Discrete-Level Broadband Excitation Signals : Binary / Ternary Chirps

Chirps (fast frequency sweeps) and chirp-like signals have become popular excitation signals in measurement techniques, particularly as the stimulus in bioimpedance measurement [1, 2]. The main advantage of chirps is the independent scalability in time and frequency domains: the duration of a pulse (usually short) and its frequency range (usually wide) can be adjusted separately. This feature is important in biomedical investigations for quick monitoring of the changing state of living tissue and cells. The classical description of chirps presumes a sinewave signal, instantaneous frequency f(t) of which changes linearly in time. Consequently, the current phase (t)=f(t)dt of this chirp is changing in proportion to the square of time. The waveform of chirp expresses as Vch(t)=sin((t))=sin(2  f(t)dt). Generation of perfect sinusoidal chirps is an intricate task. Therefore, for simplifying generation and signal processing, applying chirps of the rectangular waveform with a small number of discrete amplitude levels should be preferred, e.g., for miniaturized measurement devices. Below we deal with such kinds of discrete-level chirp-like signals, known also as pseudo-chirps [2].


Introduction
Chirps (fast frequency sweeps) and chirp-like signals have become popular excitation signals in measurement techniques, particularly as the stimulus in bioimpedance measurement [1,2].The main advantage of chirps is the independent scalability in time and frequency domains: the duration of a pulse (usually short) and its frequency range (usually wide) can be adjusted separately.This feature is important in biomedical investigations for quick monitoring of the changing state of living tissue and cells.
The classical description of chirps presumes a sinewave signal, instantaneous frequency f(t) of which changes linearly in time.Consequently, the current phase (t)=f(t)dt of this chirp is changing in proportion to the square of time.The waveform of chirp expresses as V ch (t)=sin((t))=sin(2  f(t)dt).
Generation of perfect sinusoidal chirps is an intricate task.Therefore, for simplifying generation and signal processing, applying chirps of the rectangular waveform with a small number of discrete amplitude levels should be preferred, e.g., for miniaturized measurement devices.Below we deal with such kinds of discrete-level chirp-like signals, known also as pseudo-chirps [2].
A possible architecture of the system for the EBI measurement is presented in Fig. 1 [3].It executes the cross-correlation procedure (Corr) between the response and reference signals, and thereupon calculates Fourier transform (FFT) of the cross-correlation function, i.e., the measurement of the unknown bioimpedance Ż (or any complex impedance) is accomplished in succession Corr(V z , V ref )FFT{Re(Ż(j)); Im(Ż(j))}.The result is a broadband impedance spectrum, which characterizes the difference between the EBI under study and the predetermined reference impedance Ż ref .
As a rule, the informative frequency bandwidth of the BI lies within the range from some kHz up to several MHz [1,4].The benefit of chirps is the simplicity of adjusting the sufficiently wide excitation range B exc , which covers the frequency bandwidth of the object under measurement.In general, the manner of frequency change of chirps can be various: by a power function, exponentially, piecewise linearly, etc.We will pay the main attention to the first mentioned ones or so-called power chirps, the instantaneous frequency of which is changing according to the n-th order power function with arbitrary (incl.fractional) power n.This kind of sine-wave chirps with the amplitude A can be described mathematically as where =B exc /T ch characterizes the rate of frequency change (chirping rate), T ch is the pulse duration, and B exc =f fin -f 0 with the final f fin and initial f 0 frequencies.Usually, chirps of many cycles (rotations of chirpforming vector on the complex plane) are considered.However, it was shown in [5] that chirps of a single cycle or even less ((T ch )  2π) can have the practical importance, if the very low power consumption or the extremely short Generator of chirp excitation measurement time is required (implantable devices, lab-onchips).The end frequencies, time duration and number of cycles of a chirp are strictly related.For the n-th order power chirps this relationship can be expressed as [5]     where L is the number of cycles.
The basic discrete-level pseudo-chirp is the binary one (called also signum-chirp or Non-Return-to-Zero (NRZ) chirp), waveform of which represents the signumfunction of the respective sine-wave chirp as Asgn(V ch (t)), thus having values of {± A} only -see Fig. 2a.Unlike sine-wave chirps, the spectrum of the discretelevel ones is rippling intensively.Besides, the percentage of their in-band energy is somewhat less at the equal amplitude and duration of the pulse.In practice, these drawbacks should be compensated by the simplicity of the signal processing.An additional benefit of binary chirps is their unity crest factor, and consequently, major energy compared with the sine-wave chirps of the same length.
Next, let us examine some specific spectral features of both the mentioned modifications of linear pseudochirps (n=1).

