2405/2405.01352.md

Title: Using Waste Factor to Optimize Energy Efficiency in Multiple-Input Single-Output (MISO) and Multiple-Input Multiple-Output (MIMO) Systems

URL Source: https://arxiv.org/html/2405.01352

Markdown Content: Mingjun Ying, Dipankar Shakya, and Theodore S. Rappaport This research is supported by the NYU WIRELESS Industrial Affiliates Program and the NYU Tandon School of Engineering graduate fellowship. NYU WIRELESS, Tandon School of Engineering, New York University, Brooklyn, NY, 11201

{yingmingjun, dshakya, tsr}@nyu.edu

Abstract

This paper introduces Waste Factor (W) and Waste Figure (WF) to assess power efficiency in any multiple-input multiple-output (MIMO) or single-input multiple-output (SIMO) or multiple-input single-output (MISO) cascaded communication system. This paper builds upon the new theory of Waste Factor, a systematic model for added wasted power in any cascade for parallel systems such as MISO, SIMO, and MIMO, which are prevalent in current wireless networks. Here, we also show the advantage of W compared to conventional metrics for quantifying and analyzing energy efficiency. This work explores the utility of W in assessing energy efficiency in communication channels, within Radio Access Networks (RANs).

Index Terms:

Waste Factor, Waste Figure, power efficiency, MIMO, RAN, sustainability

I Introduction

This paper presents new results for using Waste Factor (W) or Waste Figure (WF) in dB for power efficiency of parallel circuits and system configurations such as found in MISO, SIMO, and MIMO wireless systems.

W 𝑊 W italic_W has shown promise for quantifying and optimizing energy efficiency in diverse applications, including UAV communication systems [1], massively broadband systems [2], millimeter wave wireless networks, phase shifters in sub-THz phased arrays [3], wireless relay networks [4], data centers [5, 6], and circuits [6], yet its broader adoption has been somewhat limited due to its very recent development and also because the application of W in parallel systems has not been previously explored, or defined.

Inspired by Friis’s 1944 analysis of additive noise in cascaded systems [7], we adopt a similar theory and model for additive wasted power, thus providing W as a unified Key Performance Indicator (KPI) to evaluate the energy efficiency of any cascaded or paralleled system. Then, we show the application of W 𝑊 W italic_W in diverse system configurations such as MISO, SIMO, and MIMO.

II Advantages of the Waste Factor Compared to Conventional Energy Efficiency Metrics

There is a tremendous focus on energy efficiency in wireless networks.

Previously used energy metrics include the ratio between the total number of packets received at the destination node and the total energy consumption spent by the network to deliver these packets [8], and defined for Internet of things (IoT) as the ratio between ‘How many messages are received’ and ‘How many messages could have been received regarding the total energy consumption’ [9].

In [10], McCune highlights that energy efficiency in wireless links varies by more than eleven orders of magnitude and stresses the need for a uniform approach to energy efficiency across different systems. In [11], the authors study the component, equipment, and system/network levels, and find energy efficiency metrics at the component and equipment levels are well-developed, while those for system/network levels need more attention.

In [4], the consumption factor (precursor to Waste Factor) evaluates the energy efficiency of a relay network and cascaded circuits. In [3], the consumption efficiency factor (CEF) provides a quantitative metric for the trade-off between the data rate and the power consumed by a communication system using Waste Factor and provides insights for network energy efficiency with different cell sizes.

We first motivate the use of Waste Factor theory by highlighting several key advantages of W 𝑊 W italic_W over previous energy efficiency metrics:

• • Comprehensive Analysis: W 𝑊 W italic_W provides a unified assessment of power efficiency, covering every aspect of the system power usage, including signal, non-signal, and ancillary power consumption [5, 6, 12, 13].

• • System-Wide Applicability: Unlike conventional metrics focused on specific components or levels, W 𝑊 W italic_W is applicable to any source-to-sink communications architecture, including SIMO, MISO and MIMO as shown here.

• • Optimization and Strategic Design: W 𝑊 W italic_W not only offers precise insights into the sources of power waste within a system but also facilitates targeted enhancements, enabling comparative analyses and informed decision-making in early design stages [6, 5, 12, 13].

III Noise Factor and Waste Factor Fundamentals

III-A Quantifying Additive Noise with Noise Factor

Noise Factor (F) quantifies the additive noise contributed by each component in a cascade, and is crucial for analyzing signal-to-noise ratio (SNR) degradation in a cascade, where F=SNR i/SNR o 𝐹 subscript SNR 𝑖 subscript SNR 𝑜 F=\text{SNR}{i}/\text{SNR}{o}italic_F = SNR start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT / SNR start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT[7].

The total noise factor for a cascade with matched loads, from the source to sink, was derived by Friis to be [7]:

F=F 1+(F 2−1)G 1+(F 3−1)G 1⁢G 2+…+(F N−1)∏i=1 N−1 G i,𝐹 subscript 𝐹 1 subscript 𝐹 2 1 subscript 𝐺 1 subscript 𝐹 3 1 subscript 𝐺 1 subscript 𝐺 2…subscript 𝐹 𝑁 1 superscript subscript product 𝑖 1 𝑁 1 subscript 𝐺 𝑖 F=F_{1}+\frac{(F_{2}-1)}{G_{1}}+\frac{(F_{3}-1)}{G_{1}G_{2}}+\ldots+\frac{(F_{% N}-1)}{\prod_{i=1}^{N-1}G_{i}},italic_F = italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + divide start_ARG ( italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 1 ) end_ARG start_ARG italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG + divide start_ARG ( italic_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 1 ) end_ARG start_ARG italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG + … + divide start_ARG ( italic_F start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT - 1 ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N - 1 end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ,(1)

where F i subscript 𝐹 𝑖 F_{i}italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and G i subscript 𝐺 𝑖 G_{i}italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT represent the noise factor and power gain of the i t⁢h superscript 𝑖 𝑡 ℎ i^{th}italic_i start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT device, respectively. The additional noise power contributed by a device in the cascade (independent of the input noise power) is quantified by:

P additive-noise=(F i−1)⁢G i⁢N i,subscript 𝑃 additive-noise subscript 𝐹 𝑖 1 subscript 𝐺 𝑖 subscript 𝑁 𝑖 P_{\text{additive-noise}}=(F_{i}-1)G_{i}N_{i},italic_P start_POSTSUBSCRIPT additive-noise end_POSTSUBSCRIPT = ( italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - 1 ) italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,(2)

where N i subscript 𝑁 𝑖 N_{i}italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is the noise power of i t⁢h superscript 𝑖 𝑡 ℎ i^{th}italic_i start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT device input.

III-B Quantifying Additive Wasted Power with Waste Factor

Waste Factor is derived using an approach similar to Friis’s Noise Factor, yet is applied to systematically quantify the wasted power in a cascade. W 𝑊 W italic_W can be used in all types of systems that involve the transmission of information (e.g. which have a source and a sink), including circuits, wireless links, propagation channels, and data centers [6, 5, 12, 4, 2, 3].

The total power consumed by any system or cascade is directly related to the conservation of power in these four power components [4, 2, 6]: 1)P source,out subscript 𝑃 source,out P_{\text{source,out}}italic_P start_POSTSUBSCRIPT source,out end_POSTSUBSCRIPT is the power from a source that is applied at the input of the cascaded system; 2) P signal subscript 𝑃 signal P_{\text{signal}}italic_P start_POSTSUBSCRIPT signal end_POSTSUBSCRIPT is the signal power delivered to the output of the cascade or device; 3) P non-signal subscript 𝑃 non-signal P_{\text{non-signal}}italic_P start_POSTSUBSCRIPT non-signal end_POSTSUBSCRIPT is the power consumed by devices within the cascade (e.g., devices along the signal path) but which is not part of the delivered signal power and is viewed as wasted power which includes power consumed by passive or active devices or lost due to heat dissipation; 4)P non-path subscript 𝑃 non-path P_{\text{non-path}}italic_P start_POSTSUBSCRIPT non-path end_POSTSUBSCRIPT is the power consumed by devices that are not on the signal path and do not contribute to the output signal power. However, as shown in [12] and [6], quiescent power consumed by amplifiers on the signal path (when on stand-by) may be considered in P non-path subscript 𝑃 non-path P_{\text{non-path}}italic_P start_POSTSUBSCRIPT non-path end_POSTSUBSCRIPT.

