1. Motivation

A. Spread-Spectrum FM Signals

There are many applications in which a sinusoidal signal with a base or carrier frequency contains low frequency information in the phase that must be extracted. A classical example is frequency-modulated (FM) transmission of signals. In standard radio applications, the base frequency is constant but there are important cases where the base frequency is chirped or changes linearly with time.

A chirped signal may be intentionally generated in spread-spectrum communication applications. In this case, the carrier frequency is swept to spread the frequency or phase-modulated signal out over a large bandwidth. This can be useful for secure transmission because the received signal is not easily demodulated. It can also be used to reduce natural interference in transmission because excess noise in a small band will not affect the signal outside that band. A spread-spectrum transmission is also less likely to disturb other single-frequency transmissions. Pickholtz et al. [1] provide a good tutorial on the uses and basic strategies for producing and decoding spread-spectrum communications.

B. Interferometers with Accelerated Mirrors

Michelson-type interferometers, as shown in Fig. 1, can also generate phase-modulated chirped signals. An interferometer consists of a laser beam that is split into two paths or 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 $V=V_0\cos (2\pi (2\mathrm{\Delta }z/\lambda ))$, where $V_0$ is the voltage corresponding to the maximum intensity of light hitting the photodiode and $\mathrm{\Delta }z$ is the movement of the mirror. The factor of 2 in the phase takes into account that a mirror motion of distance, $\mathrm{\Delta }z$, causes an optical path length change of $2\mathrm{\Delta }z$. The signal oscillates through one full period of the cosine function whenever the path length changes by $\mathrm{\Delta }z=\lambda /2$, or $1/2$ of the wavelength of the laser light. Each time the light intensity changes through one period, the interferometer output is said to have produced an optical fringe. One optical fringe in this example, therefore, corresponds to a mirror motion of half the laser wavelength, or about 300 nm, in the case of red light. Since it is possible to resolve a small fraction of a fringe ($1/1000$ is relatively common and 1 ppm is possible), the interferometer is a very good way to measure large distances with exquisite precision.

 

Fig. 1. Schematic for a simple interferometer used to monitor the position of a moving mirror.

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In the simplest type of motion, the distance may be expressed by a term linear in time (i.e., a mirror moving with constant velocity), which produces a signal with a constant frequency. However, if the mirror is accelerated, the distance changes according to the square of the time. An accelerated mirror produces an optical signal whose frequency increases linearly with time, i.e., a chirped sinusoid.

Absolute gravity meters are an example of a practical device that uses an interferometer to directly measure distance [2]. One mirror is allowed to free fall a short distance (5–20 cm) while the other mirror is held fixed. The differential path length looks like a simple parabola in time and can be written as: $\mathrm{\Delta }z=z_0+{\nu }_{0}t+(1/2)a_{0}t$. Any additional mirror motion, $\delta z$, perhaps arising from a mechanical mirror resonance, will add to the overall path length change creating a signal of the form $V=V_0\cos ((4\pi /\lambda )(z_0+{\nu }_{0}t+(1/2)a_0{t}^2+\delta z))$. The first term introduces a constant initial phase, the second term is a constant frequency, and the third term causes a 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.

It is important to point out that higher-order terms in the time parameter also arise in these equations due to the gradient of gravity of the Earth and also because of the finite speed of light [3,4], which means that an absolute gravity meter does not produce a pure chirped signal. The vertical gravity gradient terms are small because they are proportional to $dg/dx=3\times {10}^{-6}{\mathrm{Hz}}^2$. The finite speed of light correction is even smaller and is proportion to the distance fallen divided by the speed of light. In any case, however, the presence of these additional terms do not change the basic analysis described here as long as they are known or can be modeled. The method developed in this paper is independent of the exact functional form of the chirped frequency as long as it is known or can be identified from the signal itself.

2. Signal Processing

The general form of a chirped sinusoidal signal with phase modulation, $\delta \theta$, is given by $s(t)=a\sin (\theta (t)+\delta \theta )$, where $a$ is the signal amplitude and the chirped unmodulated (carrier) phase is $\theta (t)={\theta }_0+{\omega }_{0}t+(1/2)\alpha t^2$. Here, ${\theta }_0$ is the initial phase, ${\omega }_0$ is the base angular frequency at $t=0$, and $\alpha$ is the angular acceleration.

