Optimal Requirements of a Data Acquisition System for a Quadrupolar Probe Employed in Electrical Spectroscopy

This paper discusses the development and engineering of electrical spectroscopy for simultaneous and non invasive measurement of electrical resistivity and dielectric permittivity. A suitable quadrupolar probe is able to perform measurements on a subsurface with inaccuracies below a fixed limit (10%) in a bandwidth of low (LF) frequency (100kHz). The quadrupole probe should be connected to an appropriate analogical digital converter (ADC) which samples in phase and quadrature (IQ) or in uniform mode. If the quadrupole is characterized by a galvanic contact with the surface, the inaccuracies in the measurement of resistivity and permittivity, due to the IQ or uniform sampling ADC, are analytically expressed. A large number of numerical simulations proves that the performances of the probe depend on the selected sampler and that the IQ is better compared to the uniform mode under the same operating conditions, i.e. bit resolution and medium.


Introduction.
Analogical to Digital Converter (ADC) (Razavi, 1995). Typically, an ADC is an electronic device that converts an input analogical voltage (or current) to a digital number.
A sampler has several sources of errors. Quantization error and (assuming the sampling is intended to be linear) non-linearity is intrinsic to any analog-to-digital conversion. There is also a so-called aperture error which is due to clock jitter and is revealed when digitizing a time-variant signal (not a constant value). The accuracy is mainly limited by quantization error. However, a faithful reproduction is only possible if the sampling rate is higher than twice the highest frequency of the signal. This is essentially what is embodied in the Shannon-Nyquist sampling theorem.
There are currently a huge number of papers published in scientific literature, and the multifaceted nature of each one makes it difficult to present a complete overview of the ADC models available today. Technological progress, which is rapidly accelerating, makes this task even harder. Clearly, models of advanced digitizers must match the latest technological characteristics.
Different users of sampler models are interested in different modelling details, and so numerous models are proposed in scientific literature: some of them describe specific error sources (Polge et al., 1975); others are devised to connect conversion techniques and corresponding errors (Arpaia et al., 1999) (Arpaia et al., 2003); others again are devoted to measuring the effect of each error source in order to compensate it (Björsell and Händel, 2008). Finally, many papers (Kuffel et al., 1991) (Zhang and Ovaska, 1998) suggest general guidelines for different models.
Electrical spectroscopy. Electrical resistivity and dielectric permittivity are two independent physical properties which characterize the behavior of bodies when these are excited by an electromagnetic field. The measurements of these properties provides crucial information regarding practical uses of bodies (for example, materials that conduct electricity) and for countless other purposes.
Some papers (Grard, 1990a,b) (Grard and Tabbagh, 1991) (Tabbagh et al., 1993) (Vannaroni et al. 2004) (Del Vento and Vannaroni, 2005) have proved that electrical resistivity and dielectric permittivity can be obtained by measuring complex impedance, using a system with four electrodes, but without requiring resistive contact between the electrodes and the investigated body. In this case, the current is made to circulate in the body by electric coupling, supplying the electrodes with an alternating electrical signal of low (LF) or middle (MF) frequency. In this type of investigation the range of optimal frequencies for electrical resistivity values of the more common materials is between ≈10kHz and ≈1MHz. Once complex impedance has been acquired, the distributions of resistivity and permittivity in the investigated body are estimated using well-known algorithms of inversion techniques.
Applying the same principle, but limited to the acquisition only of resistivity, there are various commercial instruments used in geology for investigating the first 2-5 meters underground both for the exploration of environmental areas and archaeological investigation (Samouëlian et al., 2005).
As regards the direct determination of the dielectric permittivity in subsoil, omitting geo-radar which provides an estimate by complex measurement procedures on radar-gram processing (Declerk, 1995) (Sbartaï et al., 2006), the only technical instrument currently used is the so-called time-domain reflectometer (TDR), which utilizes two electrodes inserted deep in the ground in order to acquire this parameter for further analysis (Mojid et al., 2003) (Mojid and Cho, 2004).