Waste Factor is defined in eq. (3) as the ratio of the total power consumed by the signal path components along a cascade P consumed,path subscript 𝑃 consumed,path P_{\text{consumed,path}}italic_P start_POSTSUBSCRIPT consumed,path end_POSTSUBSCRIPT, which is the sum of additive wasted power and the useful signal output power, to the useful signal output power [6, 5, 2, 12], and η w subscript 𝜂 w\eta_{\text{w}}italic_η start_POSTSUBSCRIPT w end_POSTSUBSCRIPT is waste factor efficiency, which is defined as the reciprocal of W 𝑊 W italic_W:

W=1 η w=P consumed,path P out=P signal+P non-signal P signal.𝑊 1 subscript 𝜂 w subscript 𝑃 consumed,path subscript 𝑃 out subscript 𝑃 signal subscript 𝑃 non-signal subscript 𝑃 signal\vspace{-3pt}W=\frac{1}{\eta_{\text{w}}}=\frac{P_{\text{consumed,path}}}{P_{% \text{out}}}=\frac{P_{\text{signal}}+P_{\text{non-signal}}}{P_{\text{signal}}}.italic_W = divide start_ARG 1 end_ARG start_ARG italic_η start_POSTSUBSCRIPT w end_POSTSUBSCRIPT end_ARG = divide start_ARG italic_P start_POSTSUBSCRIPT consumed,path end_POSTSUBSCRIPT end_ARG start_ARG italic_P start_POSTSUBSCRIPT out end_POSTSUBSCRIPT end_ARG = divide start_ARG italic_P start_POSTSUBSCRIPT signal end_POSTSUBSCRIPT + italic_P start_POSTSUBSCRIPT non-signal end_POSTSUBSCRIPT end_ARG start_ARG italic_P start_POSTSUBSCRIPT signal end_POSTSUBSCRIPT end_ARG .(3)

The total power consumption of any system P consumed,total subscript 𝑃 consumed,total P_{\text{consumed,total}}italic_P start_POSTSUBSCRIPT consumed,total end_POSTSUBSCRIPT is the sum of power consumed on the signal path P consumed,path subscript 𝑃 consumed,path P_{\text{consumed,path}}italic_P start_POSTSUBSCRIPT consumed,path end_POSTSUBSCRIPT as well as all power consumed by non-signal path P non-path subscript 𝑃 non-path P_{\text{non-path}}italic_P start_POSTSUBSCRIPT non-path end_POSTSUBSCRIPT, where W 𝑊 W italic_W is used to characterize power wasted for each component and the entire cascade on the signal path between source and sink [6, 5, 2, 4, 12, 13]:

P consumed,total subscript 𝑃 consumed,total\displaystyle P_{\text{consumed,total}}italic_P start_POSTSUBSCRIPT consumed,total end_POSTSUBSCRIPT=P source,out+P system,added absent subscript 𝑃 source,out subscript 𝑃 system,added\displaystyle=P_{\text{source,out}}+P_{\text{system,added}}= italic_P start_POSTSUBSCRIPT source,out end_POSTSUBSCRIPT + italic_P start_POSTSUBSCRIPT system,added end_POSTSUBSCRIPT +P non-signal+P non-path subscript 𝑃 non-signal subscript 𝑃 non-path\displaystyle+P_{\text{non-signal}}+P_{\text{non-path}}+ italic_P start_POSTSUBSCRIPT non-signal end_POSTSUBSCRIPT + italic_P start_POSTSUBSCRIPT non-path end_POSTSUBSCRIPT =W⁢P signal+P non-path.absent 𝑊 subscript 𝑃 signal subscript 𝑃 non-path\displaystyle=WP_{\text{signal}}+P_{\text{non-path}}.= italic_W italic_P start_POSTSUBSCRIPT signal end_POSTSUBSCRIPT + italic_P start_POSTSUBSCRIPT non-path end_POSTSUBSCRIPT .(4)

where P system,added=P signal−P source,out subscript 𝑃 system,added subscript 𝑃 signal subscript 𝑃 source,out P_{\text{system,added}}=P_{\text{signal}}-P_{\text{source,out}}italic_P start_POSTSUBSCRIPT system,added end_POSTSUBSCRIPT = italic_P start_POSTSUBSCRIPT signal end_POSTSUBSCRIPT - italic_P start_POSTSUBSCRIPT source,out end_POSTSUBSCRIPT is the power added (e.g., contributed) to the signal output power solely by the device or cascade. This leads to the practical application of W in determining the wasted power P non-signal subscript 𝑃 non-signal P_{\text{non-signal}}italic_P start_POSTSUBSCRIPT non-signal end_POSTSUBSCRIPT across a cascaded system referring to either the signal output power or input power, although it is most sensible to relate W 𝑊 W italic_W to the signal output power [6, 12],

P non-signal=(W−1)⁢P signal=(W−1)⁢P source,out⁢∏i=1 N G i,subscript 𝑃 non-signal 𝑊 1 subscript 𝑃 signal 𝑊 1 subscript 𝑃 source,out superscript subscript product 𝑖 1 𝑁 subscript 𝐺 𝑖 P_{\text{non-signal}}=(W-1)P_{\text{signal}}=(W-1)P_{\text{source,out}}\prod_{% i=1}^{N}G_{i},italic_P start_POSTSUBSCRIPT non-signal end_POSTSUBSCRIPT = ( italic_W - 1 ) italic_P start_POSTSUBSCRIPT signal end_POSTSUBSCRIPT = ( italic_W - 1 ) italic_P start_POSTSUBSCRIPT source,out end_POSTSUBSCRIPT ∏ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,(5)

where G i subscript 𝐺 𝑖 G_{i}italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT denotes the gain of the i t⁢h superscript 𝑖 𝑡 ℎ i^{th}italic_i start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT stage in the cascade, and N 𝑁 N italic_N is the total number of cascaded components. Eqs. (3), (III-B), and (5) provide methods to analyze, design, and evaluate the power efficiency of circuits and systems. Notably, W=1 𝑊 1 W=1 italic_W = 1 indicates no wasted power, while W=∞𝑊 W=\infty italic_W = ∞ implies that all power is wasted and no power is delivered to the output. Like F 𝐹 F italic_F, the parameter W 𝑊 W italic_W is independent of the signal powers. Following eq. (5), we observe F−1 𝐹 1 F-1 italic_F - 1 in (2) represents the added noise power relative to the input noise, and analogously, W−1 𝑊 1 W-1 italic_W - 1 in (5) represents the total wasted power relative to the output signal power.

W 𝑊 W italic_W for a cascade of N signal-path components can be computed from (3) and (III-B) to be given by (6) where device N is at the sink of the cascade [6, 3, 5, 2, 4]:

W=W N+(W N−1−1)G N+⋯+(W 1−1)∏i=2 N G i.𝑊 subscript 𝑊 𝑁 subscript 𝑊 𝑁 1 1 subscript 𝐺 𝑁⋯subscript 𝑊 1 1 superscript subscript product 𝑖 2 𝑁 subscript 𝐺 𝑖 W=W_{N}+\frac{(W_{N-1}-1)}{G_{N}}+\cdots+\frac{(W_{1}-1)}{\prod_{i=2}^{N}G_{i}}.italic_W = italic_W start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT + divide start_ARG ( italic_W start_POSTSUBSCRIPT italic_N - 1 end_POSTSUBSCRIPT - 1 ) end_ARG start_ARG italic_G start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_ARG + ⋯ + divide start_ARG ( italic_W start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG .(6)

Eqn. (6) resembles the cascaded Noise Factor in (1).

IV W Analysis for Parallel Communication Systems

MISO, SIMO, and MIMO are widely used architectures in wireless networks to improve system performance, capacity, and reliability. However, power efficiency analysis of these paralleled systems using W 𝑊 W italic_W has not been explored previously.

IV-A W for Multiple-Input Single-Output (MISO) System

To study MISO, we consider two transmitters (TXs) and one receiver (RX). Each transmitter communicates with the receiver over uncorrelated channels with zero-mean additive white Gaussian noise (AWGN).

IV-A 1 W for non-coherent combining MISO

In Fig. 1, TX1 has a Waste Factor W T⁢1 subscript 𝑊 𝑇 1 W_{T1}italic_W start_POSTSUBSCRIPT italic_T 1 end_POSTSUBSCRIPT with channel having W C⁢1=L C⁢1 subscript 𝑊 𝐶 1 subscript 𝐿 𝐶 1 W_{C1}=L_{C1}italic_W start_POSTSUBSCRIPT italic_C 1 end_POSTSUBSCRIPT = italic_L start_POSTSUBSCRIPT italic_C 1 end_POSTSUBSCRIPT and received power P 1 subscript 𝑃 1 P_{1}italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT into RX1 with W R⁢1 subscript 𝑊 𝑅 1 W_{R1}italic_W start_POSTSUBSCRIPT italic_R 1 end_POSTSUBSCRIPT and receiver gain G R⁢1 subscript 𝐺 𝑅 1 G_{R1}italic_G start_POSTSUBSCRIPT italic_R 1 end_POSTSUBSCRIPT. Similarly, TX2 has its own channel and provides a received signal power P 2 subscript 𝑃 2 P_{2}italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT to RX1. Using results of a lossy channel from [6], W 𝑊 W italic_W for each of these two parallel systems, when assuming non-coherent combining, allow the received powers to be combined [14] to yield a single FoM W 2∥subscript 𝑊 limit-from 2 parallel-to W_{\text{2}\mathbin{\parallel}}italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT. For two received signals r 1⁢(t)subscript 𝑟 1 𝑡 r_{1}(t)italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) and r 2⁢(t)subscript 𝑟 2 𝑡 r_{2}(t)italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) with received powers P 1 subscript 𝑃 1 P_{1}italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and P 2 subscript 𝑃 2 P_{2}italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, respectively, non-coherent combining has no phase information. The combined RX power P signal,noncoh subscript 𝑃 signal,noncoh P_{\text{signal,noncoh}}italic_P start_POSTSUBSCRIPT signal,noncoh end_POSTSUBSCRIPT in non-coherent combining at the RX antenna is typically the sum of individual powers:P signal,noncoh=P 1+P 2 subscript 𝑃 signal,noncoh subscript 𝑃 1 subscript 𝑃 2 P_{\text{signal,noncoh}}=P_{1}+P_{2}italic_P start_POSTSUBSCRIPT signal,noncoh end_POSTSUBSCRIPT = italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT[14]. The non-coherent combining assumes that the phases of the incoming signals from each TX are uniformly and identically distributed (i.i.d.), allowing for the straightforward addition of the powers[15]. Using (3) and (6), we find for a MISO system:

W 2∥noncoh superscript subscript 𝑊 limit-from 2 parallel-to noncoh\displaystyle W_{\text{2}\mathbin{\parallel}}^{\text{noncoh}}italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT=P consumed,path P signal,noncoh absent subscript 𝑃 consumed,path subscript 𝑃 signal,noncoh\displaystyle=\frac{P_{\text{consumed,path}}}{P_{\text{signal,noncoh}}}= divide start_ARG italic_P start_POSTSUBSCRIPT consumed,path end_POSTSUBSCRIPT end_ARG start_ARG italic_P start_POSTSUBSCRIPT signal,noncoh end_POSTSUBSCRIPT end_ARG =P 1⁢(W C⁢1+W T⁢1−1 G C⁢1)+P 2⁢(W C⁢2+W T⁢2−1 G C⁢2)P 1+P 2.absent subscript 𝑃 1 subscript 𝑊 𝐶 1 subscript 𝑊 𝑇 1 1 subscript 𝐺 𝐶 1 subscript 𝑃 2 subscript 𝑊 𝐶 2 subscript 𝑊 𝑇 2 1 subscript 𝐺 𝐶 2 subscript 𝑃 1 subscript 𝑃 2\displaystyle=\frac{{P_{1}\left(W_{C1}+\frac{W_{T1}-1}{G_{C1}}\right)}+{P_{2}% \left(W_{C2}+\frac{W_{T2}-1}{G_{C2}}\right)}}{P_{1}+P_{2}}.= divide start_ARG italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT italic_C 1 end_POSTSUBSCRIPT + divide start_ARG italic_W start_POSTSUBSCRIPT italic_T 1 end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT italic_C 1 end_POSTSUBSCRIPT end_ARG ) + italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT italic_C 2 end_POSTSUBSCRIPT + divide start_ARG italic_W start_POSTSUBSCRIPT italic_T 2 end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT italic_C 2 end_POSTSUBSCRIPT end_ARG ) end_ARG start_ARG italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG .(7)

Assuming the power received from each TX is proportional to some power P 𝑃 P italic_P with coefficients γ 1 subscript 𝛾 1\gamma_{1}italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and γ 2 subscript 𝛾 2\gamma_{2}italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT for the first and second transmitters, respectively, such that [P 1,P 2]=P⁢[γ 1,γ 2]subscript 𝑃 1 subscript 𝑃 2 𝑃 subscript 𝛾 1 subscript 𝛾 2[P_{1},P_{2}]=P[\gamma_{1},\gamma_{2}][ italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = italic_P [ italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ], eq. (IV-A 1) simplifies accordingly:

W 2∥noncoh=γ 1⁢(W C⁢1+W T⁢1−1 G C⁢1)+γ 2⁢(W C⁢2+W T⁢2−1 G C⁢2)γ 1+γ 2 superscript subscript 𝑊 limit-from 2 parallel-to noncoh subscript 𝛾 1 subscript 𝑊 𝐶 1 subscript 𝑊 𝑇 1 1 subscript 𝐺 𝐶 1 subscript 𝛾 2 subscript 𝑊 𝐶 2 subscript 𝑊 𝑇 2 1 subscript 𝐺 𝐶 2 subscript 𝛾 1 subscript 𝛾 2 W_{\text{2}\mathbin{\parallel}}^{\text{noncoh}}=\frac{{\gamma_{1}\left(W_{C1}+% \frac{W_{T1}-1}{G_{C1}}\right)}+{\gamma_{2}\left(W_{C2}+\frac{W_{T2}-1}{G_{C2}% }\right)}}{\gamma_{1}+\gamma_{2}}italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT = divide start_ARG italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT italic_C 1 end_POSTSUBSCRIPT + divide start_ARG italic_W start_POSTSUBSCRIPT italic_T 1 end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT italic_C 1 end_POSTSUBSCRIPT end_ARG ) + italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT italic_C 2 end_POSTSUBSCRIPT + divide start_ARG italic_W start_POSTSUBSCRIPT italic_T 2 end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT italic_C 2 end_POSTSUBSCRIPT end_ARG ) end_ARG start_ARG italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG(8)

Figure 1: A two TX and one RX MISO communication system.

Figure 2: A MISO system with M paralleled input cascade.

In Fig. 2, we envisage M parallel cascades, each cascade comprising N 𝑁 N italic_N devices, with the output signal power combined at device N+1 𝑁 1 N+1 italic_N + 1 using either coherent or non-coherent combining. To ascertain W 𝑊 W italic_W for the entire paralleled system with a single output node, we first compute the W 𝑊 W italic_W for cascade m 𝑚 m italic_m using eq. (6):

W cascade,m=W N,m+(W N−1,m−1)G N,m+⋯+(W 1,m−1)∏i=2 N G i,m.subscript 𝑊 cascade,m subscript 𝑊 𝑁 𝑚 subscript 𝑊 𝑁 1 𝑚 1 subscript 𝐺 𝑁 𝑚⋯subscript 𝑊 1 𝑚 1 superscript subscript product 𝑖 2 𝑁 subscript 𝐺 𝑖 𝑚 W_{\text{cascade,m}}=W_{N,m}+\frac{(W_{N-1,m}-1)}{G_{N,m}}+\cdots+\frac{(W_{1,% m}-1)}{\prod_{i=2}^{N}G_{i,m}}.italic_W start_POSTSUBSCRIPT cascade,m end_POSTSUBSCRIPT = italic_W start_POSTSUBSCRIPT italic_N , italic_m end_POSTSUBSCRIPT + divide start_ARG ( italic_W start_POSTSUBSCRIPT italic_N - 1 , italic_m end_POSTSUBSCRIPT - 1 ) end_ARG start_ARG italic_G start_POSTSUBSCRIPT italic_N , italic_m end_POSTSUBSCRIPT end_ARG + ⋯ + divide start_ARG ( italic_W start_POSTSUBSCRIPT 1 , italic_m end_POSTSUBSCRIPT - 1 ) end_ARG start_ARG ∏ start_POSTSUBSCRIPT italic_i = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_i , italic_m end_POSTSUBSCRIPT end_ARG .(9)

Subsequently, W 𝑊 W italic_W for the entire paralleled system using non-coherent combining at N+1 t⁢h 𝑁 superscript 1 𝑡 ℎ{N+1}^{th}italic_N + 1 start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT device is given by:

W M∥noncoh=∑i=1 M(P N,i×W cascade,i)∑i=1 M P N,i.superscript subscript 𝑊 limit-from M parallel-to noncoh superscript subscript 𝑖 1 M subscript 𝑃 𝑁 𝑖 subscript 𝑊 cascade,i superscript subscript 𝑖 1 M subscript 𝑃 𝑁 𝑖 W_{\text{M}\mathbin{\parallel}}^{\text{noncoh}}=\frac{\sum_{i=1}^{\text{M}}% \left({P_{N,i}\times W_{\text{cascade,i}}}\right)}{\sum_{i=1}^{\text{M}}{P_{N,% i}}}.italic_W start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT italic_N , italic_i end_POSTSUBSCRIPT × italic_W start_POSTSUBSCRIPT cascade,i end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_N , italic_i end_POSTSUBSCRIPT end_ARG .(10)

Here, we assume the power output coming out of each cascade is represented by [P N,1,P N,2,…,P N,M]=P⁢[γ 1,γ 2,…,γ M]subscript 𝑃 𝑁 1 subscript 𝑃 𝑁 2…subscript 𝑃 𝑁 𝑀 𝑃 subscript 𝛾 1 subscript 𝛾 2…subscript 𝛾 𝑀[P_{N,1},P_{N,2},\ldots,P_{N,M}]=P[\gamma_{1},\gamma_{2},\ldots,\gamma_{M}][ italic_P start_POSTSUBSCRIPT italic_N , 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT italic_N , 2 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_N , italic_M end_POSTSUBSCRIPT ] = italic_P [ italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_γ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ], where P 𝑃 P italic_P is a constant base power and γ i subscript 𝛾 𝑖\gamma_{i}italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are scaling factors that illustrate the relative contribution to the total received power from each cascade output. Then (10) can be denoted as:

W M∥noncoh=∑i=1 M γ i⁢W cascade,i∑i=1 M γ i.superscript subscript 𝑊 limit-from M parallel-to noncoh superscript subscript 𝑖 1 M subscript 𝛾 𝑖 subscript 𝑊 cascade,i superscript subscript 𝑖 1 M subscript 𝛾 𝑖 W_{\text{M}\mathbin{\parallel}}^{\text{noncoh}}=\frac{\sum_{i=1}^{\text{M}}{% \gamma_{i}W_{\text{cascade,i}}}}{\sum_{i=1}^{\text{M}}{\gamma_{i}}}.italic_W start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT cascade,i end_POSTSUBSCRIPT end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG .(11)

