^{1}, A. Lobo

^{1}and J. Ramos-Castro

^{2}

### Abstract

Low-noise temperature measurements at frequencies in the millihertz range are required in the laser interferometer space antenna (LISA) and LISA PathFinder missions. The required temperaturestability for LISA is around at frequencies down to 0.1 mHz. In this paper we focus on the identification and reduction in a source of excess noise detected when measuring time-varying temperature signals. This is shown to be due to nonidealities in the analog-to-digital converter (ADC) transfer curve, and degrades the measurement by about one order of magnitude in the measurement bandwidth when the measuredtemperature drifts by a few . In a suitable measuring system for the LISA mission, this noise needs to be reduced. Two different methods based on the same technique have been implemented, both consisting in the addition of dither signals out of band to mitigate the ADC nonideality errors. Excess noise of this nature has been satisfactorily reduced by using these methods when measuringtemperature ramps up to .

Support for this work came from Project No. ESP2004-01647 of Plan Nacional del Espacio of the Spanish Ministry of Education and Science (MEC). J.S. acknowledges a grant from MEC.

I. INTRODUCTION

II. TMS DESCRIPTION

III. NONIDEAL QUANTIZATION NOISE

A. SAR ADC bit error description

B. Dither signal effect on ideal ADCs

C. Dither signal effect on real ADCs

D. INL effect of real ADCs on general input signals

IV. MITIGATION OF INL ERRORS

A. Gaussian noise dither

1. Gaussian noise dither: practical implementation

B. Triangular wave dither

1. Triangular wave dither: practical implementation

V. TEST SETUP AND RESULTS

A. Test setup

B. Experimental results

VI. DISCUSSION

### Key Topics

- Temperature measurement
- 23.0
- Noise propagation
- 11.0
- Fourier transforms
- 9.0
- Wave attenuation
- 7.0
- Capacitors
- 6.0

## Figures

Conceptual drawing of the LISA constellation (graph is not to scale).

Conceptual drawing of the LISA constellation (graph is not to scale).

LTP TMS. Top: analog signal conditioning circuit: a thermistor is placed in one of the arms of a Wheatstone bridge. The bridge is fed with a square wave to avoid noise of the IA. The output of the bridge is amplified, low-pass filtered and digitized by a 16-bit SAR ADC. Bottom: the quantized signal is digitally demodulated to obtain the temperature value: samples are averaged during one polarity and during the opposite polarity. Afterward, they are subtracted and divided by 2. Details can be found in Ref. 10.

LTP TMS. Top: analog signal conditioning circuit: a thermistor is placed in one of the arms of a Wheatstone bridge. The bridge is fed with a square wave to avoid noise of the IA. The output of the bridge is amplified, low-pass filtered and digitized by a 16-bit SAR ADC. Bottom: the quantized signal is digitally demodulated to obtain the temperature value: samples are averaged during one polarity and during the opposite polarity. Afterward, they are subtracted and divided by 2. Details can be found in Ref. 10.

Noise levels of the LTP TMS: the black trace corresponds to measurements of a constant temperature. The red trace corresponds to measurements of temperature with drifts of .

Noise levels of the LTP TMS: the black trace corresponds to measurements of a constant temperature. The red trace corresponds to measurements of temperature with drifts of .

Transfer function of an ideal three-bit ADC (black trace) and of a real three-bit ADC (red trace), i.e., with nonuniform quantization steps. The INL error is the difference between dots (ideal ADC) and crosses (real ADC) integrated over the full-scale voltage [see Eq. (7)].

Transfer function of an ideal three-bit ADC (black trace) and of a real three-bit ADC (red trace), i.e., with nonuniform quantization steps. The INL error is the difference between dots (ideal ADC) and crosses (real ADC) integrated over the full-scale voltage [see Eq. (7)].

Top: quantization error functions for an ideal four-bit ADC (red trace), a nonideal ADC with (black trace) and another nonideal ADC with (blue trace). The error functions are defined as . Center: difference between the ideal ADC and the ADC with quantization errors, i.e., . Bottom: *idem* for the ADC with . Note the periodicity of the error patterns depending on which is the erroneous bit

Top: quantization error functions for an ideal four-bit ADC (red trace), a nonideal ADC with (black trace) and another nonideal ADC with (blue trace). The error functions are defined as . Center: difference between the ideal ADC and the ADC with quantization errors, i.e., . Bottom: *idem* for the ADC with . Note the periodicity of the error patterns depending on which is the erroneous bit

ADC quantization error (absolute value) as given in Eqs. (14) and (21). They are indicated by the circles on top of vertical strokes. The red dashed lines plot the Gaussian dither attenuation profile (*y*-scale from 0 to 1). Top: ideal ADC. Center: contribution of INL errors when the first bit has error with . Bottom: *idem* when and , too. Note the number of spectral components (in inverse voltage domain) falling under the Gaussian noise dither profile increases from top to bottom panels. Note that *x*-axis scales are different for and .

ADC quantization error (absolute value) as given in Eqs. (14) and (21). They are indicated by the circles on top of vertical strokes. The red dashed lines plot the Gaussian dither attenuation profile (*y*-scale from 0 to 1). Top: ideal ADC. Center: contribution of INL errors when the first bit has error with . Bottom: *idem* when and , too. Note the number of spectral components (in inverse voltage domain) falling under the Gaussian noise dither profile increases from top to bottom panels. Note that *x*-axis scales are different for and .

Top: absolute value of the input signal slope, , of a temperature measurement run. It varies between 0 and . The sensitivity at the input of the ADC is , hence temperature drifts in vary between 0 and . Bottom: STFT (or spectrogram) of the measurement in the top panel. The energy of the signal is concentrated in specific frequencies which change with time precisely following , as predicted by Eq. (28). The latter is represented by dashed superimposed traces, and are labeled by the order of the corresponding bit error (from to ). Experimental results and theoretical estimates are in excellent agreement.

Top: absolute value of the input signal slope, , of a temperature measurement run. It varies between 0 and . The sensitivity at the input of the ADC is , hence temperature drifts in vary between 0 and . Bottom: STFT (or spectrogram) of the measurement in the top panel. The energy of the signal is concentrated in specific frequencies which change with time precisely following , as predicted by Eq. (28). The latter is represented by dashed superimposed traces, and are labeled by the order of the corresponding bit error (from to ). Experimental results and theoretical estimates are in excellent agreement.

Relationship between the input signal slope, the bit error and the fundamental frequency affected by the error in the bit. When high input signal slopes are present, more bit errors will affect the measurement quality in the MBW. Clearly, LSBs show up at higher frequencies than MSBs. The shaded areas span the MBWs of LISA and LPF, as indicated. Note that LPF’s MBW is a subset of LISA’s.

Relationship between the input signal slope, the bit error and the fundamental frequency affected by the error in the bit. When high input signal slopes are present, more bit errors will affect the measurement quality in the MBW. Clearly, LSBs show up at higher frequencies than MSBs. The shaded areas span the MBWs of LISA and LPF, as indicated. Note that LPF’s MBW is a subset of LISA’s.

Top panel: Gaussian noise dither generator: the noise of an operational amplifier is highly amplified (by a factor ). The amplified Gaussian noise of the operational amplifier is band-pass filtered (the frequency cutoff corners are 100 Hz and 3 kHz). The linear spectral density of the generated noise is shown in the bottom plot. The noise of the TMS with no dither is also shown.

Top panel: Gaussian noise dither generator: the noise of an operational amplifier is highly amplified (by a factor ). The amplified Gaussian noise of the operational amplifier is band-pass filtered (the frequency cutoff corners are 100 Hz and 3 kHz). The linear spectral density of the generated noise is shown in the bottom plot. The noise of the TMS with no dither is also shown.

