Δ u$_{bw}$ ( ν$_{c}$ ) measured around a frequency ν$_{c}$ within a certain bandwidth bw is performed most conveniently in frequency space: where υ ( ν ) is the (spectral) signal density (unit: V/ √ Hz ) and bw = 1 /τ the effective bandwidth of the filter. The signal densities corresponding to 〈 x ( t ) 〉$_{τ}$ and 〈 y ( t ) 〉$_{τ}$ in the frequency domain are X$_{q}$ ( ν ) G$_{fil}$ ( ν ) and Y$_{q}$ ( ν ) G$_{fil}$ ( ν ), respectively, being X$_{q}$ ( ν ) and Y$_{q}$ ( ν ) the Fourier transforms of the quadrature signals x$_{q}$ ( t ) and y$_{q}$ ( t ), and G$_{fil}$ ( ν ) = 1 / (1+ i 2 πντ ) the (complex) gain of the filter (see figure 1). If the Q factor is low -as in air and in liquids- the signals 〈 x ( t ) 〉$_{τ}$ and 〈 y ( t ) 〉$_{τ}$ can be directly used for DSFM. In fact, for low Q factors and low tip-sample interaction the frequency shift Δ ν$_{int}$ induced by tip-sample interaction is smaller than the width of the resonance (Δ ν$_{int}$ < ν$_{0}$/Q ). Assuming the validity of the harmonic approximation for the dynamics of the cantilever, the signal 〈 x ( t ) 〉$_{τ}$ is then proportional to the frequency shift, and the signal 〈 y ( t ) 〉$_{τ}$ is proportional to the oscillation amplitude, which in air and liquids is generally used as control parameter. For high Q factors, however, the width of the resonance is smaller than the frequency shifts induced by tip-sample interaction. Moreover, high Q -factors imply that the oscillation amplitude needs a long settling time (of the order of Q/ν$_{0}$ ) to reach its steady state value[17]. In this case it is necessary to track the resonance frequency using Phase Locked Loop techniques[17, 18, 21]. This is implemented with a PI-controller that essentially adjusts the frequency of the voltage or numerically controlled oscillator (VCO or NCO, see fig. 1) so that the 〈 x ( t ) 〉$_{τ}$ component vanishes; the phase of the oscillation is then kept at − π/ 2 and the system is always at resonance. The output of the PI-controller is then directly proportional to the frequency shift, and this is the signal used in typical DSFM applications. The PI-controller represents, from an electronic point of view, a filter with a well defined bandwidth and gain. The time constant of the filters following the multiplication stages and the bandwidth and gain of the PI-controller have to be chosen so that the overall closed loop system is stable[18]. Since the precise set up of the PI-controller does not modify the essential physics to be discussed here we will assume - to simplify the discussion - an ideal controller that instantaneously transmits variations of 〈 x ( t ) 〉$_{τ}$ to the VCO. The time response of the overall system is then determined by the time constant of the filters after the multiplication stage, which induce a delay of order τ . 6 Δ u$_{bw}$ ( ν$_{c}$ ) = √ ∫ ν$_{c}$ + bw/ 2 ν$_{c}$ − bw/ 2 dν υ $^{2}$( ν ) (5)