Topic and structure of the paper.
This paper presents a discussion of theoretical modelling and moves towards a practical implementation of a quadrupolar probe which acquires complex impedance in the field, filling the technological gap noted above.
A quadrupolar probe allows measurement of electrical resistivity and dielectric permittivity using alternating current at LF (30kHz<f<300kHz) or MF (300kHz<f<3MHz) frequencies. By increasing the distance between the electrodes, it is possible to investigate the electrical properties of subsurface structures to greater depth. In appropriate arrangements, measurements can be carried out with the electrodes slightly raised above the surface, enabling completely non-destructive analysis, although with greater error. The probe can perform immediate measurements on materials with high resistivity and permittivity, without subsequent stages of data analysis.
The authors' paper (Settimi et al., 2010) proposed a theoretical modelling of the simultaneous and non invasive measurement of electrical resistivity and dielectric permittivity, using a quadrupole probe on a subjacent medium. A mathematical-physical model was applied on propagation of errors in the measurement of resistivity and permittivity based on the sensitivity functions tool (Murray-Smith, 1987). The findings were also compared to the results of the classical method of analysis in the frequency domain, which is useful for determining the behaviour of zero and pole frequencies in the linear time invariant (LTI) circuit of the quadrupole. The authors underlined that average values of electrical resistivity and dielectric permittivity may be used to estimate the complex impedance over various terrains (Edwards, 1998) and concretes (Polder et al., 2000) (Laurents, 2005), especially when they are characterized by low levels of water saturation or content (Knight and Nur, 1987) and analyzed within a frequency bandwidth ranging only from LF to MF frequencies (Myounghak et al., 2007)(Al-Qadi et al., 1995. In order to meet the design specifications which ensure satisfactory performances of the probe (inaccuracy no more than 10%), the forecasts provided by the sensitivity functions approach are less stringent than those foreseen by the transfer functions method (in terms of both a larger band of frequency f and a wider measurable range of resistivity ρ or permittivity ε r ) [see references therein (Settimi et al, 2010)] . This paper discusses the development and engineering of electrical spectroscopy for simultaneous and non invasive measurement of electrical resistivity and dielectric permittivity. A suitable quadrupolar probe is able to perform measurements on a subsurface with inaccuracies below a fixed limit (10%) in a bandwidth of LF (100kHz). The quadrupole probe should be connected to an appropriate analogical digital converter (ADC) which samples in phase and quadrature (IQ) (Jankovic and Öhman, 2001) or in uniform mode. If the quadrupole is characterized by a galvanic contact with the surface, the inaccuracies in the measurement of the resistivity and permittivity, due to the IQ or uniform sampling ADC, are analytically expressed. A large number of numerical simulations proves that the performances of the probe depend on the selected sampler and that the IQ is better compared to the uniform mode under the same operating conditions, i.e. bit resolution and medium. Assuming that the electric current injected in materials and so the voltage measured by probe are quasi-monochromatic signals, i.e. with a very narrow frequency band, an IQ downsampling process can be employed (Oppenheim et al., 1999). Besides the quantization error of IQ ADC, which can be assumed small both in amplitude and phase, as decreasing exponentially with the bit resolution, the electric signals are affected by two additional noises. The amplitude term noise, due to external environment, is modeled by the signal to noise ratio which can be reduced performing averages over a thousand of repeated measurements. The phase term noise, due to a phase-splitter detector, which, even if increasing linearly with the frequency, can be minimized by digital electronics providing a rise time of few nanoseconds. Instead, in order to analyze the complex impedance measured by the quadrupole in Fourier domain, an uniform ADC, which is characterized by a sensible phase inaccuracy depending on frequency, must be connected to a Fast Fourier Transform (FFT) processor, that is especially affected by a round-off amplitude noise linked to both the FFT register length and samples number (Oppenheim et al., 1999). If the register length is equal to 32 bits, then the round-off noise is entirely negligible, else, once bits are reduced to 16, a technique of compensation must occur. In fact, oversampling can be employed within a short time window, reaching a compromise between the needs of limiting the phase inaccuracy due to ADC and not raising too much the number of averaged FFT values sufficient to bound the round-off.
The paper is organized as follows. Section 2 defines the data acquisition system. In sec. 3, the theoretical modeling is provided for both IQ (3.1) and uniform (3.2) samplers. In sec. 4, assuming quasi-monochromatic signals, an IQ down-sampling process is employed. Besides quantization error of IQ ADC, the electric signals are affected by two additional noises: the amplitude term noise, due to external environment; and the phase term noise, due to phase-splitter detector. In sec.
5, in order to analyze the complex impedance measured by the quadrupolar probe in Fourier domain, the uniform sampling ADC is connected to a FFT processor being affected by a round-off noise. In sec. 6, the design of the characteristic geometrical dimensions of the probe is analyzed. Sec. 7 proposes a conclusive discussion. Finally, an Appendix presents an outline of the somewhat lengthy calculations.

