I Introduction
A 1000x increase in capacity and energy efficiency is one of the key requirements of fifth generation (5G) wireless communications. Millimeter wave (mmWave) communications, on one hand, are expected to enable greatly increased data rates in future wireless systems [1, 2]. On the other hand, although beamforming using large antenna arrays can overcome the high pathloss associated to frequencies at these wavelengths, using large antenna arrays with a wide bandwidth may lead to high power consumption, especially in the analog to digital converters (ADC) at the receiver, where energy efficiency may become a critical issue.
Power consumption of an ADC increases linearly with bandwidth and exponentially with the number of bits, and therefore may easily become the main constraint in mmWave multiantenna receiver technologies [3]. The receiver architectures to reduce power consumption discussed so far in the literature can be categorized in three families. Analog Combining (AC) relies on a single RadioFrequency (RF) chain and ADC. AC consumes the least power and is an attractive choice whenever more versatile digital processing is not really necessary [4]. Hybrid Combining (HC) performs combining in both the analog and the digital domains to reduce the number of RF chains and ADCs while still allowing some spatial multiplexing [5]. Digital Combining (DC) with lowresolution ADCs (for example, 14 bits) can reach a good power efficiency but at the cost of an increased quantization error [6, 7]. In this work, we study the spectral efficiency (SE) vs. energy efficiency (EE) tradeoff for quantized analog, digital and hybrid receiver architectures.
The design of energy efficient multiple antenna receivers has been studied in several works. MmWave receivers with low resolution ADCs are studied in [8, 9, 10]. The relationship between the number of ADC bits and the bandwidth is studied in [11], and MIMO with low resolution is treated in [12, 13].
Comparisons among the different architectures can be found in [14, 15]. In [14], low resolution HC beamforming algorithms are designed and the energy efficiency is thoroughly studied in comparison with DC. In [15], low resolution AC, DC and HC are compared with each other and it is shown that lowresolution DC may display a better energy efficiency than HC in some scenarios.
A significant difficulty in the study of energy efficiency in large array receivers is that ADC power values are rapidly changing. A new ADC design has been recently proposed in [16] with a power consumption one order of magnitude lower than the “state of the art representative” ADCs referenced by receiver analyses such as [11, 14, 15]. Three examples of the rapid change of Walden’s figure of merit (the energy consumption per conversion step per Hz) of ADCs [17] are given in Table I. In the present paper, we use the term High Power ADC (HPADC) to refer to the ADC power consumption values featured in mmWave receiver power consumption analyses such as [11]. We call Intermediate Power ADC (IPADC) the recently proposed ADC circuit [16], designed specifically for mmWave receivers. And we call Low Power ADC (LPADC) a best case scenario for ADC power consumption that can be inferred from the exhaustive survey of hardware designs in [18].
Scenario  Value  Generation 

HPADC  fJ/step/Hz  Receiver design state of the art [11] 
IPADC  fJ/step/Hz  Recently proposed for mmWave [16] 
LPADC  fJ/step/Hz  Ideal future projection in survey [18] 
Given this extremely dynamic parameter variation, we claim that the previous literature comparing AC, HC and DC in [14, 15] is incomplete in the sense that the results are only relevant as far as the “representative ADC” they selected remains the state of the art. In this work, we have focused on the design of a SE vs EE tradeoff comparison chart method that is easily reproducible for different values, rather than generating results only for one chosen representative ADC model. We will give example numeric results using the values of HPADC and LPADC, and the paper is complemented by a web tool where the reader may generate a chart reproducing our analysis method using any other component parameter values [19]. Our contribution extends existing receiver analyses and comparisons in the following ways:

First, we take into account engineers preference between SE and EE. We express this as a preference weight in a multiobjective utility maximization problem where the relative weight of SE vs EE is a free parameter. This gives new insights on the choice between HC and DC, complementing the EE comparison in [14].

Second, we consider both Uplink and Downlink scenarios with different numbers of antennas. Our results show that in a Downlink scenario where the receiver has fewer antennas, and using the same HPADC as in [11], DC provides the best SE vs EE tradeoff among all schemes. In Uplink, although HC gives the best standalone EE, DC comes quite close while allowing better SE, and thus the engineers’ preference on the EEvsSE tradeoff determines the ideal receiver.

Third, we show that the SE vs EE tradeoff is extremely parameterdependent by comparing charts with HPADC and LPADC. This means that existing results have limited application to the specific ADC reference chosen and cannot be considered as a universal benchmark, even when the reference is very well selected.

