On the selection of excitation signals for the fast spectroscopy of electrical bioimpedance

The typical structure of EBI spectroscopy system is given in Fig 1. An excitation signal source (generator of the excitation waveform) has inputs for setting the required timing, frequency range, and amplitude. The signal from this generator provides an input excitation for the complex impedance to be measured Ż.

Fig.1

Next block, performs both, signal conditioning (amplification, filtering, normalizing) and analog-to-digital (A/D) conversion. A part of this block (front end) is often located near the SUT for obtaining better performance.

Discrete or fast Fourier transform (DFT or FFT) of the response signal from the SUT provides the amplitude |Ż |(f) and phase Φ (f) spectra. Also the spectra of real Re Ż (f) and imaginary Im Ż (f) parts of the impedance Ż can be calculated. Multichannel synchronous detection is an alternative to the Fourier transform if the number k of frequency components is low. Later, here we use also notation X[k] for the discrete spectrum and Mag X[k] for the magnitude spectrum of spectral components k.

More exactly, to measure an unknown impedance, we can choose one of the two approaches. Either we apply a known voltage excitation across the SUT and measure the current response flowing through, or we inject a known current excitation into that object and measure the voltage response across it. We can also measure both, current and voltage, simultaneously. The last method is preferable if the excitation source is not stable enough in a required frequency range or the parasitic impedances cause too large changes on the known parameters of the excitation signal [5].

Typically, the signal-to-noise ratio (SNR) of the measured signal is considered as the main efficiency criterion of the measurement system. The SNR of the EBI spectroscopy system depends on several factors, including the frequency bandwidth and measurement time. These parameters may be as well considered as efficiency criterions.

In several cases, e.g., for implantable and wearable measurement devices, the energy consumption, and physical dimensions are important. Also the reduced complexity of the measurement system is required frequently.

In turn, the parameters as frequency bandwidth, measurement time, and SNR of measurements are mutually related.

To characterize the properties of biological objects, the α-and β-dispersion frequency ranges are to be covered, usually. In the single-cell cytometry, capacitance and conductivity of the cell membrane and conductivity of the cytoplasm influence mainly the β-dispersion area that exhibits a characteristic frequency between several kHz up to tens of MHz [5]. In the case of living tissues, the spectrum of bio-impedance shifts toward lower frequencies and the spectra of the β-dispersion range cover typically 3-4 decades within the kHz to MHz band [1], [6]. However, even if only the β dispersion area is of interest, some part of lower frequency area must be included since the α-dispersion changes the plateau, from which the β-dispersion begins.

Presentation of signals can be converted mutually between time and frequency domains by the aid of mathematical operators known as transforms. The use of Discrete Fourier Transform (DFT) or more efficient Fast Fourier Transform (FFT) for the conversion from time to frequency domain is essential in fast impedance spectroscopy. The inverse transforms (DFT^-1 and FFT^-1) apply to opposite conversions. Though the signals are classified as periodic and aperiodic ones, however, the aperiodic signals can be addressed also as periodic having infinitely long period in the time domain. In the frequency domain, such approach produces infinitely small frequency steps leading to a continuous spectrum. In signal processing practice, a signal component with the longest but finite period T_m produces a discrete spectrum with frequency steps f₁ = 1/T_m, where f₁ is the lowest frequency presented as a fundamental frequency component. A single one-shot event in one domain reflects as the periodic component in the opposite domain [7]. Even if the real signals are not periodic, DFT interprets them as periodic, which leads to the distortions of spectra (spectral leakage error). A signal x (t) is periodic if:

The smallest positive value of T_p, which satisfies the condition (1) is a period. To be sure in signal periodicity, it must be observed from infinite past to the infinite future. Fortunately, this is not necessary, if the periodicity of signals is ensured by the design of the impedance spectroscopy system.

Due to the discrete nature of signal processing, one more condition must be fulfilled for the synchronous DFT: the measurement interval T_m = N × t_s must be an integer multiple of the period T_p of the signal, where N is the length of DFT. That is, the total number of samples N used for calculation of Fourier transform, where t_s is a used sampling period. If the conditions of the synchronous measurement are not met, then a certain spectral leakage errors occur.

The spectral leakage errors can be reduced by employing special window functions in the time domain. However, this increases the complexity of signal processing and produces additional error components. Hence, if feasible, the synchronous measurement is always preferred.

The measurement interval, T_m, which corresponds to the one period of the signal, determines the lowest frequency f₁ of the spectra obtained by DFT, f₁ = 1/T_m.

Although only one period of a periodical signal fulfills the requirement of synchronous DFT (T_m = T_p), more periods allow noise reduction through averaging. When the noise is not correlated neither with the signal nor with itself across trials, averaging yields an improvement of the SNR. If σ is the amount of RMS noise in a set of p samples, the averaged RMS noise is

However, this improvement comes at the cost of increased total measurement time (observation time) T_o, T_o = T_m × p. If the noise is correlated, the improvement is less effective.

The minimal measurement interval T_m depends on the lowest frequency in the frequency range of interest. The frequency range, in turn, depends on the properties of SUT as will be explained in the next section. If the properties of SUT are varying in time, the total measurement interval should be decreased for the better conformity with the criteria of the time-invariant system.

One possibility is to decrease the observation time T_o by reducing the number of measurement samples p. However, this also decreases the SNR. Another, better solution is to minimize time delays required for the formation of the response signal. Each time, when the excitation signal is applied, some settlement time of the response signal, depending mostly on the time-constants of the SUT, is required. Thus using the excitation signal that contains all the required frequencies (a multifrequency signal) provides minimal total observation time.

However, multifrequency signals have a drawback: their crest factor (CF) - the ratio of signal’s peak value in respect to its RMS level – is often higher in comparison with a single frequency signal (sine wave). Moreover, even if the CF of the multifrequency signal is similar or even below the CF of a single sine wave (2),$$\left( \sqrt{2} \right),$$the RMS values of single multifrequency components are always less. RMS values of multifrequency components are decaying proportionally to the number of components k, as will be discussed in next sections of the paper.

The highest frequency f_k of the spectra is determined by the length N of DFT, f_k = f₁ × N/2. If the frequency bandwidth of measured signals is wider than f_k, then aliasing errors occur, because the signal power at higher frequencies mirrors at the lower frequencies. To minimize this distortion of spectra in the frequency range of interest, the sampling frequency f_s = 1/t_s should be set high enough – typically 4 to 10 times above the highest frequency (though theoretically only 2 times). Unfortunately, this also requires more high-speed DSP components and increases the number of DFT samples.

To improve the SNR, higher RMS magnitudes of the frequency components of interest (FCI) are desirable. For a graspable comparison of different waveforms, some variables must be normalized as constants. It is reasonable to normalize the amplitude to the unit value ± 1 and the duration to 1 second. Normalizing the duration of the signal against 1 second also provides a normalized presentation of the spectra since the fundamental frequency is reciprocal to the duration T_p.

In general, RMS magnitudes of the frequency components (bins) of excitation signals are not equal. The shape of a magnitude spectrum may also be decaying, rising or oscillating around some mean value. The mean value of the RMS magnitudes of a number of frequency bins that fall in the half-power limit range is used here as a figure of merit for this general case.

According to Parseval theorem, the total energy in the frequency domain must be the same as calculated in the time domain. In the case of using DFT, this relation is following:

where x[i] is a time domain signal with i running from 0 to N-1, and X[k] is its modified spectrum with k running from 0 to N/2. The modified spectrum is found by taking the DFT of the signal and dividing the first and last samples (samples 0 and N/2) by the square root of two [7].

The only signal with a single line spectrum is the sine wave. Sinusoidal signal x(t) can be described mathematically as:

where A is the amplitude and Φ is the initial phase, which specifies the waveform at time t = 0. (A cosine wave is also "sinusoidal" but with a phase-shift of π/2). If the SUT and measurement channel are linear, the output signal will also be a sinusoidal signal with the same frequency as the input. The drawback of using the single sine wave excitation is that several measurement steps at a number of frequencies are necessary and some settlement time is required in each sequential frequency step.

The infinitely short δ-pulse can be considered as an ideal excitation signal for the fast impedance spectroscopy since it covers all the frequencies in the shortest time frame. However, in practice, there is a crucial drawback – because the excitation pulses are very short ones with limited amplitudes, the energy of pulse spread over a number of frequency components, will be very low. The reason is that according to the left side of (3), the energy content of the signal is proportional to the surface area of the waveform, and in the frequency domain, this energy is divided among spectral components. In the case of very short pulses, the distribution of frequencies is dense, almost all the higher harmonics of the first frequency f₁ are present. If the amplitudes of the waveform are fixed (normalized), then the energy content of quadrangle pulses is directly proportional to their duration. If the duration of the pulses is increased, the magnitudes of frequency components are almost equal only in a shorter frequency range as illustrated in Fig. 2, and the mean value of their RMS magnitudes is nearly twenty times less compared to the RMS magnitude of a single sine wave.

Fig.2

Other types of short pulses, e.g., the chirps as short-time frequency sweeps, could also be used for EBI measurements [8]. Chirps with a single cycle or even less can be generated, too. The RMS values of the first six components of the exponentially modulated short chirp [9] (Fig. 3) are higher in comparison with the pulses shown in Fig.2, but still low since the surface area of the waveform (hatched), being proportional to the energy content of the signal, is small.

Fig.3

The further symmetrical extension of the excitation signal shown in Fig. 2, runs out into a rectangular waveform in Fig. 4. In this case, the energy content of the signal is maximal. However, the spectrum contains only odd harmonics (1^st, 3^rd, 5^th, 7^th), which are declining fast with frequency by 1/f rule. Such a simple waveform can be used, therefore, only in the cases where this type of decaying shape of the spectrum is useful. For example, a decaying part of the response current spectrum may be equalized with a similar decaying spectrum of the excitation voltage.

Fig.4

Another type of signal that covers a large number of harmonic frequencies with almost equal amplitudes is a step wave (Fig. 5). This type of excitation is often used in a time domain based impedance measurement [5], [10]. The total energy content of the signal starting from zero is near maximal, but there are also serious drawbacks. At first, most of the energy of the signal is concentrated in a DC component that is mostly undesired in EBI measurements. If the rise time of the step waveform tends to zero, the number of harmonics tends to infinity, but their magnitudes reach to zero. In practice, the rise time of the step waveform is larger than a zero moment, and the amplitudes of higher harmonics are decaying.

Fig.5

A step waveform with the rise time of 1/500 of the period T_p is shown in Fig. 5a, where the RMS magnitudes of spectral components are almost constant up to 100^th higher harmonic but very low. DC component equals 0.9985. DC component of the signal can be canceled by shifting the starting point of the waveform near to -1 (dashed line in Fig. 5a), but the RMS values of other frequency components remain the same (very small).

The energy content of the shifted waveform is proportional to a cross-hatched area, and its spectral content is similar to very short pulses.

The DC component can be canceled also by adding a second part of opposite polarity. However, the waveform resumes in near rectangular signal with a well known decaying shape of the magnitude spectrum (Fig. 4). It must be emphasized here, that DFT interprets the signal as periodic, and according to (1) all the changes of the waveform must be included in a period.

The situation described here also illustrates the drawback of time-domain impedance measurement method with a short pulse or step waveforms. In the case of unbalanced step waveforms, most of the excitation signal energy is useless for spectroscopy (concentrated in a DC component). Faster measurement cycle comes at the cost of lower SNR since the energy of harmonic spectral components is created only during the front part of the step or duration of the balanced pulses. Another reason for the low level of spectral components is that the energy spreads over many harmonic components. Averaging improves the SNR, however, this contradicts with the idea of fast measurements. Moreover, the use of longer excitation with a higher level of desired spectral components is more efficient under similar conditions since only uncorrelated noises can be effectively suppressed by averaging.

Chirps are considered here as signals of many cycles of sinusoidal oscillation with a changing „instantaneous“ frequency. A sine-wave chirp signal with an instantaneous phase θ(t) can be described mathematically as

with the „instantaneous“ frequency

In practice, however, generating of discrete signals is more convenient. The waveforms of the sinusoidal and discrete, so-called signum-chirp S_sgn (t) = sign (S_ch (t)) are shown in Fig. 6b.

Fig.6

The dependency of the mean RMS magnitudes on a number of frequency bins k of chirp signals is described in [11]. RMS magnitudes of spectral components of normalized linear sinusoidal chirp tend to the value of 1/k$$1/\sqrt{k}$$and are higher by a factor of 4/π in the case of signum chirp.

Further analysis shows some deviation (ripple) of the RMS magnitudes x of frequency components from a mean value of all used components k. The ripple in per cents is calculated as

and is presented in Fig. 7 and Fig. 8.

Fig.7

Fig.8

The advantage of signum chirp (in comparison with a sinusoidal one) is higher RMS value of its spectral components, but the ripple is also considerably higher in the cases, when the number of desired harmonic components, k > 10.

Another disadvantage of signum chirp is that some part of the energy, typically 15-20 % of total energy, always spreads into higher harmonics above k.

Another type of wideband excitation waveform with a dense distribution of frequencies is a maximum length sequence (MLBS). MLBS is a type of pseudorandom binary sequence, which can be easily generated by an n-stage linear feedback shift register (LFSR). A maximum-length LFSR produces a maximum length sequence (m sequence). It cycles through all possible N = 2_n − 1 states within the n stage shift register (except the state where all bits are zero). The MLBS period T_p = (2_n-1) × t_c, where t_c is the period of the register clock. The RMS magnitude spectrum follows the envelope of sin (f) / f function as illustrated in Fig. 9. The half-power bandwidth occurs at a frequency of approximately 0.443/t_c [12].

Fig.9

The shape of the magnitude spectrum becomes flattered as N increases, and more frequency bins fall in the half-power limit range. However, the power of each frequency component decreases. We showed in [11] that the mean RMS magnitudes of the frequency components of the MLBS signal tend to the value

where B ≈ 0.93 and k_h is a number of useful frequency bins that fall in the half-power limit range.

It may be concluded that the magnitudes of the spectral components of MLBS and signum chirp signals depend almost similarly on a number of frequency components k. Even so, there are also important differences. At first, the number of spectral components of the MLBS signal is fixed by its order n, and secondly, a shape of the spectrum is smoother with a fixed relative deviation near ± 15 % from its mean value (later defined by the half-power limit).

The excitation signals described above (except sinewave) have a dense distribution of frequencies. Spreading of the signal energy in such a way reflects in a lower power of separate frequency components. If the frequency response of SUT is smooth, as in the case of EBI measurements, the concentration of the signal energy into a limited number of components is more adequate than the dense distribution. A more detailed illustration will be given in next sections.

The accuracy of fitting the equivalent circuit model with the impedance measurement results depends on the number of frequency components. In general, the use of more frequencies improves the fitting accuracy [13]. However, the fitting accuracy also depends on the signal to noise ratio (SNR) of measurements. As already discussed in previous sections, the magnitudes of spectral components of the excitation signal are decreasing, when their number increases. It follows that the number of used spectral components has two contradictory effects on the accuracy of estimation of parameters of the equivalent circuit model. Usually, less than ten components are required in EBI spectroscopy.

The SNR is expressed through the ratio of signal and noise power P_signal /P_noise , while the power is equal to the square of RMS value. Assuming that the power density of noise is uniform in the frequency range of interest, the SNR of spectral components is proportional to the square value of their RMS levels. Since the magnitude of the impedance spectrum of biological objects decays with frequency, the spectrum of the excitation signal should have such a shape that elevates the response signal at higher frequencies. In the case, when polarizable electrodes are used, the shape of the excitation signal should provide a lower level response at lower frequencies, too.

If the noise power density is not uniform in the frequency range of interest, then the better overall SNR is obtained when the shape of the excitation spectrum follows the power density spectrum of noise. Moreover, in-vivo electrical bioimpedance (EBI) measurements must comply with the safety regulations. The safe applied current for the BF and CF type devices is limited according to the IEC60601 standard.

In EBI measurements, the required dynamic range of the impedance magnitude may reach three decades and more [1], [5]. Therefore adaption of the magnitudes of spectral components of the excitation provides better SNR. From the excitation signals described above, only the sine wave and sinusoidal multicycle chirp allow free adjustment of the magnitudes of separate frequency components.

Binary multifrequency waveforms in Fig. 10a and Fig. 11a with two discrete values (±1, if normalized) look similar to the maximum length sequence (MLBS) shown in Fig. 9a. However, a sequence of pulses in the period T_p of binary multifrequency waveforms is not distributed by the rule used for generating the MLBS waveforms.

Fig.10

Fig.11

According to the Fourier principle, periodic signals can be represented by a set of discrete frequencies. A spectrum of the periodic BMS waveform (a pattern of pulse sequence) may be composed as the sum of spectra of its L components with different durations:

Magnitude spectrum X (i) of individual waveform components i depends only on their duration (amplitudes are normalized), but their contribution into the overall magnitude spectrum of the BMS waveform also depends on their phase spectra. Phase spectra of the waveform components, in turn, depend on their polarity and position in a signal period. Choosing appropriate lengths and positions of the waveform components allow designing the BMS with different magnitudes of its spectral components, including the zero and near-to-zero values.

As a result, the shapes of the BMS spectra differ principally from the envelope of sinc-function being characteristic to MLBS. The envelope of the spectrum lines of synthesized BMS may have as decaying, as well as flat (Fig. 10b) or even rising shape (Fig.11b).

The CF of the BMS waveforms is the smallest of possible (CF = 1), and the total energy of BMS excitation is two times higher compared to the sine or multi sine wave with the same amplitude. Considering the simplicity in generating binary signals, BMS also meets the criteria of reduced energy consumption and lower complexity. The disadvantage of the BMS is that some part of the energy always spreads into higher harmonics outside the desired spectral range. Therefore, characterization of the BMS only by the smallest CF = 1 is an overestimation of its merits.

Multisine excitation retains the advantages of the sine wave while reducing the measurement time. The specific advantage of the multisine signal is that only the desired components appear in the spectra. Adding up several (k) sine waves leads to decreasing the individual spectral components in the excitation signal because its overall amplitude is limited, moreover, their RMS values decrease proportionally to CF, as we showed in [11]:

Let us indicate here that the CF of the sum of same level consequent frequency components with optimal phases can be below the corresponding value of a single sine wave (CF=2)$$\left( CF=\sqrt{2} \right)$$[4]. Our optimization algorithms provide CF below this value for a consequent frequency distribution i = 1, 2, 3, 4 …. k and k > 6, Fig. 12. The enhanced algorithm described in [14] gives CF below 1.37 at large k.

Fig.12

Let us take an example. In the case of 36 added components CF = 1.358. If the total amplitude of the sum of signals with equal amplitudes is set to 1 V, then according to (10), RMS values of all components are 122.8 mV that is 5.76 times less than RMS value of the single sine wave with the amplitude of 1 V.

In the case of very sparse distribution of frequencies, the optimization of initial phases of multisine components is ineffective, and the CF tends to its worst possible case (2k).$$\left( \sqrt{2k} \right)\,.$$However, in the case of less sparse frequency distributions as a binary logarithmic one, the results prove to be acceptable. For example, the CF of the multisine signal with binary logarithmic distribution at the number of components k = 16 (i = 1, 2, 4, 8, 16, 32, ..., 32768) tends to the value πk3/22.$$\pi \sqrt[3]{k}/2\sqrt{2}\,.$$Moreover, a multisine with the binary distribution of spectral components does not require optimization of initial phases since their zero value provides near optimal results [15].

Remark: even though some part of the BMS energy transfers to useless higher harmonics, the RMS levels of the components of well optimized BMS are higher than of the optimized multisine. A benefit of using BMS increases with the sparsity of frequency distributions. Fig. 13 illustrates RMS magnitudes of both excitation waveforms for the different frequency distributions.

Fig.13

The stair-step multifrequency waveforms (MFS) [16] and tri-level (ternary) waveforms (MFM) [17] are recently proposed for the broadband bioimpedance spectroscopy. Although these waveforms can be used as multifrequency excitation, there are also some limitations and disadvantages.

Calculations with (3) show that the total energy content of described 6-th order MFS and MFM waveforms are 61% and 83 %, respectively, in comparison with two level BMS. Taking into account that 83.6 % of the energy of the MFM signal falls into the desired frequency components, the overall energetic efficiency is 69.3 %. The typical energetic efficiency of the BMS signal depends on the distribution of frequencies and is in the range from 65 to 85 %.

Limitations of the described MFM waveforms are that only binary distribution of frequencies and decaying shapes of magnitude spectra are produced. MFS waveform provides a flat magnitude spectrum, but its total energy content is significantly lower in comparison with MFM and BMS. Furthermore, generation of MFM and MFS waveforms requires more resources in comparison with BMS.

The conducted research is not related to either human or animal use.