Data acquisition system.
In order to design a quadrupole probe [ fig. 1.a] which measures the electrical conductivity σ and the dielectric permittivity ε r of a subjacent medium with inaccuracies below a prefixed limit (10%) in a band of low frequencies (B=100kHz), the probe can be connected to an appropriate analogical digital converter (ADC) which performs a uniform or in phase and quadrature (IQ) sampling (Razavi, 1995) (Jankovic and Öhman, 2001), with bit resolution not exceeding 12, thereby rendering the system of measurement (voltage scale of 4V) almost insensitive to the electric noise of the external environment (≈1mV). IQ can be implemented using a technique that is easier to realize than in the uniform mode, because the voltage signal of the probe is in the frequency band of B=100kHz and IQ sampled with a rate of only f S =4B=400kHz, while, for example, low resolution uniform samplers are specified for rates of

5-200MHz.
With the aim of investigating the physics of the measuring system, the inaccuracies in the transfer impedance measured by the quadrupolar probe [ fig. 1.b], due to uniform or IQ sampling ADCs [ fig. 2], are provided.
If, in the stage downstream of the quadrupole , the electrical voltage V is amplified V V =A V ·V and the intensity of current I is transformed by a trans-resistance amplifier V I =A R ·I, the signals having been processed by the sampler, then: the inaccuracy Δ|Z|/|Z| for the modulus of the transfer impedance results from the negligible contributes ΔA V /A V and ΔA R /A R , respectively for the voltage and trans-resistance amplifiers, and the predominant one Δ|V V |/|V V | for the modulus of the voltage, due to the sampling, instead, the inaccuracy ΔΦ Z /Φ Z for the initial phase of the transfer impedance coincides with the one Δφ V /φ V for the phase of the voltage, due to the sampler, the initial phase of the current being null, φ I =0.

Theoretical modeling.
As concerns an IQ mode (Jankovic and Öhman, 2001), in which a quartz oscillates with a high figure of merit Q=10 4 -10 6 , the inaccuracy Δ|Z|/|Z| IQ (n,φ V ) depends strongly on bit resolution n, decreasing as the exponential function 2 -n of n, and weakly on the initial phase of voltage φ V , such (1 ) , 2 Instead, as concerns uniform sampling (Razavi, 1995), the inaccuracy Δ|Z|/|Z| U (n) for |Z| depends only on the bit resolution n, decreasing as the exponential function 2 -n of n [ fig. 4.a], (3.4) Consequently, for all ADCs, with bit resolution n: if the probe performs measurements on a medium, then the inaccuracy Δε r /ε r (f) in the measurement of dielectric permittivity ε r is characterized by a minimum limit Δε r /ε r | min (ε r ,n), interpretable as the "physical bound" imposed on the inaccuracies of the problem, which depends on both ε r and the bit resolution n, being directly proportional to the factor (1+1/ε r ), while decreasing as the exponential if the probe, with characteristic geometrical dimension L, performs measurements on a medium, with conductivity σ and permittivity ε r , working within the cut-off frequency f T =f T (σ,ε r )=σ/(2πε 0 (ε r +1)) (Settimi et al., 2010), then the absolute error E |Z| (L,σ,n) in the measurement of the modulus for the transfer impedance |Z| N (L,σ) depends on σ, L and the bit resolution n, the error being inversely proportional to both σ and L, while decreasing as the exponential function 2 -n of n [ fig. 5.b] [In fact, Z N (f,L,σ,ε r ) is fully characterized by the high frequency pole f T =f T (σ,ε r ), which cancels its denominator: the transfer impedance acts as a lowmiddle frequency band-pass filter with cut-off f T =f T (σ,ε r ), in other words the frequency equalizing Joule and displacement current. As discussed below, average values of σ may be used over the band ranging from LF to MF, therefore |Z| N (L,σ) is not function of frequency below f T (Settimi et al., 2010)].
As concerns IQ mode, with a quartz of high merit figure Q, the inaccuracy ΔΦ Z /Φ Z | IQ (n,φ V ) depends both on the bit resolution n, decreasing as the exponential function 2 -n of n, and on the voltage phase (3.5) fig. 3 As a consequence, only for uniform sampling ADCs, the inaccuracy ΔΦ Z /Φ Z (f,f S ) for the phase Φ Z must be optimized in the upper frequency f up , so when the probe performs measurements at the limit of its band B, i.e. f up =B.
Still with the aim of investigating the physics of the measuring system, when the quadrupolar probe exhibits a galvanic contact with the subjacent medium of electrical conductivity σ and dielectric permittivity ε r , i.e. h=0, and works in frequencies f lower than the cut-off frequency f T =f T (σ,ε r ) (Grard and Tabbagh, 1991), (3.10) Only if the quadrupole probe is in galvanic contact with the subjacent medium, i.e. h=0, then our mathematical-physical model predicts that the inaccuracies Δσ/σ for σ and Δε r /ε r for ε r are invariant in the linear (Wenner's) or square configuration and independent of the characteristic geometrical dimension of the quadrupole, i.e. electrode-electrode distance L (Settimi et al., 2010). If the quadrupole, besides grazing the medium, measures σ and ε r working in a frequency f much lower than the cut-off frequency f T =f T (σ,ε r ), then the inaccuracy Δσ/σ=F(Δ|Z|/|Z|,ΔΦ Z /Φ Z ) is a linear combination of the inaccuracies, Δ|Z|/|Z| and ΔΦ Z /Φ Z , for the transfer impedance, while the inaccuracy Δε r /ε r =F(Δ|Z|/|Z|) can be approximated as a linear function only of the inaccuracy Δ|Z|/|Z|; in other words, if f<<f T , then ΔΦ Z /Φ Z is contributing in Δσ/σ but not in Δε r /ε r .
As mentioned above, even if, according to Debye polarization mechanisms (Debye, 1929) or Cole-Cole diagrams (Auty and Cole, 1952), the complex permittivity of various materials in the frequency band from VLF to VHF exhibits several intensive relaxation effects and a non-trivial dependence on the water saturation  , anyway average values of electrical resistivity and dielectric permittivity may be used to estimate the complex impedance over various terrains and concretes, especially when they are characterized by low levels of water content and analyzed within a frequency bandwidth ranging only from LF to MF.