Fourth, we guarantee the universal reproducibility of our analysis with different component parameters by providing a web tool to complement the paper [19], where researchers can input different parameters and generate their own SE vs EE chart.
Additional results and further discussion can be found in the extended version of this paper in [20].
Ii System Model
We consider a pointtopoint mmWave channel with multiple input multiple output (MIMO) antenna arrays with transmit and receive dimensions and , respectively. The signal has bandwidht and a fast fading frequencyflat impulse response so that intersymbol interference is not present. Our channel model can extend simply to frequency selective systems with multicarrier modulations and cyclic prefix. We distinguish three mmWave receiver architectures with different analog or digital MIMO characteristics and ADC quantization, i.e., AC, DC and HC. For all three cases, we consider a fully digital nonquantized architecture at the transmitter. The assumption of digital transmitter is motivated by the fact that ADCs typically cause more power consumption constraints than Digital to Analog Converters. We further assume the availability of channel state information (CSI) both at the transmitter and at the receiver and we design the MIMO processing accordingly.
Iia mmWave Quantized Channel
The general received signal model is as follows:
(1) 
where the transmitter starts with a transmitted signal which may be scalar or in multiple dimensions and has power constraint . The transmitter projects onto the array using a unitary precoding matrix . The signal then passes through an random mmWave channel with distribution defined in [21] and summarized in Table II. After propagation through the channel the receiver array captures the signal and the Additive White Gaussian Noise . In AC and HC, the received signal may go through an analog combining stage where only phaseshifting and addition are permitted (each coefficient of the matrix is a phase change ). The analog signal in each RF chain is converted at the ADCs and finally goes through digital combining . We model the quantization distortion of the signal using the additive quantization noise model (AQNM) [22], in which a noise term is added before digital combining. We denote by the inverse of the signaltoquantization noise ratio, which is inversely proportional to the square of the resolution of an ADC. For a Gaussian input distribution, the values of for (where is the number of ADC bits) are listed in Table III, and for can be approximated by [12].
Macroscopic Pathloss in dB  
(Line of Sight at distance )  
Macroscopic Pathloss in dB  
(NonLOS at distance )  
Number of Scattering Clusters  
Clust. Central Angle of Arrival  
Clust. Ctrl. Ang. of Departure  
Number. of Paths per Cluster  20  
Small scale scattering per path  
Path Differential AoA  
Path Differential AoD  
Linear antenna Array 
mmWave Channel probability distribution
[21]1  2  3  4  5  

0.3634  0.1175  0.03454  0.009497  0.002499 
IiB Digital Combining (DC) Receiver
The fully digital receiver is shown in Figure 1. DC has a dedicated RF chain per antenna element, and in the channel model (1) we have that and are digital precoding and digital combining matrices, whereas analog processing is absent so that .
We compute the DC matrices using the singular value decomposition of the channel matrix
. By applying precoding and combining , we create an equivalent diagonal channel where the transmit power may be allocated across the singular values . With this formulation, the rate maximization problem with DC is given as(2) 
where the input covariance matrix that maximizes the rate is obtained using the waterfilling algorithm.
The power consumption of the DC architecture is
(3) 
where .
In this paper we use an example of component power consumptions detailed in Table IV, but other values may also be applied depending on the advances in hardware design. increases exponentially with and linearly with and with the ADC Walden’s figure of merit [23]. The power consumption of all the other components is independent of the bandwidth and the number of bits.
Finally, we define the EE of each receiver as , where is the achievable rate and represents the total power consumption.
IiC Analog Combining (AC) Receiver
The fully analog receiver is shown in Figure 2. In contrast to a DC, all receiver processing is performed in the analog domain and only one RF chain and ADC pair are needed. We have that in (1), and are digital precoding and analog combining matrices, and digital combining is absent ().
The rate maximization problem with AC is given as
(4)  
Here we may first work out the digital precoding, which only has a unitary power constraint. We choose to maximize the gain under any given value of using . Secondly, in unconstrained circumstances the optimal
would be the maximum left eigenvector (
) of , however, due to the constant amplitude constraint can only be a phase projection of it, i.e.(5) 
The power consumption of AC is evaluated as
(6) 
IiD Hybrid Combining (HC) Receiver
The hybrid receiver is shown in Figure 3. We have that in (1), , and are digital precoding, analog combining and digital combining matrices, respectively. The analog combining is obtained using Algorithm 1 from [29], and the digital precoding and combining use the Singular Value Decomposition as in the DC model, over an equivalent channel with dimensions
The achievable rate maximization with HC is given as
(7) 
Note that is upper bounded by due to the fact that has analog constraints and .
Finally, the power consumption of HC is evaluated as
(8) 
Iii SE vs EE Analysis and Examples
We plot the EE vs SE curves for each receiver as increases from 1 to 8 at increments of 1. The top of the chart corresponds to highest SE and the rightmost points correspond to highest EE. Thus, an ideal goal would be to design a receiver that is as close to the top right corner of the chart as possible. Nonetheless, this intuitive guideline is not sufficient to capture the needs of receiver design when a tradeoff between EE and SE exists. We construct a multiobjective utility optimization interpretation of the receiver selection problem that serves to discuss the interpretation of the chart. Note that this optimization is easy to solve due to the small search space, and its value is not in the result itself but in the interpretation it gives to the chart.
We consider a free parameter that represents the receiver designer’s preference between higher EE and higher SE. The “receiver utility” according to the designer’s preference can be expressed and maximized as
(9) 
where is SE maximization and is EE maximization. Since the problem is relatively easy, we can obtain the set of solutions for the entire range of parameters , where the solutions in the set correspond to all the receiver designs that are valuable for some kind of receiver designer preference. Receivers not in the set of solutions for all are receivers that would never be preferred by any designer.
We obtain SE vs EE charts for AC, HC and DC receivers with ranging from to in four scenarios, i.e., combining Downlink and Uplink with low and high SNR. We highlight the points that are utility maximization solutions in the SEvsEE charts to distinguish the most valuable receiver designs from the rest. For Uplink, we use and , and for Downlink, we use and . For high SNR we consider dB before combining, which is about m distance in LOS mmWave pathloss, and for low SNR we consider dB which is about the same distance with NLOS. We have also introduced different numbers of RF chains in HC to observe the impact of a good selection of the dimension of the hybrid system. Increasing increases SE but also the power consumption, potentially leading to a drop in EE.
We give two examples with two different sets of component parameters^{1}^{1}1Although we did not fabricate the receivers (i.e., AC, DC or HC) in hardware, the power consumption model we employ is based on hardware designs considered in recent papers, e.g., [11, 15].:

The High Power ADC (HPADC) model is based on an existing device that supports sampling at Gs/s and has been referenced in related literature such as [11]. In order to give HC Phase Shifters appropriate power consumption values, we pair the HPADC model with an existing PS model, i.e., with mW, referenced in [25]. In this example we focus on comparing the different receivers and on the impact of Downlink/Uplink in high/low SNR scenarios. The results validate the observations from [11, 14] and also provide some additional insights about HPADC.

The Low Power ADC (LPADC) model considers a likely bestcase scenario deduced from the hardware survey in [18]. Likewise, we pair this bestcase scenario for ADCs with a best case phase shifter model, with negligible power consumption mW, as in [26]. For the second example, we focus on displaying how critically parameterdependent the receiver EEvsSE tradeoff comparison is. We show how the observations from the same example change drastically and the results in [11, 14] for HPADC cannot be generalized to different ADC components in the literature.
Additional results and more examples and discussion can be found in the extended version of this paper [20].
Iiia Example 1: High power ADC characteristics
We plot the SE vs EE for the three receiver architectures with HPADC in Fig. 4. Since ADCs have high power consumption, as is increased the lines first reach upward and right, and then steer left and EE returns to the left corner while SE continues rising. This is consistent with results in [14, 20] that show that SE is monotonic in whereas EE reaches a maximum and then drops. Note that the points maximizing (9) for different values of are highlighted, so the set of highlighted points in the chart can be regarded as a collection of “all the interesting receivers”.
Note that the number of bits that maximizes SE is not necessarily the same that maximizes EE. For instance, for DC in Figure 4(a) the point in the top left corner corresponds to and maximizes SE (it is highlighted for ), whereas the rightmost point correspoinding to maximizes EE (it is highlighted for ). Notice also that the number of bits that maximizes either magnitude varies for different receivers. For example AC, HC with , and DC achieve maximum EE with and , respectively.
In Fig. 4(a) for high SNR Downlink we have a result in which DC completely outperforms all other schemes. DC with 4 or 5 bits has the best EE, and with 8 bits has the best SE. Note that only DC points are selected across all values of , always achieving the best EE vs SE tradeoff. The higher number of antennas in Uplink causes an increase in power consumption that affects DC more severely. In Fig. 4(b) the high SNR Uplink HC using with 68 bits achieves greater or equal EE than the best EE of DC. However, HC is only selected with bits for the lowest values of , whereas DC not only is chosen for maximum SE (), but also offers the best tradeoff with good SE and good EE for midrange values of ().
To observe the impact of SNR in Downlink we compare Figures 4(a) and 4(c). At high SNR spatial multiplexing offers significant rate gains, and HC SE rises significantly as grows from 2 to 10 while EE does not contract very much. The HC scheme with the best EE uses and , yet DC still manages to completely outperform HC with any number of RF chains. At low SNR, however, there is no significant spatial multiplexing advantage, and HC SE grows when steps from 2 to 4, but becomes flat thereafter, while EE keeps retracting. Nonetheless, it is at low SNR where we observe a nonDC scheme to get a point in the optimal set for the first time. This scheme is AC which is very well suited for lowSNR scenarios because it pools all the transmitter energy into the strongest propagation direction and has very low component power consumption. Notice that for HC schemes the optimal number of RF chains is SNRdependent, which is an additional handicap for the use of HC designs, since usually the same hardware design is used in all the devices at different distances in a network. The only Downlink scenario where HC achieves an EE similar to DC is with and , and yet its SE is lower.
The HC receiver works much better in Uplink, as we can see comparing Figures 4(b) and 4(d). Here we observe again that as takes values and the SE of HC grows at high SNR and becomes flat at low SNR. At high SNR, HC achieves the best EE and contributes a point with good EEvsSE tradeoff, while DC provides other good points in the tradeoff and result in the highest SE. However at low SNR is the only HC configuration that outperforms DC in a tradeoff point, and the best EE is again provided by AC. As in Downlink, note that the SNR dependence of the optimal can be an issue for HC, and a wrong choice of by a small difference makes the HC receiver worse than DC.
