Bitstream Photon Counting Chirped AM Lidar with a Digital Logic Local Oscillator

This paper introduces a new concept for the local oscillator (LO) for the Photon Counting Chirped Amplitude Modulation Lidar (PC-CAML). Rather than using a radio-frequency (RF) analog LO applied electronically either in post-detection mixing or via opto-electronic mixing (OEM) at the detector, or applied via pre-detection mixing using an optical intensity modulator as in previous systems, the new method mixes the single-bit binary counts from the photon counting detector with a single-bit binary LO using an AND binary digital logic gate. This type of LO is called the Digital Logic Local Oscillator (DLLO), and the resulting PC-CAML system is a type of bitstream lidar called bitstream PC-CAML (patent pending). The key advantage of the DLLO in the bitstream PC-CAML is that it replaces bulky, power-hungry, and expensive wideband RF analog electronics with single-bit digital logic components that can be implemented in inexpensive silicon complementary metal-oxide-semiconductor (CMOS) read-out integrated circuits (ROICs) to make the bitstream PC-CAML with a DLLO more suitable for compact lidar-on-a-chip systems and lidar array receivers than previous PC-CAML systems. This paper introduces the DLLO for bitstream PC-CAML concept, presents the initial signal-to-noise ratio (SNR) theory with comparisons to Monte Carlo simulation results, and makes suggestions for future work on this concept.


Bitstream PC-CAML with a DLLO Concept
The patented photon counting chirped amplitude modulation lidar (PC-CAML) concept has been extensively discussed previously. [ref [1][2][3][4][5][6][7][8][9][10][11][12][13] In previous PC-CAML systems, the local oscillator (LO) has been a radio-frequency (RF) analog signal applied using expensive, bulky, and power hungry RF analog electronics either in post-detection mixing or via opto-electronic mixing (OEM) at the photon counting detector, including gating of the detector, or applied via predetection mixing using an optical intensity modulator as illustrated in figure 1. [ref [1][2][3][4][5][6][7][8][9][10][11][12][13] The reader is referred to references 1-13 for more details on the theory and practice for the traditional PC-CAML system. 1 Alternatively, one can dispense with an LO in the receiver altogether by sending the wideband, high speed single-bit photon count data output from the photon counting detector directly to storage and demodulating the chirp digitally. [ref 14] However, this alternative eliminates a key advantage of the PC-CAML concept, which is the down conversion of the wide bandwidth ranging signal to the low bandwidth intermediate frequency (IF) regime where low sampling rate analog-to-digital converters (ADCs) can be used to digitize the lidar data which is then sent to storage and digital processing at a low data rate. This down sampling to a lower data rate gains in importance as the number of detectors used increases, such as for larger format array receivers, since it is far easier and less costly to move data from the ROIC to data storage and processing for multiple parallel data streams at low data rates than at high data rates. The discussion above assumes the use of Geiger-mode Avalanche Photo-Diode (Gm-APD) (also known as Single-Photon Avalanche Diode (SPAD)) photon counting detectors that output a single-bit count for each detection event, but some Gm-APD receivers output multi-bit timestamps for the counts rather than the stream of single-bit counts themselves. For such receivers, at very low count rates, the data rates for the multi-bit timestamp data may be low enough that streaming the timestamp data directly to digital storage for full digital processing of the chirped AM signal may be advantageous even for large format array receivers.
Here I present a new alternative for applying the LO in the PC-CAML concept called the Digital Logic Local Oscillator (DLLO) (patent pending 2 ). For the DLLO, the constant amplitude, singlebit output count of the photon counting detector receiving a chirped AM signal is input to an edge-triggered pulse detector which outputs a very short single-bit digital logic level pulse to one input of an AND binary digital logic gate, and the other input of the AND binary digital logic gate is connected to the single-bit digital logic level binary data stream of a chirped AM LO signal, as shown in figure 2 (a.). The wideband, high speed single-bit digital data output by the AND binary digital logic gate can be either directly sent to storage and fully digitally processed, or sent through an analog or digital low/band pass filter in the IF band, the output of which is either digitized by a low sample rate analog-to-digital converter (ADC), or digitally down sampled to a low sample rate, respectively, as shown in figure 2 (a.). The low sample rate digital data is then sent to storage and digitally processed by the usual methods for chirped AM lidar. In this concept, the AND binary digital logic gate provides the mixing of the single-bit photon count data from the Geiger-mode Avalanche PhotoDiode (Gm-APD) with the single-bit LO data.
An example circuit for the edge-triggered pulse detector is shown in figure 2 (b.). The edgetriggered pulse detector allows the count pulses input to the AND gate to be much shorter than the count pulses output by the Gm-APD, which can be longer than desired due to the Gm-APD dead-time. The edge-triggered pulse detector's output pulse can be as short as a single clock pulse as long as the rising edge of the Gm-APD's count pulse is shorter than a clock pulse, and the delay on the delayed input in the edge-triggered pulse detector circuit is shorter than a clock pulse. Other types of edge-triggered pulse circuits may be used, or the edge-triggered pulse circuit can be eliminated if the Gm-APD count pulses are short enough.

