Total Radiated Power (TRP)
Total Radiated Power (TRP) is a measure of how much power is radiated by an antenna when the antenna is connected to an actual radio (or transmitter). TRP is an active measurement, in that a powered transmitter is used to transmit through the antenna. The total received power is calculated and summed up over all possible angles (hence, it is a spherical or 3d measurement) and the result is the Total Radiated Power.
As an example, suppose that a transmitter outputs 20.0 dBm of power (or 100 mW, see decibels) when connected to a 50 Ohm load. Suppose the total received power in the far field is measured to be 17.0 dBm (which can be measured in an anechoic chamber, see measuring radiation pattern. The resultant TRP is 17.0 dBm.
Now the above paragraph might seem so stupid that you are asking yourself "isn't this just the same thing as measuring the antenna efficiency? Why can't we just figure out the transmit power of the radio, and then substract out the antenna efficiency to get the Total Radiated Power?"
The above objection is a good point. The answer is that TRP measures the radiation in an actual live system. As a result, it is a function of not just the antenna, but also the radio/transmitter and the connection between the radio and the antenna.
As an example, let's consider the previously mentioned radio that outputs 20.0 dBm of power when connected to a 50 Ohm load. We actually do not know how much power the radio will output when connected to our antenna, because the antenna impedance is not actually 50 Ohms. In fact, even though we try to design the antenna to be 50 Ohms, it will never be exactly 50 Ohms, and can be significantly different from 50 Ohms, particularly if the antenna must work over a larger range of frequencies (large bandwidth).
The loss of power due to the antenna being away from 50 Ohms is not just related to mismatch loss (i.e. non-matching impedance between radio and antenna) in this situation, but rather because the radio will not put out the same power for every impedance that it connects to. Taken to the extreme, if an open circuit or short circuit is applied across a radio terminals, the radio will output zero power. For an antenna with a VSWR of 3:1, the power output could typically swing by 3 dB (can range between 17 and 23 dBm - yes, the power can actually be higher for a mismatched antenna!).
As a result, the only way to know how the antenna and radio system will perform as a whole is to measure the Total Radiated Power (TRP). In fact, when a cell phone is certified by a wireless carrier (such as AT&T, Verizon, China Mobile, T-Mobile, etc), they do not specify the antenna efficiency, but rather the TRP for the cell phone as a whole. In practical use, this is the parameter that is important, and is strongly dependent upon both the antenna and the radio.
Suppose an antenna has a radiation pattern given by , where theta and phi are standard variables in spherical coordinates. The units of R are in Watts/Steradian (a steradian is just a two-dimensional angular unit - if you don't know what it is, don't worry about it. Just note that the surface area of a sphere is equal to 4*pi Steradians).
The total radiated power is the spherical integral of R, which means we integrate over theta and phi across every possible angle:
In Equation , the units of TRP is Watts [W]. Now, when we are measuring the Total Radiated Power in an anechoic chamber, we are actually measuring the EIRP at every angle, and then averaging it over the sphere (recall that the surface area of a sphere is 4*pi). Hence, we can calculate TRP from EIRP:
To make things a bit more complicated, note that we need to measure the vertical and horizontal (or theta and phi component) polarization power for our antenna to accurately capture the radiated power. Hence, we can rewrite equation  by breaking it down in terms of the linear polarization powers:
Note that if we want to numerically calculate TRP from a set of sampled values of the EIRP, we can approximate this with a summation as in Equation :
In Equation , note that we are sampling the EIRP at N locations along the theta axis and M locations along the phi axis (for a total of N*M point measurements).