Signal Model and Linearization for Nonlinear Chirps in ... - IEEE Xplore

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 4, APRIL 2006

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Signal Model and Linearization for Nonlinear Chirps in FMCW Radar SAW-ID Tag Request Stefan Scheiblhofer, Student Member, IEEE, Stefan Schuster, Student Member, IEEE, and Andreas Stelzer, Member, IEEE

Abstract—In this paper, we present a frequency-modulated continuous-wave radar interrogation concept, based on direct digital synthesis (DDS), that operates without the commonly necessary high-frequency DDS reference oscillator. As the generated frequency sweeps are nonlinear, standard Fourier transform methods for baseband signal evaluation cannot be applied directly. We show the corresponding signal model, derive a linearization concept on the basis of resampling, and demonstrate the linearization algorithm on simulated data, as well as on real wireless interrogations of surface acoustic-wave sensors.

Fig. 1. Interrogation setup for SAW sensors.

Index Terms—Direct digital synthesis (DDS), frequency-modulated continuous wave (FMCW) radar, linearization, nonlinear chirp.

I. INTRODUCTION EGARDING the wireless interrogation of surface acoustic-wave (SAW)-based devices, such as identification (ID) tags or temperature sensors [1], [2], the use of linear frequency-modulated continuous-wave (LFMCW) or linear frequency-stepped continuous-wave (LFSCW) radar interrogation units is of common practice [3], [4]. Fig. 1 shows the setup of an interrogation unit and a SAW sensor that is equipped with an antenna. The reflector configuration of the SAW tag is read out in the frequency domain using the inverse Fourier transform for the calculation of the individual round-trip delay times (RTDTs), which carry the code or temperature information [4]. Typically, frequency-modulated continuous-wave (FMCW) transmitters are designed to generate agile and highly linear frequency ramps [5] of the form

R

(1)

with being the starting frequency, being the slope of the ramp, and being the sweep duration. The ramp generation usually is derived from a very stable reference oscillator, as the quality of the baseband signal is highly dependent on the linearity of the frequency ramp [6], [7]. for

Fig. 2. Block schematic of the nonlinear chirp synthesizer.

The used direct digital synthesis (DDS)-based frequency synthesizer operates without a separate high-frequency DDS reference oscillator and generates fast, but nonlinear frequency chirps. As a consequence, the output of the homodyne receiver frontend consists of frequency chirps in contrast to the sinusoidal output of linear frequency modulated systems and has to be linearized before Fourier processing. This paper is organized as follows. In Section II, we examine the realized DDS system and derive a closed-form expression for the resulting transmit signal. Section III compares the FMCW radar baseband signal models for linear and generally nonlinear frequency chirps. In Section IV, a linearization scheme by means of resampling is stated and finally applied to measurement data in Section V. II. SIGNAL SYNTHESIS A. Frequency Synthesis

Manuscript received August 25, 2005; revised December 13, 2005. The authors are with the Institute for Communications and Information Engineering, Johannes Kepler University, A-4040 Linz, Austria (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.871361

Fig. 2 shows a block schematic of the realized frequency synthesizer. This specific setup is commonly known as a “fractional divider loop configuration.” Hereby the DDS is used to implement a digitally tunable high-resolution divider in a standard

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Fig. 3. Operational principle of the DDS.

phase-locked loop. In this setup, there is no need for a high-frequency DDS reference oscillator, which is usually in the range of several hundred megahertz to gigahertz for optimum DDS performance. The DDS reference clock is derived directly from the RF output signal through division by a frequency divider with ratio . The system reference is a simple quartz oscillator.

Fig. 4. Block diagram of the FMCW reader frontend and SAW tag.

