## Analytic signal demodulation of phase-modulated frequency-chirped signals |

Applied Optics, Vol. 52, Issue 9, pp. 1838-1846 (2013)

http://dx.doi.org/10.1364/AO.52.001838

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### Abstract

Both interferometers and frequency-modulated (FM) radios create sinusoidal signals with phase information that must be recovered. Often these two applications use narrow band signals but some applications create signals with a large bandwidth. For example, accelerated mirrors in an interferometer naturally create a chirped frequency that linearly increases with time. Chirped carriers are also used for spread-spectrum, FM transmission to reduce interference or avoid detection. In both applications, it is important to recover the underlying phase modulations that are superimposed on the chirped carrier. A common way to treat a chirped waveform is to fit zero crossings of the signal. For lower signal-to-noise applications, however, it is helpful to have a technique that utilizes data over the entire waveform (not just at zero crossings). We present a technique called analytic signal demodulation (ASD), which employs a complex heterodyne of the analytic signal to fully demodulate the chirped waveform. ASD has a much higher sensitivity for recovering phase information than is possible using a chirp demodulation on the raw data. This paper introduces a phase residual function,

© 2013 Optical Society of America

## 1. Motivation

### A. Spread-Spectrum FM Signals

*et al.*[1

1. R. L. Pickholtz, D. L. Schilling, and L. B. Milstein, “Theory of spread-spectrum communications: a tutorial,” IEEE Trans. Commun. **30**, 855–884 (1982). [CrossRef]

### B. Interferometers with Accelerated Mirrors

*arms*, one of which is usually

*fixed*, and the other that is reflected off of a moveable surface and is of

*variable*length. The intensity of light hitting the photodetector is proportional to the amplitude of the recombined electric fields. The signal is given by

2. I. Marson and J. E. Faller, “g-the acceleration of gravity: its measurement and its importance,” J. Phys. E **19**, 22–32 (1986). [CrossRef]

*chirp*in the frequency. The fourth term causes phase modulation in the overall chirped sinusoidal signal. It is important to recover this information because it often represents unmodeled mirror motions that can adversely affect the ability to measure gravity. While it is true that the most important parameter is the acceleration (or chirp rate) for an absolute gravity meter application, the other unmodeled mirror motions are critically important from an experimental point of view. Normally, when free-fall interferometers are built a great deal of care must be put into the optical design to minimize the effect of recoil caused by launching the object on the stationary components in the interferometer. Inevitably this introduces resonances that introduce unwanted motions in the optics that disturb the phase of the signal. It is therefore essential from a practical point of view to recover these unwanted phase residuals (mirror motions) from the raw data even though the main parameter of interest is the overall chirp of the waveform.

3. T. M. Niebauer, “The effective measurement height of free-fall absolute gravimeters,” Metrologia **26**, 115–118 (1989). [CrossRef]

4. T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, “A new generation of absolute gravimeters,” Metrologia **32**, 159–180 (1995). [CrossRef]

## 2. Signal Processing

### A. Zero-Crossing Approach

### B. Entire Waveform Approach

*et al.*[6]) with digitization of interferometer signals from an absolute gravity meter used an aliased dataset consisting of digitized voltage-time pairs. Zero-crossing times (called mean crossings) of the waveform were calculated in order to identify initial estimates of the frequency and chirp rate. The algorithm culminated with a nonlinear least-squares fit to a chirped sinusoid to reach the ultimate sensitivity required for gravity measurement. With hardware generally available in 1995, this method required about 5.3 s for 85,000 data points.

*et al.*[7

7. T. Tsubokawa and S. Svitlov, “New method of digital fringe processing in an absolute gravimeter,” IEEE Trans. Instrum. Meas. **48**, 488–491 (1999). [CrossRef]

*et al.*described a method for identification of the chirp frequency using the entire signal [8

8. T. M. Niebauer, A. Schiel, and D. van Westrum, “Complex heterodyne for undersampled chirped sinusoidal signals,” Appl. Opt. **45**, 8322–8330 (2006). [CrossRef]

*analytic signal*. The technique, therefore, is termed

*analytic signal demodulation*(ASD). This method provides an excellent method for completely demodulating the chirped frequency and bringing any phase modulation back to its original frequency centered at DC. The technique recovers phase information over the entire waveform and not just at the zero crossings of the chirped waveform.

