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M$_{Σ}$ ( t ) is a rotation matrix generating a counter-clockwise rotation with frequency ν$_{Σ}$ . In the frequency domain the signal a ( t ) is thus splitted and shifted to the sum and to the difference frequency (see figure 1). After the multiplication stages the two quadrature signals x$_{q}$ ( t ) and y$_{q}$ ( t ) are low-pass filtered over a timespan proportional to the time constant τ of the filter. The precise time domain signals are determined in appendix A. Again, two matrices M τ$_{Δ}$ ( t ) and M τ$_{Σ}$ ( t ) can be defined (relations 24 and 25 in appendix A), corresponding to a clockwise and counter-clockwise rotation with the frequencies ν$_{Δ}$ and ν$_{Σ}$ . As compared to the first matrices, these “filtered” matrices have delay angles ϕ$_{Δ}$ ( τ ) and ϕ$_{Σ}$ ( τ ) as well as multiplicative factors 1 / (1 + 4 π $^{2}$τ $^{2}$ν2 $_{Δ}$) and 1 / (1 + 4 π $^{2}$τ $^{2}$ν2 $_{Σ}$). 

Usually, in DSFM the signal entering the DSFM detection unit is at the natural frequency ν$_{0}$ , and the reference is at the same frequency ν$_{ref}$ = ν$_{0}$ . Then multiplication of the input signal with the reference signals will result in spectra around ν = 0 and ν = 2 ν$_{0}$ . In this case M τ$_{Δ}$ ( t ) = Id / 2 , with Id the identity matrix, and we find, using relations 24 and 25: 

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These signals can be conveniently represented in the frequency domain (see fig.1), where the first peak at ν = 0 is one-sided (since no negative frequency exists) and the second at ν = 2 ν$_{0}$ is two-sided. In typical DSFM applications the filter is usually set so that ν$_{0}$ >> 1 /τ , then it blocks the 2 ν$_{0}$ component and passes the signals from DC to ν$_{0}$ ≈ 1 /τ . The value of τ determines the speed and the “cleanness” of the signals. Large time constants (small bandwidth) results in clean but slow response, while, conversely, small-time constants will result in “unclean” signals – in particular with significant 2 ν$_{0}$ component - but with fast response. In our system we have found that time constants of 3 /ν$_{0}$ to 10 /ν$_{0}$ give the optimum compromise between speed and “cleanness”. 

In the frequency domain, the Fourier transforms of the signals x$_{q}$ ( t ) and y$_{q}$ ( t ) are simply multiplied by the frequency-dependent gain of the filter. Depending on the time constant of the filter, the total amount of signal may be decreased. The calculation of the total signal 