For in phase and quadrature (IQ) sampling.
As concerns IQ sampling ADCs (merit figure Q) (Jankovic and Öhman, 2001), the inaccuracy Δε r /ε r (f,n) in the measurement of permittivity ε r is characterized by an optimal working frequency f opt,IQ (f T ), close to the cut-off frequency for the modulus of the transfer impedance f T =f T (σ,ε r ), i.e. , 2 (1 ) which is tuned in a minimum value of inaccuracy Δε r /ε r | min,IQ (ε r ,n), depending both on ε r and the specifications of the sampler, in particular only its bit resolution n. The inaccuracy is directly proportional to the factor (1+1/ε r ), while decreasing as the exponential function 2 -(n-3) of n, such that Consequently, if the frequency f of the probe is much lower than the cut-off frequency f T (σ,ε r ) for the transfer impedance, then the inaccuracy Δσ/σ(n) for σ is a constant, and the inaccuracy Δε r /ε r (f,n) for ε r shows a downward trend in frequency, as (f T /f) 2 , such that both the inaccuracies decrease as exponential functions of n, the first inaccuracy as 1/2 n-2 while the second one as 1/2 n-1 , i.e. [ fig. 6] (3.14) Even if the frequency f is much higher than f T (σ,ε r ), it could be proven that the inaccuracies Δσ/σ(f,n) and Δε r /ε r (f,n) do not deviate too much from an upward trend in frequency as a parabolic line (f/f T ) 2 , with a high gradient for the σ measurement and a low gradient for the ε r measurement, still decreasing as exponential functions of n, the first inaccuracy as 1/2 n-2 while the second one as Only if f is lower than f T , then the measurements could be optimized for ε r and σ, and might require that the inaccuracy Δε r /ε r (n,f) for ε r is below the prefixed limit Δε r /ε r | fixed (10%) within the frequency band B=100kHz, choosing a minimum bit resolution n min,IQ (f T ,ε r ,), which depends on both f T and ε r and increases as the logarithmic function log 2 of both the ratios f T /B and 1/ε r , i.e. min, 2 2 2 1 1 log 2 log log (1 ) Referring to the IQ sampling ADCs, the inaccuracies Δσ/σ in the measurement of the electrical conductivity σ and Δε r /ε r for the dielectric permittivity ε r were estimated for the worst case, when the inaccuracies Δ|Z|/|Z| IQ (n,φ V ) in the measurement of the modulus and ΔΦ Z /Φ Z | IQ (n,φ V ) for the phase of the transfer impedance respectively assume the mean and the maximum values, i.e.

Δ|Z|/|Z|
To conclude, for the IQ mode, the optimal and minimum values of the working frequency of the quadrupolar probe interact in a competitive way. The more the quadrupole analyzes a subjacent medium characterized by a low electrical conductivity, with the aim of shifting the optimal frequency into a low band, the more the low conductivity has the self-defeating effect of shifting the minimum value of frequency into a higher band. In fact, the probe could work around a low optimal frequency, achievable in measurements of transfer impedance with a low cut-off frequency, typical of materials characterized by low conductivity. Instead, the more the minimum value of the working frequency is shifted into a lower band, the more the minimum bit resolution for the sampler has to be increased; if the medium was selected, then increasing the inaccuracy for the measurement of the dielectric permittivity, or, if the inaccuracy was fixed, shifting the cut-off frequency into a high band, i.e. selecting a medium with high conductivity. Finally, while the authors' analysis shows that the quadrupole could work at a low optimal frequency, if the transfer impedance is characterized by a low cut-off frequency, in any case and in accordance with the more traditional results of recent scientific publications referenced (Grard, 1990, a-b) (Grard and Tabbagh, 1991) (Tabbagh et al., 1993) (Vannaroni et al. 2004)(Del Vento and Vannaroni, 2005), the probe could perform measurements in an appropriate band of higher frequencies, centered around the cut-off frequency, where the inaccuracy for the measurements of conductivity and permittivity were below a prefixed limit (10%).

