Frequency Synchronization Analysis in Digital lock-in Methods for Bio-impedance Determination

Transcription

1 .478/msr-4-47 Freqency ynchronization Analysis in Digital lock-in Methods for Bio-impedance Determination obert Brajkovič, Tomaž Žagar and Dejan Križaj niversity of Ljbljana, Faclty of Electrical Engineering, Tržaška 5, Ljbljana, lovenia Geodetic nstitte of lovenia, Jamova cesta, Ljbljana, lovenia, The lock-in method is one of the most freqently sed methods for reconstrction of measred signals and as sch freqently applied in the (bio)impedance method to determine the modls and phase of the (bio)impedance. n implementation of the method in a (bio)impedance measrement device one has to consider possible non synchronized freqencies of the reference and the analyzed signals as well as potential sorces of noise. n this work we analyzed these errors theoretically and experimentally. We show that both amplitde and phase errors depend on the relative difference of the freqencies of the reference and investigated signal as well as the nmber of integration periods. Theoretically, these errors vanish dring the determination of the (bio)impedance modls and phase. n practical implementation the inaccracies appear at points of very low determined signal amplitdes de to the limited accracy of analog to digital converters and are distribted arond these points de to other sorces of noise inherent in implementation of the measrement device. Keywords: mpedance, bioimpedance measrements, freqency synchronization, lock-in method.. NTODCTON N PHYCAL EXPEMENT we often analyze periodic signals of small amplitdes that are bried in noise and other signal distrbances with amplitdes (energy) that might be significantly larger than the analyzed signal itself. Becase conventional filtering techniqes do not provide satisfactory reslts in sch cases, special filtering techniqes are reqired in order to differentiate signal from noise. One of the most freqently sed methods ses the so called lock-in approach (also known as phase sensitive detection) [], [7]. Phase sensitive detectors reqire (pre) knowledge of a period of analyzed periodic signal. A reference signal is created in accordance with an expected measred signal and is compared with it. Depending on the implementation of the lock-in method there are several possibilities of generation of a reference signal as well as its synchronization with the measred signal []. n this investigation we will analyze the problems of synchronization or rather non-synchronization of the external reference signal and the measred signal in particlar in the frame of its sage in the (bio)impedance measrement method. n cases where a signal is accompanied by broad band noise it is feasible to generate reference signal externally instead of sing excitation signal as reference. n this case the phase sensitive detector responds also to harmonic components of the excitation signal even thogh the fndamental component is well defined []. Bioimpedance as defined in the electrical circit theory is essentially a transfer impedance, determined as a ratio of a voltage and a crrent complexor which are not necessarily related to the same nodes [5]. However, the techniqe assmes that the same crrent flows throgh all the nodes. This method is particlarly of interest in stdies of impedance methods in biological systems in order to redce or even eliminate the difficlties related to the complex electrode-tisse electrical phenomena []. The discssed topics will be analyzed theoretically as well as experimentally sing a portable battery powered bioimpedance measring device that is crrently nder development [8]. Depending on the reqired application bioimpedance measring devices can vary significantly. Here or goal was to develop a low cost portable device with a freqency range from Hz to 4 khz to be sed in BA analysis. The device is based on sage of a low cost microcontroller with bilt-in AD/DA converters, which are sed to generate the sinsoidal driving (inpt) signal and to perform acqisition of the system response. The data is transferred sing a WiFi protocol to the laptop compter on which the digital lock-in method is performed and the bioimpedance is determined [8]. everal problems might arise from sing low-cost, low-resoltion AD/DA converters not optimized for real-time data acqisition. For example sinsoidal driving signal is limited by the bit resoltion where the data acqisition is prone to jitter and other sorces of noise. nternal or intrinsic sorces of noise are inherent in a circit while external sorces refer to the noise present in the signals applied or to the noise introdced into the circit by some other means [], [7]. Another problem is the prereqisite for signal generation based on exact whole nmber of points in a period [4]. Also, a clock freqency of a microcontroller may vary in time [6], which in or case reslts in slight adjstment of a freqency of the driving signal. Freqency drift describes the change in accracy of a given freqency to environmental changes sch as temperatre, hmidity, pressre or simply changes over long periods of time [8]. These circmstances together with sorces of noise inherent in the electronic design and device realization reqire detailed analysis of the error related to not precisely tned freqencies of the reference and the driving signal. Here we will first briefly describe the fndamental principle of the lock-in method that is based on the properties of the correlation fnction. Then we will show 343

