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Consider an original idealized narrow-band model for one matched spatial mode and one polarization of waves. In the classical Rayleigh-Jeans law regime, an available thermal power can be written $P=k_{B}T\Delta \nu$, where $\Delta \nu$ is the effective bandwidth and $T$ the equivalent input noise temperature. Assume a linear detector with stable positive gain and absorb its bandwidth, Boltzmann constant and response factor into $g$. Then the output is $S=g(T+T_r)$, with input-referred receiver noise temperature $T_r$. These are model assumptions, not a reconstruction of the historical electronics.

Two references with known noise temperatures $T_h\ne T_c$ giveso the two-load radiometer calibration yieldsSubtracting the reference equations eliminates the unknown receiver term; substituting back determines it. An arbitrary stable output offset can likewise be eliminated from the interpolation for $T_s$, although separating that offset from $T_r$ then requires additional information.

For deliberately chosen inputs, set $T_h=300\mathrm{K}$, $T_c=4.2\mathrm{K}$, $T_r=40\mathrm{K}$ and $g=2mV/K$. These are equivalent noise temperatures; physical load thermometry must be converted consistently at the observing frequency. The reference outputs are $S_h=680\mathrm{mV}$ and $S_c=88.4\mathrm{mV}$. A sky-input output $S_s=90\mathrm{mV}$ gives $T_s=5.0\mathrm{K}$. If independent estimates assign $1.4\mathrm{K}$ to atmosphere, $0.5\mathrm{K}$ to antenna loss and $0.1\mathrm{K}$ to ground pickup in this lumped additive model, the residual is $3.0\mathrm{K}$.

The subtraction exposes what a noise budget must achieve. A $1\mathrm{K}$ error in the assigned atmospheric term becomes a $1\mathrm{K}$ error in the residual, even with exact detector calibration. Multiplicative attenuation requires a different correction: if an isothermal atmosphere has transmission $t$, its ideal contribution is $t{T}_{\mathrm{cosmic}}+(1-t)T_{\mathrm{atm}}$ in the same classical equivalent-temperature convention. Recovering $T_{\mathrm{cosmic}}$ then requires both subtracting atmospheric emission and dividing by $t$. The additive toy budget applies when its terms already refer to the same receiver input.

There is also a conditioning issue. If $S_h-S_c$ is small, a fixed output error strongly changes the inferred temperature. Widely separated reference temperatures improve the scale determination only if the receiver remains linear over that range; measuring the reference repeatedly tests stability rather than proving linearity. The experiment therefore motivates a model, a calibration and a set of independent checks, each with its own assumptions.

In 1965, Arno Penzias and Robert Woodrow Wilson reported measurements with a 20-foot horn-reflector antenna at $4.080\mathrm{GHz}$. They manually switched a traveling-wave maser receiver between the horn and a helium-cooled reference. From $6.7\mathrm{K}$ zenith antenna temperature, atmospheric and antenna/ground contributions left an excess $3.5\pm 1.0\mathrm{K}$.^[ref]

Elevation scans estimated the atmospheric term. Taping horn seams produced no measurable change; a nearby transmitter tested ground response. These controls addressed distinct possible contaminations.^[ref]

The comparison does not make all nuisance contributions disappear. A reference can establish a response scale, but atmospheric emission and pickup through unwanted antenna directions are also real incident radiation. In his later account, Robert Woodrow Wilson describes these competing contributions as central to a radio astronomy measurement.^[ref]

The Princeton group proposed primordial thermal radiation as an interpretation and called for more wavelength measurements. Their separate $3\mathrm{cm}$ radiometer had not yet produced results.^[ref] A single-frequency excess therefore does not by itself establish a complete Planck spectrum. Reproducibility across independent instruments, observing directions and frequencies tests the interpretation further.

For a stable radiometer with affine output $S=g(T+T_r)$, let $g>0$ be the fixed response coefficient and $T_r$ the receiver noise temperature. Two reference loads with known equivalent input noise temperatures $T_c<T_h$ give outputs $S_c$ and $S_h$. Subtracting their outputs eliminates $T_r$; dividing source and reference differences eliminates $g$:This two-load radiometer calibration assumes unchanged response and receiver noise temperature between readings, with the loads and source referred to the same input. It also eliminates an additional constant output offset. Physical load temperatures must first be converted to the applicable equivalent input noise temperatures when the classical thermal approximation fails. The Penzias-Wilson microwave background experiment gives an original worked example.