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

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<EMAIL_ADDRESS>
###### 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,
which systematically models added wasted power in any cascade for parallel
systems such as MISO, SIMO, and MIMO systems, 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) for power
efficiency of parallel circuits and system configurations such as found in
MISO, SIMO, and MIMO wireless systems. $W$ 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 $W$ 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 $W$ over previous energy efficiency metrics:

  * •

Comprehensive Analysis: $W$ 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, $W$ is applicable to any source-to-sink communications
architecture, including SIMO, MISO and MIMO as shown here.

  * •

Optimization and Strategic Design: $W$ 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 $F=\text{SNR}_{i}/\text{SNR}_{o}$ [7]. The
total noise factor for a cascade with matched loads, from the source to sink,
was derived by Friis to be [7]:

$F=F_{1}+\frac{(F_{2}-1)}{G_{1}}+\frac{(F_{3}-1)}{G_{1}G_{2}}+\ldots+\frac{(F_{N}-1)}{\prod_{i=1}^{N-1}G_{i}},$
(1)

where $F_{i}$ and $G_{i}$ represent the noise factor and power gain of the
$i^{th}$ device, respectively. The additional noise power contributed by a
device in the cascade (independent of the input noise power) is quantified by:

$P_{additive-noise}=(F-1)G_{i}N_{i}.$ (2)

where $N_{i}$ is the noise power of $i^{th}$ 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. $W$ 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)$P_{source,out}$ is the power from a source that is applied at the input of
the cascaded system; 2) $P_{signal}$ is the signal power delivered to the
output of the cascade or device; 3) $P_{non-signal}$ 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)$P_{non-path}$ 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 $P_{non-path}$.

Waste Factor is defined in eq. (3) as the ratio of the total power consumed by
the signal path components along a cascade $P_{consumed,path}$, 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 $\eta_{\text{w}}$ is waste factor
efficiency, which is defined as the reciprocal of $W$:

$\vspace{-3pt}W=\frac{1}{\eta_{\text{w}}}=\frac{P_{consumed,path}}{P_{\text{out}}}=\frac{P_{signal}+P_{non-
signal}}{P_{signal}}.$ (3)

The total power consumption of any system $P_{consumed,total}$ is the sum of
power consumed on the signal path $P_{consumed,path}$ as well as all power
consumed by non-signal path $P_{non-path}$, where $W$ 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]:

$\displaystyle P_{consumed,total}$
$\displaystyle=P_{source,out}+P_{system,added}$ $\displaystyle+P_{non-
signal}+P_{non-path}$ $\displaystyle=WP_{signal}+P_{non-path}.$ (4)

where $P_{system,added}=P_{signal}-P_{source,out}$ 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
$P_{non-signal}$ across a cascaded system referring to either the signal
output power or input power, although it is most sensible to relate $W$ to the
signal output power [6, 12],

$P_{non-signal}=(W-1)P_{signal}=(W-1)P_{source,out}\prod_{i=1}^{N}G_{i},$ (5)

where $G_{i}$ denotes the gain of the $i^{th}$ stage in the cascade, and $N$
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, $W=1$ indicates no wasted power, while $W=\infty$ implies
that all power is wasted and no power is delivered to the output. Like $F$,
the parameter $W$ is independent of the signal powers. Following eq. (5), we
observe $F-1$ in (2) represents the added noise power relative to the input
noise, and analogously, $W-1$ in (5) represents the total wasted power
relative to the output signal power.

$W$ 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]:

$W=W_{N}+\frac{(W_{N-1}-1)}{G_{N}}+\cdots+\frac{(W_{1}-1)}{\prod_{i=2}^{N}G_{i}}.$
(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 $W$ 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-A1 W for non-coherent combining MISO