For uniform sampling.
As concerns uniform sampling ADCs (Razavi, 1995), if the frequency f of the probe is lower than the cut-off frequency f T for the modulus |Z|(f,L) of the transfer impedance, i.e. f T =f T (σ,ε r ), then, the higher the bit resolution n, the more the optimal working frequency f opt,U (f T ,f S ,n), which minimizes inaccuracy Δε r /ε r (f,f S ,n) in the measurement of the permittivity ε r , approximately depends on the cut-off frequency f T (σ,ε r ) and the specifications of the sampler, in particular the sampling rate f S and n, increasing with both f T and f S , while decreasing as the exponential function 2 -n/3 of n, such Moreover, the higher the bit resolution n, the more the inaccuracy Δε r /ε r (f,f S ,n) for ε r can not go down beyond a minimum limit of inaccuracy Δε r /ε r | min,U (ε r ,n), which approximately depends on both ε r and n, being directly proportional to the factor (1+1/ε r ), while decreasing as the exponential function 2 -(n-1) of n, similarly to IQ sampling. The minimum value of inaccuracy Δε r /ε r | min,U (ε r ,n) in uniform mode is higher than the minimum inaccuracy Δε r /ε r | min,IQ (ε r ,n) corresponding to the IQ Finally, the minimum value of frequency f U,min (f T ,n), which allows an inaccuracy Δε r /ε r (f,f S ,n) below a prefixed limit Δε r /ε r | fixed (10%), depends both on f T (σ,ε r ) and n, being directly proportional to f T , while decreasing as an exponential function of n, such that [ fig. 8] min, , (3.20) To conclude, also in the uniform mode, the optimal frequency and the band of frequency of the quadrupolar probe interact in a competitive way. In fact, an ADC with a high bit resolution is characterized by a low sampling rate, for which, having selected the subjacent medium to be analyzed, the higher the resolution of the sampler used, with the aim of shifting the optimal frequency of the quadrupolar probe into a low band, the more the low sampling rate has the selfdefeating effect of narrowing its frequency band. Moreover, a material with low electrical conductivity is usually characterized by low dielectric permittivity. Having designed the ADC, the more the quadrupole measures a transfer impedance limited by a low cut-off frequency, the more it can work at a low optimal frequency, even if centered in a narrow band. Finally, having selected the medium to be analyzed and designed the sampler, the more the frequency band of the probe is widened, the more the inaccuracy of the measurements is increased.

Noisy IQ Down-Sampling.
In signal processing, down-sampling (or "sub-sampling") is the process of reducing the sampling rate of a signal. This is usually done to reduce the data rate or the size of the data (Oppenheim et al., 1999) (Andren and Fakatselis, 1995).
The down-sampling factor, commonly denoted by M, is usually an integer or a rational fraction greater than unity. If the quadrupolar probe injects electric current into materials at a RF frequency f, then the ADC samples at a rate fs fixed by: being M preferably, but not necessary, a power of 2 to facilitate the digital circuitry Since down-sampling reduces the sampling rate, one must be careful to make sure the Shannon-Nyquist sampling theorem criterion is maintained. If the sampling theorem is not satisfied then the resulting digital signal will have aliasing. To ensure that the sampling theorem is satisfied, a lowpass filter is used as an anti-aliasing filter to reduce the bandwidth of the signal before the signal is down-sampled; the overall process (low-pass filter, then down-sample) is called decimation. Note that the anti-aliasing filter must be a low-pass filter in down-sampling. This is different from sampling a continuous signal, where either a low-pass filter or a band-pass filter may be used.
A practical scheme to select the sampling rate is to launch two time sequences as in fig. 2.a.. Now there is a problem due to timing. If the quadrupole would work at a fixed frequency f, then the proper relationship between the rate fs and the down-sampling factor M could be easily found.
Instead, if the probe is performing a sort of electrical spectroscopy, then an enable signal for the sampling and holding circuit (S&H) must be generated. The time frame should be such that the sequences n·M·T for the sample I and n·M·T+T/4 for the sample Q could be obtained, corresponding to the period T of the maximum working frequency. So the rate fs would be ensured as a M factor sub-multiple of frequency f. A possible conceptual scheme of this implementation is shown in fig.   2.b.
Besides the quantization error of IQ ADC, which can be assumed small both in amplitude and phase, as decreasing exponentially with the bit resolution n, the electric signals are affected by two additional noises. The amplitude term noise, due to external environment, is modelled by the signal to noise ratio SNR = 30dB which can be reduced performing averages over one thousand of repeated measurements (A = 10 3 ). The phase term noise, due to a phase-splitter detector, which, even if increasing linearly with the frequency f, can be minimized by digital electronics providing a rise time of few nanoseconds (τ = 1ns). In analytical terms: With respect to the ideal case involving only a quantization error, the additional noise both in amplitude, due to external environment, and especially in phase, due to phase-shifter detector, produce two effects: firstly, both the curves of inaccuracy Δε r /ε r (f) and Δσ/σ(f) in measurement of the dielectric permittivity ε r and electric conductivity σ are shifted upwards, to values larger of almost half a magnitude order, at most; and, secondly, the inaccuracy curve Δε r /ε r (f) of permittivity ε r is narrowed, even of almost half a middle frequency (MF) decade. So, both the optimal value of frequency f opt , which minimizes the inaccuracy Δε r /ε r (f) of ε r , and the maximum frequency f max , allowing an inaccuracy Δε r /ε r (f) below the prefixed limit Δε r /ε r | fixed (10%), are left shifted towards lower frequencies, even of half a MF decade. Instead, the phase-splitter is affected by a noise directly proportional to frequency, which is significant just from MFs; so, the minimum frequency f min , allowing Δε r /ε r (f) below Δε r /ε r | fixed (10%), remains almost invariant at LFs [ fig. 9

Fast Fourier Transform (FFT) processor and round-off noise.
In mathematics, the Discrete Fourier Transform (DFT) is a specific kind of Fourier transform, used in Fourier analysis. The DFT requires an input function that is discrete and whose non-zero values have a limited (finite) duration. Such inputs are often created by sampling a continuous function. Using the DFT implies that the finite segment that is analyzed is one period of an infinitely extended periodic signal; if this is not actually true, a window function has to be used to reduce the artefacts in the spectrum. In particular, the DFT is widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal. A key enabling factor for these applications is the fact that the DFT can be computed efficiently in practice using a Fast Fourier Transform (FFT) algorithm.
It is important to understand the effects of finite register length in the computation. Specifically, arithmetic round-off is analyzed by means of a linear-noise model obtained by inserting an additive noise source at each point in the computation algorithm where round-off occurs. However, the effects of round-off noise are very similar among the different classes of FFT algorithms (Oppenheim et al., 1999).
Generally, a FFT processor which computes N samples, represented as n FFT +1 bit signed fractions, is affected by a round-off noise which adds to the inaccuracy for transfer impedance, in amplitude (Oppenheim et al., 1999) and in phase (Dishan, 1995)(Ming andKang, 1996), So, maximizing the register length to n FFT =32, the round-off noise is entirely negligible. Once that n FFT <32, if the number of samples is increased N>>1, then the round-off noise due to FFT degrades the accuracy of transfer impedance, so much more in amplitude (5.1) how much less in phase (5.2).
One can overcome this inconvenience by iterating the FFT processor for A cycles, as the best estimate of one FFT value is the average of A FFT repeated values. The improvement is that the inaccuracy for the averaged transfer impedance, in amplitude and phase, consists on the error of quantization due to the uniform sampling ADC (3.4)-(3.7) and the round-off noise due to FFT (5.1)-(5.2), the last term being decreased of A , i.e.: and especially in phase So, the round-off noise due to FFT is compensated. The quantization error due to ADC decides the accuracy for transfer impedance: it is constant in amplitude, once fixed the bit resolution n, and can be limited in phase, by an oversampling technique f S >>f.
In As comments on eqs. (5.9)-(5.11), a low number of samples N min , corresponding to the Shannon-Nyquist limit, shortens the time window (5.9). An high oversampling ratio lowers the phase inaccuracy although it raises the samples number hold by uniform ADC and especially the cycles number iterated by FFT; however, even a minimal oversampling ratio R O,min limits the phase inaccuracy with the advantage of not raising to much the samples number hold by ADC (5.10) and especially the cycles number iterated by FFT (5.11).
The quadrupole (frequency band B) exhibits a galvanic contact with the subjacent nonsaturated medium (terrestrial soil or concrete with low permittivity ε r = 4 and high resistivity σ S ≈ 3.334·10 -4 S/m, σ C ≈ 10 -4 S/m). It is required that the inaccuracy Δε r /ε r (f,f S ,n) in the measurement of ε r is below a prefixed limit Δε r /ε r | fixed (10-15%) within the band B (100kHz

Characteristic geometrical dimensions of the quadrupolar probe.
In this section, we refer to Vannaroni's paper (Vannaroni et al., 2004) which discusses the dependence of TX current and RX voltage on the array and electrode dimensions. The dimensions of the quadrupolar probe terminals are not critical in the definition of the system because they can be considered point electrodes with respect to their separating distances. In this respect, the separating distance to consider are either the square array side [ fig. 12.a] or the spacing distance for a Wenner configuration [ fig. 12.b]. The only aspect that could be of importance for the practical implementation of the system is the relationship between electrode dimensions and the magnitude of the current injected into the ground. Current is a critical parameter of the mutual impedance probe in that, in general, given the practical voltage levels applicable to the electrodes and the capacitive coupling with the soil, the current levels are expected to be quite low, with a resulting limit to the accuracy that can be achieved for the amplitude and phase measurements. Furthermore, low currents imply a reduction of the voltage signal read across the RX terminals and more stringent requirements for the reading amplifier.
In the Appendix it is proven that, having fixed the input resistance R in of the amplifier stage and selecting the minimum value of frequency f min for the quadrupole [ fig. 1], which allows an inaccuracy in the measurement of the dielectric permittivity ε r below a prefixed limit ( Moreover, known the minimum bit resolution n min for the uniform or IQ sampling ADC, which allows an inaccuracy for permittivity ε r below the limit 10%, the electrode-electrode distance L(r,n min ) can also be defined, as it depends only on the radius r(R in ,f min ) and the bit resolution n min , the distance being directly proportional to r(R in ,f min ) and increasing as the exponential function 2 n min of n min [ fig. 13 Finally, the radius r(R in ,f min ) remains invariant whether the probe assumes the linear (Wenner's) or the square configuration, while, having also given the resolution n min , then the distance L S (r,n min ) in the square configuration must be smaller by a factor (2-2 1/2 ) compared to the corresponding distance

Conclusive discussion.
This paper has discussed the development and engineering of electrical spectroscopy for simultaneous and non invasive measurement of electrical resistivity and dielectric permittivity. A suitable quadrupolar probe is able to perform measurements on a subsurface with inaccuracies below a fixed limit (10%) in a bandwidth of low (LF) frequency (100kHz  (12)]. The principal advantages are: firstly, the minimum value f min (n) of the frequency, which allows an inaccuracy for permittivity ε r below 10%, is slightly lower when the probe is connected to an IQ rather than a uniform ADC, other operating conditions being equal, i.e. the resolution and the surface; and, secondly, the inaccuracy Δσ/σ for conductivity σ, calculated in f min (n), is smaller using IQ than with uniform sampling, being Δσ/σ| IQ <Δσ/σ| U of almost one order of magnitude, under the same operating conditions, in particular the resolution. As a minor disadvantage, the optimal frequency f opt , which minimizes the inaccuracy for ε r , is generally higher using IQ than uniform sampling, being f opt,IQ > f opt,U of almost one middle frequency (MF) decade, at most, under the same operating conditions, in particular the surface [ fig. 6].
Instead the uniform mode is specified by two degrees of freedom, the resolution of bit n U and the rate of sampling f S , compared to the IQ mode, characterized by one degree of freedom, the bit resolution n IQ . Consequently the quadrupolar probe could be connected to a uniform ADC with a sampling rate f S sufficiently fast to reach, at a resolution (for example n U =8) lower than the IQ's (i.e. n IQ =12), the same prefixed limit (i.e. 10%) of inaccuracy in the measurement of the dielectric permittivity ε r , for various media, especially those with low electrical conductivities σ [tab.1].
Assuming that the electric current injected in materials and so the voltage measured by probe are quasi-monochromatic signals, i.e. with a very narrow frequency band, an IQ down-sampling process can be employed. Besides the quantization error of IQ ADC, which can be assumed small both in amplitude and phase, as decreasing exponentially with the bit resolution, the electric signals are affected by two additional noises. The amplitude term noise, due to external environment, is modeled by the signal to noise ratio which can be reduced performing averages over a thousand of repeated measurements. Since, in this paper, the conceptual and technical problem for designing the "heart" of the instrument have already been addressed and resolved, a further paper will complete the technical project, focusing on the following two aims.
First aim: the implementation of hardware which can handle numerous electrodes, arranged so as to provide data which is related to various depths of investigation for a single measurement pass; consequently, this hardware must be able to automatically switch transmitting and measurement pairs.
Second aim: the implementation of acquisition configurations, by an appropriate choice of transmission frequency, for the different applications in which this instrument can be profitably used.