The angular frequency is the derivative of the phase, $\omega (t)={\omega }_0+\alpha t$. It has a constant base frequency that is linearly increasing with time. The frequency, $f=(\omega /2\pi )$, has the same linear dependence on time, $f(t)=f_0+kt$, where $f=0$ is the base frequency and $k$ is the chirp or frequency sweep rate.

In spread-spectrum applications, the exact parameters of the chirp rate itself are not particularly interesting but they are required to demodulate the transmitted phase-modulated signal.

In an interferometric ballistic absolute gravimeter or gradiometer, the chirp rate is the most important parameter of the signal because it is proportional to the gravitational acceleration of the falling mirror. The phase fluctuations represent noise in the measurement.

Both applications result in the same mathematical problem and can be solved using the same signal processing methods.

There are two general techniques for processing chirped sinusoidal signals to find the base frequency and the chirp rate. The most common technique is to time zero crossings of the chirped sinusoid and then fit these zero crossings to a parabolic equation. The advantage of this approach is that it is simple and does not require a lot of data. It was used almost exclusively in the past due to insufficient digitization speed and digital memory. However, recently, these constraints have relaxed considerably.

The other approach is to sample the entire waveform typically at equal time intervals and then apply either a transform or some type of filter to recover the signal. A least-square fit of the data to a theoretical model is a special case of this method.

A. Zero-Crossing Approach

Timing zero crossings was the early method of choice when electronic circuits first became available to detect a threshold and coincidently count a stable clock to provide accurate and precise timing. Because each zero crossing of a sinusoid corresponds to a constant phase, the method provides a direct mapping of distance (equal phase intervals) and time for interferometers. The most important advantage is that there is no need to capture every zero crossing, which means that the data rate can be quite small even for high-frequency signals. For example, in a free-fall ballistic gravity meter the interferometer output signal chirps from DC to 6 MHz in about 0.2 s and in the process creates about a million zero crossings. Thirty years ago only about 100 of these zero crossings were captured and stored by the electronics, which was sufficient to make gravity determinations at about one part in a billion. As the speed of electronics increased, it became possible to record more and more zero crossings. Currently, it is possible to time the occurrence of every zero crossing. However, even at this maximum sampling rate, the analog signal bandwidth required to correctly time zero crossings is much higher than the effective sampling rate from zero crossings. This means that high-frequency noise in the analog signal aliases into the signal bandwidth for the zero-crossing method. Thus, the zero-crossing method will normally have a lower SNR compared to a method that uses the full bandwidth of the detector.

A benefit of using zero crossings is that they provide a direct connection to phase; they can be linearly least-squares fit to a parabola for chirped waveforms. Any deviations in the fit of the phase to the time are therefore immediately interpretable as phase errors (or spurious mirror motions in an interferometer). This is normally a valid interpretation because the error on a zero crossing is more likely to be caused by a phase error than by an amplitude error (for cases where the amplitude noise is not excessive). The inverse is true at the maximum of the sinusoid where phase errors are suppressed and amplitude errors are maximized.

One problem with using zero crossings is that if there is a large amount of noise, it can cause the signal to trigger the timer circuitry at a much earlier or later phase than the real occurrence of the zero crossing. High noise can even cause the discriminator circuit to double trigger and generate invalid data. Zero-crossing systems tend to be stable for quiet signals but have trouble dealing with noisy signals (e.g., amplitude noise of 10% or more). Dramatically better results can be obtained with high noise signals when the entire waveform is digitized and least-squares fit to a model chirp function. This is because averaging over all phases of the chirped sinusoid provides a much tighter constraint on the overall chirp rate than timing only zero crossings.

B. Entire Waveform Approach

A general approach to the signal processing is to collect data over the entire waveform. In 1978, Murata [5] used an oscilloscope to capture the analog interferometer signal from an absolute gravity meter. He made sensitive time measurements on the oscilloscope to calculate the chirp rate resulting from the acceleration of a freely falling mirror.

Early efforts (Parker 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.

Full digitization of the waveform, became available later when faster analog-to-digital converters and computers with larger memory depths became more widely available. According to Nyquist criterion, the sampling rate must be at least twice the bandwidth of the signal. In some cases, this restriction can be relaxed but for the purpose of this paper, it will be assumed that the digitization rate is at twice the Nyquist frequency so that no aliasing exists. In the example of the free-fall gravity meter above, a minimum sampling frequency of 12 MHz is required, which means that at least two million samples are required for each measurement compared to the 100 used with the zero-crossing method, and the 85,000 used by Parker [6]. This method is increasing in popularity as computer speeds have increased and the cost of memory has decreased.

In 1999, Tsubokawa et al. [7] described full digitization of the interferometer fringe signal from an absolute gravity meter at 200 MHz. They computed zero crossings from the fringe signal and then used these to calculate the parabolic phase versus time relationship in a similar manner as done with traditional zero crossings. The advantage of this method is that they could avoid a nonlinear least-squares but the disadvantage is that the method gives a much heavier weight to the data at zero phase compared to data at other points on the sinusoid. It essentially ignores data at the extrema of the sinusoid.

Niebauer et al. described a method for identification of the chirp frequency using the entire signal [8]. The method employed a combination of complex demodulation followed by a nonlinear least-squares fit. The advantage of this approach is that the entire waveform is utilized equally and not just data near the zero phase as in the zero-crossing method. This more full utilization of all of the data provides a higher SNR than the zero-crossing method that uses only the data near zero crossings.

A deficiency in these earlier papers is that they do not present a method for recovering residual phase information. Parker [6] noted that, “… some additional complexity is involved in extracting residual position from the digitized fringe signal.” The difficulty is that either amplitude or a phase error will cause a difference between the raw data and the chirped model at each data point. Phase errors produce a much larger deviation of the raw signal and the model near a zero crossing of the chirped sinusoid than at the extrema. Some methods calculate the phase errors only near zero crossings to avoid this ambiguity.

Clearly it is an advantage to recover residual phase information at every point on the waveform independent of the phase of the carrier. This is even more advantageous in cases like spread-spectrum FM where the phase modulation riding on top of the chirp is actually the transmitted signal.

The spread-spectrum FM application would naturally suggest the use of a demodulation algorithm rather than a technique that utilize data near a zero crossing to detect phase modulation riding on top of the chirped waveform. The difficulty with a straight complex demodulation of a frequency chirp is that while it successfully collapses the bandwidth of the carrier for positive frequencies, it actually broadens the chirp in the negative frequencies. This can introduce an artificial background noise that obscures phase variations.

The excess noise can be avoided by first removing the negative frequencies of the chirp before complex demodulation. A signal with only positive frequency components is commonly known as the 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.

Bedrosian [9] discussed single sideband and double sideband demodulation using an analytic signal but the ASD method given here to recover a phase-modulated signal riding on top of a chirped carrier has not been described in the literature.

Recovering signals from a chirped source in general requires a two-stage algorithm. The first stage identifies the base frequency and chirp rate of the signal. This can be accomplished through prior knowledge of the transmitted signal or it may be determined from the signal itself. The second stage is to demodulate the chirped signal in order to determine phase variations in the frequency chirped signal. Knowledge of the base frequency and chirp rate found in the first stage is utilized in the second stage.

1. Identification of the Chirp (The Model)

The first step in signal processing is to determine the essential parameters, ${\theta }_0$, ${\omega }_0$, and $\alpha$, of the unmodulated chirp. In spread-spectrum communication prior knowledge of the sweep frequency may be available. The only issue is to synchronize the chirp with the received signal. This can be made easier with markers in the signal.

If the received signal is from an interferometer, however, the base frequency and chirp rate typically must be determined from the received signal itself. This may also be true if one wants to demodulate a spread-spectrum transmission with an unknown chirp.

Signal identification can be accomplished quickly with an iterative application of a chirped frequency complex demodulation. An estimate for the chirp rate and base frequency can be obtained from the power spectrum of the received signal or the complex demodulated signal. Complex demodulation with the correct chirp and base frequency collects and collapses the power in the positive frequencies of the chirped waveform to a single frequency at DC. The process can be iterated to improve the chirp and base frequency parameters until the entire chirp is removed from the positive frequency components.

Once the signal power has been collapsed into one bin at DC, the base frequency and chirp rate for the model are close enough that a nonlinear least-squares fit in the time domain will converge. This provides an even better value of the base frequency and chirp rate than is possible in the frequency domain.

This method of finding initial estimates for the base frequency and chirp rate followed by a nonlinear least-squares fit is reminiscent of the two-step approach used by Parker [6] except that complex demodulation is easier and faster to carry out with a standard fast Fourier transform (FFT) and does not require finding zero crossings (mean crossings) in the digital domain.