In Fig. 1, TX1 has a Waste Factor $W_{T1}$ with channel having $W_{C1}=L_{C1}$
and received power $P_{1}$ into RX1 with $W_{R1}$ and receiver gain $G_{R1}$.
Similarly, TX2 has its own channel and provides a received signal power
$P_{2}$ to RX1. Using results of a lossy channel from [6], $W$ 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
$W_{\text{2}\mathbin{\parallel}}$. For two received signals $r_{1}(t)$ and
$r_{2}(t)$ with received powers $P_{1}$ and $P_{2}$, respectively, non-
coherent combining has no phase information. The combined RX power
$P_{signal,noncoh}$ in non-coherent combining at the RX antenna is typically
the sum of individual powers:$P_{signal,noncoh}=P_{1}+P_{2}$ [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:

$\displaystyle W_{\text{2}\mathbin{\parallel}}^{noncoh}$
$\displaystyle=\frac{P_{consumed,path}}{P_{signal,noncoh}}$
$\displaystyle=\frac{{P_{1}\left(W_{C1}+\frac{W_{T1}-1}{G_{C1}}\right)}+{P_{2}\left(W_{C2}+\frac{W_{T2}-1}{G_{C2}}\right)}}{P_{1}+P_{2}}.$
(7)

Assuming the power received from each TX is proportional to some power $P$
with coefficients $\gamma_{1}$ and $\gamma_{2}$ for the first and second
transmitters, respectively, such that
$[P_{1},P_{2}]=P[\gamma_{1},\gamma_{2}]$, eq. (IV-A1) simplifies accordingly:

$W_{\text{2}\mathbin{\parallel}}^{noncoh}=\frac{{\gamma_{1}\left(W_{C1}+\frac{W_{T1}-1}{G_{C1}}\right)}+{\gamma_{2}\left(W_{C2}+\frac{W_{T2}-1}{G_{C2}}\right)}}{\gamma_{1}+\gamma_{2}}$
(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 $N$
devices, with the output signal power combined at device $N+1$ using either
coherent or non-coherent combining. To ascertain $W$ for the entire paralleled
system with a single output node, we first compute the $W$ for cascade $m$
using eq. (6):

$W_{cascade,m}=W_{N,m}+\frac{(W_{N-1,m}-1)}{G_{N,m}}+\cdots+\frac{(W_{1,m}-1)}{\prod_{i=2}^{N}G_{i,m}}.$
(9)

Subsequently, $W$ for the entire paralleled system using non-coherent
combining at ${N+1}^{th}$ device is given by:

$W_{\text{M}\mathbin{\parallel}}^{noncoh}=\frac{\sum_{i=1}^{\text{M}}\left({P_{N,i}\times
W_{cascade,i}}\right)}{\sum_{i=1}^{\text{M}}{P_{N,i}}}.$ (10)

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

$W_{\text{M}\mathbin{\parallel}}^{noncoh}=\frac{\sum_{i=1}^{\text{M}}{\gamma_{i}W_{cascade,i}}}{\sum_{i=1}^{\text{M}}{\gamma_{i}}}.$
(11)

####  IV-A2 W for coherent combining MISO

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

$P_{signal,coh}=\left|\sqrt{P_{1}}+\sqrt{P_{2}}\right|^{2},$ (12)

where $\theta_{1}$ and $\theta_{2}$ are the phases of $r_{1}(t)$ and
$r_{2}(t)$ respectively.

Considering coherent combining in a M TX single RX MISO system, we assume the
power output of $i^{th}$ TX as $P_{Ti}$, and the power received from
transmitter $i$ at the RX before combining as $P_{i}$, and the power into the
RX after coherent combining at the RX antenna as $P_{signal,coh}$. Here we
assume
$[P_{1},P_{2},\ldots,P_{M}]=P[\gamma_{1},\gamma_{2},\ldots,\gamma_{M}]$, 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 $P$.

