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An antenna aperture describes the collecting area associated with a receiving antenna. The antenna effective area need not equal its geometric projected area.

The antenna effective area converts incident energy flux into available received power for a specified direction and matched polarization of an electromagnetic wave. In a lossless free-space link, a transmitter of antenna gain $G$ at range $r$ and a receiver of effective area $A_{\mathrm{eff}}$ have $P_r/P_t=G{A}_{\mathrm{eff}}/(4\pi r^2)$. Impedance or polarization mismatch reduces the delivered power.

The Friis transmission equation describes a far-field free-space antenna link with aligned directions, matched polarizations and negligible additional loss. It follows by multiplying the transmitter's axial energy flux $P_t{G}_t/(4\pi d^2)$ by the receiver's antenna effective area ${\lambda }^2{G}_r/(4\pi )$. A receiving impedance mismatch multiplies the available received power by $1-|{\Gamma }_L{|}^2$. Wavelength scaling must state which gains, apertures and impedances are held fixed.

Antenna gain in direction $(\theta ,\varphi )$ compares the far-field power per solid angle with an isotropic radiator using the same accepted input power:It includes radiation efficiency; for the lossless dipole $P_{\mathrm{in}}=P_{\mathrm{rad}}$. The antenna effective area is defined for incidence from a specified direction with matched polarization and conjugate impedance matching:Its units are square metres and it need not equal the physical metal area.

For ordinary reciprocal antennas, electromagnetic reciprocity makes the coupling between two matched ports the same when transmitter and receiver are exchanged. Free-space transfer in one direction is proportional to $G_1{A}_2$, while the reverse transfer is proportional to $G_2{A}_1$. Thus $A/G$ must be a universal factor at that wavelength, independent of antenna or direction. The supplied relation fixes it asEquivalently, thermal detailed balance prevents an angular receiving pattern from collecting more radiation than its reciprocal emitting pattern would return in equilibrium. Reciprocity and the specified matching are the assumptions behind this statement; it is not a general theorem for active nonreciprocal devices.

The transmitting antenna is matched to its input transmission line, so its real terminal electrical resistance is $Z_i$. The identical receiving antenna has the same electrical impedance at the operating frequency. Its available received power for a matched load is the incident energy flux times its antenna effective area:The real receiving load is $Z_t$, so its voltage-wave reflection coefficient relative to $Z_i$ is ${\Gamma }_L=(Z_t-Z_i)/(Z_t+Z_i)$. The fraction of available power delivered to the output line isConsequently, neglecting path losses, mismatch of polarization of an electromagnetic wave and line losses,This is the Friis transmission equation with a receiving impedance-mismatch factor. For fixed $P_i,G_a,d,Z_i,Z_t$, it scales as ${\lambda }^2$. That scaling is conditional on holding antenna gain fixed: a fixed physical aperture can instead have $G_a\propto {\lambda }^{-2}$, and any wavelength dependence of the impedances also enters the expression. The matched-output check $Z_t=Z_i$ gives $P_t=P_i\mathcal{F}$.

The Johnson-Nyquist noise formula gives the open-circuit mean-square voltage in a small frequency band $d\nu$. When a source electrical resistance $R$ drives a load $R$ with impedance matching, the load sees half the source voltage, so the available noise power isThe factor of four in the open-circuit formula is therefore not an extra factor in the delivered power.

The transmitter's antenna gain $G$ makes the axial spectral energy flux at range $d$Multiplication by the receiving antenna effective area $A$ gives $d{P}_r=k_B{T}_{t}GAd\nu /(4\pi d^2)$. Define the equivalent receiving antenna temperature $T_A$ by the available matched-load noise $d{P}_r=k_B{T}_{A}d\nu$. ThenThis is the transmitter-induced equivalent noise temperature of the receiver port. It does not determine the actual physical temperature of its internal electrical resistance, or any additional receiver and background noise. The calculation assumes the usual far-field link, negligible path loss beyond geometric spreading, matched polarization of an electromagnetic wave at the two antennas and the stated effective collecting area. No additional one-half polarization factor is needed for the matched polarization of radiation from one transmitting antenna.