Complex demodulation is effective for identification of the chirp rate but it is not optimal for recovering residual phase information. The reason for this is that although complex demodulation collapses the power in the positive frequencies, it essentially doubles the chirp (the bandwidth) for the negative frequency components. This inverse behavior for positive and negative frequency components can be understood intuitively if one views a positive (negative) chirped frequency as a vector accelerating counterclockwise (clockwise) in the complex plane. Complex demodulation with a chirped frequency is equivalent to changing to a rotating reference frame that stops the positive frequency in the complex plane but causes the negative frequency to accelerate at twice the original rate in the opposite direction.

If the chirp occupies most of the available bandwidth in the digital domain, these doubly chirped negative frequencies after complex demodulation can become aliased across the entire digital spectrum and cannot later be removed with a low-pass filter. The negative frequency components tend to create a pseudo-white background noise after complex demodulation.

This problem can be avoided by first removing the negative frequency components prior to complex demodulation. This method, described in detail later, will be referred to as ASD. ASD provides a much lower noise level for complex demodulation with a chirped frequency.

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

It is helpful to visualize the steps of the analysis by applying it to a simulated signal. A chirped frequency appears as a block or band of raised frequencies, as shown in Fig. 2. This spectrum was made from a sinusoidal signal with unity amplitude in the time domain whose base frequency started at 5 kHz and had a chirp rate of $100\mathrm{kHz}/\mathrm{s}$ lasting about 0.33 s.

 

Fig. 2. Simulated chirped waveform with large phase modulation.

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The simulated signal also includes phase modulation $\delta \theta (t)={\varphi }_m\sin (2\pi f_{m}t)$, with a modulation index of ${\varphi }_m=0.1\mathrm{rad}$ at a modulation frequency of $f_m=10\mathrm{kHz}$. The waveform was sampled at $f_s=100\mathrm{kHz}$. The large phase modulation is responsible for the variation in amplitude seen in the spectrum. When the modulation index is much smaller ($<0.001$), the frequency response appears as a solid block of frequencies.

Let the lowest and highest frequencies in the power spectrum be given by $f_L$ and $f_H$. They can be estimated by the edges of the band of raised frequency components. The frequencies provide estimates for the chirp rate and base frequency. The chirp rate is given approximately by $k=((f_H-f_L)/{\tau }_{\mathrm{obs}}^2)$, where ${\tau }_{\mathrm{obs}}^2$ is the observation time or the time corresponding to the data segment being scrutinized. The base frequency estimate is $f_0=(1/2)(f_H+f_L)-(1/2)k{\tau }_{\mathrm{obs}}^2$.

One can verify that the correct parameters have been identified with the spectrum by using them to complex demodulate the signal. This is done by performing a complex multiplication, or by heterodyning, the chirped signal with, $H(\theta )=e^{-i\theta }$, where the phase is given by $\theta (t)=2\pi (f_{0}t+(1/2)k{t}^2)$. There is no need for an initial constant offset phase as it is simply an overall multiplicative constant with unit amplitude and does not affect the ability of the heterodyne to demodulate or remove the chirp from the received signal.

The complex heterodyne shifts the spectrum of the signal left by the base frequency and collapses all of the signal power in the positive frequency band from the chirp to DC. If the chirp parameter is not close enough to the actual chirp rate, the spectrum of the complex demodulated signal, $C(t)=s(t)e^{-i\theta (t)}$, will again have a band of frequencies (albeit normally a much reduced band) centered on zero frequency. The correction to the base frequency and chirp rate can be recalculated using the above formulae. These new values can simply be added to initial estimates because phase changes are linear in the exponential. Once the correct base frequency and chirp rate has been identified, complex demodulation concentrates all of the power in the chirp into the zero frequency bin at DC. Typically, only a few iterations are required to completely remove the base frequency and chirp rate from the complex demodulated signal.

Once the base frequency and chirp have been removed, the phase modulation that still remains in the complex demodulated signal is restored to its original frequency. Figure 3 shows a successfully complex demodulation of the chirped signal in Fig. 2. The phase modulation at 10 kHz is now visible in the spectrum of the complex demodulated signal at its correct frequency location.

 

Fig. 3. Phase plot of the complex demodulated signal.

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Although the demodulated signal shown in Fig. 3 correctly demodulates the signal and successfully recovers the phase modulation, the background noise is rather high and limits the detectability of phase modulation below a modulation index of about 0.01. The reason for the high level of background noise in the plot is due to the fact that the complex demodulation using a chirped frequency doubly chirps the negative frequencies creating a pseudo-white noise spectrum background. We will show in the next section that the large noise background is eliminated when the negative frequency components are removed from the signal before demodulation.

One can improve the values obtained for the base frequency and chirp rate ($f_0$ and $k$) by using a least-squares fit in the time domain. The least-squares procedure will in general find a much better estimate for than is possible using a complex heterodyne because the discrete Fourier transform has granularity of one bin or $\delta f=1/T_{\mathrm{obs}}$, where $T_{\mathrm{obs}}$ is the time span of the digitized signal. It is a common misconception that one cannot resolve frequency better than this limit but it is not a limitation in the time domain. For example, in an absolute gravity meter, a chirped signal is observed of about 0.2 s (when the object free-fall drops about 20 cm). Each frequency bin in an FFT over this time span is $\delta f=5\mathrm{Hz}$. Equating the phase corresponding to the smallest detectable chirp to a corresponding phase change in the interferometer interrogating a free-falling mirror, $2\pi \delta f{T}_{\mathrm{obs}}=2\pi \delta a{T}_{\mathrm{obs}}^2/\lambda$, we find that the granularity of the FFT corresponds to an acceleration error of $\delta a=2\times {10}^{-5}\mathrm{m}/{\mathrm{s}}^2$. This error is about 1000 times larger than can be achieved with a least-squares fit of the same data in the time domain!

It is convenient to construct a model having two quadrature components (a sum of a sine and a cosine terms), $M(t)=A\cos ({\omega }_{0}t+(1/2)\alpha t^2)+B\sin ({\omega }_{0}t+(1/2)\alpha t^2)$. The least-squares fit is linear in the coefficients for the two amplitudes, $A$ and $B$. The base frequency and chirp rate, however, occur inside the phase of the sine and cosine and require a nonlinear least-squares fit. This requires starting estimates for the base frequency and chirp, ${\omega }_0=2\pi f_0$ and $\alpha =2\pi k$, that come from the first stage of the analysis. The nonlinear least squares will converge as long as the complex demodulation in the first stage is seen to combine all of the chirped power into the DC frequency bin (the lowest frequency bin).

The nonlinear least-squares identification stage of the signal processing converges quickly and is easy to code for digital data. One might ask why complex demodulation was used at all since a least-squares fit in the time domain ultimately provides a better result. The reason is that the complex heterodyne provides values for the initial velocity and chirp rate within one frequency bin of the FFT, which is precisely the same precision required for the initial estimates of the parameters for the nonlinear least-squares algorithm to converge. These two techniques, therefore, are quite complementary.

The least-square residual is formed by subtracting the sampled data from the model function, $r(t)=s(t)-M(t)$. This provides an estimate for the amplitude error or amplitude modulation for the received signal but is not very useful for recovering phase errors or demodulating FM or phase modulation.

3. Analytic Signal Demodulation (ASD)

We have shown in Fig. 3 that phase modulation riding on top of a chirped carrier is recovered by complex demodulating the signal with a matched chirped frequency.

Recall that phase modulation represents the signal that is being transmitted in a spread-spectrum FM application. Or in the case of a signal from an interferometer, the phase information is directly interpretable as unmodeled mirror motions. In either application, the phase information is quite important.

The downside of using complex demodulation with a chirped frequency is the high background noise. The noise is caused by the expansion of the bandwidth of the negative frequencies in the chirp caused by the complex demodulation. In our simulated example, these negative frequencies create a pseudo-white noise background that limits signal detection for a modulation index less than about 0.01. This problem can be avoided and a much lower overall noise can be achieved if the negative frequencies are first removed from the received signal prior to complex demodulation. This is the motivation for ASD.

A signal with frequency components of only one sign (positive or negative) is commonly called the analytic signal. The analytic signal is complex because the frequency content in any real signal has both positive and mirrored negative frequency components. One way to form a complex analytic signal, $s_a(t)$, is to remove the negative frequencies with the help of the Hilbert transform, $\mathcal{H}(s)$. The Hilbert transform rotates the negative frequency components by 90 deg and the positive frequency components by $-90\mathrm{deg}$. Thus, if the Hilbert transform of a function is added to itself, the negative frequencies will be eliminated and the positive frequencies will be doubled. Mathematically, this is expressed as, $s_a=s-i\mathcal{H}(s)$. Clearly, only the negative frequency components of the Hilbert transform are necessary to create an analytic signal since there is no advantage to doubling the positive frequency content. Thus, one can use the analytic signal of the Hilbert transform (with negative frequencies) to produce the analytic signal.

The Hilbert transform of a real signal is in general a complicated procedure and must be done in the frequency domain. It is much simpler and computationally more efficient to use the Hilbert transform of the known model, $M$, identified in the first step of the analysis, to remove most of the negative frequency content in the real signal. This creates an approximation to the analytic signal having only positive frequency components.

The closer the 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.

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

Only the negative frequency components of the model are needed for creating the analytic signal as discussed above. The analytic signal of the model with negative frequency components is given by $M_{a^{-}}=(1/2)(A+iB)e^{-i\theta }$.

The approximated analytic received signal is therefore $\tilde{s}=s-i\mathcal{H}(M_a)$, or $\tilde{s}=s(t)-(1/2)(A+iB)e^{-i\theta }$.

Note that this is also easily understood (without referencing the Hilbert transform) as the received signal with the negative frequencies of the chirped model removed.

If the model is a good approximation to the received signal, the received signal without its negative frequency components will be approximately the same as the analytic signal of the model with only positive frequencies except that the phase will still have the modulated phase fluctuations, $\delta \theta$, or $\tilde{s}\cong (1/2)(A-iB)e^{i(\theta +\delta \theta )}$.

This expression makes it clear that a complex demodulation of the approximated analytic signal will demodulate the phase modulations, without doubling the chirp rate of any negative frequencies (because they are removed prior to complex demodulation).

This suggests that it is useful to define a phase residual function as $R_{\theta }=2\frac{(s(t)-\frac{1}{2}(A+iB)e^{-i\theta })e^{-i\theta }}{A-iB}$, assuming a received signal, $s({\theta }_r)=A_r\cos ({\theta }_r)+B_r\sin ({\theta }_r)$. We assume that the amplitudes of the model parameters are nearly identical with those of the received signal. Thus, the coefficients of the received signal can be written as $A_r=A+\delta A$ and $B_r=B+\delta B$. Furthermore, the phase residual of the received signal is also nearly equal to the phase of the model plus the phase fluctuations, $\delta \theta$, impressed upon the chirp or, ${\theta }_r=\theta +\delta \theta$. With these assumptions, the phase residual can be expressed exactly as

C. Residual Comparisons Using the ${\mathsf{R}}_{\mathsf{\theta }}$ Function with a Complex Demodulation

Figure 4 shows the fully demodulated phase residuals obtained using the ASD technique. The plot shows the spectrum of the phase (at each point in time) computed from the phase residual function, $R_{\theta }$.

 

Fig. 4. Phase residual function, $R_{\theta }$, recovers the demodulated signal.

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One can see the much lower noise level ($5\times {10}^{-4}$) obtained using the phase residual function, $R_q$, compared to a complex heterodyne in Fig. 3. The background noise in this example is a factor of about 20 smaller due to the removal of the negative frequencies in the raw data (analytic signal) prior to the complex demodulation. This is consistent with the error calculation that states that the error is proportional to the modulation index (0.1 in this example). As mentioned above, the error on the phase decreases as the modulation index decreases for the single sideband technique. It is at first glance curious that the main noise source for the ASD method is caused by the modulation index but this is because the negative frequencies of the unknown phase modulation are not removed before demodulation. Thus, the ASD technique automatically increases in sensitivity as the phase-modulated signal amplitude decreases so that the phase modulation is always recoverable. The background noise level for the complex demodulation, on the other hand, is constant and independent of modulation index. For example, a modulation index as low as ${10}^{-5}$ is easily detectable using ASD whereas a minimum modulation index of 0.02 is necessary for a complex heterodyne to detect the modulation described in our simulated example.

The low noise background of the ASD is very important for interferometer applications where mirror motions result in a very small modulation index. For example, in the absolute gravimeter application given earlier, the noise level of the interferometer is typically 1 nm, which corresponds to a modulation index of only 0.003 (with an effective wavelength of about 300 nm). The gravimeter application requires use of the ASD in order to correctly interpret the phase residuals as mirror motion because a simple complex demodulation produces too much background noise.