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

$W_{\text{M}\mathbin{\parallel}}^{coh}=\frac{\sum_{i=1}^{\text{M}}\left[{\gamma_{i}\left(W_{Ci}+\frac{W_{Ti}-1}{G_{Ci}}\right)}\right]}{\left|\sum_{i=1}^{\text{M}}{\sqrt{\gamma_{i}}}\right|^{2}}.$
(13)

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

$W_{\text{M}\mathbin{\parallel}}^{coh}=\frac{\sum_{i=1}^{\text{M}}\left({P_{N,i}\times
W_{cascade,i}}\right)}{\left|\sum_{i=1}^{\text{M}}{\sqrt{P_{N,i}}}\right|^{2}}=\frac{\sum_{i=1}^{\text{M}}\left({\gamma_{i}W_{cascade,i}}\right)}{\left|\sum_{i=1}^{\text{M}}{\sqrt{\gamma_{i}}}\right|^{2}}.$
(14)

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

$W_{\text{MISO}}=W_{N+1}+\frac{(W_{\text{M}\mathbin{\parallel}}-1)}{G_{N+1}}.$
(15)

where $W_{N+1}$ and $G_{N+1}$ represents the Waste Factor and gain of the
${(N+1)}^{th}$ 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 $D_{0,1}$, is characterized by a Waste Factor $W_{0}$ and a gain
$G_{0}$. 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-B1 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
$[P_{N,1},P_{N,2},\ldots,P_{N,M}]=P[\gamma_{1},\gamma_{2},\ldots,\gamma_{M}]$,
then Waste Factor for the dashed box using non-coherent combining at its
output is:

$W_{\text{M}\mathbin{\parallel}}^{noncoh}=\frac{\sum_{i=1}^{\text{M}}\left({P_{N,i}\times
W_{cascade,i}}\right)}{\sum_{i=1}^{\text{M}}{P_{N,i}}}=\frac{\sum_{i=1}^{\text{M}}\left({\gamma_{i}\times
W_{cascade,i}}\right)}{\sum_{i=1}^{\text{M}}{\gamma_{i}}},$ (16)

and
$G_{\text{M}\mathbin{\parallel}}^{noncoh}=\left(\sum_{i=1}^{\text{M}}{P_{N,i}}\right)/\left(\sum_{i=1}^{\text{M}}{P_{0,i}}\right)$
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-B2 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 $D_{0,1}$, and
$[P_{N,1},P_{N,2},\ldots,P_{N,M}]=P[\gamma_{1},\gamma_{2},\ldots,\gamma_{M}]$.
Then Waste Factor for the system in the dashed box is:

$W_{\text{M}\mathbin{\parallel}}^{coh}=\frac{\sum_{i=1}^{\text{M}}\left({P_{N,i}\times
W_{cascade,i}}\right)}{\left|\sum_{i=1}^{\text{M}}{\sqrt{P_{N,i}}}\right|^{2}}=\frac{\sum_{i=1}^{\text{M}}\left({\gamma_{i}W_{cascade,i}}\right)}{\left|\sum_{i=1}^{\text{M}}{\sqrt{\gamma_{i}}}\right|^{2}},$
(17)

and
$G_{\text{M}\parallel}^{{coh}}={\left(\sum_{i=1}^{M}\sqrt{P_{N,i}}\right)^{2}}/{\left(\sum_{i=1}^{M}P_{0,i}\right)}$
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:

$W_{\text{SIMO}}=W_{\text{M}\mathbin{\parallel}}+\frac{(W_{0}-1)}{G_{\text{M}\mathbin{\parallel}}}.$
(18)

where $G_{\text{M}\mathbin{\parallel}}$ 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-C1 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:

$\begin{bmatrix}P_{R1}^{noncoh}&P_{R2}^{noncoh}\end{bmatrix}=\begin{bmatrix}P_{T1}&P_{T2}\end{bmatrix}\begin{bmatrix}W_{C11}^{-1}&W_{C12}^{-1}\\\
W_{C21}^{-1}&W_{C22}^{-1}\end{bmatrix}.$ (19)

where $P_{Ri}^{noncoh}$ is the non-coherently combined power at the antenna of
RX $i$. The total signal-path power consumption of the system before the
receiver using non-coherent combining is:

$P_{consumed,path}^{noncoh}=\sum_{i=1}^{2}{\left({P}_{Ri}^{noncoh}W_{2\parallel}^{noncoh}\right)},$
(20)

where $W_{2\parallel}^{noncoh}$ 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:

$W_{\text{2I2O}}^{1,noncoh}=\frac{\sum_{i=1}^{2}{\left({P}_{Ri}^{noncoh}W_{2\parallel}^{noncoh}\right)}}{\sum_{i=1}^{2}{\left({P}_{Ri}^{noncoh}\right)}}.$
(21)

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

Letting $[P_{R1}^{noncoh},P_{R2}^{noncoh}]=P^{noncoh}[\gamma_{1},\gamma_{2}]$,
then (21) can be written as:

$W_{\text{2I2O}}^{1,noncoh}=\frac{\sum_{i=1}^{2}{\left({\gamma}_{i}W_{2\parallel}^{noncoh}\right)}}{\sum_{i=1}^{2}{\left({\gamma}_{i}\right)}}.$
(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 ($W_{\text{2I2O}}^{noncoh}$) is calculated by cascading the first-
stage $W_{\text{2I2O}}^{1,noncoh}$ and
$W_{\text{2}\mathbin{\parallel}}^{noncoh}$ in the dotted box from the RX side
based on (10):

$W_{\text{2I2O}}^{noncoh}=W_{\text{2}\mathbin{\parallel}}^{noncoh}+\frac{(W_{\text{2I2O}}^{1,noncoh}-1)}{G_{\text{2}\mathbin{\parallel}}^{noncoh}}.$
(23)

Figure 5: General structure of the RU and UE.

####  IV-C2 W for coherent combining MIMO

The received power of each RX using coherent combining is:

$\begin{bmatrix}P_{R1}^{coh}&P_{R2}^{coh}\end{bmatrix}=\left(\begin{bmatrix}\sqrt{P_{T1}}&\sqrt{P_{T2}}\end{bmatrix}\begin{bmatrix}\sqrt{W_{C11}^{-1}}&\sqrt{W_{C12}^{-1}}\\\
\sqrt{W_{C21}^{-1}}&\sqrt{W_{C22}^{-1}}\end{bmatrix}\right)^{2},$ (24)

where the square here performs an element-wise square.

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

$P_{consumed,path}^{coh}=\sum_{i=1}^{2}{\left({P}_{Ri}^{coh}W_{2\parallel}^{coh}\right)},$
(25)

where $P_{Ri}^{coh}$ is the coherently combined power at the antenna of RX
$i$, and $W_{2\parallel}^{coh}$ 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:

$W_{\text{2I2O}}^{1,coh}=\frac{\sum_{i=1}^{2}{\left({P}_{Ri}^{coh}W_{2\parallel}^{coh}\right)}}{\sum_{i=1}^{2}{\left({P}_{Ri}^{coh}\right)}}.$
(26)

Letting $[P_{R1}^{coh},P_{R2}^{coh}]=P^{coh}[\gamma_{1},\gamma_{2}]$, then
(26) can be written as:

$W_{\text{2I2O}}^{1,coh}=\frac{\sum_{i=1}^{2}{\left({\gamma}_{i}W_{2\parallel}^{coh}\right)}}{\sum_{i=1}^{2}{\left({\gamma}_{i}\right)}}.$
(27)

To encapsulate the entire power efficiency of the system, including the
receivers, the complete Waste Factor for a 2I2O system using coherent
combining ($W_{\text{2I2O}}^{coh}$) is calculated by cascading the first stage
$W_{\text{2I2O}}^{1,coh}$ and $W_{\text{2}\mathbin{\parallel}}^{coh}$ from
receiver side based on (14):

$W_{\text{2I2O}}^{coh}=W_{\text{2}\mathbin{\parallel}}^{coh}+\frac{(W_{\text{2I2O}}^{1,coh}-1)}{G_{\text{2}\mathbin{\parallel}}^{coh}}.$
(28)

####  IV-C3 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
$P$,

$[P_{R1},P_{R2},\ldots,P_{R\text{M}}]=P[\gamma_{1},\gamma_{2},\ldots,\gamma_{\text{M}}],$
(29)