IV-A 2 W for coherent combining MISO

For coherent combining, the receiver must be phase-synchronized with signals r 1⁢(t)subscript 𝑟 1 𝑡 r_{1}(t)italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) and r 2⁢(t)subscript 𝑟 2 𝑡 r_{2}(t)italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ). If r 1⁢(t)subscript 𝑟 1 𝑡 r_{1}(t)italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) and r 2⁢(t)subscript 𝑟 2 𝑡 r_{2}(t)italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) are phase-aligned and have powers P 1 subscript 𝑃 1 P_{1}italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and P 2 subscript 𝑃 2 P_{2}italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, the combined signal power at the RX antenna P signal,coh subscript 𝑃 signal,coh P_{\text{signal,coh}}italic_P start_POSTSUBSCRIPT signal,coh end_POSTSUBSCRIPT is given by the square of the magnitude of the vector sum of the two signals [14]:

P signal,coh=|P 1+P 2|2,subscript 𝑃 signal,coh superscript subscript 𝑃 1 subscript 𝑃 2 2 P_{\text{signal,coh}}=\left|\sqrt{P_{1}}+\sqrt{P_{2}}\right|^{2},italic_P start_POSTSUBSCRIPT signal,coh end_POSTSUBSCRIPT = | square-root start_ARG italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG + square-root start_ARG italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,(12)

where θ 1 subscript 𝜃 1\theta_{1}italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and θ 2 subscript 𝜃 2\theta_{2}italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are the phases of r 1⁢(t)subscript 𝑟 1 𝑡 r_{1}(t)italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) and r 2⁢(t)subscript 𝑟 2 𝑡 r_{2}(t)italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) respectively.

Considering coherent combining in a M TX single RX MISO system, we assume the power output of i t⁢h superscript 𝑖 𝑡 ℎ i^{th}italic_i start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT TX as P T⁢i subscript 𝑃 𝑇 𝑖 P_{Ti}italic_P start_POSTSUBSCRIPT italic_T italic_i end_POSTSUBSCRIPT, and the power received from transmitter i 𝑖 i italic_i at the RX before combining as P i subscript 𝑃 𝑖 P_{i}italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, and the power into the RX after coherent combining at the RX antenna as P signal,coh subscript 𝑃 signal,coh P_{\text{signal,coh}}italic_P start_POSTSUBSCRIPT signal,coh end_POSTSUBSCRIPT. Here we assume [P 1,P 2,…,P M]=P⁢[γ 1,γ 2,…,γ M]subscript 𝑃 1 subscript 𝑃 2…subscript 𝑃 𝑀 𝑃 subscript 𝛾 1 subscript 𝛾 2…subscript 𝛾 𝑀[P_{1},P_{2},\ldots,P_{M}]=P[\gamma_{1},\gamma_{2},\ldots,\gamma_{M}][ italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ] = italic_P [ italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_γ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ], where we show that the received power from each TX has a relationship with each other, and all are related to the ratio of some P 𝑃 P italic_P.

Using eq. (3) and (12), and following eq. (8), W for this coherent combining MISO can be reformulated as:

W M∥coh=∑i=1 M[γ i⁢(W C⁢i+W T⁢i−1 G C⁢i)]|∑i=1 M γ i|2.superscript subscript 𝑊 limit-from M parallel-to coh superscript subscript 𝑖 1 M delimited-[]subscript 𝛾 𝑖 subscript 𝑊 𝐶 𝑖 subscript 𝑊 𝑇 𝑖 1 subscript 𝐺 𝐶 𝑖 superscript superscript subscript 𝑖 1 M subscript 𝛾 𝑖 2 W_{\text{M}\mathbin{\parallel}}^{\text{coh}}=\frac{\sum_{i=1}^{\text{M}}\left[% {\gamma_{i}\left(W_{Ci}+\frac{W_{Ti}-1}{G_{Ci}}\right)}\right]}{\left|\sum_{i=% 1}^{\text{M}}{\sqrt{\gamma_{i}}}\right|^{2}}.italic_W start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT [ italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT italic_C italic_i end_POSTSUBSCRIPT + divide start_ARG italic_W start_POSTSUBSCRIPT italic_T italic_i end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT italic_C italic_i end_POSTSUBSCRIPT end_ARG ) ] end_ARG start_ARG | ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT square-root start_ARG italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .(13)

Based on (13), considering a general MISO system in Fig. 2 with a single output, W for the entire parallel system using coherent combining is:

W M∥coh=∑i=1 M(P N,i×W cascade,i)|∑i=1 M P N,i|2=∑i=1 M(γ i⁢W cascade,i)|∑i=1 M γ i|2.superscript subscript 𝑊 limit-from M parallel-to coh superscript subscript 𝑖 1 M subscript 𝑃 𝑁 𝑖 subscript 𝑊 cascade,i superscript superscript subscript 𝑖 1 M subscript 𝑃 𝑁 𝑖 2 superscript subscript 𝑖 1 M subscript 𝛾 𝑖 subscript 𝑊 cascade,i superscript superscript subscript 𝑖 1 M subscript 𝛾 𝑖 2 W_{\text{M}\mathbin{\parallel}}^{\text{coh}}=\frac{\sum_{i=1}^{\text{M}}\left(% {P_{N,i}\times W_{\text{cascade,i}}}\right)}{\left|\sum_{i=1}^{\text{M}}{\sqrt% {P_{N,i}}}\right|^{2}}=\frac{\sum_{i=1}^{\text{M}}\left({\gamma_{i}W_{\text{% cascade,i}}}\right)}{\left|\sum_{i=1}^{\text{M}}{\sqrt{\gamma_{i}}}\right|^{2}}.italic_W start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT italic_N , italic_i end_POSTSUBSCRIPT × italic_W start_POSTSUBSCRIPT cascade,i end_POSTSUBSCRIPT ) end_ARG start_ARG | ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT square-root start_ARG italic_P start_POSTSUBSCRIPT italic_N , italic_i end_POSTSUBSCRIPT end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT ( italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT cascade,i end_POSTSUBSCRIPT ) end_ARG start_ARG | ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT square-root start_ARG italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .(14)

We see for MISO in general, using W M∥subscript 𝑊 limit-from M parallel-to W_{\text{M}\mathbin{\parallel}}italic_W start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT for either non-coherent (11) or coherent (14) combining, we can calculate W of a MISO system shown in Fig. 2 as:

W MISO=W N+1+(W M∥−1)G N+1.subscript 𝑊 MISO subscript 𝑊 𝑁 1 subscript 𝑊 limit-from M parallel-to 1 subscript 𝐺 𝑁 1 W_{\text{MISO}}=W_{N+1}+\frac{(W_{\text{M}\mathbin{\parallel}}-1)}{G_{N+1}}.italic_W start_POSTSUBSCRIPT MISO end_POSTSUBSCRIPT = italic_W start_POSTSUBSCRIPT italic_N + 1 end_POSTSUBSCRIPT + divide start_ARG ( italic_W start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT - 1 ) end_ARG start_ARG italic_G start_POSTSUBSCRIPT italic_N + 1 end_POSTSUBSCRIPT end_ARG .(15)

where W N+1 subscript 𝑊 𝑁 1 W_{N+1}italic_W start_POSTSUBSCRIPT italic_N + 1 end_POSTSUBSCRIPT and G N+1 subscript 𝐺 𝑁 1 G_{N+1}italic_G start_POSTSUBSCRIPT italic_N + 1 end_POSTSUBSCRIPT represents the Waste Factor and gain of the (N+1)t⁢h superscript 𝑁 1 𝑡 ℎ{(N+1)}^{th}( italic_N + 1 ) start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT single output device.

IV-B W for Single-Input Multiple-Output (SIMO) System

Consider a SIMO system as depicted in Fig. 3. Waste Factor for the system within the dashed box is analogous to the MISO structure. Here, we need to consider coherent combining or non-coherent combining for the output of the parallel cascade in the dashed box of Fig. 3. Waste factor for a SIMO system is derived using a cascade comprising two main components. The first component denoted as D 0,1 subscript 𝐷 0 1 D_{0,1}italic_D start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT, is characterized by a Waste Factor W 0 subscript 𝑊 0 W_{0}italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and a gain G 0 subscript 𝐺 0 G_{0}italic_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. The second component includes the devices within the dashed box of Fig. 3, and here we need the gain and Waste Factor of the dashed box.

