Today, there is a great need for ensuring the energy-efficient design of circuits and systems, yet the electrical engineering field has lacked a clear, well-defined, unified key performance indicator (KPI) for quantifying the power efficiency of any device or cascade of devices or systems. The industry has such a metric for quantifying additive noise along a cascade and this is known as the noise factor (F) or noise figure (NF) when expressed in dB. This standardized approach to quantifying additive noise now dons the specification sheets of virtually all receiving devices or receivers and is used as a key figure of merit (FoM) in the research and design communities when creating new devices or exploring system concepts.

This article develops a FoM for the comparison of wasted power along a cascade. This new FoM, called the power waste factor, or simply the *waste factor* (W) or the *waste figure* (WF) expressed in dB, is derived using a similar mathematical modeling approach taken by Harald Friis in 1944 to create noise figure. This article will show that the waste figure is a useful FoM for comparing and contrasting the power efficiency of any circuit or cascade of circuits or systems. Just as noise figure characterizes the additive noise of a cascade, waste figure characterizes the additive wasted power along a cascade. Waste figure is a handy metric to determine and compare consumed power along a cascade and is useful in identifying design choices that optimize power consumption while providing a measure of the power efficiency of any circuit or cascade. In an era where energy efficiency is more important than ever, the waste factor and waste figure are new metrics for accomplishing power-efficient circuits and system designs. The waste factor (W), also called waste figure (WF) defined as WF (dB) = 10 log (W), merely requires knowledge of the device efficiency and device gain. Following the mathematical modeling approach taken by Harald Friis where he developed noise figure,^{1} this article offers a standardized framework for evaluating power consumption in any wired or wireless device, system or network. This framework may have utility in standards bodies to reduce global power consumption.

The waste figure provides a simple analytical formulation and mathematical model for the additive power wasted along virtually any cascade. As shown here, W enables extremely general analysis, with application to circuits, transceivers, channels and data centers. Virtually anything that forms a cascade of devices or systems for information flow may be characterized and quantified by W. Using the mathematical formulation of W in circuit and system analysis, it becomes possible to characterize specific power efficiency performance levels while gaining insight into design approaches that assure minimal power consumption. Furthermore, W allows a standardized way to interpret and analyze power consumption in any device or cascade, making it a powerful analysis tool as well as the basis of a learning model for artificial intelligence (AI) and machine learning (ML) design and control for optimized power efficiency. While some authors have recently applied the waste figure to analyze the power efficiency and to make design tradeoffs in unmanned aerial vehicle (UAV) cellular infrastructure systems,^{2} millimeter wave wireless network^{3} phase shifters in sub-THz phased arrays and data centers,^{4 }the waste figure remains relatively unknown and undiscovered. However, it offers a standard FoM for evaluating the power efficiency of any circuit or cascade.

This paper introduces the mathematical derivation of the waste figure and shows that its mathematical basis is strikingly similar to noise figure. The definition of waste figure is a simple and powerful metric based on the efficiency (*η*) and gain (G) of the various elements of a cascade. While the power efficiency may be referred to either the input or output signal power of the cascade, it is most sensible to consider the waste factor as related to the signal output power of a device or cascade, P_{signal,out}. The result of the waste figure for a cascade will be derived and it will be shown that the mathematical expressions for the waste figure of a cascaded system have a nearly identical form to Friis’ original noise figure expression. The canonical results when deriving the waste figure for a passive load will be shown, which yields mathematically similar results as compared to the noise figure as well as key results when using W to analyze the power efficiency of a transmitter (TX)-receiver (RX) link with a lossy channel as part of the cascade. It will be shown that the waste figure may be used in circuit design to indicate the most power-efficient cascades while illuminating the most critical components that dominate power consumption and thus energy efficiency.

As shown subsequently, waste factor denotes the *total power consumed by a circuit or cascade*, divided by the* signal power out of the circuit or cascade*, and represents a new way to consider wasted power or power efficiency along a cascade. By definition, W is always greater than or equal to 1, where W=1 (WF=0 dB) denotes an optimally power-efficient circuit or cascade, where all the power consumed by the circuit or cascade is found to be in the usable signal output power. Conversely, if W equals infinity, it means there is no power output delivered from the cascade while power is being consumed (e.g., a dummy load). The article shows that waste factor is the inverse of power efficiency in standard passive circuit theory and the inverse of the original definition of total power added efficiency (PAE) for DC-powered amplifiers.^{12} The waste factor may be used to readily quantify the total wasted power and efficiency of a cascade of matched, linear devices.

Examples will be shown to illustrate how waste figure can characterize the power efficiency of various types of cascades, including a cascade of a TX, a propagation channel and a RX, a homodyne transmitter, and even a data center (we show here how W offers improvement over a popular data center power efficiency metric). These examples allow engineers to make simple approximations regarding the impact and placement of power-efficient components in a radio network operating in a lossy channel. The virtue of using W is that a standard metric may now be used to demonstrate intuitive design choices. This will allow engineers to conduct power efficiency studies on a circuit or cascade, or any type of source-to-sink link.

**FUNDAMENTALS OF NOISE FIGURE (NF) AND WASTE FIGURE (WF)**

This section shows the duality between the derivations of noise figure (NF) and waste figure (WF), which are used to evaluate additive noise and additive wasted power in any circuit or cascade, respectively.

**Noise Factor (F): Quantifying Additive Noise in Cascades**

Noise factor (F) defines the additive noise along a cascade which causes degradation of SNR along the cascade. The noise factor, F, was defined by Harald Friis in 1944^{1} as the ratio of the input SNR to output SNR, where *F=SNR _{i}/SNR*

_{o}. When F is expressed in dB terms, it is referred to as the noise figure, where a value of 0 dB indicates that no added noise and no degradation in SNR occurs along a device or a cascade of devices. Friis’ formula is widely used to calculate the overall F of cascaded devices, where each device has its own individual F and power gain, G. Once the total F is calculated for a cascade, it can be used to determine the overall noise power contribution of the entire cascade.

Friis was interested in modeling how the input noise level would be amplified and intensified in a receiver and created a mathematical model that considered the additional noise power contributed by components moving from the source (input) to the sink (detector). He was aware that there existed a nominal input thermal noise level to the cascade of *kTB *that could be measured as well as an output noise level that could be measured. Thermal white noise power is defined by: *N=kTB*, where *N *is the noise power available at the output of a thermal noise source,* k*=1.380×10^{-23} J/K is Boltzmann’s constant, *T* is the temperature in Kelvin and *B* is the noise bandwidth in Hz. At the input of a cascade, Friis was able to turn a signal on and off that could be added to the noise and detected at the output of the cascade. In this manner, he was able to adjust and measure the SNR at the input to the cascade relative to the output of the cascade. Since he had control of the cascade input signal power level and could measure the noise power at the output of the cascade relative to the signal and noise powers at the input, he defined noise figure relative to the input noise power of the cascade as shown in **Equation 1**:

From Equation 1, F for a cascaded system with matched loads is given by **Equation 2**, where the first component F_{1} is closest to the source, (e.g., the antenna or front-end of a RX). Friis found Equation 2 simply by systematically applying Equation 1 at each successive stage in a cascade realizing that each component had output noise *N*_{o} which included an additive noise component related to F.

In Equation 2, F_{i} represents the noise factor of the *i*_{th} device and

represents the power gain of the *i _{th} *device (linear, not in dB). For a given noise input power,

*N*

_{i}and noise output power,

*N*

_{o}, Friis showed that F (defined in Equation 1) was directly related to the additive noise contributed by the device or cascade alone as defined in

**Equation 3**:

From Equation 3, it is clear that (F-1)*GNi *represents the additive noise power contributed by the component or cascade, above and beyond the input noise power and as referred to the input. F = 1 (0 dB) means there is no additional noise power contributed by the cascade other than what was applied to the input and then amplified with ideal noiseless amplifiers.

**Waste Factor (W): A New Metric for the Analysis and Comparison of Power Efficiency**

Waste factor (W) or waste figure (WF in dB) characterizes the power efficiency of a cascaded system by modeling the power wasted (e.g., power consumed but not delivered as output power) in each of the components along a cascade. Like the mathematical modeling method used by Friis to derive noise figure, the power wasted by a device or cascade may be analyzed by modeling the progressive useful signal power transferred along a cascade relative to the total power consumed by the components of the cascade. The derivation of the waste figure assumes that all power that is not transferred along the cascade as signal output is simply “wasted power” since such wasted power is not being passed along the cascade to the next stage as a useful signal.

**This is a new way of defining power efficiency.** In this way, it becomes clear that instead of additive noise being accumulated at each stage of a cascade as in noise figure, the amount of wasted power (e.g., power not delivered in the signal to the successive stage) from each of the components accumulates from the input to the output. Since it is easy and customary to measure the power consumption of an entire cascade of devices (e.g., the total power consumed by a system) and also easy to measure the signal power at the output in watts (e.g.,* P*_{out} = *GP*_{in}), the waste factor is defined relative to the output of a device or cascade and is given by **Equation 4**:

Where:

*η*_{w} = *waste factor efficienc*y and it equals 1/W

*P _{consumed} *= total power consumed by the device or cascade

*P*_{out} = output signal power

*P _{non-signal} *= total consumed power minus the signal power delivered to the output, which is the wasted power of the device.

As mentioned previously, the waste factor for an active component is the inverse of the total PAE, a key metric for assessing power amplifier efficiency as shown by A. A. Sweet.^{12} The original PAE metric quantifies how effectively an RF amplifier with a DC power supply transforms the total consumed power, encompassing both the DC power supply and RF input power, into RF output power and is defined as^{12}

Inspection of Equation 4 (also see Equation 11) shows *PAE*^{#1} is identical to the reciprocal of W. In 1993, Walker^{13} introduced a second definition of PAE, given as

which is now widely regarded as the industry standard for evaluating the efficiency of RF power amplifiers. It is easy to show from Equation 4 that W is related to *PAE*^{#2} such that

which is nearly identical to *PAE*^{#1} when gain is large.

Similar to how Friis defined noise figure as NF = 10log_{10}(F) (dB), the waste figure is defined as WF = 10log_{10}(W) (dB). It was shown^{3,4,7,14} and derived previously that the waste factor of a cascade is expressed as **Equation 5**:

In Equation 5, *W*_{i} represents the waste factor of the *i*_{th} device and

represents the power gain of the *i*_{th} device (linear, not in dB) and the *N*_{th} device is the closest device to the information sink (e.g., the output of the cascade). Note the mathematical similarity between Equation 2 and Equation 5, except Equation 2 increases from source to sink and Equation 5 increases from sink to source.