The ASD technique was shown here on only one simulated signal. We have done many simulations using different chirped signals and have varied the base frequency, chirp rate (bandwidth), the modulation index, the phase modulation frequency, and many other parameters, and the result is always consistent with our error calculations. We have also used this technique to recover spurious mirror motions in digitized interferometer signals from absolute gradiometers with a very narrow bandwidth and gravity meters with a very wide bandwidth that would not have been visible using pure complex demodulation.

In general, we have found the residual function, $R_{\theta }$, to be an extremely valuable tool for recovering phase modulations from chirped sinusoidal signals. This function will recover phase modulation from a wide variety of signals with narrow or wide bandwidths. It will also work correctly for more complicated broadband signals as long as the phase, $\theta (t)$, or the frequency dependence is known.

3. Conclusion

There are special applications that produce a phase-modulated signal impressed on a chirped base carrier frequency. Spread-spectrum communications use this technique for a variety of purposes. It can be useful for secure transmission because it is difficult to demodulate these signals without prior information about the frequency behavior of the carrier. It can also be used in cases where spreading out the signal over a large bandwidth may reduce interference from natural sources. It also reduces the possibility that the signal will interrupt transmissions sent on a single carrier frequency.

Interferometers with an accelerated mirror are another source of signals with a chirped frequency. Currently most absolute gravimeters drop a mirror in a vacuum to measure gravity. This produces a very wideband chirp, from DC to 6 MHz in 0.2 s, and yet contains valuable information in the phase having extremely small modulation index (small mirror motions) of ${10}^{-3}$ or smaller. Signal processing for this application requires a method with very high fidelity and low noise. Absolute gradiometers also use interferometry to interrogate the differential motion of two mirrors in free fall. This application produces a signal that is very narrowband but still contains interesting phase information (mirror motions). Although the chirp rate is very low, where the base frequency varies by less than 1 Hz over the entire measurement lasting 0.1–0.2 s, it is necessary to identify the chirp as well as mirror motions that are riding on top of this chirp.

The first step in the signal processing is to identify the chirp. A chirped waveform has a power spectrum that looks like a flat band of frequencies. The beginning and ending frequencies in the bandwidth can be used to estimate the base frequency and the chirp rate. A complex demodulation of the signal using a chirped frequency with these parameters will then compress the signal spectrum to a single frequency bin at DC. The method can be iterated to improve the chirp parameters until all of the power of the demodulated signal is concentrated into the lowest frequency bin of its FFT. A further refinement is possible in the time domain, where a chirped sinusoidal model is fit in a least-squares sense to the received signal.

Finally, a method called the ASD, uses the 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.

The ASD method can be accomplished using a phase residual function, $R_{\theta }$, that both removes the negative frequencies in the received signal and complex demodulates the analytic signal in one step. This function compresses the positive frequency band without introducing the corresponding problem of doubling the chirp for the negative frequencies. The phase of the residual function is equivalent to the phase modulation that was originally added to the chirped carrier. This corresponds to the transmitted signal in spread-spectrum communications with a chirped carrier. In interferometer applications, the phase residual function can be directly interpreted as mirror motion.

The advantage of ASD is that it avoids generation of noise due to the negative frequencies of the chirp that are not demodulated but rather doubly chirped in a simple complex demodulation. The method is similar to other single-sideband methods used for nonchirped transmissions.

The ASD method was compared to a straight complex demodulation of the received signal on a simulated chirped signal with a modulated signal added to the phase. Figure 3 shows the demodulated phase modulated signal using complex demodulation and Fig. 4 shows the improvement (10 dB) in noise level achieved using the ASD method. This simulation demonstrates how an anomalously large background noise, shown in Fig. 3, is created by the negative frequency components that are not demodulated but in fact are spread out over the entire spectrum by the complex demodulation with a chirp. Many different simulations have been carried out that are all consistent with the noise calculations provided for this method.

The ASD method also can be used for other less obvious applications. It will demodulate phase from any broadband carrier as long as the phase is known either beforehand or can be identified by the waveform itself. For example, one could transmit a spread-spectrum signal using a carrier having a pseudo-random frequency sequence with added phase information (voice or video) that could later be demodulated using ASD where the step of identification of the carrier is eliminated with prior knowledge of the carrier frequency.