Inherent Single-bit Binary Received Signal for PC-CAML
The constant amplitude output of the Gm-APD makes its output inherently single-bit binary, i.e., there is either a count, or there is no count. Thus, the bitstream PC-CAML may operate with either a chirped sinusoidal or chirped square wave transmitted waveform without requiring any additional thresholding and/or limiting in the receiver to convert a sinusoid to a square wave, as needed in the bitstream radar when operating with a transmitted chirped sinusoidal signal.

DLLO Waveforms
The DLLO for the bitstream PC-CAML may be either a single-bit chirped square wave or a single-bit sampled chirped sinusoid. The advantage of having a single-bit sampled chirped sinusoidal DLLO instead of a single-bit chirped square wave DLLO is that whether the signal is a chirped square wave or a chirped sinusoid, mixing the signal and the DLLO will not produce harmonics in the IF spectrum. For a single-bit sampled chirped sinusoidal DLLO, the single-bit sample pulses may be, for example, pulse width modulated (PWM) or pulse density modulated (PDM) by the amplitude of the analog chirped sinusoid. The single-bit sampling of the chirped sinusoidal DLLO waveform may be uniform or non-uniform.
In the case of uniform sampling, the single-bit sampling rate must be many times the sampling rate of an equivalent multi-bit digitizer to retain the same signal-to-quantization-noise ratio (SQNR ]. In addition, when the number of 1's in the random sampling of the LO exceeds half of the total number of available samples, the apparent modulation depth of the chirped sinusoidal LO diminishes, going to zero when all of the samples are filled with 1's, for the same reason that this happens to the chirped sinusoidal signal that is single-bit sampled by the Gm-APD when the Gm-APD approaches saturation as explained in section 2.1. Initial Monte Carlo simulations verified the significant reduction of the IF signal's SNR due to the additional LO noise and apparent modulation depth loss. Therefore, I do not consider random single-bit sampling of the chirped sinusoidal LO to be worthwhile to investigate further.
Therefore, this paper discusses just a single-bit chirped square wave DLLO.

Harmonics and their Mixing Products
At signal levels well below receiver saturation with a chirped sinusoidal signal, the bitstream PC-CAML will be mixing a randomly sampled chirped sinusoidal modulation of the received signal count rate with a chirped square wave LO. Just as for the linear response receiver based CAML with a chirped sinusoidal signal and chirped square wave LO, this does not produce harmonics in the mixed signal's intermediate frequency (IF) power spectrum.
However, when the transmitted waveform is a chirped square wave or when the receiver starts to saturate so that it "squares off" the received waveform, the bitstream lidar will be mixing a chirped square wave signal and a chirped square wave LO which will produce odd-order harmonics and their mixing products.
These odd-order harmonics and their mixing products also occur in the bitstream radar, for which both the signal and LO are always chirped square waves. Bjorndal, et. al., have shown three ways of dealing with these odd-order harmonics to mitigate their effects in bitstream radar by suppressing them or even by utilizing them for improved range resolution [ref [15][16][17][18], and which can also be used in the bitstream PC-CAML: 1. Known pseudorandom or random dithering of the chirp signal's phase with or without noise shaping to spread out the harmonics over the spectrum 2. Transmitted waveform additional delay to shift the harmonics beyond the intermediate frequency (IF) corresponding to the desired unambiguous range 3. Detection at a harmonic frequency for improved range resolution.
As Bjorndal points out, dithering the chirp signal's phase can spread out the energy of the harmonics over the power spectrum, but this increases the noise floor. [ref 18, pp. 12, 56, 62-65]