Top: quantization error (circles on strokes) for a real ADC with a bit error in , , and the triangular wave dither profile with amplitude (red dashed lines with *y*-scale from 0 to 1). Center: same as above for an error in bit , . Bottom: same as above but with instead of . Note that -axis scales are different for and .

Top: quantization error (circles on strokes) for a real ADC with a bit error in , , and the triangular wave dither profile with amplitude (red dashed lines with *y*-scale from 0 to 1). Center: same as above for an error in bit , . Bottom: same as above but with instead of . Note that -axis scales are different for and .

Top: triangular wave dither signal and signal coming from the measurement chain (dashed line). Bottom: triangular wave generator. A 12-bit DAC is commanded by an eight-bit up-and-down counter.

Top: triangular wave dither signal and signal coming from the measurement chain (dashed line). Bottom: triangular wave generator. A 12-bit DAC is commanded by an eight-bit up-and-down counter.

Experiment setup scheme. “Analog electronics” is the analog signal conditioning circuit of the TMS and PS stands for the programmable power supply used to control the power dissipated in the heater to generate the desired temperature profile.

Experiment setup scheme. “Analog electronics” is the analog signal conditioning circuit of the TMS and PS stands for the programmable power supply used to control the power dissipated in the heater to generate the desired temperature profile.

Top: design and experimental (achieved with the feedback temperature control shown in Fig. 12) temperature profiles. Bottom: time derivative of the signals shown in the top plot. Generated slopes are: 0.5, 1, 2, 4, 8, and .

Top: design and experimental (achieved with the feedback temperature control shown in Fig. 12) temperature profiles. Bottom: time derivative of the signals shown in the top plot. Generated slopes are: 0.5, 1, 2, 4, 8, and .

Temperature linear spectral density for different configurations: 16-bit ADC with no dither, 16-bit ADC with Gaussian dither, 16-bit ADC with triangular wave dither, and 24-bit Sigma-Delta ADC with no dither. Temperature input signals are ramps with slopes of 1, 4, 8, and from top to bottom, respectively. As can be seen in absence of dither, the noise of the ADC increases (in amplitude and bandwidth) with respect to measurements at a constant temperature (see Fig. 3) with the slope of the input signal. Gaussian noise dither satisfactorily attenuates this effect for slopes up to , though causing the floor noise of the measurements to slightly increase. Triangular wave dither works satisfactorily for slopes up and the floor noise remains untouched. The 24-bit Delta-Sigma ADC works also rather well for the tested slopes (see text in Sec. V B for details).

Temperature linear spectral density for different configurations: 16-bit ADC with no dither, 16-bit ADC with Gaussian dither, 16-bit ADC with triangular wave dither, and 24-bit Sigma-Delta ADC with no dither. Temperature input signals are ramps with slopes of 1, 4, 8, and from top to bottom, respectively. As can be seen in absence of dither, the noise of the ADC increases (in amplitude and bandwidth) with respect to measurements at a constant temperature (see Fig. 3) with the slope of the input signal. Gaussian noise dither satisfactorily attenuates this effect for slopes up to , though causing the floor noise of the measurements to slightly increase. Triangular wave dither works satisfactorily for slopes up and the floor noise remains untouched. The 24-bit Delta-Sigma ADC works also rather well for the tested slopes (see text in Sec. V B for details).

Attenuation achieved by the Gaussian noise dither and the triangular wave dither. The gain shown for each bit corresponds to the lowest frequency introduced by the bit error. Gaussian noise dither with (3 mV) mitigates the error in the bits while the triangular wave with (155 mV) does so up to .

Attenuation achieved by the Gaussian noise dither and the triangular wave dither. The gain shown for each bit corresponds to the lowest frequency introduced by the bit error. Gaussian noise dither with (3 mV) mitigates the error in the bits while the triangular wave with (155 mV) does so up to .

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