For a given signal input power to a cascade, Pin, and cascade signal output power, Pout, W is a FoM that represents a measure of total additive power that is wasted (e.g., not part of the delivered signal power) at the output of a cascade. W quantifies the amount of consumed power that is wasted (e.g., consumed power that is not found in the output signal power) relative to the total signal power delivered at the output of the cascade, independent of the signal power applied at the input of the cascade (see Equations 8 and Equation 9). Thus, W defines a relationship between added wasted power of a device or cascade, vis à vis the delivered signal power in a similar way that Friis defined in Equation 3. This is shown in Equation 6:
From Equation 6, it is clear that (W - 1)Pout represents the total additive wasted power contributed by a component or by all the components within the cascade and is conveniently referred to the output. W=1 (0 dB) means there is no wasted power contributed by a component or by the cascade and the cascade has perfect 100 percent efficiency with all the power consumed by the cascade found in the output signal power. Similarly, W = infinity means all the power is wasted in the cascade, with no output signal power. Such formulation provides an intuitive way to understand power waste at each stage of the cascade and allows engineers to use W to compare the power efficiency of different devices and systems in a new way, just as noise figure did for noise analysis 80 years ago.
Figure 1 Power decomposition of a cascaded communication system or device.5-7
The mathematical basis of waste factor and waste figure was derived originally by Murdock and Rappaport5,6,7,14 by noting that the power consumption of any device in a circuit or system may be decomposed into four distinct power consumption components which may be summed together to determine the total consumed power of any device or network. The four power components include (i) the signal path power delivered to the output by a device on the signal path within a circuit or cascade (e.g., the output signal power that is delivered to the sink), (ii) the power consumed by a device which is on the signal path (e.g., a component on the cascade), but which is not provided in the output signal path (this is called the additive “wasted” power of a component, since as described previously, the component is on the cascade but such power is not delivered to the output signal of the device), (iii) all other power that is consumed by components that are not on the signal path or which are not associated with a device on the signal-path cascade and (iv) the signal power delivered to the input of the device or cascade by a source that is connected to the input. Figure 1 and Table 1 describe the four types of power that make up any circuit or system.
The non-path powers are not considered in the derivation of waste figure since these components are not found in a cascade along a signal path, just as noise figure does not consider non-path components as noise contributors. However, the total power consumption of both the signal path and non-path components is easily found using W and the power components of Table 1 using upcoming Equation 9. While waste figure characterizes a device or cascade on a signal path, the basic principles of power superposition and waste factor may be useful in analyzing non-path wasted power and power consumption using similar mathematical models and new figures of merit. It may be that non-path powers include the quiescent power drain of cascaded components that are in a sleep/off state or for the case when impedances are high-Z or greatly mismatched. This remains an open area for definition and usage.