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

$W_{\text{MIMO}}^{1}=\frac{\sum_{i=1}^{\text{M}}{\left({\gamma}_{i}W_{\text{M}\parallel}\right)}}{\sum_{i=1}^{\text{M}}{\left({\gamma}_{i}\right)}}.$
(30)

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

$W_{\text{MIMO}}=W_{\text{M}\mathbin{\parallel}}+\frac{(W_{\text{MIMO}}^{1}-1)}{G_{\text{M}\mathbin{\parallel}}}.$
(31)

where $G_{\text{M}\mathbin{\parallel}}$ 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:

$G_{\text{M}\mathbin{\parallel}}=\frac{\sum_{i=1}^{\text{M}}P_{Ri}G_{Ri}}{\sum_{i=1}^{\text{M}}P_{Ri}}.$
(32)

##  V Waste Factor Analysis in Communication Systems

Here, we discuss the calculation of the $W$ for a RAN depicted in Fig. 5.
Waste Factor, $W_{C}=L_{C}$, 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
$W_{C}^{\text{eff}}={L_{C}}/\left({G_{\text{TX}}^{\text{ant}}G_{\text{RX}}^{\text{ant}}}\right)=1/G_{C}^{\text{eff}}$
to simplify the overall channel loss, where $G_{\text{TX}}^{\text{ant}}$ and
$G_{\text{RX}}^{\text{ant}}$ 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 $P_{source,out}$. 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 Section IV-B, where $W$ for the system is
calculated based on the performance of individual components within each
parallel cascade.

$W_{\text{M}\mathbin{\parallel}}^{\text{RU}}=\frac{\sum_{i=1}^{M}{P_{T,i}\times\left(W_{\text{Ant}}+\frac{W_{\text{PA}}-1}{G_{\text{Ant}}}+\frac{W_{\text{PS}}-1}{G_{\text{PA}}G_{\text{Ant}}}\right)}}{\sum_{i=1}^{M}{P_{T,i}}},$
(33)

where $P_{T,i}$ represents the transmit power from the $i^{th}$ antenna
element. Next, we determine W for the entire RU, which includes all parallel
cascades and the mixer, denoted as $W_{\text{RU}}$, incorporating the overall
gain of the paralleled structure in the RU:

$W_{\text{RU}}=W_{\text{M}\mathbin{\parallel}}^{\text{RU}}+\frac{(W_{\text{Mix}}-1)}{G_{\text{M}\mathbin{\parallel}}^{\text{RU}}},$
(34)

where
$G_{\text{M}\mathbin{\parallel}}={\sum_{i=1}^{M}{P_{T,i}}}/{\left(P_{source,out}G_{\text{Mix}}\right)}$
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:

$W_{\text{RU}}=W_{\text{Ant}}+\frac{W_{\text{PA}}-1}{G_{\text{Ant}}}+\frac{W_{\text{PS}}-1}{G_{\text{PA}}G_{\text{Ant}}}+\frac{W_{\text{Mix}}-1}{G_{\text{PS}}G_{\text{PA}}G_{\text{Ant}}}.$
(35)

Extending the analysis to the receiver side alone, the user equipment (UE) can
be treated as a MISO system, and $W_{\text{UE}}$ 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 Section IV-A:

$W_{\text{UE}}=W_{\text{Mix}}+\frac{W_{\text{PS}}-1}{G_{\text{Mix}}}+\frac{W_{\text{LNA}}-1}{G_{\text{Mix}}G_{\text{PS}}}+\frac{W_{\text{Ant}}-1}{G_{\text{Mix}}G_{\text{PS}}G_{\text{LNA}}}$
(36)

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

$W_{\text{RAN}}=W_{\text{UE}}+\frac{W_{C}^{\text{eff}}-1}{G_{\text{UE}}}+\frac{W_{\text{RU}}-1}{G_{C}^{\text{eff}}G_{\text{UE}}}.$
(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 in alignment with [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=216.81pt]{Figures/WF_28GHz_BSVar.pdf}
\put(33.0,25.0){\includegraphics[width=124.30447pt]{Figures/UEBSPosition.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 Factory 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.