Appendix.
A series of two spherical capacitors with radius r and spacing distance L>>r is characterized by the electrical capacitance 0 0 with ε 0 the dielectric constant in vacuum.
A quadrupolar probe, with four spherical electrodes of radius r and separating distance L>>r, is arranged in the Wenner's configuration, with total length L tot =3L, which is specified by the pairs of transmitting electrodes T 1 and T 2 at the ends of quadrupole and the reading electrodes R 1 and R 2 in the middle of probe. The quadrupolar probe is characterized by a capacitance almost invariant for the pairs of electrodes T 1,2 and R 1,2 , The charge Q of the electrodes being equal, the electrical voltage across the pair T 1 and T 2 approximates the voltage between R 1 and R 2 , As regards the equivalent capacitance circuit which schematizes the transmission stage of quadrupole [ fig. 14.a], if the effect of the capacitance C, across the electrodes, is predominant relative to the shunted capacitances C T1 and C T2 , describing the electrical coupling between the transmitting electrodes and the subjacent medium, then, the probe, working in the frequency f, injects in the medium a minimum bound for the modulus of the current |I| min , |I| min increasing linearly with f.
Concerning the equivalent capacitance circuit which represents the reception stage of the quadrupolar probe [ fig. 14.b], the effect of the capacitance C, across the electrodes, is predominant even relative to the shunted capacitances C R1 and C R2 , describing the coupling between the reading electrodes and the subjacent medium, Notice that ΔV min,R1,R2 is independent of f, as |I| min is directly and |Z| min inversely proportional to f.
As a first finding, the analogical digital converter (ADC), downstream of probe, must be specified by a bit resolution n, such that: Instead, if the quadrupole exhibits a galvanic contact with a medium of electrical conductivity σ and dielectric permittivity ε r , working in a band lower than the cut-off frequency, i.e. f T =f T (σ,ε r )=1/2π·σ/ε 0 (ε r +1), then it measures the transfer impedance in modulus As final findings, the voltage amplifier, downstream of the probe, must be specified by an input resistance R in larger than both the transfer impedance, i.e. (A.12) and the reactance associated to the capacitance C, which is characterized by a maximum value in the minimum of frequency f min , i.e.
2 2 min min 0 0 min 0 min .13) Tables and captions.   Tab             a subsurface of dielectric permittivity around ε r ≈ 4 and low electrical conductivity σ, like a nonsaturated terrestrial soil (σ ≈ 3.334·10 -4 S/m) or concrete (σ ≈ 10 -4 S/m). In the hypothesis that the probe is connected to a sampler of bit resolution n, ranging from 8 bit to 24 bit: (a) semi-logarithmic plot for the "physical bound" imposed on the inaccuracies, i.e. Δε r /ε r | min (ε r ,n), as a function of the resolution n; (b) semi-logarithmic plot for the absolute error E |Z| (L 0 ,σ,n) for the transfer impedance in modulus below its cut-off frequency, as a function of the resolution n. . Bode's diagrams for the inaccuracies Δε r /ε r (f) and Δσ/σ(f) in the measurement of the dielectric permittivity ε r and the electrical conductivity σ, plotted as functions of the frequency f, for non-saturated terrestrial soil (a) or concrete (b, c) analysis. The probe is connected to uniform or IQ or samplers (in the worst case, when the internal quartz is oscillating at its lowest merit factor Q ≈ 10 4 ), of bit resolution n=12 (a, c) or n=8 (b), which allow inaccuracies in the measurements below a prefixed limit, 15% referring to (a, b) and 10% for (a, c), within the frequency band B=100kHz  which allows an inaccuracy in the measurement of the dielectric permittivity below a prefixed limit (10%), as a function of the bit resolution n [the cut-off frequency of the transfer impedance fixed as f T =f T (ε r ,σ) corresponding to the subsurface defined by the measurements (ε r ,σ)]. quadrupole is connected to an IQ sampler of bit resolution n = 12 and it is affected by an additional noise both in amplitude, due to the external environment (signal to noise ratio SNR = 30 dB, averaged terms A =10 3 ), and in phase, due to a phase-splitter detector specified by (rise time, τ = 1ns) or (phase inaccuracy, ΔΦ Z /Φ Z = 0.2°). The noisy probe allows inaccuracies Δε r /ε r (f) and

Figures and captions.
Δσ/σ(f) in the measurement of permittivity ε r and conductivity σ below a prefixed limit, 15% referring to (a) and 10% for (b), within the frequency band B=100kHz [Tab. 2].