Aliasing
Dithering can also be used to address the issue of aliasing. As Bjorndal states "If we sample, either by transmitting a sampled bitstream, or sample on the receiver without an anti-aliasing filter then anything above the sample rate is going to fold down...Avoiding aliasing on the receiver side is best done with an analog low-pass filter and a multi-bit ADC ..., while aliasing on the transmitter can be mitigated with intentional dithering..." [ref 18, p.56] Use of an antialiasing filter with a multi-bit ADC in the wideband signal path on the receiver, however, complicates the receiver, and dithering on the transmitter complicates the transmitter.
Fortunately, at signal levels well below saturation, the quantum randomness of the photon stream and noise counts in the PC-CAML provides nonuniform random sampling of the transmitted waveform, which can eliminate the aliasing inherent in uniform sampling. The Nyquist sampling theorem applies only to uniform sampling. For example, Beutler showed that any spectral distribution is alias-free if randomly sampled by a Poisson point process. [ref 19] However, for the bitstream PC-CAML, as the signal level increases into receiver saturation so that all the available clock time interval samples start filling up with ones, the sampling becomes more like uniform sampling and aliasing can occur. Gatt, et. al., show that the lidar signal counts from a photon counting Gm-APD have a negative binomial distribution rather than a Poisson distribution due to the speckle diversity. [ref 20, pp. 3263-3265] (Note that the form of the negative binomial distribution used in Gatt, et. al., is also called the Gamma Poisson (mixture) distribution. [ref 21 -22]). They also model the noise counts with a Poisson distribution. The resulting signal plus noise counts for the Gm-APD based lidar have a Negative Binomial Plus Poisson (NBPP) distribution resulting from the convolution of the negative binomial and Poisson distribution given in Gatt,et. al.. [ref 20,p. 3264] When the counts per matched filter impulse response time are much less than one, the counts distribution is approximately Poisson [ref 20], so the alias-free property of Poisson sampling holds in this limit.
The author is not aware of any prior work showing that an NBPP point process has the alias-free property in general. Determining whether or not the NBPP point process provides alias-free random sampling in general is beyond the scope of this paper and is suggested for future work.
However, the counts from the Gm-APD can be viewed as the result of an additive random sampling (ARS) process. An ARS process is defined by having sample times given by t n = t n-1 +  n , where n is an integer, and { n } is a family of independent identically distributed positive random variables. This says that the current count sample time is the previous count sample time plus a random delay time, which is the case for the Gm-APD counts for PC-CAML. An important result for our purposes from the thesis of King Chuen Lo is that an ARS process is alias-free for any distribution of { n } for a sampling duration T larger than T a , where T a depends on the probability density function of . Thus, the average count rate for an ARS process can be much lower than the Nyquist sampling rate for the signal bandwidth if a sufficient number of additive random counts are accumulated, but the clock rate must be at least twice the signal bandwidth for alias-free sampling.

Dead-Time and Count Pulse Rise Time Limitations on Signal Bandwidth
One might think that the chirp signal's maximum frequency would be limited to no more than 1/2t d , where t d = the Gm-APD's dead-time. As Redman, et. al., demonstrated, however, it is the rise-time of the count pulse, not the dead-time, that sets the upper frequency limit for PC-CAML with pre-or post-detection mixing. [ref [2][3][4].
On the other hand, for OEM mixing by modulating the bias voltage of the Gm-APD directly with a square-wave LO signal, the highest useful chirp frequency is limited by the Gm-APD deadtime. [ref 2-4, 8-9, 13] If, however, the LO is applied by modulating the gate open duration of the Gm-APD such that the gate open duration is proportional to the amplitude of a sinusoidal LO, then the dead-time limit on the highest chirp frequency can be eliminated by setting the longest gate open duration, Dt g , and the shortest time sample interval between starts of opening the gate, t gs , such that t gs -Dt g > t d , where t d is the dead-time. [ref 11, p. 11802] In this case, the highest chirp frequency is limited by the smallest gate duration that can be applied to the Gm-APD.
Luo's analysis does not take into account the finite rise-time of the edges of the count pulses. If the clock time interval, , is much larger than the count pulse rise-time, then the signal must be band-limited within [− 1/2∆ , 1/2∆] as stated by Luo. If, however,  is less than or equal to onehalf the count pulse rise-time, then the signal bandwidth must be less than or equal to half the bandwidth of the count pulse's rising edge. When the count pulse rise-time equals 2, so that the count pulse rising edge is sampled at the Nyquist rate, then the signal must be band-limited within [-1/2t r ,1/2t r ] = [−1/4∆ , 1/4∆], where t r is the count pulse rise-time. Gatt 20, p. 3264] For well designed photon counting lidars, in order to prevent receiver saturation, transmitter power and/or receiver throughput control will be used to enforce m s << M since the minimum of M is 1, and if m s were not much smaller than 1, the receiver would be nearly saturated. Also for well designed photon counting lidars, measures such as using narrow band optical filters and low dark count rate Gm-APDs, will be taken to insure that m n <<1, where m n is the average number of noise photoelectrons per matched filter impulse response time, otherwise, the arm probability will be too low for practical operation. (See reference 20 for a more detailed discussion of m n and the arm probability.) Therefore, for a well designed bitstream PC-CAML with transmitter power and/or receiver throughput control, and low noise count rate, the signal plus noise will be well approximated by Poisson distributed signal plus noise, and the signal spectrum will be alias-free for any accumulation duration if the signal's bandwidth is within the band limits set by the clock time interval or the count pulses' rise-time as discussed above. In addition, since the counts form an ARS process, even for high count rates, but count rates still well below receiver saturation, the spectrum will be alias-free for a sufficient number of accumulated counts if the signal bandwidth, count pulse rise-time, and clock time interval meet the requirements discussed above.

Detection at a Harmonic Frequency for Improved Range Resolution
When using a chirped square wave signal, detection at a harmonic frequency for improved range resolution can be done for processing bitstream PC-CAML data in the same manner as described by Bjorndal, et. al., for bitstream radar [ref [16][17][18], but further investigation into that technique is beyond the scope of this paper and is suggested for future work. If the harmonics are to be used for improved range resolution, the additional bandwidth of the harmonics to be used for this purpose must be included in the signal bandwidth limits for alias-free sampling discussed above.

Transmitted Waveform Delay for Shifting Harmonics Beyond the Unambiguous Range
Since dithering increases the noise floor and is not needed for anti-aliasing for a well designed bitstream PC-CAML operating well below receiver saturation, and since detection at a harmonic frequency is beyond the scope of this paper, in this paper, just the ability of a transmitted waveform delay to shift the frequencies of the harmonics beyond the desired unambiguous range frequency for the bitstream PC-CAML with a DLLO is demonstrated with simulation results. The additional delay on the transmitted waveform is equivalent to an advance of the LO waveform. Therefore, the additional transmitted waveform delay can be implemented by starting the LO waveform earlier than the start of the transmitted waveform and extending the LO's chirp duration and frequency along the chirp's temporal slope for complete overlap with the round-trip delayed received chirp signal's duration.

Bitstream PC-CAML with a DLLO SNR Theory
The purpose of this paper is to introduce the new bitstream PC-CAML with a DLLO concept and to show through simulation results how it works. The purpose of this paper is not to develop a comprehensive theory of operation for the new concept, so the initial SNR theory presented herein has a limited range of applicability to the new concept, and an improved theory needs to be developed in future work.
The initial electrical power signal-to-noise ratio (SNR) theory used herein for the bitstream PC-CAML with a DLLO concept is derived from the SNR theory for photon counting Gm-APDs developed by Gatt  Note that the clock time interval equals the matched filter impulse response time and the deadtime in this theory and in the simulations discussed in section 3.
Also note that formulating the SNR theory in terms of the signal and noise counts output by the Gm-APD per matched filter impulse response time after having been subjected to the dead-time and arm probability constraints as done in equation (1.), rather than in terms of the signal and noise photoelectrons per matched filter impulse response time prior to the dead-time and arm probability constraints as done by Gatt et. al., has the practical advantage for comparison to experimental results of using the noise only count rates and signal plus noise count rates that would be measured in an experiment without having to have an auxiliary measurement of the incident photon rates and the Gm-APD's photon detection efficiency. (Note that the average signal only count rate in an experiment is easily calculated by subtracting the measured average noise only count rate from the measured average signal plus noise count rate.) However, for use in system performance modeling, the original formulation of the SNR theory by Gatt et. al. is probably more useful since the photon rates will be calculated by the mathematical models and the Gm-APD's photon detection efficiency will be specified in the system design parameters.
This SNR theory does not include the effects of energy loss from the fundamental IF frequency to higher order odd harmonics and their mixing products for the chirped square wave modulation waveforms in the bitstream PC-CAML. This SNR theory also assumes 100% modulation depth for the PC-CAML signal.

Sinusoidal Modulation Depth Loss Near Saturation
As the average number of signal counts output by the Gm-APD per available clock time interval increases towards 1, the apparent modulation depth of a sinusoidal chirped AM waveform decreases due to the Gm-APD being able to output at most only a single count per dead-time which is set equal to the clock time interval in the simulations (see section 3). For the sinusoidal signal, when the average count rate is so high that a count is output for every clock time interval, the single-bit count data stream looks like that of a constant, unmodulated signal corresponding to a modulation depth of zero and therefore, the SNR goes to zero.

Saturation Effects on Chirped Square Wave Signals
It must be noted that the number of available clock time intervals for a signal count (i.e., for a 1) for a fully modulated chirped square wave signal is half that of a chirped sinusoidal signal. This is because even with the negative binomial signal fluctuation statistics, there are no signal counts in the low areas of the square wave since the signal is exactly zero for those areas. (There are noise counts in those areas, but here we are discussing just the clock time intervals available for signal counts.) Thus, for the fully modulated chirped square wave, half of the clock time intervals will be 0 and only half of the clock time intervals will be available for a signal count.
Therefore, for a fully modulated chirped square wave signal, as the mean number of signal counts output by the Gm-APD per clock time interval available for a signal count increases towards 1, the signal becomes more like a deterministic chirped square wave signal that is uniformly sampled at the clock rate, and the floor of the power spectrum of the mixer output becomes dominated by sidelobes and perhaps aliasing, depending on the sampling rate and signal bandwidth, rather than dominated by noise for the low noise rates of a well designed PC-CAML. These effects of Gm-APD receiver saturation on the chirped sinusoidal and chirped square wave forms are illustrated in figures 6 and 7 below with results from the Monte Carlo simulations. These effects, however, are not included in the initial SNR theory represented by equation (1.).

Different Effects of Speckle Diversity on Sinusoidal and Square Wave Signals
As shown by the Monte Carlo simulation results in section 3, figure 6 (a.), the value of the speckle diversity, M, affects the SNR for the chirped sinusoidal signal as predicted by the SNR theory. However, the simulation results in figure 6 (b.) show that the value of M makes no difference in the SNR for the chirped square wave signal in contrast to the theory's predictions. The following is an untested conjecture to explain these results, but more work on the SNR theory is clearly needed.
For a fully modulated chirped square wave where the signal is high, if any signal is detected, the output level of the Gm-APD is the same constant high level regardless of amplitude fluctuations on the input optical signal. Where the signal is zero, there are no fluctuations since there are no signal counts in those areas, just noise counts. Although the photon arrival rates will vary with the speckle amplitude fluctuations in the high areas of the chirped square wave signal, this may just look like additive random sampling of the high areas of the chirped square wave with a different distribution of random samples depending on M, which does not change the SNR. Therefore, for the chirped square wave signal this causes the effect of the speckle induced signal amplitude fluctuations on the SNR to be nearly eliminated.
If this is true, then if the chirped square wave were not fully modulated so that the small, but non-zero, signal level in the low areas of the square wave would be subject to amplitude fluctuations, then the amplitude fluctuations over the whole signal would be detected as photon arrival rate fluctuations, which would reduce the SNR with increasing amplitude fluctuations corresponding to decreasing M.
For the chirped sinusoidal signal there is some signal everywhere except at the exact nulls of the fully modulated chirped sinusoidal waveform so that the speckle induced amplitude fluctuations cause photon arrival rate fluctuations at almost all points on the signal, and these fluctuations are higher for lower M causing the SNR to be lower for lower M just as seen in the Monte Carlo results and theory predictions for the chirped sinusoidal signal case.