2 how this is implemented in the (bio)impedance method. Next, non-synchronization is analyzed mathematically and experimentally. Finally, we demonstrate how the nonsynchronization inflences determination of the (bio)impedance.. THEOETCAL NVETGATON Correlation between two signals can be sed as a measre for similarities between the two signals [5]. n case two signals have the same period T the average crosscorrelation is determined as T / ( ϕ, T ) = f ( t) f ( t+ ϕ) dt () T T / ignals f ( t ) and f ( t ϕ) + may differ bt are related by having the same shape and the freqency (period) of the inpt signal. They are shifted in time, which is denoted throgh a phase difference ϕ. A schematic description of the lock-in method based on correlation techniqes is presented in Fig.. Clearly, if the signals are not correlated the cross-correlation fnction is zero and is non-zero otherwise. ignal response depends on the degree of correlation between the measred signal and reference []. = () N where is the center freqency and is the bandwidth. n case the measred (and the reference) signals are sinsoidal as they are in the (bio)impedance method the inpt signal can be written as where, and ( t) sin( t+ ϕ ) (3) ϕ are the amplitde, freqency and the phase of the signal, respectively. Likewise, the reference signal is written as r( t) sin( t+ ϕ ) (4) Mltiplication of the two signals yields ( t) r( t) cos( [ ] t+ ϕ ϕ) cos( [ + ] t+ ϕ + ϕ) The first term with [ ] freqencies and the second term with [ ] (5) is attribted to the low + to high freqencies of the mltiplied signal. n case of eqal freqencies of the reference and inpt signal ( = ),the mltiplication reslt is simplified to Fig.. Basic principle of the lock-in method... (ingle) phase sensitive detection Phase sensitive detection is based on two operations: synchronos demodlation and integration. First, the inpt signal is mltiplied with the reference one and then it is integrated (low pass filtered) reslting in a DC signal ν ( t) as shown in Fig.. v ( ) ( ) ( ) t = = t r t = cos cos t+ + ( ) ( ϕ ϕ ) where = ϕ ϕ. A well-known reslt is obtained when the reslt is averaged over all measred periods NT v( t) dt NT yielding only a DC component (6) v( t) cos( ) = (7) Fig.. chematic description of the phase sensitive detection. De to the properties of the correlation fnctions, the phase sensitive detection behaves as a band pass filter with respect to its reference freqency []. The bandwidth of the filter depends on the integration time (nmber of integrated cycles N ). f the integration is carried ot over N periods of the sinsoidal pertrbation, the eqivalent filter selectivity is given by [4] The otpt expresses two properties: linearity and sensitivity. The otpt is maximal when the reference and the inpt signals are in phase. f the amplitdes of the signals are fixed, the otpt depends on the cosine of the = ϕ ϕ. This is named phase detection phase difference []. n case the term cos( ) is constant the otpt is linear and depends only on the amplitdes and which is known as amplitde detection. n case the freqencies of the reference and inpt signals differ, the mltiplication yields 344

3 v( t) cos( ) <> t cos( t+ t+ ϕ + ϕ) (8) enables both the in-phase and qadratre components of a coherent signal to be measred simltaneosly []. with = +. ntegration of both parts does not reslt in a DC component proportional to cos( ) bt in a mch more complex term depending on the ratio, the phase difference = ϕ ϕ and the nmber of integration periods N : v ( t) <> sin( π N ) sin+ π N + sin(4π N + π N + ) sin 4π N Non-synchronization between the reference and the inpt signal ths yields phase and amplitde determination errors. For instance, the phase determination errors (de to solely non-synchronized freqencies of the signals) can be calclated as η[ ] arccos( v( t) ) = <> (9) = and are presented in Fig.3 for several different phase differences. Phase error is dependent on term arccos( v( t) ) <>, which approaches limit vale of 9 with increasing ratio. Fig.4. chematic presentation of the two phase lock-in system. For synchronized inpt and reference signals ( = ) the otpts are and ( t) cos( ) = () ( t) sin( ) = () where ( ) t = denotes the 'in phase' or real and ( ) t = the 'qadratre' or imaginary component. The magnitde of the inpt signal can be determined from = ( + ) () = = = eliminating the phase dependence. The phase is then determined as = = = tan ( ) (3) = For non-synchronized freqencies we get Fig.3. Phase difference determination depending on o o o N= 5 and phase differences =, 45, 9. for ( t) <> sin( π N ) sin+ π N + sin(4π N + π N + ) sin 4π N (4).. Two phase sensitive detection The two phase techniqe of the lock-in method in the (bio)impedance method