In the development of noise figure, the output power of the input source was due to thermal noise at the receiver input. For waste figure, the output power of a signal source is the signal input power to the cascade or device. When considering the four power sources of Table 1, it becomes clear that for any cascade or device along the signal path, which will be called a “system,” the total power consumed by just the system carrying the signal (Pconsumed), is equal to the total power consumed by the system, including the input signal power, minus the input signal power to the system, in order to refer W to the output of the system alone, regardless of input signal levels (assuming linearity). This is expressed in Equation 7:
where Psystem-added is the signal power added by the system and the signal output power of the system is Psignal = Psource,out + Psystem-added. From Equation 7, the total power consumption of the system along the signal path, independent of the input signal power, Psource,out, is given by the superposition of Equation 7 and is shown in Equation 8:
The use of W in Equation 6 and Equation 8 provides a convenient method to compute the total and wasted power consumed by a device or cascade. Using Equation 8 and then by including the non-path powers, Pnon-path, contributed by all components that are off the signal cascade (such as displays, heat sinks, power supplies, etc.), the total power consumption of any system may be found as shown in Equation 9 using superposition of all power types in Table 1:7,14
The only factors that must be known for every device on the signal path along the chain of a cascaded network are gain and efficiency. Gain has the standard definition for the signal path components in the cascade that carry information to later stages of the cascade and the efficiency is defined as the reciprocal of the waste factor (W) in Equation 4. These two formulations are shown in Equation 10:
As noted earlier, the waste factor efficiency, ηw, as defined in Equation 4 and Equation 10, is the reciprocal of W. For passive devices, 1/W = ηw is identical to the traditional definition of efficiency when relating output power to input power
In the special case of active components powered by a DC source, W is equal to
which is equal to the reciprocal of PAE#1 12 as shown in Equation 11 from Equation 8:
RF, microwave and circuit engineers characterize amplifiers using PAE.10-13 While Equation 11 shows total PAE12 (PAE#1) is the inverse of the waste factor for the case of an amplifier that uses a DC power supply, a more customary definition for PAE,13
is easily shown to be related to W as:
which is nearly identical to Equation 11 when G is large. To consider the power wasted along a cascade of active devices, the reader can use Equation 5 which is proved in Equations 22 to 24 by simply using 
to characterize power efficiency and power consumption using the waste factor of a cascade, similar to how noise factor is used to quantify additive noise and SNR degradation for a cascade.
It should be clear that waste factor efficiency accounts for the proportion of signal power output when compared to the total power consumed, on the signal path cascade where the total power consumed includes both output signal power and all non-signal (e.g., wasted) power, independent of the input to the device or cascade. From Equations 4, 8 and 10, it can be seen from superposition that Pnon-signal (additive wasted power in a device or cascade) and the total power consumed by the device or cascade are related to W by Equation 12 and Equation 13:
Fundamentally, Equation 4 and Equation 10 show that W is defined as the inverse of efficiency.3-7 Defining the waste factor in this manner provides a FoM that may be used for any device or cascade and relates the waste factor to additive wasted power, just as noise figure did for additive noise power. This is shown in Equation 14.
Because W characterizes the useful signal power delivered to the output of any device or cascade relative to the entire power consumed along the device or cascade, it is most sensibly calculated referred to the output. Employing the concept of waste factor allows one to quantify power dissipation within a device (e.g., wasted power, since some power is not contained in the signal that is carried forward) which from Equations 13 and 14 and Table 1 can be expressed in Equation 15:
Figure 2 A general cascaded communication system with a pair of devices.
It is clear from Equation 15 that the input power to the cascade, (e.g., Psource,out) is not required to be known when using W, since W is a characteristic of the cascade or circuit, unrelated to input or output signal powers, as the cascade is assumed to be linear and matched. When a device along the cascade is turned off (e.g., Psignal = 0 or Psource,out = 0), W is still defined for a device or cascade, just as Friis was able to turn off the input signal power and still define F for a cascade, even when F = SNRi/SNRo was infinite or undefined.
The formulation of W for a cascaded system as shown in Equation 5 is proved using the simple cascade of two devices shown in Figure 2.
Using Figure 2, the waste factor for the cascade signal path (where Pnon-path from auxiliary components is neglected since they are not on the cascade) can be calculated as illustrated using a cascade of two devices. First, the total power consumed at the output terminal of network 2 can be defined using Equation 4 or Equation 10 as shown in Equation 16:
Then, the power consumed at the output terminal of network 1 can be defined in Equation 17:
Here, Pconsumed,1 denotes the total power consumption at the output terminal of device 1, which includes both the signal power applied to the input as well as the additional power consumed and contained in the signal that is transmitted to the subsequent device. It also includes the power wasted (e.g., not contained in the signal) by device 1, itself. When Psource,out (the input signal power) is subtracted from the total consumed power using Equations 7, 16 and 17, the standalone power consumption of device 1 is defined by Equation 18:
Applying the same approach to device 2, the standalone power consumption of device 2 is defined in Equation 19:
