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10
Aspects of Using Chirp Excitation for
Estimation of Bioimpedance Spectrum
Toivo Paavle, Mart Min and Toomas Parve
Th. J. Seebeck Dept. of Electronics
Tallinn University of Technology,
Estonia
1. Introduction
Short frequency swept signals, known as chirps, are widely used as excitation or stimulus
signals in various areas of engineering as radar and sonar techniques, acoustics and
ultrasonics, optical and seismological studies, but also in biomedical investigations,
including bioimpedance measurement and impedance spectroscopy (Müller & Massarani,
2001; Misaridis & Jensen, 2005; Barsoukov & Macdonald, 2005; Nahvi & Hoyle, 2009).
Signal processing in the chirpbased applications is often combined with pulse compression
via crosscorrelation procedure and Fourier analysis. In this chapter, a similar approach is
proposed for estimation of the frequency response (the impedance spectrum) of electrical
bioimpedance. An advantage of the chirpbased method is that the characteristics of a
biological object can be obtained in a wide frequency range during a very short
measurement cycle, which nearly eliminates the influence of lowfrequency biological
processes (heart beating, breathing, pulsation of blood) to the result of measurement.
The changes of a spectrum monitored at sequent time intervals by means of the Fourier
Transform (known as spectrogram) is an informative base for interpreting the processes in
biological objects. Furthermore, the signal treatment by the crosscorrelation yields a better
noise immunity for the measurement system (Barsoukov & Macdonald, 2005), and adds
some alternatives for estimation of the bioimpedance properties and behavior.
2. Basics of bioimpedance measurement
2.1 Object of measurement
The impedance of living tissues or, in general, of arbitrary biological matter (electrical
bioimpedance, EBI) can be characterized by its electrical equivalent, which, in turn, can
be represented as the frequencydependent complex vector Ż(jω) = Re(Ż(jω)) +
jIm(Ż(jω)) = Z(ω) exp(jΦz(ω)), where ω = 2πf, Z(ω) = (Re(Ż(jω))2 + Im(Ż(jω))2)½, and
Φz(ω) = arctg(Im(Ż(jω))/Re(Ż(jω))).
On the other hand, the bioimpedance is not a constant value, but a function, which is
changing in time due to numerous biological processes in the living tissue, and as a matter
of fact, is a function of frequency and time altogether as Ż=Z(jω,t).
Fourier Transform – Signal Processing
238
Injection of the excitation current (stimulus) Iexc(t) with known parameters into biological
object courses the response voltage Vz(t), analysis of which enables to estimate the
impedance spectrum Ż(ω) of the object being under the investigation.
Fig. 1 illustrates the path of excitation current through the cells of a tissue, where rext
corresponds to the extracellular resistance, and the resistive components rint together with
intercellular capacitances Cc constitutes the intracellular impedance (Grimnes & Martinsen,
2008; Min & Parve, 2007).
Fig. 1. Formation of the electrical bioimpedance of tissue
Very often, the response signal is analyzed using the Fourier Transform F(Vz(t)) to get
information about the frequencydependent state and changes of the biological matter. Here,
the impedance spectrum of EBI manifests as Ż(jω)=F(Vz(t))/F(Iexc(t)), i.e., the Fourier
Transform of the response voltage determines the impedance spectrum of the object oneto
one thanks to the predetermined parameters of the excitation signal.
Usually, in theoretical considerations and simulations, the bioimpedance is substituted by a
certain RCcircuit. Naturally, the accuracy of such approximation depends on the number
and configuration of components. In Fig. 2a, a 5element circuit is shown, where R0
corresponds to the extracellular resistance rext of a tissue, while R1, C1, R2 and C2 stand for
the intracellular parameters. The respective Bode diagram has two real poles fp1, fp2, and
two zeros fz1, fz2 (Fig. 2b), spread typically over the frequency range from some kHz up to
several MHz (Nebuya et al 1999; Pliquett et al, 2000; Grimnes & Martinsen, 2008).
The Laplace transform of the 5element EBI (Fig. 2a) can be expressed as (Paavle et al 2008)
(
)
(
)
[]
()()
011 22
02 11 1 22 11 22
11
() (1 ) (1 ) 1 1
RsCR sCR
Zs R sC sC R sC sC R sC R sC R
+
+
=+++ ++ + (1)
V
z
I
exc
R
0
R
1
R
2
C
2
C
1
f
p1
f
z1
f
p2
f
z2
log(Ż/R
0
)
log
f
(a) (b)
Fig. 2. (a) 5element model and (b) Bode diagram of the bioimpedance
In the simplest case, where C2→0 (basic 3element EBI):
low and high
frequency
mostly high
frequency
paths for current
Iexc Iexc
Cc Cc
rint
rext
Vz
Aspects of Using Chirp Excitation for Estimation of Bioimpedance Spectrum
239
(
)
()
011
10 1
1
() 1
RsCR
Zs sC R R
+
=++
(2)
For adequate estimation of the bioimpedance, the spectrum of excitation signal should cover
the frequency range of the object Ż as much as possible. For that reason, several types of
broadband excitation signals (e.g., Maximal Length Sequence (Gawad et al, 2007), multisine
(Sanchez et al 2011)) are preferable. However, in this chapter, we will focus on the use of
chirp excitation due to several advantages of it: wide and flat amplitude spectrum (Nahvi &
Hoyle, 2009) together with independent scalability in the time and frequency domain.
2.2 Essential of chirp excitation
2.2.1 Variety of chirps
A sinewave based chirp with current phase θ(t) can be described mathematically as
()
(
)
()
(
)
ch () sin sin 2 dVt A t A
f
tt=θ= π
∫ (3)
where A is the amplitude, and f(t) = (dθ(t)/dt)/2π is instantaneous frequency of the chirp
signal.
A definite class of chirps has the instantaneous frequency f(t), which changes accordingly to
some power function of the nth order (power chirps). Their specific quantity is chirping rate
β=(ffin–fst)/Tchn, where fst and ffin are the initial and final frequencies, respectively, and Tch is
the duration of the pulse. Chirps of this type can be expressed as
()
(
)
(
)
1
ch st
() sin 2 β/1
n
Vt A ft t n
+
=π+ + (4)
The instantaneous frequency of a chirp can be increasing (upchirps, β>0) or decreasing
(downchirps, β<0) quantity. Besides, sometimes it is practical to generate chirps with a
symmetrical bidirectional frequency change (bidirectional or double chirps). In this case, the
actual duration of the pulse is 2Tch, and the sign of chirping rate alters at t =Tch, causing
mirrored waveform of the pulse against that moment.
Fig.3 sketches waveforms of different chirps with the equal pulse duration and almost equal
frequency range. A very basic linear (n=1) chirp with fst=0 can be described in accordance
with the expression (4) as Vch(t)=Asin(2πffint2/2Tch). It is depicted in Fig. 3a.
Fig. 3b shows the waveform of quadratic (n=2) downchirp. However, the rule of frequency
change can be arbitrary. For example, in some specific measurements, excitation with the
exponential chirping rate β =( ffin /fst) t/Tch is appropriate (Darowicki & Slepski, 2004). The
waveform of sinusoidal exponential chirps (see Fig. 3c) is described as
(
)
(
)
ch ch
sin 2 (β1) / ln(β)
st
Vt fT=π − (5)
Generation of a perfect sinewave chirp requires quite complicated hardware, which can
cause problems, especially in onchip solutions. That is why socalled signumchirps
(known also as pseudo, binary or NonReturntoZero (NRZ) chirps) are often
implemented (Figs. 3d and 3e; the latter one depicts a binary chirp with bidirectional change
Fourier Transform – Signal Processing
240
of frequency). This kind of chirps can be defined by the signumfunction of respective sine
wave chirps as Vch(t)=Vsgn(t) = sign(Vsin(t)), which have binary values +A and –A only.
(a) (b) (c)
(d) (e) (f)
Fig. 3. Examples of chirp waveforms with the equal maximal frequency and duration, Vch(t)
vs. time: (a) linear sinewave chirp; (b) quadratic downchirp; (c) exponential chirp;
(d) signumchirp; (e) signumchirp with bidirectional (downup) run of frequency;
(f) ternary chirp with shortened duty cycle by 30°
Rectangular waveforms simplify signal processing: both generation of excitation and
processing of the response will be substantially simpler. Especially simple is calculating of
the correlation function or deconvolution in the time domain – shifting and multiplication
with a reference signal having only {+1, 1}, or {+1, 0, 1} values (Rufer et al, 2005).
Additional advantage of signumchirps is their unity crest factor (i.e., the peak amplitude
ratio to the rootmeansquare (RMS) value of the signal), and major energy compared with
the sinewave chirps of the same length. Unfortunately, using of rectangular signals causes
the worse purity of the spectrum due to the accompanying higher harmonic components.
Suppression of the particular harmonic component in the spectrum of rectangular signals
can be achieved by shortening the duty cycle of the signal by a certain degree αd per every
quarterperiod (Parve & Land, 2004). Signumchirps, modified in this way, are called
ReturntoZero (RZ) or ternary chirps. It means that Vsgn(t) returns to zero, if the value of
current phase falls into the intervals 2nπ – αr<θ(t)<2nπ+αr or (2n+1)π – αr <θ(t)< (2n+1)π+αr,
where αr = παd /180 in radians, and n=0, 1, 2, … (Fig. 3f). For explanation let us remember
that spectra of a rectangular signal can be declared as the Fourier series of odd harmonics:
1
cos cos3 cos(2 1)
44
( ) sin sin 3 sin(2 1)
13 21
rr r
i
i
AA
ttt it
i
∞
=
⎡
⎤
αα −α
⎡⎤
ω = ω+ ω+ = − ω
⎢
⎥
⎢⎥
ππ−
⎣⎦
⎣
⎦
∑
…F (6)
It follows from (6) that the kth harmonic (k=2i−1, and i=1, 2, 3, …) is absent from the series
F(ωt), when kαr=±(2n+1)π/2, and n=0, 1, 2, … , because of cos(kαr)=0. Consequently, in the
case of αr = π/6, the (3+6n)th harmonics are removed, and for αr = π /10, the (5+10n)th
harmonics are removed, etc. (Paavle et al, 2007).
2.2.2 Shorttime chirps (titlets)
Commonly, referring to chirps, signals of many cycles (rotations by 2π of the chirp
generating vector) are considered (multicycle chirps). It does not need to be so, and the
Aspects of Using Chirp Excitation for Estimation of Bioimpedance Spectrum
241
chirps with a single cycle or even less (θ(Tch) ≤ 2π) can be generated and used, too. For such
kind of ultra short chirps, the neologism “titlets” or Minimal Length Chirps (MLC) have
been used (Min et al, 2011a).
The titlets can be very effective excitation signals in applications, where broadband
excitation is necessary, but the minimal power consumption or extremely fast measurement
are required at the same time. Because of the prospective use of titlets, a special attention to
their properties will be paid below. The diagram in Fig. 4 explains the forming of a single
cycle linear sinewave chirp.
θ
sin θ
cos θ
sin θ(t)
t
t
1
0
1
θ
fin
=2π
θ
st
=0
θ
i
=
θ(t
i
)
t=0
f
t=0 t
i
T
ch
=2/f
fin
f
fin
f
st
= 0
f(t
i
)
nonuniformly
rotatin
g
p
haso
r
fin
ft /2=
2/2 fin
ff =
Fig. 4. Genesis of the linear singlecycle sinewave chirp
Voltage
Time, μs
n=1; L=½
n=1, L=¼
n=2; L=2×¼
exponential, L=1, f
st
=10 kHz
0.0 5.0 10.0 15.0 20.0 25.0
1
0
1
μs
Fig. 5. Examples of titlet waveforms with ffin=100 kHz (fst=0 for power titlets)
Considering (4), we can state that the equality of phases θ(t) = 2π(fstTch+βTch/(n+1) = 2πL is
valid for the t =Tch. Thereby, the number of cycles L can be an integer or fractional quantity.
It allows us to derive the relationship between the length of the chirp and its parameters. For
the nth order power chirps this relationship expresses as (Min et al, 2011a)
(
)
(
)
ch fin
1/ st
TLn n
ff
=+ + (7)
Thus, for the singlecycle linear chirp with fst=0, the pulse duration Tch=2/ffin. It follows for
this case that the change of polarity (θ(t) = π) occurs at fin
2/t
f
=, while the instantaneous
frequency is fin
2/2ff= (see Fig. 4).
Fourier Transform – Signal Processing
242
Similarly, considering (5), one can show that for exponential chirps the duration of a pulse is
(
)
ch fin st fin st
ln / /( )TL
ff ff
=⋅ − (8)
Simulated waveforms of some typical titlets with L ≤ 1 are shown in Fig. 5.
2.3 Signal analysis and bioimpedance measurement by using Fourier Transform
Spectral analysis is the irreplaceable method as for the study of the behavior of excitation
signals in the frequency domain as well as for processing the response signal in the
bioimpedance measurements.
It is obligatory to remember that in the modeling and processing of chirp signals the
selected sampling rate fs must follow the Nyqvist criterion over the whole frequency range,
i.e., the condition fs ≥ 2ffin must be fulfilled. The sampling rate determines the total number
of samples Nch during a chirped pulse as Nch = Tch fs. Yet, the simulation or real processing
time Ttot can be longer than Tch. In this case, the zero padding is used at t>Tch. The length of
the input array for the Fast Fourier Transform (FFT) is N=Ttotfs ≥ Nch, and it determines the
frequency resolution Δf (difference between two successive frequency bins) of the FFT
processing, whereas Δf = fs/N according to the uncertainty principle (Vaseghi, 2006). The
acquired positive frequency range of the FFT processing in Hz is 0, …, NΔf/2 by steps of Δf.
2.3.1 Principles of bioimpedance measurements
There are several methods to calculate the frequency spectrum of the bioimpedance
response. A typical way is the implementation of two FFTchannels for the response and
excitation signals separately (Min et al, 2011a). Nevertheless, in the following discussion we
will focus mainly on the structure and modeling of a specific bioimpedance measurement
system, which incorporates the crosscorrelation procedure together with following FFT
processing (Paavle et al, 2008, Min et al, 2009, see Fig. 6).
Fig. 6. Basic structure of the measurement system
In such a system, the crosscorrelation function (CCF) is calculated: (a) between the response
Vz(t) and excitation Vch(t) at the unity gain in the reference channel (Vref(t)= Vch(t), Żref =1),
or (b) between the response and predefined reference signal Vref(t) (reference channel
includes the known impedance Żref ≠1). In the latter case, the system works as the matched
filter, enabling detection of mismatches between the impedance vectors Ż and Żref more
precisely together with somewhat better noise reduction. The source noise ns(t) and object
noise nz(t) can be taken into account in simulations as shown in Fig. 6.
Pxy(f)
Φxy(f)
Ż
Iexc
Vz
Reference
Żref
FFT
I
V
Vch
Vref
rxy Pxy(f)
ns(t) nz(t)
Chirp
generator
Cross
correlator
Calculation
of
EBI components
Aspects of Using Chirp Excitation for Estimation of Bioimpedance Spectrum
243
In general, calculation of the CCF proceeds as
(
)
xy z ref z ref
() (), () () ( ),rCorrVtVtVtVt
τ
==+τ (9)
where τ is a variable delay (lag) and the overline denotes averaging.
Using broadband chirp excitation, the waveform rxy(τ) of CCF is similar to the sin(x)/x type
sincfunction affected by the nature of Vz(t) and is strongly compressed close to τ = 0.
The CCF includes both the information about the amplitude level and phase shift of the
bioimpedance vector. Thanks to this fact, only a single FFTblock is necessary to obtain the
complete information about the object. Otherwise, when using the direct Fourier Transform
of the response, another FFT channel is required to establish the basis for phase evaluations.
In accordance with the crosscorrelation theorem (generalized WienerKhinchin theorem),
the correlation function is equivalent to the inverse Fourier Transform of the complex cross
power spectrum density Pxy(f) with the magnitude and phase spectral components
22
xy xy xy
( ) (Re( ( ))) (Im( ( )))Pf Pf Pf=+
and Φxy(f)=arctg(Im(Pxy(f))/ Re(Pxy(f))), respectively
(Vaseghi, 2006). Thereby, the phase component Φxy(f) specifies the phase difference between
the vectors Vz and Vref, and magnitude presents the geometric mean of the power spectral
densities of the signals Vz(t) and Vref(t) (McGhee et al, 2001).
2.3.2 Modeling of the measurement system
Accordingly to the main architecture of measurement system (Fig. 6) and model of EBI (Fig.
2a), a special PCmodel (Fig. 7) was developed for verifying the theoretical conceptions
(Paavle et al, 2008).
Fig. 7. Model of the measurement system
The crosscorrelation function is calculated by the cumulative adder for every mth lag τm as
[] [] [ ]
ch 1
xy z ref m
ch 0
1,
N
k
rm VkV k
N
−
=
=⋅+τ
∑ (10)
where k = 0 … Nch–1, and Nch is the number of samples per length of the chirp pulse
(McGhee et al, 2001).
Chirp
generator
Vexc
VzFFT
Windowing
(optional)
I
exc
Ż
R
0
R
1
R
2
C
2
C
1
Vch
V/I Żref
Cor
r
(
V
z
,
V
ref
)
Pxy(f) Φxy(f)
×
Delay
τ = var
Correlato
r
Discrete control
of the lag τm
Vref
Calculations
Σ ÷Nch
reset
at next m
Fourier Transform – Signal Processing
244
If the uniform delay step is Δτ, then for the overall delay interval (observation time)
tobs =τmax – τmin (τmin ≤τm≤ τmax) the number of necessary computing cycles is M +
1 with
M = tobs/Δτ. As a result, we acquire an array of correlation values rxy[m] with m = 0 … M.
Supposing that the selected Δτ and τmin are integer multiples of the sampling interval 1/fs,
the mth delay expresses as τm = fs(τmin+mΔτ).
In the following Fourier Transform, the array rxy of length M+1 represents the input stream
for the FFTblock, while in accordance with the uncertainty principle (see above), the
frequency resolution of spectra becomes Δf =1/tobs over the frequency range f = 0 … 1/(2Δτ).
3. Spectral features and energy of chirps
Next, let us take a look at the results of direct Fourier Transform of distinct chirp pulses. The
direct FFT of different excitation signals enables to compare their amplitude and power
spectra together with estimation of the energy properties keeping in view the requirements
for the bioimpedance measurement.
Important and desired spectral properties of excitation signals to improve the quality of
wideband measurement are:
• flat amplitude spectrum with minimal fluctuation (ripple) together with the absence of
overshoots inside the generated (excitation) bandwidth Bexc =ffin  fst;
• steep dropdown of the amplitude spectrum outside the bandwidth Bexc;
• maximal energyefficiency, i.e., the ratio between the energy lying within the generated
(excitation) bandwidth Bexc and total energy of the signal.
For some specific applications, these properties must coincide with minor power
consumption and with shortness of signal to ensure quick measurement.
L=100 L=10 L=1
Frequency, kHz
L=1000
L=100
L=10
L=1
Frequency, Hz
L=1000
20log(V(f )/V(f )
i0
dB
10
10
30
50
0.03
0.02
0.01
0
ch
Magnitude V (f )
0.0 20.0 40.0 60.0 80.0 100.0 kHz 1E3 1E4 1E5 Hz
(a) (b)
Fig. 8. Voltage spectral density of linear chirps 0…100 kHz (Δf=50 Hz): (a) linear scaling at
the unity amplitude of chirp pulses; (b) normalized logarithmic scaling
Fig. 8 depicts the amplitude spectra of linear sinewave chirps of different length in the
linear and in the normalized logarithmic scales. Thereby, the base of normalizing is the
amplitude V(f0), i.e., the magnitude at the lowest frequency bin f0 of the Fourier
Transform. It is obvious that long chirps assure the more fitting shape of spectra, except a
certain rippling caused by the Gibbs effect. The latter can be suppressed by using a kind of
windowing in the time domain (often a boxcartype window function is used for this
purpose), but in this case, the total energy of the signal decreases.
Aspects of Using Chirp Excitation for Estimation of Bioimpedance Spectrum
245
Fig. 9 shows the frequency run and respective amplitude spectra of different titlets. It
appears that the desired shape of the spectrum, including flatness, satisfying dropdown
outside the bandwidth (slopes from –20 dB/dec to –80 dB/dec were observed) and
admissible overshoots can be achieved by a proper selection of the type and length of titlet.
Moreover, additional correction and shaping of spectra can be attained by windowing of
titlets in the time domain (see below in Sect. 3.2). In Fig.10, waveforms and amplitude
spectra of some rectangular chirps (see Sect. 2.2.1) with L=10 are shown. Naturally, for this
kind of chirps, the rippling of spectra is noticeable.
n=1; L=½
n=1; L=1
n=2; L=1
n=3; L=¼
20 dB/dec
n=1; L=2×¼
80 dB/dec
20log(V(fi)/V(f0)
40 dB/dec
Frequency, Hz
Time,
μ
s
n=1; L=½ n=1; L=1
n=3; L=¼
n=1; L=2×¼
n=2; L=1
Freq.,MHz
0.1
0.05
0
0 5 10 15 20 25 μs
10
10
30
50
1E3 1E4 1E5 Hz
(a) (b)
Fig. 9. (a) Frequency run and (b) amplitude spectra of different titlets with Bexc=0…100 kHz
Time,
μ
s
V
ch
(t)
V
ch
(t)
d
=0°
d
=30°
d
=0°
d
=30°
d
=18°
Frequency, Hz
αα
α
α
α
1E3 1E4 1E5 Hz
20log(V(f )/V(f )
i 0
10
10
30
50
1
0
1
1
0
1
0 40 80 120 160 200 μs
0 40 80 120 160 200 μs
(a) (b)
Fig. 10. (a) Waveforms of binary and ternary chirps with αd=30°; (b) amplitude spectra of
binary and ternary chirps with αd =30° and αd =18° (Δf=100 Hz)
3.1 Energy of chirps
3.1.1 Sinewave chirps
In principle, the total energy of a chirp signal at the unity load (Rload=1Ω) expresses in time
domain as
ch 2
tot ch
0
() ,
T
EVtdt=∫ (11)
which leads to Etot = (A2/2)Tch for the long term chirps of sinusoidal waveform.
According to the Parseval’s theorem, the total energy in the frequency domain is the same as
the energy in the time domain (Vaseghi, 2006). For chirps with the finite length, a certain
Fourier Transform – Signal Processing
246
part of signal energy falls outside the useful (generated) bandwidth caused by higher
frequency components. For that reason, it is necessary to distinguish the total energy of a
generated chirp pulse, which varies proportionally with Tch, and the useful energy Eexc,
falling inside the chirp bandwidth Bexc. Typically, Eexc< Etot, while absolute values of both
quantities depend on the chirp length, waveform and spectral nature. To characterize the
percentage of the useful energy, we employ the term of energyefficiency as δE=Eexc/Etot. For
sinewave chirps with L→∞ the ratio δE →1.
1
10
100
1000
10000
1 10 100 1000
92
94
96
98
100
1 10 100 1000
Number of cycles L
Number of cycles L
Energyefficiency δ
E
, %
Energy of chirp E
tot
, nJ
(a) (b)
Fig. 11. (a) Total energy and (b) energyefficiency of linear sinewave chirps
Type of chirp (titlet) L fst, Hz Tch, μsPavg, mW Etot, nJ δE, %
¼ 0 5.0 0.32 1.6 55.0
½ 0 10.0 0.38 3.8 90.6
1 0 20.0 0.41 8.3 93.5
10 0 200.0 0.47 94.4 97.8
linear
100 0 2000 0.49 982 99.3
½ 0 15.0 0.29 4.4 93.9
1 0 30.0 0.33 10.0 95.7
quadratic
100 0 3000 0.464 1393 99.5
cubic 1 0 40.0 0.28 11.1 96.4
1 1 115.1 0.135 15.6 97.8
exponential 10 1 1151 0.235 271 99.5
doublequadratic 2×¼ 0 2×7.5 0.23 3.4 93.1
1 0 20.0 1.0 20.0 84.1
NRZ signumchirp 1000 0 20e3 1.0 20e3 85.1
RZ chirp (18º short.) 1000 0 20e3 0.8 16.0e3 93.1
RZ chirp (30º short.) 1000 0 20e3 0.67 13.3e3 92.1
Table 1. Energy and average power of different chirp pulses with fst≈0 and ffin=100 kHz
Considering expressions (11) and (4) or (5), the analytical description of the chirp energy
becomes very complicated. So it is reasonable to calculate the respective quantities
numerically. Nevertheless, a good approximation of total energy and energyefficiency of
arbitrary chirp signal can be obtained using the results from the Fourier Transform as
follows (Paavle et al, 2010):
Aspects of Using Chirp Excitation for Estimation of Bioimpedance Spectrum
247
max
fin
st
1
22
Echi chi
0
()/ ()
N
N
iN i
Vf Vf
−
==
δ=
∑∑
(12)
where V(fi) is the value of the amplitude spectrum at ith frequency bin, Nst and Nfin are the
numbers of frequency bins, corresponding to the fst and ffin, respectively. Nmax is the total
number of frequency bins, and the divisor in (12) corresponds to the total energy Etot. Of
course, this method enables to calculate the partial energy for any frequency interval inside
the range of 0 to NmaxΔf.
The curves in Fig. 11 present the dependence of the energy and energyefficiency of linear
sinewave chirps on the number of chirp cycles graphically. A selection of power and
energy parameters for different chirps and titlets, obtained from the FFTprocessing by
using the expression (12), are converged into Table 1, where the average power
ch 2
av
g
ch tot ch
ch 0
1() /
T
PVtdtET
T
==
∫ presumes the 1 kΩ load.
3.1.2 Binary signumchirps
A specific feature of signumchirps is the gradually decreasing amplitude spectrum by step
width of 2Bexc as shown in Fig. 12 (Min et al, 2009). Let us analyze this phenomenon.
The fundamental harmonic of a regular rectangular signal with amplitude A1 = (4/π) A has a
rootmeansquare (RMS) value 1/2 4 /( 2)AA=π
, energy E1 = (A12/2) Tch = (8/π2) A2 Tch,
and power W1 = E1/Tch= (8/π2) A2, which creates a constant value power spectral density
(PSD) w1 =W1/Bexc, V2/Hz, within the bandwidth Bexc of fundamental harmonic:
(
)
22
1exc
8/ /wAB=π (13)
The amplitudes of kth higher harmonics are of Ak = A1 /k. Therefore, the power of every kth
higher odd harmonic (k =3, 5, 7, …) is equal to W1 /k2, being spread over the frequency range
Bk =kffin − kfst =kBexc. Due to the fact that higher harmonics have k times wider bandwidth
than the fundamental (first) harmonic has, the PSD for higher harmonics can be expressed
through (12) as wk= (w1/k2)/k = w1/k3 (Min et al, 2009).
The total power of generated signal Vch(t) is gradually distributed over its whole frequency
range, theoretically from fst to ∞, see Fig. 12. The PSD ph of every gradual hth level (h= 1, 2, 3,
4, …) of the spectrum, beginning from the first one p1, is the sum of power spectral densities
of the fundamental and higher harmonics, w1 and wk, which contribute into the given level h.
Into the PSD p1 of the first level contribute all the signal components p1= w1+ w3+ w5+ w7+ …,
but into the second level mere the higher harmonics p2 = w3+ w5+ w7+ …, (k = 3, 5, 7, …), and
into the third level only the harmonics beginning from k = 5, p3=w5+ w7+ …, etc.
Generally,
()
3
h1 21
hm
ih
pw i
+
−
=
≈−
∑ (14)
in which i is an integer beginning from h, that is: i = h, (h+1), (h+2), (h+3), …, (h+m), where m
is the number of higher odd harmonics taken into account.
Fourier Transform – Signal Processing
248
The partial power P1= p1Bexc= Pexc of the first level (h=1) is the useful excitation power
()
13
exc 1 21
m
ih
PW i
+
−
=
≈−
∑ (15)
The power Pout falling outside Bexc is lost. The lost power can be found as a summed up
power of higher harmonics within the next levels of spectrum (h= 2, 3, 4, etc., see Fig. 10):
()( )
13
out 1
1
21/21
m
i
PW i i
+
=
≈−−
∑ (16)
where W1 = w1Bexc in (15) and (16) is the power of fundamental harmonic. As the minimal
frequency for every level is kfst, then the equations (1416) are approximate ones, which
become absolutely exact only for fst=0. In practice, when the excitation bandwidth Bexc
exceeds one decade significantly, e.g., ffin / fst > 30, these equations are exact enough for
engineering calculations.
The total chirp energy is Etot= Eexc + Eout = ( Pexc+Pout ) Tch and the useful excitation energy in it
is Eexc= Pexc Tch . From (15) and (16) follows the role of useful energy δE = Eexc /Etot.
Considering fst=0, the ratio for the energyefficiency expresses as
() ()
max max
32
E
11
21 / 21
hh
ii
ii
−−
==
⎛⎞⎛⎞
δ= − −
⎜⎟⎜⎟
⎝⎠⎝⎠
∑∑
(17)
A
0
A
2
≈ A
1
/4.5
A
3
≈
A
1
/
8.4 A
4
≈A
1
/12.5
A
1
=4 A
0
/
NRZ signumchirp
sinewave chirp
Magnitude, V(f)
Frequency, kHz
2B
exc
π
0.04
0.02
00 100 200 300 400 500 kHz
Fig. 12. Amplitude spectra of 1000cycles sinewave chirps and signumchirps with fst=0 and
ffin=100 kHz (A=1)
Taking into account all the higher harmonics (m→∞ and (max h)→∞), the sums in (17) will
obtain limit values, which can be found via Riemann zeta function (Dwight, 1961):
1
() x
n
xn
∞
−
=
ζ=
∑
(18)
Aspects of Using Chirp Excitation for Estimation of Bioimpedance Spectrum
249
Riemann’s mathematics gives us the following limit values for the sums in (17):
22
1
(2 1) / 8
i
i
∞−
=
−=π
∑, (19)
3
1
(2 1) 7 (3)/ 8
i
i
∞−
=
−=ζ
∑, (20)
where ζ(3)≈1.202 is termed the Apéry’s constant (Dwight, 1961). Now, using equations (19)
and (20), we can express the part of useful energy in the linear NRZ signumchirp pulse as
2
E7(3)/
δ
=
ζ
π (21)
Fig. 13. (a) Amplitude spectrum of signumchirp with fst=50 kHz and ffin=100 kHz (A=1;
L=1000); (b) explanatory diagram of forming the amplitude spectrum at fst≠0
When ffin / fst >> 30 and m >> 1, the value of δE = 0.852 and the spectrum has practically
uniform energy distribution over the Bexc. The relative decrement of every hth step of the
averaged amplitude can be calculated from (14) as δh= (Eexc /Eh)1/2=A1/Ah. The values for
some lower order levels are δ2 ≈ 4.51, δ3 ≈ 8.44, and δ4 ≈ 12.47 − see the approximations given
in Fig. 12. For the next levels, a rough approximation δh ≈ 4(h−1) +1 is appropriate for
engineering calculations (Min et al, 2009).
Magnitude, V(f)
Frequency, kHz
sinewave chirp
NRZ signumchirp
Result of 32tap moving averaging
3fst–ffin
fst
3r
d
harmonics
3r
d
and 5
th
harmonics
5
th
harmonics
(a)
Bexc
0 fst .. ffin ..3fst 5fst 3ffin 7fst 5ffin 7ffin f
5
th
harmonics 5fst…5ffin
7
th
harmonics 7fst…7ffin
Summary spectrum
3r
d
harmonics 3fst…3ffin
Fundamental harmonics fst…ffin
Magnitude
(b)
Bexc Bzero
Combination of higher harmonics
Fourier Transform – Signal Processing
250
The shape of amplitude spectrum becomes more complicated, if the initial frequency fst≠0.
Though, the spectrum retains the gradual character, the levels do not attenuate
monotonically. An example of this kind of spectrum is shown in Fig. 13a, where fst= 50 kHz
and ffin=100 kHz. In general, the height and location of spectral levels can be various,
depending on the ratio between fst and ffin.
The forming of spectra as the sum of its harmonic components is explained by the draft
diagram in Fig. 13b. Fulfillment of the condition 3fst > ffin produces the frequency area
Bzero = 3fst –ffin next to the excitation bandwidth, where the spectrum has almost zero
amplitude. In this particular case, the spectral density within the excitation bandwidth is
determined only by the fundamental harmonics from fst to ffin. It can be shown, considering
the Eqs.(13) –(16), that here the energyefficiency δE=8/π2.
3.1.3 Ternary chirps
Caused by the absence of particular harmonics (see Sect. 2.2.1), the position and span of
amplitude levels of ternary chirps (RZ chirps) differ from the levels of the respective binary
chirp. Fig. 14a enables to compare the amplitude spectra of several rectangular chirps with
L=1000 at various durations of the zerolevel state αd (shortening).
Let us pay attention to the changed stretch of the amplitude levels. For example, in the case
of αd =30º, the average amplitude of 2nd level (h=2) stretches uniformly from ffin up to the
5ffin. As the 3rd harmonic are absent, this 2nd level is formed by the amplitudes of the 5th and
higher harmonics up to the frequency 5ffin.
(a) (b)
Fig. 14. (a) Amplitude spectra of ternary chirps with ffin=100 kHz (32tap moving averaging
has been applied for smoothing); (b) energyefficiency of ternary chirps vs. zerolevel state
(shortening)
Average power of ternary chirps is less than it of respective binary chirps, but the
percentage of their useful energy is surprisingly high – over 90% mostly (see also Table 1).
Actually, it is possible to generate ternary chirps with any value of the zerostate, but only a
few values produce the removing of particular harmonics (see Eq. 6 in Sect. 2.2.1).
Nevertheless, we can analyze the energy properties of arbitrary ternary chirp. Employing
the equation (12), the maximal energyefficiency δE ≈93.4% was observed at
α
d ≈22.5º, which
does not cause disappearance of any odd harmonic component. Dependence of the energy
efficiency on the shortening αd is plotted in Fig. 14b.
Shortening
α
d, deg
Energyefficiency, %
NRZ;
α
d=0º
Magnitude, V(f)
Frequency, kHz
RZ;
α
d=18º
RZ;
α
d=30º
Aspects of Using Chirp Excitation for Estimation of Bioimpedance Spectrum
251
3.2 Windowing of chirps
In the use of some shorttime chirps, the problem of flatness of the amplitude spectrum
arises (see Fig. 9b). For the singlecycle sinewave chirp, the maximum overshoot of
normalized voltage spectral density is about +7.4 dB inside the chirp bandwidth. To
improve the flatness of spectrum, a kind of additional windowing of chirp pulses should be
used (in fact, every finite chirp pulse can be dealt as one inside the rectangular window,
however we consider this case as the unwindowed one). As a rule, the windowing
accompanies with some loss of total energy and power of signals, but still the δE can be
rather high. On the other hand, frequently the spectral density attains steeper dropoff
outside the chirp bandwidth due to windowing. Hence, the optimal choice of windows
presumes a certain tradeoff always.
In this work, several typical (Hanning, Hamming, Nuttall, etc.) and some specific window
functions Fwin(t) were under study (Barsoukov & Macdonald, 2005). For example, a
convenient shaping of amplitude spectra was achieved by implementing window functions
Fwin(t)=sin2(πt /Tch), which can be dealt as a particular case of the Tukey window with the
squaredsine lobes and with the tapering time Tch /2. Almost perfect shaping was attained
using a nonsymmetrical windowing in the form of Fwin(t)=(t/Tch)a with the selectable
exponent a (usually a =2…8) (Paavle et al, 2010). Some windowing results both in the time
and frequency domain are shown in Fig. 15.
(a) (b)
Fig. 15. Effect of windowing to the singlecycle sinewave chirp, fst=0, ffin=100 kHz, Tch=20 μs:
(a) windowed waveforms; (b) normalized spectra of the windowed waveforms
Usually, the deviation about ± 3 dB of the amplitude spectrum inside the Bexc is considered
as satisfactory for spectral flatness. There are several ways to achieve such the requirement.
Using the nonsymmetric windowing makes the deviation even less than ± 1 dB accessible
(see Fig.15b), but the quantity of useful energy reduces noticeably due to the plain slope of
the spectrum at f >ffin: for the case, as shown in Fig. 15, the energyefficiency δE=73.1% was
observed in simulations. Substantially higher δE was achieved by using of Nuttall
(δE=92.3%) and Hanning (δE=95.3%) windows, but the deviation of amplitude spectra inside
the Bexc is from 0 to 6 dB and from +0.2 to 6 dB, respectively (Paavle et al, 2010). The impact
of the squaredsine windowing was almost the same as it of the Hanning window.
As a rule, windowing of the generated signal demands some additional power consumption
and reduces the total energyefficiency. However, in the case of very short signals this
Time,
μ
s
unwindowed (rect.window)
Nuttall
squaredsine
(Tukey)
nonsymm., a=5.2
Voltage
20log(V(fi)/V(f0)
+7.4 dB max
6.3 dB @ ffin
unwindowed
nonsymm.,a = 5.2
Nuttall Tukey
Frequency, Hz
0
+1
dB
–
1dB f
fin
Fourier Transform – Signal Processing
252
drawback can be overcome using the lookup table, which stores the externally calculated
values of windowed titlets and which is loaded into the FPGA (Min et al, 2011a).
4. Simulation examples
The following examples refer to the linear and quadratic sinewave chirps and are the
results of simulation using the modeling structure in Fig. 7. In all the cases, the chirp
excitation from f = 0 to 100 kHz and amplitude A=1 V was implemented. The basic model of
EBI was a 5element impedance Ż1, consisting of R0=1 kΩ, R1=200 Ω, R2=100 Ω, C1=30 nF,
C2=20 nF and having the following corner frequencies in the Bode diagram (see Fig. 2b):
fp1=2.9, fz1=26.5, fp2=45.2, and fz2=79.6 kHz.
(c) (d)
Fig. 16. Examples of the timefrequency analysis of the EBI (sinewave chirp excitation with
L=1000): (a) waveforms of excitation and response; (b) normalized correlation functions;
(c, d) crosspower spectral density and phase spectra at variations of the EBI components
Fig. 16 demonstrates simulation results in the case of multicycle chirp excitation with pulse
duration of Tch=20 ms (1000 cycles). Fig. 16a shows the response voltage Vz(t) on the
background of the nonwindowed excitation signal. Fig. 16b shows crosscorrelation
functions in a stretched time scale at the Ż= Ż1, and at the unity value impedance Ż= Ż0 =1
(in fact, the latter case corresponds to the autocorrelation function of the excitation signal).
Results of the Fourier Transform of the CCF are shown in Figs. 16c and 16d, where one can
watch the swing of spectral curves, if the capacitive components of EBI change by +50%. In
addition, Fig. 16c demonstrates, how the boxcartype windowing (here squaredsine Tukey
window with the tapering time of 0.05Tch) reduces the influence of the Gibbs effect.
Intensive fluctuation of spectra, caused by the concurrent higher harmonics, can be more
disturbing for signumchirp excitation (see Figs. 10b, 12, and 13a). This problem can be
diminished by proper selection of correlation parameters, e.g., the shorter tobs with rougher
frequency resolution smoothes spectrograms essentially (Paavle et al, 2008).
V(t)
Time, ms
Ż=Ż0=1
V
z
at Ż=Ż1
L=1000
Phase Φzr(f), deg
kHz
Ż0
Ż1
C1
*
=1.5 C1
C2
*
=1.5 C2
Fre
q
uenc
y
, kHz
(b)
(a)
Ma
g
nitude.

Pzr
(
f
)
, V
2
/Hz
Ż0, windowed
Ż0
Ż1 C1
*
=1.5 C1
C2
*
=1.5 C2
6e
4
4e
4
2e
4
0
Frequency, kHz
Lag,
μ
s
autocorrelation rxx
(Ż=Ż0) crosscorrelation rxy
(Ż=Ż1)
Correlation
Aspects of Using Chirp Excitation for Estimation of Bioimpedance Spectrum
253
Fig. 17. The use of double quartercycle quadratic chirp excitation (L=¼, n=2, Tch=2×7.5 μs,
fst =0, ffin=100 kHz): (a) signal waveform; (b) variation of instantaneous frequency;
(c) amplitude spectrum of the chirp pulse; (d) phase spectra at matched filtering (Żref = Ż1)
and at changing of the EBI components +5%
For interpreting the variations of frequency responses, the phase spectra should be
preferred. First, it is less affected by the spectral fluctuations. Secondly, the phase spectra
enable to distinguish changing of object parameters somewhat more clearly than the
amplitude ones – compare the respective curves in Figs. 16c and 16d. Nevertheless,
substantially better resolution of the EBI for the diagnosing purposes can be achieved by
using matched filtering, where instead of Żref=1 in the reference channel, a predetermined
Żref≈Żx is used, approximately equal to the impedance Żx under study (see Fig. 6). Joining
matched filtering together with analysis of phase spectra allows applying the full scale of
some degrees only and permits very good sensitivity for detection of small changes of the
object parameters.
Simulation results in implementing of the bidirectional quartercycle quadratic chirp
excitation for detection of tiny deviations of the EBI vector are presented in Fig. 17. The
waveform of excitation pulse and the corresponding frequency run are shown in Figs. 17a
and 17b, respectively. Fig. 17c depicts the normalized amplitude spectrum of the excitation
signal and Fig. 17d manifests deviation of phase spectra, when the value of single
components increases +5% over their initial quality.
(d)
Phase deviation
ΔΦz(f), deg
Frequency, kHz
1.05C1
matched, Ż1=Żref
1.05R1
1.05R2
1.05C2
(c)
Time,
μ
s
Vexc, V
Freq., kHz
(
b
)
0
50
100
(a)
Time,
μ
s
20log(V(fi)/V(f0)
–80 dB/dec
Frequency, Hz
Fourier Transform – Signal Processing
254
Another advantage of the matched filtering is reducing the impact of additive noise,
especially the impact of source noise ns(t). Fig. 18a depicts the noisy input signal, where the
generated chirp is affected by the Gaussian noise with the zero mean and variance σN2 – see
Fig. 6. The source noise causes some error in the phase spectra at the higher frequencies (Fig.
18b). This error depends on the deviation of component parameters of the equivalent circuit
(here C1 was changed), and on the signaltonoise ratio (SNR=Pavg/σN2), but as a rule, the
error remains small even at considerably high level of the source noise.
(a) (b)
Fig. 18. Linear singlecycle chirp (Tch=20 μs) at the presence of additive source noise:
(a) noisy input signal; (b) spectra of phase differences at 1.05C1 and at different input signal
tonoise ratio (SNR) levels
As the noise from object (nz(t) in Fig. 6) affects the crosscorrelation procedure non
symmetrically, then suppression of noise effect is of importance and serves special attention
even at the higher SNR values (Min et al, 2011b).
5. Conclusion
The advantages of using chirps as excitation signals in bioimpedance measurement are their
wide and almost flat amplitude spectrum together with the independent scalability both in
the time and frequency domain – we can choose the frequency range and duration of the
excitation pulse almost independently from each other. These features enable to
accommodate the generation of excitation signals with the expected properties of the object
to be estimated comparatively simply.
It was shown that the shortening of chirp pulses retains the general benefits of chirps – their
flat amplitude spectrum within the predetermined frequency range, which permits their
implementation in energyefficient measuring instruments. More than 90% of the generated
excitation energy falls into the desired bandwidth even in the case of very short excitation
pulses required for providing ultra quick measurement and analysis of dynamic objects.
These requirements are obligatory for investigation of objects with rapidly changing
parameters (e.g., for identification of fast moving bioparticles such as cells and droplets in
highthroughput microfluidic systems), and in the devices, in which the low power
consumption is important (e.g., wearable units and medical implants). However, shortening
of chirp excitation and measurement time should not be excessive. It must be as short as
possible to avoid significant impedance changes during the Fourier analysis, but as long as
possible to enlarge the excitation energy and to obtain a better signaltonoise ratio.
Am
p
litude, V
SNR≈0 dB pure signal
Time,
μ
s
Phase deviation, de
g
Frequenc
y
, kHz
SNR≈6 dB
SNR
≈
0dB
pure signal
Aspects of Using Chirp Excitation for Estimation of Bioimpedance Spectrum
255
The described measurement method, based on the sequentially performed crosscorrelation
and Fourier Transform, enables the joint timefrequency spectral analysis of bioimpedance.
As the time domain crosscorrelation function includes the full information about the
complex vector Ż(ω), only a single FFTblock is necessary for the Fourier analysis and the
complete frequency domain determination of the object. Moreover, the crosscorrelation
procedure assures substantial suppression of noise introduced by the object.
Applying the modification of system architecture with the crosscorrelation based matched
filtering enables to avoid the impact of noise in much higher degree then a simple cross
correlation. In addition, such the matched filtering permits to increase the sensitivity of
measurement, especially through the analysis of phase spectra. Identification and
interpretation of relative deviations of the object parameters through the detected tiny phase
shifts is of great importance, which in turn can give valuable information about the state and
processes in the biological objects.
6. Acknowledgment
This work was supported by the European Union through the European Regional
Development Fund, Estonian targetfinanced project SF0142737s06 and by Enterprise
Estonia through the ELIKO Competence Center.
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