The input to a lock-in amplifier has the form Vi(t) = Vi sin(ωt + θi) where Vi, ω, θi are the amplitude, frequency and phase of the input signal respectively. This signal is multiplied by a reference signal of the same frequency ω, amplitude Vr and phase θr. If the multiplied signal is fed to a low pass filter of cut-off frequency ω, then the final output signal is
1
\(\rm \frac{1}{2} V_i V_r \cos \left(\theta_i-\theta_r\right)\)
2
\(\rm V_i V_r\left[\cos (\theta_i-\theta_r)-\cos \left(\frac{1}{2} ω t+\theta_i+\theta_r\right)\right]\)
3
ViVr sin(θi - θr)
4
\(\rm V_i V_r\left[\cos (\theta_i-\theta_r)+\cos \left(\frac{1}{2} ω t+\theta_i+\theta_r\right)\right]\)