Binary chirps with f 0 = 0
A specific feature of binary chirps is the gradually decreasing amplitude and power spectra by plateaus h (h=1, 2, …; Fig. 3).This effect and the energy distribution of binary chirps along the frequency spectrum, considering f 0 =0, was discussed by the authors in [6] as follows.
The fundamental harmonic (k=1) of a rectangular signal with levels ± A has the root-mean-square (RMS) value ) 2 π /( 4A and (presuming the unity load) the power In the case of rectangular-wave chirps the fundamental harmonics from f 0 to f fin are spread over the whole bandwidth B exc , creating a constant value power spectral density (PSD) w 1 =W 1 /B exc , V 2 /Hz.The power of every k th higher odd (k=3, 5, …) harmonic W k =W 1 /k 2 is spread over the frequency range B k =kf fin  kf 0 =kB exc with PSD of w k = W 1 /(B exc k 3 ) -see Fig. 3a.
All the odd harmonic components contribute a certain part of their power for the first plateau (h=1) of the power spectrum with the width of B exc (excitation power P exc ).It can be expressed as the sum The power of every next plateau (h>1) with the width of 2B exc is formed by the harmonics from (2h-1) th and higher, which expresses as (Fig. 3b) The total power of all spectral plateaus outside the generated bandwidth is For the first spectral plateau, the PSD p 1 =P 1 /B exc , while for all the next plateaus p h =P h /(2B exc ).The values of p 1 /p h =2P 1 /P h can be calculated by ( 3)-( 5) -some of the results are shown in Table 1 and in Fig. 3c.The total energy of the generated chirp signal is E tot =(P exc +P out )T ch .However, it is essential to know not only the full energy, but also the amount of the energy E exc , within the generated bandwidth (useful energy).Energyefficiency of the generated excitation signal can be expressed as the ratio of  E =E exc /E tot =P exc /(P exc +P out ).Using (3)-( 5) yields [6]     where  denotes Riemann's zeta-function.
The theoretical percentage of the useful energy by ( 6) is  E 0.853.In practice, the energy-efficiency of a signal can be evaluated from the results of FFT-processing as a)

W/Hz
where |V ch ( i )| is the value of the amplitude spectrum at i th frequency bin  i , N 0 and N fin are the numbers of frequency bins, corresponding to the f 0 and f fin , respectively.N max is the total number of frequency bins of the FFT processing.Some specific values for the chirps with length of 1000 cycles are presented in Table 2.

Binary chirps with f 0 >0
When the initial frequency of a chirp f 0 ≠0, then the shape of spectra becomes more complicated.Though the spectrum retains the gradual character, the shrinking of plateaus is not always monotonic.It depends on the ratio between f 0 and f fin .In a certain case, one or more frequency sub-bands B z with zero magnitude appear into a spectrum.
The forming of such a spectrum as the sum of its harmonics is explained by the draft diagram in Fig. 4a.It is evident that fulfillment of the condition 3f 0 > f fin produces the frequency area B z,1 = 3f 0 -f fin next to the excitation bandwidth, where the spectrum has almost zero magnitude.In this specific case, the spectral density within the excitation bandwidth is determined only by the fundamental harmonic components from f 0 to f fin .In general, the frequency area above f fin can include more than a single zero sub-band, which separate spectral blocks of particular harmonics.Fulfilling of the condition causes the zero-valued area from the frequency (2i-1)f fin up to (2i+1)f 0 , i.e., B z,i =(f fin +f 0 ) -2i(f fin -f 0 ) (Figs. 4b, 4c).
The condition (8) with i=1 has the more practical importance, assuring moderately wide, but distinctly separated excitation bandwidth -see simulated and smoothed spectrum in Fig. 4b, where f 0 =50 kHz and f fin =100 kHz.This condition with i 1 simplifies evaluation of the energy-efficiency because the energy within the excitation bandwidth is always equal to the energy of first harmonic components from f 0 to f fin only (see the discussion in the previous chapter) Thus, in the case of binary chirps with 3f 0 > f fin the theoretical energy-efficiency is constantly δ E ≈0.81, i.e., it is equal to that of the common rectangular signal.Let us note that at i is equivalent to the f 0 =f fin .

Ternary chirps
A method to improve the spectral properties of rectangular signals (incl.rectangular chirps) is shortening their duty cycle by a certain amount, thus modifying the binary signal to a ternary one (see Figs. 1b and 4) [7].The shortening can be characterized by the relative time t/T ch or by the equivalent phase angle  (for binary signals  =0) per a quarter of a period (or cycle), within of which the value of signal remains on the zero state (Fig. 5).Let us explain the effect of shortening by Fourier series, which expresses for the rectangular-wave signal with the magnitude ±A as the sum of odd harmonics As it follows from (10), the k th harmonic (k=2i-1 with i=1, 2, 3, …) is absent from the series F(t), when k =(2i+1)π/2, and i=0, 1, 2, …, because of cos(k)=0.For example,  =π/6, causes removing of the (3+6i) th harmonics, =π/10 removes the (5+10i) th harmonics, etc.This is valid for the rectangular chirps, too, but in this case the time duration of every quarter-cycle is changing in accordance with the instantaneous frequency (Fig. 5b).Hence, in the generation of ternary chirps V ch (t)=sgn(sin((t))) one must follow the current phase: if 2kπ-<θ(t)<2kπ+ or (2k+1)π-<θ(t)< (2k+1)π+, (k=0, 1, 2, …), then V ch (t)=0 (Fig. 6).Assuming numerical processing with sampling frequency f samp , the current phase  i (i=0, 1, 2,…,T ch f samp -1) in degrees according to (1) expresses as

Fractional-power (sub-linear) chirps
The previous examples indicated almost constant (on the average) spectral density within the excitation bandwidth (Figs.3,4,6), but in biomedical studies it is sometimes practical to apply the stimulus with the spectral density rising with frequency.This effect can be achieved by using of sub-linear power chirps with the exponent n, slightly less than unity, for example, n=0.8 to 0.9.

Conclusions
The chirp-wave excitation signal is highly convenient in bioimpedance measurements and impedance spectroscopy due to the possibility of fast and broadband identification of objects.Though the best measurement quality can be achieved by implementing the sine-wave chirp stimulus, the advantage of rectangular-wave signals is the simplicity of generation and processing of signals.
Aside the simpler hardware and software, another essential factor is energy consumption, which is extremely important in the development of miniaturized devices.That is why the energy-efficiency of rectangular-wave chirps were analyzed in this work.It was shown, that the useful energy of discrete-level chirps is sufficiently high, exceeding 90% of the generated energy for ternary ones.
Depending on the aim and practicable needs, the shape of spectra can be simply modified by changing the initial frequency and duty cycle of generated chirps or selecting an appropriate mathematical basis of the signal.
The drawback of rectangular-wave chirps is their intensely rippling spectra, which can disturb the interpreting of measurement results.A way to overcome this issue is to focus on the phase spectra, in which the effect of rippling is diminished largely.This opportunity is more closely treated in the earlier works of authors [3].

Fig. 1 .
Fig. 1.Core structure of the 2-channel BI measurement system

Fig. 2 .
Fig. 2. Waveform of: (a) binary chirp; (b) ternary chirp Another important class of pseudo-chirps is ternary (trinary) chirps or Return-to-Zero (RZ) chirps with the possible values of {+A; 0; -A} -see Fig. 2b.Unlike sine-wave chirps, the spectrum of the discretelevel ones is rippling intensively.Besides, the percentage of their in-band energy is somewhat less at the equal amplitude and duration of the pulse.In practice, these drawbacks should be compensated by the simplicity of the signal processing.An additional benefit of binary chirps is their unity crest factor, and consequently, major energy compared with the sine-wave chirps of the same length.Next, let us examine some specific spectral features of both the mentioned modifications of linear pseudochirps (n=1).

Fig. 3 .
Fig. 3. (a) Power distribution of harmonic components; (b) power of spectral plateaus h; (c) simulated power spectrum at f fin =100 kHz

Fig. 5 .
Fig. 5. Sketched ternary signals: (a) period of a signal with constant frequency; (b) cycle of a chirp

Table 1 .
The ratio of the PSD plateaus p 1 /p h

Table 2 .
Energy and power of chirps (A=1 V, R load =1 k)