Finally, if we wish to add a constant device power consumption ruler we can superimpose it with diagonal dotted lines. In Downlink, only AC is viable under W, whereas above W DC outperforms the other schemes. In Uplink, only AC is viable under W, HC presents a better choice in the range W and DC outperforms all schemes above W. Surprisingly, DC is a better receiver for both smaller devices (UE with W) and large devices (macro cell BS with W), while HC is better for midrange power devices (such as a pico cell BS with W). In devices below 1 W AC seems to be the only viable option.
IiiB Example 2: Low power ADC characteristics
Note that all the observations made about Figure 4 apply only to receivers fabricated with the component parameters in Table IV and the HPADC in Table I. A significant contribution of our analysis chart is its versatility to reproduce the analysis with changed parameters. In Figure 5 we represent the same analysis performed with the LPADC model and W power consumption for phase shifters, as a sort of “future expected values”. Since ADCs have lower power consumption, as is increased the lines progress to the top right and, differently from HPADC, EE always increases with in LPADC. Also, now the number of bits that maximizes SE for a receiver also maximizes EE.
In Fig. 5(a) for high SNR Downlink we have a result in which DC completely outperforms all schemes and there is no tradeoff. DC with bits is significantly above in terms of both EE and SE. For the Uplink in Fig. 5(b) there is a slight tradeoff in which points are very close. HC with achieves slightly better EE with slightly worse SE, and DC with is chosen for some value of .
To observe the impact of SNR in Downlink we compare Figures 5(a) and 5(c). We observe again the saturation of HC SE as grows from 2 to 10 at low SNR although it grows at high SNR. Moreover, DC remains the dominant receiver in most tradeoff preferences except if one sets , where AC is chosen with a tiny gain in EE and a huge drop in SE. Cases such as AC here exemplify why a tradeoff utility maximization interpretation can lead to new insights on the receiver selection problem.
Looking at the Uplink we compare the high and low SNR cases in Figures 5(b) and 5(d), respectively. Here we observe again the saturation of HC with an increase of in low SNR and the issue that the optimal is SNRdependent. In addition, mismatched ’s are outperformed by DC. Furthermore, AC offers slightly better EE with a huge SE drop, HC contributes a couple of points in the tradeoff, and DC contributes most of the points in the EEvsSE tradeoff.
Finally, with regard to power consumption, the same observations we made for HPADC apply here.
Iv Conclusions
In this work, we studied and compared the spectral and energy efficiencies of Analog, Digital and Hybrid combining schemes. We considered receivers operating at mmWave and equipped with low resolution analog to digital converters (ADC). We developed a multiobjective optimization formulation, where the preference between SE and EE is weighted by a free parameter. This allows researchers/designers to identify the best receiver scheme depending on their needs.
We considered both Uplink and Downlink scenarios, and our results showed that in the Downlink, where the receiver is equipped with fewer antennas, DC outperforms other schemes, whereas in the Uplink there is a tradeoff between HC and DC. Moreover, in low SNR scenarios AC achieves the highest EE, at the expense of severe drops in SE.
We also argued that the EE vs SE comparison among different receiver designs is extremely parameter dependent, and results based on one chosen set of parameter values do not represent a complete picture due to the fact that ADCs and other component circuits are rapidly improving. To illustrate this, we produced comparison charts with low power and high power ADCs and showed that the SE vs EE tradeoff results differ significantly. Therefore, ADCs and other component parameters directly affect the choice of the most appropriate combining scheme. To address the effect of this critical parameter dependency, we developed an easily reproduced analysis method in chart form and provided a web tool to complement this paper, where the readers may obtain their own SE vs EE tradeoff charts by plugging in the set of parameter values for the hardware components they have available.
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