IV-B 1 W for non-coherent combining SIMO

Consider non-coherent combining at the output of paralleled cascades, and the input power of each cascade is independent. Assume [P N,1,P N,2,…,P N,M]=P⁢[γ 1,γ 2,…,γ M]subscript 𝑃 𝑁 1 subscript 𝑃 𝑁 2…subscript 𝑃 𝑁 𝑀 𝑃 subscript 𝛾 1 subscript 𝛾 2…subscript 𝛾 𝑀[P_{N,1},P_{N,2},\ldots,P_{N,M}]=P[\gamma_{1},\gamma_{2},\ldots,\gamma_{M}][ italic_P start_POSTSUBSCRIPT italic_N , 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT italic_N , 2 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_N , italic_M end_POSTSUBSCRIPT ] = italic_P [ italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_γ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ], then Waste Factor for the dashed box using non-coherent combining at its output is:

W M∥noncoh=∑i=1 M(P N,i×W cascade,i)∑i=1 M P N,i=∑i=1 M(γ i×W cascade,i)∑i=1 M γ i,superscript subscript 𝑊 limit-from M parallel-to noncoh superscript subscript 𝑖 1 M subscript 𝑃 𝑁 𝑖 subscript 𝑊 cascade,i superscript subscript 𝑖 1 M subscript 𝑃 𝑁 𝑖 superscript subscript 𝑖 1 M subscript 𝛾 𝑖 subscript 𝑊 cascade,i superscript subscript 𝑖 1 M subscript 𝛾 𝑖 W_{\text{M}\mathbin{\parallel}}^{\text{noncoh}}=\frac{\sum_{i=1}^{\text{M}}% \left({P_{N,i}\times W_{\text{cascade,i}}}\right)}{\sum_{i=1}^{\text{M}}{P_{N,% i}}}=\frac{\sum_{i=1}^{\text{M}}\left({\gamma_{i}\times W_{\text{cascade,i}}}% \right)}{\sum_{i=1}^{\text{M}}{\gamma_{i}}},italic_W start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT italic_N , italic_i end_POSTSUBSCRIPT × italic_W start_POSTSUBSCRIPT cascade,i end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_N , italic_i end_POSTSUBSCRIPT end_ARG = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT ( italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT × italic_W start_POSTSUBSCRIPT cascade,i end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ,(16)

and G M∥noncoh=(∑i=1 M P N,i)/(∑i=1 M P 0,i)superscript subscript 𝐺 limit-from M parallel-to noncoh superscript subscript 𝑖 1 M subscript 𝑃 𝑁 𝑖 superscript subscript 𝑖 1 M subscript 𝑃 0 𝑖 G_{\text{M}\mathbin{\parallel}}^{\text{noncoh}}=\left(\sum_{i=1}^{\text{M}}{P_% {N,i}}\right)/\left(\sum_{i=1}^{\text{M}}{P_{0,i}}\right)italic_G start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT = ( ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_N , italic_i end_POSTSUBSCRIPT ) / ( ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT 0 , italic_i end_POSTSUBSCRIPT ) is calculated as the ratio of the sum of the total output power of the devices in the dashed box using non-coherent combining to the sum of the input powers.

IV-B 2 W for coherent combining SIMO

If we consider coherent combining at each output of the parallel cascade, and assume each cascade has an independent input power from device D 0,1 subscript 𝐷 0 1 D_{0,1}italic_D start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT, and [P N,1,P N,2,…,P N,M]=P⁢[γ 1,γ 2,…,γ M]subscript 𝑃 𝑁 1 subscript 𝑃 𝑁 2…subscript 𝑃 𝑁 𝑀 𝑃 subscript 𝛾 1 subscript 𝛾 2…subscript 𝛾 𝑀[P_{N,1},P_{N,2},\ldots,P_{N,M}]=P[\gamma_{1},\gamma_{2},\ldots,\gamma_{M}][ italic_P start_POSTSUBSCRIPT italic_N , 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT italic_N , 2 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_N , italic_M end_POSTSUBSCRIPT ] = italic_P [ italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_γ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ]. Then Waste Factor for the system in the dashed box is:

W M∥coh=∑i=1 M(P N,i×W cascade,i)|∑i=1 M P N,i|2=∑i=1 M(γ i⁢W cascade,i)|∑i=1 M γ i|2,superscript subscript 𝑊 limit-from M parallel-to coh superscript subscript 𝑖 1 M subscript 𝑃 𝑁 𝑖 subscript 𝑊 cascade,i superscript superscript subscript 𝑖 1 M subscript 𝑃 𝑁 𝑖 2 superscript subscript 𝑖 1 M subscript 𝛾 𝑖 subscript 𝑊 cascade,i superscript superscript subscript 𝑖 1 M subscript 𝛾 𝑖 2 W_{\text{M}\mathbin{\parallel}}^{\text{coh}}=\frac{\sum_{i=1}^{\text{M}}\left(% {P_{N,i}\times W_{\text{cascade,i}}}\right)}{\left|\sum_{i=1}^{\text{M}}{\sqrt% {P_{N,i}}}\right|^{2}}=\frac{\sum_{i=1}^{\text{M}}\left({\gamma_{i}W_{\text{% cascade,i}}}\right)}{\left|\sum_{i=1}^{\text{M}}{\sqrt{\gamma_{i}}}\right|^{2}},italic_W start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT italic_N , italic_i end_POSTSUBSCRIPT × italic_W start_POSTSUBSCRIPT cascade,i end_POSTSUBSCRIPT ) end_ARG start_ARG | ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT square-root start_ARG italic_P start_POSTSUBSCRIPT italic_N , italic_i end_POSTSUBSCRIPT end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT ( italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT cascade,i end_POSTSUBSCRIPT ) end_ARG start_ARG | ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT square-root start_ARG italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,(17)

and G M∥coh=(∑i=1 M P N,i)2/(∑i=1 M P 0,i)G_{\text{M}\parallel}^{\text{coh}}={\left(\sum_{i=1}^{M}\sqrt{P_{N,i}}\right)^% {2}}/{\left(\sum_{i=1}^{M}P_{0,i}\right)}italic_G start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT = ( ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT square-root start_ARG italic_P start_POSTSUBSCRIPT italic_N , italic_i end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT 0 , italic_i end_POSTSUBSCRIPT ) is the ratio of the total output power using coherent combining of the devices in the dashed box to the sum of the input powers. Based on eq. (16) and (17), the overall Waste Factor for the general SIMO system can be expressed as:

W SIMO=W M∥+(W 0−1)G M∥.subscript 𝑊 SIMO subscript 𝑊 limit-from M parallel-to subscript 𝑊 0 1 subscript 𝐺 limit-from M parallel-to W_{\text{SIMO}}=W_{\text{M}\mathbin{\parallel}}+\frac{(W_{0}-1)}{G_{\text{M}% \mathbin{\parallel}}}.italic_W start_POSTSUBSCRIPT SIMO end_POSTSUBSCRIPT = italic_W start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT + divide start_ARG ( italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - 1 ) end_ARG start_ARG italic_G start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT end_ARG .(18)

where G M∥subscript 𝐺 limit-from M parallel-to G_{\text{M}\mathbin{\parallel}}italic_G start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT is different depending on whether coherent or non-coherent combing at the output of parallel cascades.

Figure 3: A SIMO system with M paralleled output cascade.

IV-C W for Multiple-Input Multiple-Output (MIMO) System

IV-C 1 W for non-coherent combining MIMO

For a 2-Input 2-Output (2I2O) MIMO system in Fig. 4, the received power of each RX using non-coherent combining is:

[P R⁢1 noncoh P R⁢2 noncoh]=[P T⁢1 P T⁢2]⁢[W C⁢11−1 W C⁢12−1 W C⁢21−1 W C⁢22−1].matrix superscript subscript 𝑃 𝑅 1 noncoh superscript subscript 𝑃 𝑅 2 noncoh matrix subscript 𝑃 𝑇 1 subscript 𝑃 𝑇 2 matrix superscript subscript 𝑊 𝐶 11 1 superscript subscript 𝑊 𝐶 12 1 superscript subscript 𝑊 𝐶 21 1 superscript subscript 𝑊 𝐶 22 1\begin{bmatrix}P_{R1}^{\text{noncoh}}&P_{R2}^{\text{noncoh}}\end{bmatrix}=% \begin{bmatrix}P_{T1}&P_{T2}\end{bmatrix}\begin{bmatrix}W_{C11}^{-1}&W_{C12}^{% -1}\ W_{C21}^{-1}&W_{C22}^{-1}\end{bmatrix}.[ start_ARG start_ROW start_CELL italic_P start_POSTSUBSCRIPT italic_R 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT end_CELL start_CELL italic_P start_POSTSUBSCRIPT italic_R 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ] = [ start_ARG start_ROW start_CELL italic_P start_POSTSUBSCRIPT italic_T 1 end_POSTSUBSCRIPT end_CELL start_CELL italic_P start_POSTSUBSCRIPT italic_T 2 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] [ start_ARG start_ROW start_CELL italic_W start_POSTSUBSCRIPT italic_C 11 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_CELL start_CELL italic_W start_POSTSUBSCRIPT italic_C 12 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_W start_POSTSUBSCRIPT italic_C 21 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_CELL start_CELL italic_W start_POSTSUBSCRIPT italic_C 22 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ] .(19)

where P R⁢i noncoh superscript subscript 𝑃 𝑅 𝑖 noncoh P_{Ri}^{\text{noncoh}}italic_P start_POSTSUBSCRIPT italic_R italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT is the non-coherently combined power at the antenna of RX i 𝑖 i italic_i. The total signal-path power consumption of the system before the receiver using non-coherent combining is:

P consumed,path noncoh=∑i=1 2(P R⁢i noncoh⁢W 2∥noncoh),P_{\text{consumed,path}}^{\text{noncoh}}=\sum_{i=1}^{2}{\left({P}{Ri}^{\text{% noncoh}}W{2\parallel}^{\text{noncoh}}\right)},italic_P start_POSTSUBSCRIPT consumed,path end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT italic_R italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT ) ,(20)

where W 2∥noncoh W_{2\parallel}^{\text{noncoh}}italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT represents Waste Factor for a 2TX paralleled system together with the channel, which is the same as (8). Before the power goes into the receivers, we define the first stage Waste Factor of the 2I2O system using non-coherent combining in the dashed box of Fig. 4:

W 2I2O 1,noncoh=∑i=1 2(P R⁢i noncoh⁢W 2∥noncoh)∑i=1 2(P R⁢i noncoh).W_{\text{2I2O}}^{1,\text{noncoh}}=\frac{\sum_{i=1}^{2}{\left({P}{Ri}^{\text{% noncoh}}W{2\parallel}^{\text{noncoh}}\right)}}{\sum_{i=1}^{2}{\left({P}_{Ri}^% {\text{noncoh}}\right)}}.italic_W start_POSTSUBSCRIPT 2I2O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 , noncoh end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT italic_R italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT italic_R italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT ) end_ARG .(21)

Figure 4: A two-input and two-output MIMO system.

Letting [P R⁢1 noncoh,P R⁢2 noncoh]=P noncoh⁢[γ 1,γ 2]superscript subscript 𝑃 𝑅 1 noncoh superscript subscript 𝑃 𝑅 2 noncoh superscript 𝑃 noncoh subscript 𝛾 1 subscript 𝛾 2[P_{R1}^{\text{noncoh}},P_{R2}^{\text{noncoh}}]=P^{\text{noncoh}}[\gamma_{1},% \gamma_{2}][ italic_P start_POSTSUBSCRIPT italic_R 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT , italic_P start_POSTSUBSCRIPT italic_R 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT ] = italic_P start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT [ italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ], then (21) can be written as:

W 2I2O 1,noncoh=∑i=1 2(γ i⁢W 2∥noncoh)∑i=1 2(γ i).W_{\text{2I2O}}^{1,\text{noncoh}}=\frac{\sum_{i=1}^{2}{\left({\gamma}{i}W{2% \parallel}^{\text{noncoh}}\right)}}{\sum_{i=1}^{2}{\left({\gamma}_{i}\right)}}.italic_W start_POSTSUBSCRIPT 2I2O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 , noncoh end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG .(22)

To encapsulate the entire power efficiency of the system, including the receivers, the complete Waste Factor for a 2I2O system using non-coherent combining (W 2I2O noncoh superscript subscript 𝑊 2I2O noncoh W_{\text{2I2O}}^{\text{noncoh}}italic_W start_POSTSUBSCRIPT 2I2O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT) is calculated by cascading the first-stage W 2I2O 1,n⁢o⁢n⁢c⁢o⁢h superscript subscript 𝑊 2I2O 1 𝑛 𝑜 𝑛 𝑐 𝑜 ℎ W_{\text{2I2O}}^{1,noncoh}italic_W start_POSTSUBSCRIPT 2I2O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 , italic_n italic_o italic_n italic_c italic_o italic_h end_POSTSUPERSCRIPT and W 2∥noncoh superscript subscript 𝑊 limit-from 2 parallel-to noncoh W_{\text{2}\mathbin{\parallel}}^{\text{noncoh}}italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT in the dotted box from the RX side based on (10):

W 2I2O noncoh=W 2∥noncoh+(W 2I2O 1,noncoh−1)G 2∥noncoh.superscript subscript 𝑊 2I2O noncoh superscript subscript 𝑊 limit-from 2 parallel-to noncoh superscript subscript 𝑊 2I2O 1 noncoh 1 superscript subscript 𝐺 limit-from 2 parallel-to noncoh W_{\text{2I2O}}^{\text{noncoh}}=W_{\text{2}\mathbin{\parallel}}^{\text{noncoh}% }+\frac{(W_{\text{2I2O}}^{1,\text{noncoh}}-1)}{G_{\text{2}\mathbin{\parallel}}% ^{\text{noncoh}}}.italic_W start_POSTSUBSCRIPT 2I2O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT = italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT + divide start_ARG ( italic_W start_POSTSUBSCRIPT 2I2O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 , noncoh end_POSTSUPERSCRIPT - 1 ) end_ARG start_ARG italic_G start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT noncoh end_POSTSUPERSCRIPT end_ARG .(23)

Figure 5: General structure of the RU and UE.

IV-C 2 W for coherent combining MIMO

The received power of each RX using coherent combining is:

[P R⁢1 coh P R⁢2 coh]=([P T⁢1 P T⁢2]⁢[W C⁢11−1 W C⁢12−1 W C⁢21−1 W C⁢22−1])2,matrix superscript subscript 𝑃 𝑅 1 coh superscript subscript 𝑃 𝑅 2 coh superscript matrix subscript 𝑃 𝑇 1 subscript 𝑃 𝑇 2 matrix superscript subscript 𝑊 𝐶 11 1 superscript subscript 𝑊 𝐶 12 1 superscript subscript 𝑊 𝐶 21 1 superscript subscript 𝑊 𝐶 22 1 2\begin{bmatrix}P_{R1}^{\text{coh}}&P_{R2}^{\text{coh}}\end{bmatrix}=\left(% \begin{bmatrix}\sqrt{P_{T1}}&\sqrt{P_{T2}}\end{bmatrix}\begin{bmatrix}\sqrt{W_% {C11}^{-1}}&\sqrt{W_{C12}^{-1}}\ \sqrt{W_{C21}^{-1}}&\sqrt{W_{C22}^{-1}}\end{bmatrix}\right)^{2},[ start_ARG start_ROW start_CELL italic_P start_POSTSUBSCRIPT italic_R 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT end_CELL start_CELL italic_P start_POSTSUBSCRIPT italic_R 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ] = ( [ start_ARG start_ROW start_CELL square-root start_ARG italic_P start_POSTSUBSCRIPT italic_T 1 end_POSTSUBSCRIPT end_ARG end_CELL start_CELL square-root start_ARG italic_P start_POSTSUBSCRIPT italic_T 2 end_POSTSUBSCRIPT end_ARG end_CELL end_ROW end_ARG ] [ start_ARG start_ROW start_CELL square-root start_ARG italic_W start_POSTSUBSCRIPT italic_C 11 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_ARG end_CELL start_CELL square-root start_ARG italic_W start_POSTSUBSCRIPT italic_C 12 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_ARG end_CELL end_ROW start_ROW start_CELL square-root start_ARG italic_W start_POSTSUBSCRIPT italic_C 21 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_ARG end_CELL start_CELL square-root start_ARG italic_W start_POSTSUBSCRIPT italic_C 22 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_ARG end_CELL end_ROW end_ARG ] ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,(24)

where the exponent 2 denotes an element-wise square of each term in the resulting matrix.

The total signal-path power consumption of the system before the coherently combined received power going into RX is given by:

P consumed,path coh=∑i=1 2(P R⁢i coh⁢W 2∥coh),P_{\text{consumed,path}}^{\text{coh}}=\sum_{i=1}^{2}{\left({P}{Ri}^{\text{coh% }}W{2\parallel}^{\text{coh}}\right)},italic_P start_POSTSUBSCRIPT consumed,path end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT italic_R italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT ) ,(25)

where P R⁢i coh superscript subscript 𝑃 𝑅 𝑖 coh P_{Ri}^{\text{coh}}italic_P start_POSTSUBSCRIPT italic_R italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT is the coherently combined power at the antenna of RX i 𝑖 i italic_i, and W 2∥coh W_{2\parallel}^{\text{coh}}italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT represents Waste Factor for a 2-TX paralleled system together with the channel, which is the same as (13) with M = 2. Before the power goes into the receivers, we define the first-stage Waste Factor of the 2I2O system using coherent combining as:

W 2I2O 1,coh=∑i=1 2(P R⁢i coh⁢W 2∥coh)∑i=1 2(P R⁢i coh).W_{\text{2I2O}}^{1,\text{coh}}=\frac{\sum_{i=1}^{2}{\left({P}{Ri}^{\text{coh}% }W{2\parallel}^{\text{coh}}\right)}}{\sum_{i=1}^{2}{\left({P}_{Ri}^{\text{coh% }}\right)}}.italic_W start_POSTSUBSCRIPT 2I2O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 , coh end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT italic_R italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT italic_R italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT ) end_ARG .(26)

Letting [P R⁢1 coh,P R⁢2 coh]=P coh⁢[γ 1,γ 2]superscript subscript 𝑃 𝑅 1 coh superscript subscript 𝑃 𝑅 2 coh superscript 𝑃 coh subscript 𝛾 1 subscript 𝛾 2[P_{R1}^{\text{coh}},P_{R2}^{\text{coh}}]=P^{\text{coh}}[\gamma_{1},\gamma_{2}][ italic_P start_POSTSUBSCRIPT italic_R 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT , italic_P start_POSTSUBSCRIPT italic_R 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT ] = italic_P start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT [ italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ], then (26) can be written as:

W 2I2O 1,coh=∑i=1 2(γ i⁢W 2∥coh)∑i=1 2(γ i).W_{\text{2I2O}}^{1,\text{coh}}=\frac{\sum_{i=1}^{2}{\left({\gamma}{i}W{2% \parallel}^{\text{coh}}\right)}}{\sum_{i=1}^{2}{\left({\gamma}_{i}\right)}}.italic_W start_POSTSUBSCRIPT 2I2O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 , coh end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG .(27)