is most often implemented as shown in Fig.4. This method eliminates the difficlties of phase dependency in phase sensitive detection. The response is therefore maximal when the inpt and reference signals are broght in-phase at the phase sensitive detector inpt. This incorporates adjsting the setting of the phase shifter very careflly to = ϕ ϕ =, which is in particlar sensitive in case of the noisy inpt signal with a low amplitde. This phase sensitivity is redced by the sage of a pair of phase sensitive detectors operated in qadratre. This arrangement and ( t) <> cos( π N ) cos π N cos(4π N+ π N + ) cos 4π N (5) n case the amplitde and the phase of the inpt signal is determined sing the same procedre as for synchronized signals (() and (3)), this reslts in phase and amplitde determination errors. 345

4 The relative amplitde error is determined as = <> ξ[ %] = %. = t is presented in Fig.5 for N= 5 (chosen prely for figre clarity reasons). Clearly, the error is zero at = and non-zero in other cases. [ ] δ = = <> <> ( t ) ( ) <> t <> = arccos ( ) arcsin ( ) (5) representing the difference between the in-phase and the qadratre component. Fig.7 demonstrates this difference that depends on the term and on the nmber of periods and the phase difference between the inpt and the reference signals. The difference is zero only in one case when the freqencies are synchronized. This principle can be sed to accrately synchronize the signals. Fig.5. elative amplitde determination error depending on for N= 5. The phase determination error is determined as [ ] = <> = tan ( ) tan ( ) = <> = (4) = <> λ and is shown in Fig.6. t expresses a linear increase with, repeated in accordance with amplitde minima shown in Fig.5. Fig.6. elative phase determination error depending on N= 5. The phase determination error is different in case of the two-phase lock-in method where the in-phase and the qadratre component are sed for phase determination. n case =, the real component of the two-phase system is proportional to cos( ϕ ) and imaginary to sin( ϕ ) and ( t ) ( ) = t = arccos ( ) = arcsin ( ). n case we analyze the term for Fig.7. Phase difference between real and imaginary component of a two phase lock-in system. 3. MPEDANCE DETEMNATON 3.. Theoretical analysis and nmerical reslts n the previos chapter we described the se of a lock-in method for a single inpt signal. n the present chapter we analyze the impedance method for which we analyze two signals: the inpt signal in the (bio)impedance method is the crrent i( t ), while the otpt signal is voltage ( t ) in accordance with the measrement scheme shown in Fig.. For theoretical analysis we assme both signals are ideal sinsoidal, having the same freqency, bt differ in phase and amplitde. Both signals are analyzed sing the lock-in method as described, to obtain the amplitde and as well as the phase and. As shown in the previos chapter both amplitde and phase depend on the freqency of the reference signal and the nmber of periods of the signals sed. However, in case both signals are compared to the same reference signal (as they are), then the shape of the amplitde variation with the reference signal is the same for both signals as presented in Fig.8. The discrepancies occr only in points with very small amplitdes (in theory at amplitde zeros) that are de to ronding errors. Fig.9 presents phase dependence on the freqency of the reference signals for both signals as well as its difference. The phase difference is eqal, disregarding the freqency of the reference signal except in points of zero amplitdes. Nmerical analysis was carried ot at N=, = V, = V, = A and phase difference = =,7 for better presentation. 346

5 Fig.8. Amplitde (, ) and impedance magnitde ( Z ) determination depending on for N=. Fig.. Measrement scheme for (bio)impedance determination and impedance nder test. The synchronization test was performed on the C circit with nominal vales of = 33Ω, C= nf in = kω at signal freqency f = khz. Freqency f of reference sine signal sed in digital lock-in was generated in the range from 8 Hz to Hz ( f = f + f ). ignal reconstrction and impedance determination was analyzed in the reference freqency range and presented in dependency on the ratio. eslts of impedance magnitde and phase determination are shown in Fig. and Fig.. Fig.9. Phase ( ϕ, ϕ ) and impedance phase (ϕ ) determination depending on for N=. 3.. Experimental analysis and measrement reslts Experimental verification was performed on previosly described prototype device sing several test impedances (Fig.). Device design is based on physical separation of the measrement and the data parts shown in Fig.. The measrement component is sed for acqiring voltage and crrent wave forms from device nder test and is based on sage of a low cost microcontroller with bilt-in AD/DA converters [9]. Data are sent to the compter for frther reconstrction, analysis and presentation. The commnication between the data part (compter) and measrement part (device) is established throgh an Ad-Hoc WiFi connection [7]. Data part ses a digital lock-in method (algorithm) for reconstrction of the amplitde and phase of the excitation and otpt signals and frther determination of the magnitde and phase of the (bio)impedance. The freqency of the excitation and otpt signals is determined more exactly in a post-processing step from digitized signals. Freqency of the reference sine signal sed in digital lock-in method shold in theory be eqal to the freqency of the excitation signal in order to avoid errors in determination of the amplitde and phase of the excitation and otpt signals [3]. Fig.. Amplitdes of a crrent and voltage signals depending on the ratio. Fig.. Phase of a crrent and voltage signal and its difference depending on the ratio. 347

6 4. DCON Althogh the general idea behind phase sensitive detection or lock-in method is straightforward, additional isses shold be considered when it comes to the actal implementation. These inclde freqency synchronization, limited accracy of AC/DC converters, freqency drift, jitter, extrinsic/intrinsic noise, etc. Freqency drift and jitter are two different sorces of error for a timing device. Jitter refers to variation in the otpt signal or its precision. A timing sorce with low jitter will otpt a relatively consistent freqency. Drift, in contrast, describes the change in accracy of a given freqency over environmental changes sch as temperatre, hmidity, pressre or simply changes over long periods of time. Low DAC-resoltion, jitter and stochastic noise will affect signal-to-noise ratio of each measred signal, which can be improved by averaging. However, in case fast impedance measrements are reqired, this is possible only by redcing the nmber of sampled periods per freqency. n this case the inflence of jitter cannot be canceled completely. The lock-in method is typically sed to improve the signal-to-noise ratio; in (bio)impedance the techniqe is sed to perform signal reconstrction determination of impedance phase and modls. everal modern bioimpedance systems separate the measrement and data processing parts. n sch cases the inpt and the reference signals are not generated from the same clock sorce so their freqencies may differ. Hence, a method of synchronization of a reference and inpt signal may be reqired to avoid the potential emergence of digital phase locked loops. n order to test the freqency synchronization, the best method seems to be to compare the in-phase and the qadratre component of the inpt signal as shown in Fig.7. Only when the freqencies are synchronized the phase difference is zero. ignificant errors in phase and amplitde of a signal can occr in case the reference and inpt signal are not synchronized. ch errors depend on the relative freqency difference and nmber of integration cycles. From the viewpoint of rejecting asynchronos noise and achieving better signal-to-noise ratio in signal reconstrction it is crcial to redce signal bandwidth (). The price to be paid, however, is an excessively long data acqisition time, dring which the stability condition might be violated. f we look from the viewpoint of error determination de to nonsynchronized freqency of the reference and the analyzed signal, we notice that error sensitivity (slope) is increased with the nmber of integrated periods. The synchronization accracy is ths more important at longer integration times. f we want to avoid the errors de to non-integer nmber of integration cycles, care mst be taken to flfill the fs eqation k = N, where k is nmber of samples, N is f nmber of sampled periods, f is signal freqency and is the sampling freqency. f the vales of the variables are whole nmbers, each sample has its opposite pair and the average vale of alternate components will be zero. Otherwise an error is entered into the measrement system f that is more prononced for small nmber of integrated periods. The impedance modls is obtained by determination of the qotient of the amplitdes of both signals while the impedance phase is determined from the phase difference. The ratio of amplitdes yields the same reslts regardless of the freqency of the reference signal. n theory it is ths not important to have absoltely eqal freqency of the reference and analyzed signals becase the ratio between the amplitdes is the same. The same holds for the phase difference between the signals which (in theory) does not depend on the freqency of the reference signal. n both cases, however, the reslts are incorrect in points of zero or very small determined signal amplitdes (in theory at amplitde zeros) that are de to ronding errors. These deviations are dependent on the nmber of integrated periods and the phase difference between the signals. n nmerical analysis sing pre sine signals, deviations are seen as error spikes (Fig.8 and Fig.9). f, however, we inclded in analysis a qantization (nmber of bits) error and asynchronos noise (not shown in figres), the errors are dispersed arond the spikes (Križaj D, data not shown). The same otcome was observed in experimental analysis, where the signal was created with only bit resoltion and the data acqisition was prone to jitter and other sorces of noise. Experimental analysis was carried ot in the freqency range from khz to 4 khz. Variable was varied in the same ratio regardless of the analyzed signal freqency so impedance modls and phase determination had a similar corse at higher freqencies, bt different vales. Clock freqency of a microcontroller may vary in time and introdce freqency deviations in signal reconstrction, which in trn also affect the form of the determination error. n theory amplitde zeros of both signals appear in exactly the same points, which is not the case in experimental analysis where the errors are becoming more prononced at higher vales of. The averaging method might be interesting to increase the accracy of magnitde and phase determination or to verify the correctness of the method. 