Single-bit Quantization Noise
Lastly, noise due to single-bit quantization must be calculated to use in the SNR theory of equation ( For a single-bit, N bits =1, the quantization noise per 1-bit count is then 1/10 0.778 = 1/6. Therefore, the average quantization noise per clock time interval is the average number of signal plus noise counts per clock time interval times 1/6, which is given by the following in the simulations: m q_n_counts_avg = m spn_counts_avg /6 (3.) where m q_n_counts_avg = the average number of quantization noise counts per clock time interval in the simulations m spn_counts_avg = the average number of signal plus noise counts per clock time interval in the simulations.
In the initial SNR theory, equation (1.), m qncounts is set equal to the value of m q_n_counts_avg for the simulations for which the SNR theory and simulation results are being compared.
Development of an SNR theory that includes the effects of quantization noise, aliasing, the harmonics and their mixing products, less than 100% signal modulation depth, and Gm-APD saturation for the bitstream PC-CAML that is applicable for both chirped sinusoidal and chirped square wave signals is beyond the scope of this paper and is suggested for future work.

Monte Carlo Simulation Results Compared to Initial SNR Theory
The Note that the value of the target range used in the simulations is chosen so that the resulting round-trip delay time makes the intermediate frequency (IF) for the target return signal exactly equal to some frequency sample in the power spectrum computed in the simulations to avoid complications in computing the SNR for comparison to the theory due to the target range peak straddling two frequency samples.
The random Poisson distributed noise counts and negative binomial distributed signal counts per clock time interval are given by equations (6.) and (7.), respectively: (6.) (7.) where In Mathcad ® , the function if(logical expression, A, B) means that if the logical expression is true, then the result is A, otherwise the result is B In Mathcad ® , the symbol ^ represents the Boolean logical AND i = row index over clock time intervals j = column index over trials Note: The "if" statements in equations (6.) and (7.) enforce the restriction of having at most one count output by the Gm-APD per dead-time, which equals the clock time interval in the simulations.
In Mathcad ® , rnbinom(m,n,p) returns a vector of m random numbers having the negative binomial distribution: The following expression in Mathcad ® performs the logical AND on the single-bit signal plus noise counts and the single-bit LO data streams to produce the single-bit mixer output: where = the signal plus noise counts j = column index over trials, and the superscript <j> indicates the j th column of a 2D array.
The single-bit mixer output is Hann windowed to reduce sidelobes, and zero padded to eight times its original length. The Hann window is given by (12.) The Hann windowed and zero padded single-bit mixer output is digitally band pass filtered by a super Gaussian filter that filters out the DC peak and high frequencies prior to down sampling to prevent aliasing. The digital super Gaussian band pass filter for the zero padded signal is defined in the frequency domain by the following in the Mathcad ® simulations: (13.) where ceil() = the round up to the next highest integer function ii = index over the frequency bins  The super Gaussian band pass filter function is multiplied by the complex fast Fourier transform (cfft) of the Hann windowed and zero padded single-bit mixer output. The real part of the inverse cfft (icfft) of the resulting product is the filtered mixer output which is then down sampled by a factor of 32 as follows: (15.) where Mixed_filt_ds = the filtered and down sampled Hann windowed and zero padded mixer output Mixed_filt = the filtered Hann windowed and zero padded mixer output prior to down sampling The power spectrum of the filtered and down sampled Hann windowed and zero padded singlebit mixer output is computed in Mathcad ® for each trial. The resulting power spectra are averaged together to produce the mean power spectrum for all the trials.
The mean noise floor of this mean power spectrum is computed over the portion of the spectrum between the target signal's fundamental IF and the third harmonic of that frequency. The value of this mean noise floor is the denominator in computing a simulation's mean SNR.
The peak value of the mean power spectrum at the fundamental IF minus the value of the mean noise floor is used as the numerator in computing a simulation's mean SNR.
Note that since the simulated data are quantized to one bit, the single-bit quantization noise is inherently included in the simulated power spectra.    The simulation results for the chirped sinusoidal signal and the SNR theory are in good agreement up to an average signal counts per clock time interval, m s_counts_avg , of about 0.6 for M=1 and 0.7 for M=1E+06. As discussed in section 2.1, above these levels the saturation effects of power loss to the harmonics and their mixing products, and of modulation depth loss make the simulations' SNR results rollover with increasing m s_counts_avg . The SNR theory over estimates the SNR for these high signal levels since the effects of power loss to the harmonics and their mixing products, and of modulation depth loss are not included in the SNR theory.