9. E. Bedrosian, “The analytic signal representation of modulated waveforms,” Proc. IRE **50**, 2071–2076 (1962). [CrossRef]

#### 1. Identification of the Chirp (The Model)

#### 2. Simulation of a Chirped Waveform with an FM Modulated Signal

*model*having two quadrature components (a sum of a sine and a cosine terms),

#### 3. Analytic Signal Demodulation (ASD)

*analytic signal*,

*model*,

*most*of the negative frequency content in the real signal. This creates an approximation to the analytic signal having only positive frequency components.

*model*is to the real data the better the approximation will be. That is one reason why it can be useful (although it is not absolutely necessary) to improve the model using a least-squares fit in the time domain prior to employing ASD.

*model*function using exponential functions makes the frequency domain more explicit: where

*model*removed.

*model:*The second term is given by The amplitude of the error can be evaluated explicitly: It is not too surprising that the error is proportional to the modulation index,

### C. Residual Comparisons Using the R θ Function with a Complex Demodulation

## 3. Conclusion

*model*is fit in a least-squares sense to the received signal.

*model*determined in the first step to remove negative frequency information from the received signal. This produces an approximate analytic signal, which is then complex demodulated.

## References

1. | R. L. Pickholtz, D. L. Schilling, and L. B. Milstein, “Theory of spread-spectrum communications: a tutorial,” IEEE Trans. Commun. |

2. | I. Marson and J. E. Faller, “g-the acceleration of gravity: its measurement and its importance,” J. Phys. E |

3. | T. M. Niebauer, “The effective measurement height of free-fall absolute gravimeters,” Metrologia |

4. | T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, “A new generation of absolute gravimeters,” Metrologia |

5. | I. Murata, “A transportable apparatus for absolute measurement of gravity,” Bulletin of the Earthquake Research Institute |

6. | P. R. Parker, M. A. Zumberge, and R. L. Parker, “A new method forfringe-signal processing in absolute gravity meters,” Manuscripta Geodaetica |

7. | T. Tsubokawa and S. Svitlov, “New method of digital fringe processing in an absolute gravimeter,” IEEE Trans. Instrum. Meas. |

8. | T. M. Niebauer, A. Schiel, and D. van Westrum, “Complex heterodyne for undersampled chirped sinusoidal signals,” Appl. Opt. |

9. | E. Bedrosian, “The analytic signal representation of modulated waveforms,” Proc. IRE |

**OCIS Codes**

(060.4080) Fiber optics and optical communications : Modulation

(060.5060) Fiber optics and optical communications : Phase modulation

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(120.5060) Instrumentation, measurement, and metrology : Phase modulation

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: January 3, 2013

Manuscript Accepted: February 5, 2013

Published: March 13, 2013

**Citation**

T. M. Niebauer, "Analytic signal demodulation of phase-modulated frequency-chirped signals," Appl. Opt. **52**, 1838-1846 (2013)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-52-9-1838

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### References

- R. L. Pickholtz, D. L. Schilling, and L. B. Milstein, “Theory of spread-spectrum communications: a tutorial,” IEEE Trans. Commun. 30, 855–884 (1982). [CrossRef]
- I. Marson and J. E. Faller, “g-the acceleration of gravity: its measurement and its importance,” J. Phys. E 19, 22–32 (1986). [CrossRef]
- T. M. Niebauer, “The effective measurement height of free-fall absolute gravimeters,” Metrologia 26, 115–118 (1989). [CrossRef]
- T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt, and F. Klopping, “A new generation of absolute gravimeters,” Metrologia 32, 159–180 (1995). [CrossRef]
- I. Murata, “A transportable apparatus for absolute measurement of gravity,” Bulletin of the Earthquake Research Institute 53, 49–130 (1978).
- P. R. Parker, M. A. Zumberge, and R. L. Parker, “A new method forfringe-signal processing in absolute gravity meters,” Manuscripta Geodaetica 20, 173–181 (1995).
- T. Tsubokawa and S. Svitlov, “New method of digital fringe processing in an absolute gravimeter,” IEEE Trans. Instrum. Meas. 48, 488–491 (1999). [CrossRef]
- T. M. Niebauer, A. Schiel, and D. van Westrum, “Complex heterodyne for undersampled chirped sinusoidal signals,” Appl. Opt. 45, 8322–8330 (2006). [CrossRef]
- E. Bedrosian, “The analytic signal representation of modulated waveforms,” Proc. IRE 50, 2071–2076 (1962). [CrossRef]

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