5. CONCLON ynchronization errors of the lock-in method, also known as phase sensitive detection, were analyzed by theoretical investigation, experimentally sing an approach based on comparison of the investigated signal to a reference signal by correlation. When the freqency of the reference signal is eqal to the freqency of the investigated signal, the amplitde and phase of the investigated signal can be determined simply from the reslt of integration of the prodct of the reference and the investigated signal (correlation). The otpt is proportional to the amplitde and cosine of the phase difference between the signals. f the freqencies of the signals differ, which can occr in case the freqency of the investigated signal is not exactly known, an 348

7 error in amplitde and phase determination occrs. The amont of this error depends on the relative difference between the freqencies of the signals and the nmber of periods sed in integration. The phase error is zero only at synchronized freqencies when the correlation fnction is maximal. n search for freqency synchronization we do not know in advance the expected maximal vale of correlation so phase error is a better sign of non-synchronization between the signals. When the lock-in method is sed in the (bio)impedance measrement method, two (crrent and voltage) signals are concomitantly analyzed sing the in-phase and the qadratre components. The impedance modls is obtained by determination of the qotient of the amplitdes of both signals while the impedance phase is determined from the phase difference. t is shown that in this case the freqency synchronization is not mandatory so the impedance phase and modls are correctly determined whilst disregarding signals that are not freqency synchronized. Experimental verification (a nmerical investigation) shows that de to limited accracy of the AC/DC converters and other sorces of noise, a determination error necessarily exists in points of very small convoltion amplitdes. n smmary, or stdy may aid in nderstanding the sorces of impedance determination error when the lock-in method is tilized. n particlar, or approach highlights the errors cased by the non-synchronized freqencies of the reference and the analyzed signals. EFEENCE [] Blair, D.P., ydenham, P.H. (975). Phase sensitive detection as a means to recover signals bried in noise. Jornal of Physics E: cientific nstrments, 8, [] Bckingham, M.J. (985). Noise in Electronic Devices and ystems. Ellis Horwood Ltd. [3] Darren, W. (7). mplementing digital lock-in amplifiers sing the dspc DC. AN5. Microchip Technology. AppNotes/5A.pdf [4] Gabrielli, C. (98). dentification of electrochemical processes by freqency response analysis. Technical report 4/83. olartron nstrmentation Grop. [5] Grimnes,., Martinsen, O. (). Bioimpedance and Bioelectricity Basics, Academic Press. [6] Horowitz, P., Hill, W. (996). The Art of Electronics, econd edition. Cambridge niversity Press. [7] acopini, E., mith, B., tefanini, G., Carsotto,. (983). Digital techniqes applied to phase-sensitive detection. Jornal of Physics E: cientific nstrments, 6, [8] Križaj, D., Baloh, M., Brajkovič,., Žagar, T. (3). Design and development of a portable WiFi enabled BA device. Jornal of Physics: Conference eries, 434, 6. [9] NXP emicondctors N.V. (4). LPC769/68/67/66/65/63/63. Prodct data sheet. 68_67_66_65_64_63.pdf [] McAdams, E.T., Jossinet, J. (995). Tisse impedance: A historical overview. Physiological Measrement, 6, A-A3. [] Meade, M.L. (98). Advances in lock-in amplifiers. Jornal of Physics E: cientific nstrments, 5, [] cofield, J.H. (994). A freqency-domain description of a lock-in amplifier. American Jornal of Physics, 6 (), [3] tanford esearch ystems. Abot lock-in amplifiers. Application note #3. downloads/pdfs/applicationnotes/abotlas.pdf [4] Tarchanidis, K.N., oilemes, A.T. (). Digitally synthesized alternative crrent sinsoidal waveform for resistive sensor measrement. Jornal of Engineering cience and Technology eview, 4 (), [5] Tab, H., chilling, D.L. (97). Principles of Commnication ystems. McGraw-Hill. [6] owing Networks. (). WiFi N-XV-D v.3 8/8/. Data sheet. files/resorces/wifly-n-xv-d.pdf [7] Wilmshrst, T.H. (99). ignal ecovery from Noise in Electronic nstrmentation. CC Press. [8] aof, A. (3). Minimize freqency drift in crystals. Electronic Design, minimize-freqency-drift-crystals. eceived Jly, 4. Accepted October 3,