For the chirped square wave signal, the simulation results and the theory are qualitatively different in that the simulation results for M = 1 and M = 1E+06 do not differ significantly even for higher signal levels as they do for the SNR Theory results. See the earlier discussion of a tentative conjecture to explain this in subsection 2.3. Even so, for the chirped square wave signal, the SNR values for the simulation results for both low and high M are within about 2 dB of the initial SNR theory values for high M over the range of m s_counts_avg plotted.
At m s_counts_avg above about 0.8, the SNR values for the chirped square wave signal simulation results exceed those of the SNR theory, whereas the SNR values for the chirped sinusoidal signal simulation results roll over to approach zero as m s_counts_avg approaches 1.
As explained in section 2.1, the latter behavior is due to the clock time interval samples all being filled with 1's as m s_counts_avg approaches 1 for the chirped sinusoidal signal. This looks like a loss in modulation depth for the sinusoidal signal until at m s_counts_avg = 1, the modulation depth goes to 0 and the signal looks like a constant level signal with no modulation.
As explained in section 2.2, for the fully modulated chirped square wave signal, however, the clock time interval samples where the zero's of the chirped square wave are located are never filled by 1's for any signal level, and those clock time interval samples can only have an occasional 1 due to noise at the low noise levels in the simulations and in a well designed PC-CAML. Therefore, as m s_counts_avg approaches 1, only the clock time interval samples corresponding to the high levels of the chirped square wave become filled with 1's, and this causes the waveform to approach the uniformly sampled deterministic chirped square wave form as m s_counts_avg approaches 1. This results in the floor of the power spectrum of the chirped square wave signal plus noise becoming dominated by sidelobes and perhaps aliasing, depending on the signal bandwidth and clock rate, as m s_counts_avg approaches 1. This is illustrated by the graph in figure 7 of the raw power spectra prior to filtering and down sampling for the simulated PC-CAML's stochastic chirped square wave signal for m s_counts_avg = 1, m n_counts_avg = 1E-4, and M = 1E+06 ANDed with a deterministic chirped square wave LO (red trace), and a noiseless, deterministic chirped square wave signal multiplied by a deterministic chirped square wave LO (blue trace). Clearly, the power spectrum, including the floor between the spectral peaks, for the mixer outputs with the stochastic chirped square wave signal for m s_counts_avg = 1 ANDed with the chirped square wave LO is nearly identical to that of the noiseless, deterministic chirped square wave signal multiplied by the chirped square wave LO.
Note that a well designed PC-CAML system using transmitter power and/or receiver throughput control to prevent saturation would operate at the lower signal levels where the initial SNR theory agrees with the simulation results, and where the spectral floor is dominated by noise. Also note that for the average signal counts per clock time interval regime of 0.05 to 0.7, the theory and simulation results predict SNRs of about 10-25 dB for the additive noise level of m n_counts_avg = 1E-4 used in this study. This range of SNRs is sufficient for many lidar applications. These SNR results will vary with the mean number of additive noise counts per clock time interval. For the number of clock time intervals accumulated of 2 13 used in the simulations, the mean total number of noise counts accumulated over the chirp duration of 4.096 ms in the simulations is about 0.82, corresponding to an average noise count rate of about 200 kHz for m n_counts_avg = 1E-4. This mean noise count rate is on the high side compared to the dark noise count rates of many Gm-APD receivers, but can be comparable to the dark noise plus solar background noise count rates in some scenarios.
As a spot check to verify that the SNR theory and simulation results agree at higher noise levels for a signal level below saturation, the simulation was run for a chirped sinusoidal signal at two much higher noise levels for m s_counts_avg = 0.5 with the results shown in table 1. The results shown in table 1 indicate that the initial SNR theory probably provides reasonable SNR estimates for a chirped sinusoidal signal at a level well below saturation even with high noise levels that are well above the noise count rates expected for a well designed PC-CAML.
Note that prior to adding the quantization noise term, the SNR theory over estimated the SNR compared to the simulation results for low m s_counts_avg (less than about 0.2), but after addition of the quantization noise terms, the SNR theory is in good agreement with the simulation results for these low values of m s_counts_avg . Therefore, it is important to include the quantization noise caused by the single-bit quantization inherent in the bitstream PC-CAML with DLLO concept.   Figure 8 shows the mean power spectrum without (a.) and with (b.) an additional 59.99 ns delay on the transmitted chirped square wave signal. In the latter case, the frequencies plotted have all been digitally shifted by subtracting the IF corresponding to the additional transmitted waveform delay to return the fundamental IF peak corresponding to the target range to its original position without the additional delay. These simulation results demonstrate the ability of the additional delay on the transmitted waveform to move the harmonics beyond the desired unambiguous range.

