＞ 0$ , and the pilot length that maximizes the sum SE τ^∗> 10, which is in accordance with the first case in Proposition 1. However, when the transmit power p=10 dB, it is seen in the figure that for b> 1. The sum SE is a monotonically decreasing function of the pilot length τ. This observation is also understandable because this means ${S^{'}}_{1} (N) ＜ 0$ , which leads to the pilot length τ^∗that maximizes the sum SE being equal to N=10.In both cases, it is important to notice that τ^∗ increases with decreasing b, which implies that longer training length is needed in low-resolution massive MIMO systems in order to improve the accuracy of channel estimation.

Figure 6

Figure 6 Uplink sum SE per cell versus the pilot length τ,where M=100 and N=10

The pilot length τ^∗that maximizes the sum SE for different b and p_d is given in Tab.2. We note that the pilot length τ^∗varies with b and p_d, and basically τ^∗decreases with increasing b and p_d, which agrees with our argument in Proposition 1 and the simulated result in Fig.6. Another interesting observation is that τ^∗ for b=1 is quite different from the others. Although τ^∗decreases when p_d is increasing, it converges to 19 instead of 10(the number of users). This result indicates that unlike the infinite precision cases^[36], increasing the transmit power cannot make the pilot length converge to the number of users for the extreme case with one-bit quantizers. In other words, the accuracy of the channel estimation will not improve unlimitedly even in the high SNR regime.

Table 2 The pilot length τ^∗for different transmit power p^d and quantization bits b

b				p_d (dB)
b	−15	−10	0	10	20	30	40
1	53	35	21	19	19	19	19
2	45	27	12	10	10	10	10
3	43	25	10	10	10	10	10
∞	42	24	10	10	10	10	10

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Next we study the effect of several system parameters, especially the number of BS antennas and the quantization bits, on the total power consumption in Eq. (36). The ADCs are responsible for a large part of the power consumption and distortion noise. Plus, the computational complexity of baseband operations is proportional to the number of quantization bits. Thus Fig.7 gives the total power consumption as a function of b. As stated in Lemma 1, Fig.7 shows that P(b, M)increases monotonically with the quantization bits which are reasonable as a higher level of hardware quality leads to higher power consumption. Another interesting observation is that when the number of BS antennas becomes larger, the increase of total power consumption with quantization bits accelerates. The relation between M and b under fixed total power consumption is plotted in Fig.8 and it can be seen that M decreases with b which is in accordance with Eq. (39).

Figure 7

Figure 7 The total power consumption versus the number of quantization bits b,for M=50,200,400

Fig.9 depicts a 3-D graph of uplink sum SE R_SUM in terms of b and M. We see that R_SUM is monotonically increasing with b and also increasing with M. For a maximum power consumption γ, we plot the curves where P(b, M)=γ with γ={60, 64, 68}respectively. The corresponding M and b that maximize the sum SE are represented with triangle points. As proofed before, the maximum R_SUM is found when P(b, M)=γ. It is seen from the figure that for a maximum power consumption of γ={60, 64, 68}, the maximum sum SE is achieved at M={159, 200, 235}and b={5, 5, 5}, which is clearly a massive MIMO with low-resolution ADCs setup.

Figure 8

Figure 8 The number of BS antennas M versus quantization bits b under different total power consumption constraint

Figure 9

Figure 9 Uplink sum SE as a function of M and b(3-D graph).The green,magenta,and blue lines represent the curves where P(b,M)=γ with γ ={60,64,68}. The triangle points are the corresponding M and b that maximize the sum SE

Once we have the value of M^∗ and b^∗ for a given γ, we can also obtain the EE(in bit/J), which is defined as the ratio between the average sum SE(in bit/s)R_SUM(b^∗, M^∗)and the average total power consumption γ(in J/s). Fig.10 shows the EE as a function of the power consumption constraint γ. It can be seen that the EE first increases and then decreases as the maximum power consumption grows from 60 to 80.When the pilot length equals 10 and 30, the maximum EE is achieved at γ^∗={67, 69}, which corresponds to M^∗={226, 245}and b^∗={5, 5}respectively. This observation indicates that we should choose the appropriate γ to maximize EE, which gives us further insight on how to design an energy efficient massive MIMO system.

Ⅵ. CONCLUSION

This paper analyzed the pilot length and how to select the number of antennas at BS and the ADC resolution to maximize the sum SE while keeping the total power consumption under a threshold in quantized massive MIMO systems. We first derived a tight approximation expression of the up link sum SE for MRC receivers with MMSE channel estimates, capitalizing on which the pilot length which maximize the sum SE is put forward. Subsequently, we used a realistic model of total power consumption which clearly describes the effect of several system parameters(such as M and b)on the power consumption. Based on it, we formulated an optimization problem and pointed out how to find the value of M and b. Our result shows that a longer pilot length is required in massive MIMO systems with low-resolution ADCs. In particular, for the extreme case with one-bit quantizers, the efficient way to obtain the maximum SE is to assign more training time instead of increasing the transmit power. Numerical results show that the ADC resolution corresponding to about 5 bits is a better choice considering the significance of deploying lowcost massive MIMO systems.

Figure 10

Figure 10 Energy efficiency as a function of total pow consumption constraint γ for the pilot length τ={10,30}

APPENDIX

Although the true distribution of ${\hat{g}}_{n}$ can not be determined due to the quantization, we approximate it as Gaussian as the worst-case by using the result of Eqs. (7)and(8)and replacing ${\hat{g}}_{n}$ with an auxiliary random vector w_n according to

{\hat{g}}_{n} = {\tilde{σ}}_{n} b, w_{i} = {[b]}_{i} ~ C N (0, 1), (49)

where σ_n is the variance of the channel estimate ${\hat{g}}_{n}$ . According to the orthogonality principle, it is described as

{\tilde{σ}}_{n}^{2} = β_{n} - σ_{e_{m n}}^{2} = \frac{α^{2} τ p_{t} β_{n}^{2}}{α^{2} τ p_{t} β_{n} + α^{2} + σ_{t}^{2}} . (50)

Applying[32, Lemma 1], we begin by approximating R_n with

{\tilde{R}}_{n} = 1 b (1 + \frac{p_{d} α^{2} E {{‖ g_{n} ‖}^{4}}}{E {S_{\hat{G}}}}), (51)

where

E {S_{\hat{G}}} = α^{2} p_{d} \sum_{i = 1, i \neq n}^{N} E {{| {\hat{g}}_{n}^{H} {\hat{g}}_{i} |}^{2}} +

α^{2} p_{d} \sum_{i = 1}^{N} E {{‖ {\hat{g}}_{n} ‖}^{2}} σ_{e_{m n}}^{2} + α^{2} + E {{‖ {\hat{g}}_{n} ‖}^{2}} +

α (1 - α) E {{\hat{g}}_{n}^{H} d i a g (p_{d} G G^{H} + I_{M}) {\hat{g}}_{n}} . (52)

According to Ref. [22], the expectation of squared norm terms ${‖ {\hat{g}}_{n} ‖}^{2}$ is

E {{‖ {\hat{g}}_{n} ‖}^{2}} = {\tilde{σ}}_{n}^{2} M, (53)

and the expectation of squared norm terms ${| {\hat{g}}_{n}^{H} {\hat{g}}_{i} |}^{2}$ is

E {{‖ {\hat{g}}_{n} ‖}^{2}} = {\tilde{σ}}_{n}^{4} {\tilde{σ}}_{i}^{2} M, (n \neq i) . (54)

The random variable ${‖ {\hat{g}}_{n} ‖}^{2}$ is gamma-distributed with shape M and scale β_n and can be denoted by ${‖ {\hat{g}}_{n} ‖}^{2} ~ Γ (M, β_{n})$ . Then, we obtain

E {{‖ {\hat{g}}_{n} ‖}^{4}} = {\tilde{σ}}_{n}^{4} (M^{2} + M) . (55)

Subsequently, we compute the last term in Eq. (52)as follows

E {{\hat{g}}_{n}^{H} d i a g [p_{d} (\hat{G} + Ξ) {(\hat{G} + Ξ)}^{H} + I_{M}] {\hat{g}}_{n}}

= E {\sum_{m = 1}^{M} {| {\hat{g}}_{m n} |}^{2} (1 + p_{u} \sum_{i = 1}^{N} σ_{e_{m n}}^{2} + p_{d} \sum_{i = 1, i \neq n}^{N} {| {\hat{g}}_{m i} |}^{2} +

p_{d} {| {\hat{g}}_{m n} |}^{2})}

= (1 + p_{d} \sum_{i = 1}^{N} σ_{e_{m n}}^{2}) \sum_{m = 1}^{M} E {{| {\hat{g}}_{m n} |}^{2}} +

p_{d} \sum_{m=1}^{M} \sum_{i = 1, i \neq n}^{N} E {{| {\hat{g}}_{m n} |}^{2}} {{| {\hat{g}}_{m i} |}^{2}} + p_{d} \sum_{m = 1}^{M} E {{| {\hat{g}}_{m n} |}^{4}} . (56)

Employing $E {{| {\hat{g}}_{m n} |}^{2}} = {\tilde{σ}}_{n}^{2}$ and $E {{| {\hat{g}}_{m n} |}^{4}} = 2 {\tilde{σ}}_{n}^{4}$ to compute Eq. (56), we get

E {{\hat{g}}_{n}^{H} d i a g [p_{d} [p_{d} (\hat{G} + Ξ) {(\hat{G} + Ξ)}^{H} + I] {\hat{g}}_{n}]}

M {\tilde{σ}}_{n}^{2} (1 + p_{u} \sum_{i = 1}^{N} β_{i} + p_{d} σ_{n}^{2}) . (57)

Substituting Eqs. (53), (54)and(57)into Eq. (52)yields

E {S_{\hat{G}}} = α M {\tilde{σ}}_{n}^{2} (p_{d} \sum_{i = 1}^{N} β_{i} + p_{d} σ_{n}^{2} - 2 α p_{d} {\tilde{σ}}_{n}^{2} + 1) . (58)

Applying Eqs. (50), (55), (58)into Eq. (51)and simplifying the equation, we finally obtain the desired result in Eq. (14).

The authors have declared that no competing interests exist.

作者已声明无竞争性利益关系。

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