## References

  * [1] Y. Xing _et al._ , “High altitude platform stations (haps): Architecture and system performance,” in _2021 IEEE 93rd Vehicular Technology Conference (VTC-Spring)_ , 2021, pp. 1–6.
  * [2] J. N. Murdock and T. S. Rappaport, “Consumption factor: A figure of merit for power consumption and energy efficiency in broadband wireless communications,” in _2011 IEEE GLOBECOM Workshops (GC Wkshps)_ , 2011, pp. 1393–1398.
  * [3] O. Kanhere _et al._ , “A power efficiency metric for comparing energy consumption in future wireless networks in the millimeter-wave and terahertz bands,” _IEEE Wireless Communications_ , vol. 29, no. 6, pp. 56–63, 2022.
  * [4] J. N. Murdock and T. S. Rappaport, “Consumption factor and power-efficiency factor: A theory for evaluating the energy efficiency of cascaded communication systems,” _IEEE Journal on Selected Areas in Communications_ , vol. 32, no. 2, pp. 221–236, 2014.
  * [5] M. Ying _et al._ , “Waste factor: A new metric for evaluating power efficiency in any cascade,” in _2023 IEEE Global Communications Conference (GLOBECOM)_ , 2023, pp. 6735–6740.
  * [6] T. S. Rappaport, M. Ying, and D. Shakya, “Waste figure and waste factor: New metrics for evaluating power efficiency in any circuit or cascade,” _Microwave Journal_ , vol. 67, no. 5, pp. 54–84, May 2024.
  * [7] H. Friis, “Noise figures of radio receivers,” _Proceedings of the IRE_ , vol. 32, no. 7, pp. 419–422, July 1944.
  * [8] D. Boyle _et al._ , “Energy-efficient communication in wireless networks,” _ICT-Energy Concepts for Energy Efficiency and Sustainability_ , 2017.
  * [9] T. Hoßfeld _et al._ , “Analytical model for the energy efficiency in low power iot deployments,” in _2022 IEEE 8th International Conference on Network Softwarization (NetSoft)_. IEEE, 2022, pp. 19–24.
  * [10] E. McCune, “A comprehensive view of wireless link energy efficiency,” in _2019 IEEE Global Communications Conference (GLOBECOM)_ , 2019, pp. 1–5.
  * [11] T. Chen, H. Kim, and Y. Yang, “Energy efficiency metrics for green wireless communications,” in _2010 International Conference on Wireless Communications & Signal Processing (WCSP)_. IEEE, 2010, pp. 1–6.
  * [12] T. S. Rappaport _et al._ , “Waste factor and waste figure: A unified theory for modeling and analyzing wasted power in radio access networks for improved sustainability,” _(submitted) IEEE IEEE Open Journal of the Communications Society_ , April 2024.
  * [13] T. S. Rappaport and N. Piovesan, “Waste factor, a figure of merit for sustainability: Holistic analysis of energy and throughput trade-offs in radio access networks,” in _2024 Wireless Communications and Networking Conference (WCNC) Tutorial 14_ , April 2024.
  * [14] S. Sun _et al._ , “Millimeter wave multi-beam antenna combining for 5G cellular link improvement in new york city,” in _2014 IEEE International Conference on Communications (ICC)_ , 2014, pp. 5468–5473.
  * [15] T. S. Rappaport, _Wireless communications: principles and practice_. Cambridge University Press, updated 2nd edition, 2024.
  * [16] D. Shakya _et al._ , “Dense urban outdoor-indoor coverage from 3.5 to 28 GHz,” in _2022 IEEE International Conference on Communications (ICC)_ , 2022, pp. 932–937.
  * [17] ——, “Radio propagation measurements and statistical channel models for outdoor urban microcells in open squares and streets at 142, 73, and 28 GHz,” _IEEE Transactions on Antennas and Propagation_ , 2024.
  * [18] G. R. MacCartney and T. S. Rappaport, “Millimeter-wave base station diversity for 5G coordinated multipoint (CoMP) applications,” _IEEE Transactions on Wireless Communications_ , vol. 18, no. 7, pp. 3395–3410, 2019.