Simulation Results for the Transmitted Waveform Delay to Shift the Harmonics
This technique works because adding a delay to the transmitted waveform spreads the frequency difference between the fundamental IF and its n th harmonic since the n th harmonic has n times the chirp bandwidth making its frequency slope over the same waveform duration n times steeper.

Conclusion
The concept and initial performance modeling and simulation results for the new bitstream PC-CAML with a DLLO system were presented in this paper. The results of the initial SNR theory and Monte Carlo simulations presented herein indicate that the bitstream PC-CAML with a DLLO performs as expected for low signal levels.
At high signal levels nearing saturation, the SNR for the simulation results for a chirped sinusoidal signal rolls off due to energy losses to higher order harmonics and to reductions in the apparent modulation depth of the received AM waveform, neither of which are included in the initial SNR theory presented herein. For a fully modulated chirped square wave signal, at high levels near saturation, however, the SNR continues to increase as the mixer output approaches that for a deterministic chirped square wave signal, and the floor of the power spectrum of the mixer output becomes dominated by sidelobes and perhaps aliasing, depending on the signal bandwidth and sample rate. Methods for removing or reducing aliasing developed for bitstream radars can be applied to bitstream PC-CAML if necessary.
The issue of the higher order harmonics generated by the chirped square wave modulation can be mitigated by using the same methods for this used in bitstream radar. The effectiveness for bitstream PC-CAML with a DLLO of one of these methods, that of applying an additional delay on the transmitted waveform to move the harmonics to frequencies higher than the intermediate frequency of the desired unambiguous range, was demonstrated herein with simulation results.
The key advantages of the bitstream PC-CAML with a DLLO are that it can be implemented in the unit cells of a photon counting lidar receiver's ROIC by adding AND binary digital logic gates, and the received signal and LO just consist of streams of single-bit binary data, eliminating the need for bulky, power-hungry, and expensive wideband RF analog electronics. The DLLO single-bit binary waveform data can be computed prior to operation and stored in a buffer in each ROIC unit cell, or in a single buffer and distributed to each ROIC unit cell in realtime during operation. The DLLO data buffer can be a circular buffer for continuous repetition of the DLLO single-bit binary waveform data. Multi-GHz clocks and AND digital logic gates are readily implemented in inexpensive silicon CMOS ROICs, and Multi-tens-of-GHz can be attained with more expensive technologies. These advantages make the bitstream PC-CAML with a DLLO more suitable for compact lidar-on-a-chip systems and lidar array receivers than previous PC-CAML systems.