Frequency in hz in view of the above we can write dp

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frequency in Hz. In view of the above we can write dP n = hf e hf/KT - 1 df hf KT -→ KTdf for power available over an infinitesimal bandwidth df . Hence, over a finite bandwidth B in the frequency range f KT/h , where dP n is independent of f , the available noise power of a resistor is P n = KTB. It is convenient to define a noise phasor V n , with a mean-square denoted as | V n | 2 , such that P n = KTB | V n | 2 8 R B. The noise phasor V n represents a co-sinusoid that carries the same average power as noise voltage v ( t ) over a unit bandwidth 8 , and can be treated as a regular voltage phasor 8 Any bandwidth unit can be employed here, e.g., Hz, kHz, or even μ Hz, as long as the same unit is used to express B in the formula for P n . 120
4 Radiation, antennas, links, imaging in circuit calculations in superposition with phasors defined for the same frequency f . Noise phasors V n and V m due to di ff erent resistors in a circuit are assumed to have (by definition) independent random phases so that | V n + V m | 2 = | V n | 2 + | V m | 2 is true (see Example 1 below). So far we talked about resistor noise, avoiding the subject in the title of this section, namely, antenna noise. It turns out, however, that all the formulae above for P n , dP n , as well as noise phasor V n , also apply to the noise output of antennas immersed in blackbody radiation. To see how and why, assume that an antenna, terminated by a matched load R - jX , is in thermal equilibrium with a background described by Planck’s blackbody distribution E ( f, T ) df = 8 π c 3 hf 3 e hf/KT - 1 df hf KT -→ 8 π c 3 KTf 2 df for electromagnetic energy density E . The antenna load resistance R at the blackbody temperature T will produce and deliver an average noise power KTB to its matched antenna over a bandwidth B , only be radiated out into the noisy background (assuming a lossless antenna for simplicity). This, of course, would lead to a cooling of the load unless the antenna picks up and delivers an equal amount of noise power, KTB , back to the load. Since a net heat exchange is not possible between systems at equal temperatures — the load and the background, in this case — it follows that the antenna must indeed pick up an available noise power of KTB from incident blackbody radiation over a bandwidth B [e.g., Burgess , 1941]. The available noise power of antennas within a bandwidth B is always expressed as P n = | V n | 2 8 R B = KTB whether or not the antenna is actually immersed in blackbody radiation. Hence, T in the formula above represents an equivalent antenna temperature which accounts for the radiation picked up from the environment, a sum of a large number independent random co-sinusoids with a mean square amplitude | V n | 2 in each unit bandwidth. We envision an open circuit voltage phasor V 1 + V 2 + V 3 + · · · where each component is a random-phased voltage phasor due to an independent ra- diation source, sources like radio stars, galaxies, our galactic core, etc., (all of which constitute signals for a radio astronomer), as well as atmospheric lightning and thermal emissions from atmospheric gases, and so on. The mean-squared value of the large sum

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