An LNA designer quickly finds that the source impedance for minimum noise figure is generally not the same as the impedance for maximum gain.

Noise parameters
A two-port's noise is characterized by four parameters: NFmin, the optimum source reflection coefficient Gamma_opt (magnitude and angle), and the noise resistance Rn. Noise figure for any source reflection Gamma_s is:

F = Fmin + (4 Rn/Z0) |Gamma_s - Gamma_opt|^2 / ((1 - |Gamma_s|^2) |1 + Gamma_opt|^2)

Rn indicates sensitivity: a large Rn means NF degrades quickly as you move away from Gamma_opt.

Gain matching
Maximum gain needs a conjugate match at input and output. Because Gamma_opt and the conjugate input match differ, a noise-matched LNA has some input mismatch, which is undesirable for the preceding filter or antenna.

Common techniques
1. Plot noise circles and available gain circles on the Smith chart, and choose a compromise source impedance.
2. Inductive source degeneration in FET designs moves the input impedance's real part closer to 50 ohms while having little effect on Gamma_opt, letting noise match and input match approach each other.
3. Balanced amplifiers with quadrature hybrids provide good input match while each amplifier is noise-matched, at the cost of more parts and the hybrid's loss before the amplifiers.
4. Feedback can widen bandwidth and improve stability at some noise cost.

Stability first
Check stability factors (k and mu) over a wide frequency range, not only your band. LNAs with high gain transistors may oscillate out of band. Resistive loading on the drain/collector side or in the bias network is a standard fix.

Losses before the transistor, including the input matching network's own loss, add directly to NF, so use high-Q components there.