B. Generation of the Nonlinear Sweep The DDS core (see Fig. 3) generates a sinusoidal output signal using a -bitderived from its frequency reference wide phase accumulator. Every clock cycle the phase accumulator is incremented by the value of the frequency tuning word (FTW), mapped to amplitude information via a lookup table, and converted to an analog signal using a digital-to-analog concan be used verter. The delta frequency tuning word to modulate the output frequency. The output frequency according to the FTW (for ) calculates to

If the DDS reference would be driven by a fixed frequency oscillator, the FTW sweep would result in a linear frequency chirp at the DDS output. In the current case the DDS reference itself is not fixed, as it is derived from the frequency chirping signal . Therefore, the frequency sweep becomes nonlinear. Combining (4) and (5), we derive an implicit expression for the output frequency

(6)

(2) Therefore, the resulting divider value , including the prescaler value , in the phase-locked loop calculates to

that can be solved, assuming initial zero phase without loss of generality, to

(7)

(3) which yields a closed-loop output frequency of

This nonlinear square-root frequency function can closely be approximated by developing it into an th-order Taylor series (4) with being the phase-locked loop reference frequency. The utilized synthesizer Analog Devices AD9956 [8] offers an automatic frequency sweeping capability. In this mode, the synthesizer sweeps the FTW proportional to the phase of the DDS reference input between a programmable start and stop , as shown in Fig. 3. For small frequency FTW using the increments per DDS reference clocks, this can be linearly modeled by (5) with being the start FTW, being the slope of the function, including the frequency divider of ratio , and being the phase of the closed-loop output signal with frequency .

(8)

with the coefficients , which will be used for the derivation of the linearization scheme. III. SIGNAL MODEL Here, we show the baseband signal model for target interrogation with linear frequency chirps in comparison to the polynomial approximation of nonlinear chirps, both for the case of homodyne reception. For better understanding, Fig. 4 shows a simplified block diagram of the utilized FMCW reader frontend. This homodyne concept uses a 90 -hybrid coupler for transmit–receive separation.

SCHEIBLHOFER et al.: SIGNAL MODEL AND LINEARIZATION FOR NONLINEAR CHIRPS IN FMCW RADAR SAW-ID TAG REQUEST

A. Signal Model for the Linear Chirp

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and yields a transmit signal

An LFMCW transmitter generates an interrogation signal, according to (1), of the form (15) (9)

Using binomial coefficients, the phase of the time-delayed receive signal can be written as

with being the amplitude, being the starting frequency, being the slope of the linear ramp. Regarding a single and target, the return signal is a time-delayed version of the transmit signal damped by a factor

(16)

(10)

After mixing with the attenuated and time-delayed return signal from the target and subsequent low-pass filtering, the IF signal results in

where is the target RTDT. Due to the homodyne principle, the IF signal at the mixer output is generated by multiplying transmit and receive signals. Removing the high-frequency sum terms by low-pass filtering yields an IF signal of the form (11) If the RTDT is small compared to the sweep time, with , the term can be omitted in comparison to . The response signals of multiple targets simply superpose, thus a -target setup yields

(17) or noted in permutated order

(18) Interpreted in terms of phase and frequency components,

(12)

with being the amplitude of each signal component. The IF signal, therefore, consists of a sum of cosines with the frequencies

(19) with

(13) As can be seen from (12), the RTDT of the targets can easily be estimated using Fourier transform methods when LFMCW interrogation is used.

The same approach is applicable to the th-order polynomial approximation of the nonlinear frequency waveform (8). The —representing the argument of the coactual phase value sine transmit function—is given by

the function hereby represents the frequency and represents the phase component of the IF signal. It is important to note that, compared to the linear model (12), the frequency in general depends on not only linear, but with higher order terms. However, if the nonlinear frequency course is short-term linear and, hence, does not deviate from a linear course with regard to the RTDT interval remarkably, these higher order terms can be neglected. Therefore, with respect to the given nonlinear frequency course, for short RTDTs compared to the measurement time , becomes approximately independent from as follows:

(14)

(20)

B. General Signal Model for the Nonlinear Chirp

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Fig. 5. Frequency measurement of a 300-s up–down chirp.

Fig. 6. FFT spectra of a simulated 50-Hz cosine and a frequency chirp, from 50 to 100 Hz, N = 1000 pts, sampling frequency f = 1000 Hz.

In this case, the frequency term represents a frequency chirp of order and the relationship between RTDT and IF frequency is given by

(21)

with a maximum relative frequency error in the order of 10 . The deviations from the theoretical values result from the slight approximation error due to the quadratic model, as well as the dynamics of the phase-locked loop, especially the loop filter, that impact the frequency response. Assuming the second-order model (22), the IF signal according to (20) is

C. Second-Order Model for the Nonlinear DDS Chirp For the considered ramp parameters—a sweep bandwidth of 100 MHz and a frequency sweep duration of several hundred microseconds—the frequency course can be characterized using a second-order polynomial

(22) with being the start frequency, being the slope, and being the curvature of the frequency ramp, with a high degree of accuracy. Here, and represent the polynomial coefficients and of (8). For verification of model (22), frequency measurements on the prototype system were taken using the technique described in [9]. Fig. 5 presents an agile 300- s upchirp from 2.4 to 2.5 GHz and a downchirp of the same rate generated with the DDS system. The linear course with the ramp’s initial slope is shown to point out the dominating quadratic nature of the DDS ramp. The slope and curvature component of the upchirp have been calculated using linear regression and show very good agreement to the calculated theoretical results and Hz/s Hz/s Hz/s Hz/s

(23)

with a constant phase term . Compared to (12), a quadratic chirped frequency transmit signal generates an IF output signal, which is not of constant frequency anymore. The new relationship between RTDT and IF frequency is given by (24) Therefore, the IF signal for multiple targets consists of a sum of linear frequency chirps. IV. CHIRP RESAMPLING AND LINEARIZATION The standard fast Fourier transform (FFT) is not suitable for estimating the frequency content of chirped frequency signals, as it basically correlates the signal under test with complex exponential functions of constant frequency. The Fourier transform of a frequency chirp will not contain a sharp peak—as expected for the pure sine wave—but the signal energy will smear over a broader frequency range, dependent on the curvature of the quadratic chirp (22) (see Fig. 6). The main idea for the linearization of a chirp is to interpret it as a time-distorted version of a sinusoid. This can be done in the following two ways.


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If a chirp signal is sampled with sampling intervals proportional to its phase, the resulting discrete dataset represents a pure sinewave. This approach has to be realized in hardware and significantly increases the system complexity. An implementation can be found in [7]. The reverse process samples the chirp with constant sampling frequency and digitally shifts the samples on the timeline afterwards. As in the above case, the time shift of the samples is proportional to the phase of the chirp. After the de-chirping process, the original samples are spaced equidistant in phase—therefore, representing a sinewave—but nonequidistant in time. One possible way to calculate the spectrum of that data is to apply the nonuniform discrete Fourier transform (NDFT) [10] that operates on nonuniform sampled data with sub-FFT performance. Our approach linearizes the data before the transformation, therefore, a standard FFT is applicable. A. Calculation of the Resampling Scheme Considering the linear chirped frequency IF signal with known slope, we need to calculate new sampling instants spaced equidistant in phase. The actual phase of a time-equidistant sampled cosine, respectively, chirp according to the signal models (12) and (23) is given by (25) (26) with and being the sampling intervals. The constant phase terms have been neglected without loss of generality. Calculation of the phase increments per sample for both models yields (27) (28) with being the number where is in the range of 1 to , we can now deof samples. By setting termine the series of modified sampling intervals , which results in a complete linearization of the chirped IF signal

(29) hereby is the equidistant sampling interval of the corresponding cosine. Additionally, the sampling instants can be calculated directly by solving (30)

for , which yields the vector of absolute sampling times for to

(31)

Fig. 7. FFT spectra of a 50-Hz cosine and a linearized frequency chirp from 50 to 100 Hz, N = 1000 pts, f = 1000 Hz, L = 100.

It is important to note that the sampling instants do not depend on the target RTDT , thus, this linearization is independent from the target configuration as long as previously made assumptions hold. The same procedure can be used to calculate the resampling scheme for higher order polynomial models (19) even though the corresponding equations might only be solved numerically. B. Interpolation The linearization of the sampled data uses interpolation in a two-stage process. Based on the uniform sampled chirp signal, are to be calculated, as they repthe samples at the times resent the de-chirped dataset on equidistant sampled points. First, the band-limited real dataset is sinc interpolated by an integer factor [11] that is chosen corresponding to the vector of sampling times. Since most of the calculated sampling indo not comply with an integer ratio of the original stants sampling interval, the resultant real data vector of the first step is then interpolated linear in the second step to minimize deviations from the exact value. In Fig. 7, the simulated 50–100-Hz frequency chirp has been linearized with the described method. It shows excellent match with the ideal cosine of 50 Hz. V. MEASUREMENT RESULTS To prove both the signal model (23) and the linearization technique, we took wireless measurements of a seven-digit SAW ID tag using the prototype FMCW system depicted in Fig. 4. The SAW tag is fabricated on LiNbO with alumina metallization and is 6 mm 2 mm in size, with corresponding reflector RTDTs of 0.6–2 s. Fig. 8 illustrates the comparison between FFT evaluation of the raw baseband data and the FFT evaluation of the linearized data. The spectrum of the original chirped IF signal clearly shows the energy smearing effect due to the nonstationary frequency content. According to (24), this spreading effect gets worse for larger RTDTs, as the absolute frequency deviation linearly increases with . Simultaneously, the amplitude decreases as the total peak energy stays constant. After

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= 801

=

Fig. 8. Spectral response of a seven-digit SAW ID tag N pts and f : MHz, readout distance 0.7 m, effective isotropic radiated power 10 mW, antenna gain 10 dBi.

25

Fig. 10. Spectrogram of the linearized baseband signal.

+

VI. CONCLUSION In this paper, we have presented a DDS-based FMCW radar frontend that generates nonlinear frequency chirps and operates without the commonly used high-frequency DDS reference oscillator. We derived a closed-form expression for the nonlinear frequency course and stated a general baseband signal model for the according polynomial approximation of order . We stated a linearization method that is applicable to short-term linear frequency courses and allows standard FFT methods for signal evaluation to be applied. Finally, the performance of both the radar system and linearization algorithm was shown on real measurement data of a SAW ID tag.

REFERENCES

Fig. 9. Spectrogram of the raw baseband signal.

linearizing the data with the interpolation technique, the spectral energy is reshaped, which results in clear and narrow spectral peaks, as expected for a linear frequency-modulated request signal. The maximum relative frequency deviation compared to an LFMCW request is in the order of 10 , corresponding to 200-ps deviation in time, mainly due to neglecting the higher order terms of in (19). This is negligible compared to the 10-ns spacing of the SAW ID tag’s coding grid. The spectrograms of both raw and resampled data illustrate the effect of the resampling process in the time domain. The light-gray lines represent the IF frequency progression of each target response in time. As the model (23) states, the target frequencies in Fig. 9 rise with increasing measurement time with a slope dependent on the target RTDT. The spectrogram of the linearized data in Fig. 10 shows constant frequencies over the measurement interval that are suitable for FFT evaluation.