To encapsulate the entire power efficiency of the system, including the receivers, the complete Waste Factor for a 2I2O system using coherent combining (W 2I2O coh superscript subscript 𝑊 2I2O coh W_{\text{2I2O}}^{\text{coh}}italic_W start_POSTSUBSCRIPT 2I2O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT) is calculated by cascading the first stage W 2I2O 1,coh superscript subscript 𝑊 2I2O 1 coh W_{\text{2I2O}}^{1,\text{coh}}italic_W start_POSTSUBSCRIPT 2I2O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 , coh end_POSTSUPERSCRIPT and W 2∥coh superscript subscript 𝑊 limit-from 2 parallel-to coh W_{\text{2}\mathbin{\parallel}}^{\text{coh}}italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT from receiver side based on (14):

W 2I2O coh=W 2∥coh+(W 2I2O 1,coh−1)G 2∥coh.superscript subscript 𝑊 2I2O coh superscript subscript 𝑊 limit-from 2 parallel-to coh superscript subscript 𝑊 2I2O 1 coh 1 superscript subscript 𝐺 limit-from 2 parallel-to coh W_{\text{2I2O}}^{\text{coh}}=W_{\text{2}\mathbin{\parallel}}^{\text{coh}}+% \frac{(W_{\text{2I2O}}^{1,\text{coh}}-1)}{G_{\text{2}\mathbin{\parallel}}^{% \text{coh}}}.italic_W start_POSTSUBSCRIPT 2I2O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT = italic_W start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT + divide start_ARG ( italic_W start_POSTSUBSCRIPT 2I2O end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 , coh end_POSTSUPERSCRIPT - 1 ) end_ARG start_ARG italic_G start_POSTSUBSCRIPT 2 ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT coh end_POSTSUPERSCRIPT end_ARG .(28)

IV-C 3 W for general MIMO system

If we assume proportional combined powers of each RX, which means that the power after coherent or non-coherent combining, the received power of each RX has a relationship with each other, and all are related to the ratio of some P 𝑃 P italic_P,

[P R⁢1,P R⁢2,…,P R⁢M]=P⁢[γ 1,γ 2,…,γ M],subscript 𝑃 𝑅 1 subscript 𝑃 𝑅 2…subscript 𝑃 𝑅 M 𝑃 subscript 𝛾 1 subscript 𝛾 2…subscript 𝛾 M[P_{R1},P_{R2},\ldots,P_{R\text{M}}]=P[\gamma_{1},\gamma_{2},\ldots,\gamma_{% \text{M}}],[ italic_P start_POSTSUBSCRIPT italic_R 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT italic_R 2 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_R M end_POSTSUBSCRIPT ] = italic_P [ italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_γ start_POSTSUBSCRIPT M end_POSTSUBSCRIPT ] ,(29)

then (21) and (27) can be extended to:

W MIMO 1=∑i=1 M(γ i⁢W M∥)∑i=1 M(γ i).W_{\text{MIMO}}^{1}=\frac{\sum_{i=1}^{\text{M}}{\left({\gamma}{i}W{\text{M}% \parallel}\right)}}{\sum_{i=1}^{\text{M}}{\left({\gamma}_{i}\right)}}.italic_W start_POSTSUBSCRIPT MIMO end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT ( italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT ( italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG .(30)

Eventually, W 𝑊 W italic_W for the generalized MIMO system is:

W MIMO=W M∥+(W MIMO 1−1)G M∥.subscript 𝑊 MIMO subscript 𝑊 limit-from M parallel-to superscript subscript 𝑊 MIMO 1 1 subscript 𝐺 limit-from M parallel-to W_{\text{MIMO}}=W_{\text{M}\mathbin{\parallel}}+\frac{(W_{\text{MIMO}}^{1}-1)}% {G_{\text{M}\mathbin{\parallel}}}.italic_W start_POSTSUBSCRIPT MIMO end_POSTSUBSCRIPT = italic_W start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT + divide start_ARG ( italic_W start_POSTSUBSCRIPT MIMO end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - 1 ) end_ARG start_ARG italic_G start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT end_ARG .(31)

where G M∥subscript 𝐺 limit-from M parallel-to G_{\text{M}\mathbin{\parallel}}italic_G start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT is the gain of the paralleled RXs, which can be calculated based on the ratio of the total output power of RXs to the input power of RXs:

G M∥=∑i=1 M P R⁢i⁢G R⁢i∑i=1 M P R⁢i.subscript 𝐺 limit-from M parallel-to superscript subscript 𝑖 1 M subscript 𝑃 𝑅 𝑖 subscript 𝐺 𝑅 𝑖 superscript subscript 𝑖 1 M subscript 𝑃 𝑅 𝑖 G_{\text{M}\mathbin{\parallel}}=\frac{\sum_{i=1}^{\text{M}}P_{Ri}G_{Ri}}{\sum_% {i=1}^{\text{M}}P_{Ri}}.italic_G start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_R italic_i end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT italic_R italic_i end_POSTSUBSCRIPT end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT M end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_R italic_i end_POSTSUBSCRIPT end_ARG .(32)

V Waste Factor Analysis in Communication Systems

Here, we discuss the calculation of the W 𝑊 W italic_W for a RAN depicted in Fig. 5. Waste Factor, W C=L C subscript 𝑊 𝐶 subscript 𝐿 𝐶 W_{C}=L_{C}italic_W start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT = italic_L start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT, is crucial for assessing the energy efficiency of communication channels to include mobile users[6, 4, 13, 12]. When considering links with antenna gain, we define an effective channel Waste Factor W C eff=L C/(G TX ant⁢G RX ant)=1/G C eff superscript subscript 𝑊 𝐶 eff subscript 𝐿 𝐶 superscript subscript 𝐺 TX ant superscript subscript 𝐺 RX ant 1 superscript subscript 𝐺 𝐶 eff W_{C}^{\text{eff}}={L_{C}}/\left({G_{\text{TX}}^{\text{ant}}G_{\text{RX}}^{% \text{ant}}}\right)=1/G_{C}^{\text{eff}}italic_W start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT eff end_POSTSUPERSCRIPT = italic_L start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT / ( italic_G start_POSTSUBSCRIPT TX end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ant end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT RX end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ant end_POSTSUPERSCRIPT ) = 1 / italic_G start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT eff end_POSTSUPERSCRIPT to simplify the overall channel loss, where G TX ant superscript subscript 𝐺 TX ant G_{\text{TX}}^{\text{ant}}italic_G start_POSTSUBSCRIPT TX end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ant end_POSTSUPERSCRIPT and G RX ant superscript subscript 𝐺 RX ant G_{\text{RX}}^{\text{ant}}italic_G start_POSTSUBSCRIPT RX end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ant end_POSTSUPERSCRIPT represent the antenna gains at the transmitter and receiver, respectively.

V-A Waste Factor Evaluation in RAN Site Chain

We first study a single TX at the base station (BS), specifically the radio unit (RU). This RU is composed of a digital-to-analog converter (DAC), a mixer, M phase shifters (PSs), M power amplifiers (PAs), and M antennas, with the power output from the DAC denoted as P source,out subscript 𝑃 source,out P_{\text{source,out}}italic_P start_POSTSUBSCRIPT source,out end_POSTSUBSCRIPT. This configuration is analyzed as a SIMO system, where each parallel cascade in the dashed box includes a PS, a PA, and an antenna.

Let us assume non-coherent combining at the receiver, which implies that the incoming signals are combined based on their power levels without considering their phase information. Refer to SectionIV-B, where W 𝑊 W italic_W for the system is calculated based on the performance of individual components within each parallel cascade.

W M∥RU=∑i=1 M P T,i×(W Ant+W PA−1 G Ant+W PS−1 G PA⁢G Ant)∑i=1 M P T,i,superscript subscript 𝑊 limit-from M parallel-to RU superscript subscript 𝑖 1 𝑀 subscript 𝑃 𝑇 𝑖 subscript 𝑊 Ant subscript 𝑊 PA 1 subscript 𝐺 Ant subscript 𝑊 PS 1 subscript 𝐺 PA subscript 𝐺 Ant superscript subscript 𝑖 1 𝑀 subscript 𝑃 𝑇 𝑖 W_{\text{M}\mathbin{\parallel}}^{\text{RU}}=\frac{\sum_{i=1}^{M}{P_{T,i}\times% \left(W_{\text{Ant}}+\frac{W_{\text{PA}}-1}{G_{\text{Ant}}}+\frac{W_{\text{PS}% }-1}{G_{\text{PA}}G_{\text{Ant}}}\right)}}{\sum_{i=1}^{M}{P_{T,i}}},italic_W start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT RU end_POSTSUPERSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_T , italic_i end_POSTSUBSCRIPT × ( italic_W start_POSTSUBSCRIPT Ant end_POSTSUBSCRIPT + divide start_ARG italic_W start_POSTSUBSCRIPT PA end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT Ant end_POSTSUBSCRIPT end_ARG + divide start_ARG italic_W start_POSTSUBSCRIPT PS end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT PA end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT Ant end_POSTSUBSCRIPT end_ARG ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_T , italic_i end_POSTSUBSCRIPT end_ARG ,(33)

where P T,i subscript 𝑃 𝑇 𝑖 P_{T,i}italic_P start_POSTSUBSCRIPT italic_T , italic_i end_POSTSUBSCRIPT represents the transmit power from the i t⁢h superscript 𝑖 𝑡 ℎ i^{th}italic_i start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT antenna element. Next, we determine W for the entire RU, which includes all parallel cascades and the mixer, denoted as W RU subscript 𝑊 RU W_{\text{RU}}italic_W start_POSTSUBSCRIPT RU end_POSTSUBSCRIPT, incorporating the overall gain of the paralleled structure in the RU:

W RU=W M∥RU+(W Mix−1)G M∥RU,subscript 𝑊 RU superscript subscript 𝑊 limit-from M parallel-to RU subscript 𝑊 Mix 1 superscript subscript 𝐺 limit-from M parallel-to RU W_{\text{RU}}=W_{\text{M}\mathbin{\parallel}}^{\text{RU}}+\frac{(W_{\text{Mix}% }-1)}{G_{\text{M}\mathbin{\parallel}}^{\text{RU}}},italic_W start_POSTSUBSCRIPT RU end_POSTSUBSCRIPT = italic_W start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT RU end_POSTSUPERSCRIPT + divide start_ARG ( italic_W start_POSTSUBSCRIPT Mix end_POSTSUBSCRIPT - 1 ) end_ARG start_ARG italic_G start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT RU end_POSTSUPERSCRIPT end_ARG ,(34)

where G M∥=∑i=1 M P T,i/(P source,out⁢G Mix)subscript 𝐺 limit-from M parallel-to superscript subscript 𝑖 1 𝑀 subscript 𝑃 𝑇 𝑖 subscript 𝑃 source,out subscript 𝐺 Mix G_{\text{M}\mathbin{\parallel}}={\sum_{i=1}^{M}{P_{T,i}}}/{\left(P_{\text{% source,out}}G_{\text{Mix}}\right)}italic_G start_POSTSUBSCRIPT M ∥ end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_T , italic_i end_POSTSUBSCRIPT / ( italic_P start_POSTSUBSCRIPT source,out end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT Mix end_POSTSUBSCRIPT ) is the gain of the parallel cascade in the dashed box of RU.

For a system with an identical paralleled cascade, the transmit power from each antenna element is the same, and W in (34) can be simplified to:

W RU=W Ant+W PA−1 G Ant+W PS−1 G PA⁢G Ant+W Mix−1 G PS⁢G PA⁢G Ant.subscript 𝑊 RU subscript 𝑊 Ant subscript 𝑊 PA 1 subscript 𝐺 Ant subscript 𝑊 PS 1 subscript 𝐺 PA subscript 𝐺 Ant subscript 𝑊 Mix 1 subscript 𝐺 PS subscript 𝐺 PA subscript 𝐺 Ant W_{\text{RU}}=W_{\text{Ant}}+\frac{W_{\text{PA}}-1}{G_{\text{Ant}}}+\frac{W_{% \text{PS}}-1}{G_{\text{PA}}G_{\text{Ant}}}+\frac{W_{\text{Mix}}-1}{G_{\text{PS% }}G_{\text{PA}}G_{\text{Ant}}}.italic_W start_POSTSUBSCRIPT RU end_POSTSUBSCRIPT = italic_W start_POSTSUBSCRIPT Ant end_POSTSUBSCRIPT + divide start_ARG italic_W start_POSTSUBSCRIPT PA end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT Ant end_POSTSUBSCRIPT end_ARG + divide start_ARG italic_W start_POSTSUBSCRIPT PS end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT PA end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT Ant end_POSTSUBSCRIPT end_ARG + divide start_ARG italic_W start_POSTSUBSCRIPT Mix end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT PS end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT PA end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT Ant end_POSTSUBSCRIPT end_ARG .(35)

Extending the analysis to the receiver side alone, the user equipment (UE) can be treated as a MISO system, and W UE subscript 𝑊 UE W_{\text{UE}}italic_W start_POSTSUBSCRIPT UE end_POSTSUBSCRIPT is calculated assuming the same components across all parallel cascades in a single UE using non-coherent combining, based on eq. (11) for non-coherent combing MISO in SectionIV-A:

W UE=W Mix+W PS−1 G Mix+W LNA−1 G Mix⁢G PS+W Ant−1 G Mix⁢G PS⁢G LNA subscript 𝑊 UE subscript 𝑊 Mix subscript 𝑊 PS 1 subscript 𝐺 Mix subscript 𝑊 LNA 1 subscript 𝐺 Mix subscript 𝐺 PS subscript 𝑊 Ant 1 subscript 𝐺 Mix subscript 𝐺 PS subscript 𝐺 LNA W_{\text{UE}}=W_{\text{Mix}}+\frac{W_{\text{PS}}-1}{G_{\text{Mix}}}+\frac{W_{% \text{LNA}}-1}{G_{\text{Mix}}G_{\text{PS}}}+\frac{W_{\text{Ant}}-1}{G_{\text{% Mix}}G_{\text{PS}}G_{\text{LNA}}}italic_W start_POSTSUBSCRIPT UE end_POSTSUBSCRIPT = italic_W start_POSTSUBSCRIPT Mix end_POSTSUBSCRIPT + divide start_ARG italic_W start_POSTSUBSCRIPT PS end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT Mix end_POSTSUBSCRIPT end_ARG + divide start_ARG italic_W start_POSTSUBSCRIPT LNA end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT Mix end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT PS end_POSTSUBSCRIPT end_ARG + divide start_ARG italic_W start_POSTSUBSCRIPT Ant end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT Mix end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT PS end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT LNA end_POSTSUBSCRIPT end_ARG(36)

Finally, Waste Factor for the entire RAN cascade, including a RU, a wireless channel, and a UE is derived using eq. (6):

W RAN=W UE+W C eff−1 G UE+W RU−1 G C eff⁢G UE.subscript 𝑊 RAN subscript 𝑊 UE superscript subscript 𝑊 𝐶 eff 1 subscript 𝐺 UE subscript 𝑊 RU 1 superscript subscript 𝐺 𝐶 eff subscript 𝐺 UE W_{\text{RAN}}=W_{\text{UE}}+\frac{W_{C}^{\text{eff}}-1}{G_{\text{UE}}}+\frac{% W_{\text{RU}}-1}{G_{C}^{\text{eff}}G_{\text{UE}}}.italic_W start_POSTSUBSCRIPT RAN end_POSTSUBSCRIPT = italic_W start_POSTSUBSCRIPT UE end_POSTSUBSCRIPT + divide start_ARG italic_W start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT eff end_POSTSUPERSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT UE end_POSTSUBSCRIPT end_ARG + divide start_ARG italic_W start_POSTSUBSCRIPT RU end_POSTSUBSCRIPT - 1 end_ARG start_ARG italic_G start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT eff end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT UE end_POSTSUBSCRIPT end_ARG .(37)

VI Simulation and Results Discussion

Simulations were conducted within a 28 GHz Coordinated Multi-Point (CoMP) communication system to assess the effectiveness of using WF for a MIMO system. The network featured 512 UEs, with the number of BSs varying from 1 to 20, the simulation cell has a radius of 1 km, and each BS encompasses a radius of 200 m. It is assumed that only users within these ranges are served. Refer to Section V, BSs and UEs using non-coherent combining were positioned at heights of 15 and 1.5 meters, respectively, with a separation of 300 m between the nearest BSs. Antenna gains for BSs and UEs were set to 26 dB and 6 dB, respectively, and the bandwidth is 400 MHz [16]. This configuration represents a typical urban microcell (UMi) environment, aiming for an SNR of 10 dB at the UEs. The transmit power was dynamically adjusted to meet the SNR requirement at each UE, with a cap of 100 Watts for each BS. The path loss of the wireless channel, incorporating a line-of-sight (LOS) path loss exponent of 2.27 and a shadow fading standard deviation of 8.15 dB, was derived from urban propagation research conducted by NYU WIRELESS [17]. Additionally, UEs situated within the coverage areas of multiple BSs received combined power from all such BSs, demonstrating the collaborative transmission feature of the CoMP system [18].

Simulation results, shown in Fig. 6, demonstrated that as the number of BSs increased, the WF decreased. This indicates that network densification improves the overall system power efficiency. Additionally, when the number of BSs is small, the slope of Fig. 6 is steeper, indicating that WF drops (improves) significantly in a CoMP scenario with increasing BS density, WF drops significantly as the number of BSs increases. However, as the number of BSs continues to increase, the WF reaches a lower limit. The results suggest that a combination of network densification and strategic component optimization can lead to better system design.

\begin{overpic}[width=224.03743pt]{Figures/WF_28GHz_BSVar.pdf} \put(35.0,22.0){\includegraphics[width=115.63243pt]{Figures/UEBSPositions.pdf}% } \end{overpic}

Figure 6: Waste Figure with different base station numbers in a 28 GHz coordinated multi-point communication network.

VII Conclusions

This paper introduces the Waste Factor theory for complex paralleled architectures like MISO, SIMO, and MIMO. The work compares W and WF with conventional energy efficiency metrics, highlighting advantages in providing a comprehensive and flexible approach to power efficiency analysis. By extending the analysis of power waste to include the impact of communication channels, RAN chains, and paralleled systems, this work offers a complete framework for assessing the power efficiency of modern wireless networks. Future directions include applying W to evaluate power consumption in various beamforming structures and RF chain on/off strategies, as well as exploring ring structure network energy consumption.

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