# Portable and Highly Versatile Impedance Meter for Very Low Frequency Measurements
## 2 Materials and Methods
### Digital Signal Elaboration Software
For the correct operation of QLSA, records of the sampled signals need to be continuously fed to its internal engine. Samples relative to M input signals must be fed to QLSA in the form of an array of interleaved samples with a length exactly equal to M \(\times\)\(N_{R}\), where N\({}_{\text{R}}\) is the record length that must be a power of 2 and must be selected upon initialization. In NB mode, QLSA allows the estimation of the power spectra of all input channels, together with the cross spectra among them, in a frequency interval from \(f_{min}\) to\(f_{max}\), with \(f_{min}\) and \(f_{max}\) defined by the user, and with a frequency resolution \(\Delta\)\(f_{NB}\) given by:  
\[\Delta f_{NB}=\frac{f_{max}-f_{min}}{N_{R}}\;;\;f_{max},\;f_{min}<\frac{f_{CK}} {2} \tag{18}\]More importantly, in NB mode, the Equivalent Noise Bandwidth (ENB) of the filter used for the estimation of the power spectrum across each frequency within the interval in Equation (18) can be set independently of the sampling frequency \(f_{CK}\) and the record length \(N_{R}\). In particular, as discussed in detail in [36], QLSA can perform spectral estimation for several different ENBs at the same time with:  
\[ENB_{i}=\frac{1-p_{i}}{1+p_{i}}\times\frac{f_{CK}}{N_{R}};\;0<p_{i}<1 \tag{19}\]  
where \(p_{i}\) is a set of values defined by the user before starting the measurements. The ability to set estimation filters across each frequency with a very narrow ENB is a key requirement in the case of our application. When a sinusoidal signal is at the input of a spectrum analyzer, one obtains a non-zero output not only at the frequency of the input signal but also at other frequencies depending on the frequency response of the filter used for the estimation [42]. Typically, we obtain the maximum in the response at the frequency of the input signal, and we may want to employ filters with very narrow ENB to improve the signal-to-noise ratio. When more than one sinusoid is present at the input, the response corresponding to the different sinusoids superimposes and, in some cases, the presence of a sinusoid with a large amplitude may completely mask the presence of another sinusoid at a close frequency with a much smaller amplitude. This is indeed the case in the application we are interested in and, therefore, great care must be paid to selecting the most proper frequency response for the estimation filter. It is worth noting that the time required to complete a measurement is, roughly, inversely proportional to the ENB of the filter being employed, so one must always find a compromise between selectivity and measurement time. In this respect, the fact that QLSA allows the estimation of spectra using a number of different ENBs at the same time represents an important feature since one can stop the measurement as soon as the measurements corresponding to the ENB that allow sufficient selectivity become available. In order to better understand this aspect and to justify the need for the carrier removal block in Figure 3, we reported some spectra estimations obtained by feeding synthetic data to QLSA. The spectra reported in Figure 4 were obtained setting a standard sampling frequency of \(f_{CK}\) = 48 kHz, a carrier frequency \(f_{C}\) = 10 kHz; the frequency of the test signal \(\;v_{VD}\) (Figure 1) was set to \(f_{LF}\) = 100 mHz and its amplitude was set to 10 mV. In this condition, and with a zero DC component, the power spectra evaluated by QLSA in NB mode in a frequency interval of 1 Hz across the carrier, with \(N_{R}\) = 214 and for a few different NEBs, are reported in Figure 4a. Because of the modulation performed by M\({}_{2}\) in Figure 1, we obtain two sinusoidal components at frequencies \(f_{C}\) - \(f_{LF}\) and \(f_{C}\) + \(f_{LF}\). As we had anticipated, we obtain a non-zero spectrum over a rather wide interval across the frequency of the sinusoids at the input of the system. This  
Figure 3: Simplified diagram illustrating the interaction among the different software routines for signal generation, acquisition, and elaboration.