[1] A. Pohl, “A review of wireless SAW sensors,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 47, no. 2, pp. 317–332, Mar. 2000. [2] F. Schmidt and G. Scholl, “Wireless SAW identification and sensor systems,” in Advances in Surface Acoustic Wave Technology, Systems and Applications, C. Ruppel and T. Fjeldly, Eds. London, U.K.: World Sci., 2001, vol. 2, pp. 277–325. [3] R. Peter and C. S. Hartmann, “Passive long range and high temperature ID systems based on SAW technology,” in Proc. Sensor Conf., Nuremberg, Germany, May 13–15, 2003, vol. 1, pp. 335–340. [4] A. Stelzer, S. Schuster, and S. Scheiblhofer, “Readout unit for wireless SAW sensors and ID-tags,” in Proc. 2nd Int. Acoust. Wave Dev. for Future Mobile Commun. Syst. Symp., Chiba, Japan, Mar. 3–5, 2004, pp. 37–44. [5] T. Musch, I. Rolfes, and B. Schiek, “A highly linear frequency ramp generator based on a fractional divider phase-locked-loop,” IEEE Trans. Instrum. Meas., vol. 48, no. 2, pp. 634–637, Apr. 1999. [6] M. Pichler, A. Stelzer, P. Gulden, and M. Vossiek, “Influence of systematic frequency-sweep nonlinearity on object distance estimation in FMCW/FSCW radar systems,” in Proc. 33rd Eur. Microw. Conf., Munich, Germany, Oct. 6–10, 2003, pp. 1203–1206. [7] M. Vossiek, P. Heide, M. Nalezinski, and V. Mágori, “Novel FMCW radar system concept with adaptive compensation of phase errors,” in Proc. 26th Eur. Microw. Conf., Prague, Czech Republic, Sep. 9–13, 1996, pp. 135–139. [8] AD9956—2.7 GHz DDS-based AgileRF™ synthesizer Analog Devices, Norwood, MA, Datasheet, Rev. A, Sep. 2004 [Online]. Available: http://www.analog.com


[9] A. Stelzer, K. Ettinger, J. Höftberger, J. Fenk, and R. Weigel, “Fast and accurate ramp generation with a PLL-stabilized 24 GHz SiGe VCO for FMCW and FSCW applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, vol. 2, pp. 893–896. [10] A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput., vol. 14, pp. 1368–1393, 1993. [11] R. E. Crochier and L. R. Rabiner, Multirate Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1983, pp. 35–39.

Stefan Scheiblhofer (S’03) was born in Linz, Austria, in 1979. He received the M.Sc. degree in mechatronics from Johannes Kepler University, Linz, Austria in 2003, and is currently working toward the Ph.D. degree at the Institute for Communications and Information Engineering, Johannes Kepler University. His primary research interests concern advanced radar system concepts, development of the associated signal-processing algorithms, statistical signal processing, and their application to SAW devices for ID and temperature measurement purposes.

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Stefan Schuster (S’03) was born in Linz, Austria, in 1978. He received the Dipl.-Ing. (M.Sc.) degree in mechatronics from Johannes Kepler University Linz, Austria, and is currently working toward the Ph.D. degree at the Institute for Communications and Information Engineering, Johannes Kepler University. His research interests include all types of signal processing, especially focused on radar signal processing, as well as RF system design.

Andreas Stelzer (M’00) was born in Haslach an der Mühl, Austria, in 1968. He received the Diploma Engineer degree in electrical engineering from the Technical University of Vienna, Vienna, Austria, in 1994, and the Dr.techn. degree (Ph.D) in mechatronics (with honors sub auspiciis praesidentis rei publicae) and Habilitation degree from the Johannes Kepler University Linz, Linz, Austria, in 2000 and 2003, respectively. In 1994, he joined Johannes Kepler University, as a University Assistant. Since 2000, he is with the Institute for Communications and Information Engineering, Johannes Kepler University. In 2003, he became an Associate Professor with the Johannes Kepler University. His research focuses on microwave sensors for industrial applications, RF and microwave subsystems, electromagnetic compatibility (EMC) modeling, digital signal processing (DSP), and microcontroller boards, as well as high-resolution evaluation algorithms for sensor signals. Dr. Stelzer is member of the Austrian Engineering Society (OVE). He serves as an associate editor for IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS.