Signal Statistical Mechanics

Abstract

We are interested in determining the physics bound for the detection of signals in modern digital radio frequency (RF) hardware. Classical signal theory (Kalman filters) requires that the signal-to-noise power ratio (SNR) $>10$, but this is not the physics bound. Instead, the physics bound is much more complicated. Because an important application is radar, we ask whether, in a time interval of 1 $\mathsf{\mu }$s, a signal is present within the noise of the receiver baseband. For radar, this would be the pulse return reflection. For our analysis, we use the Keysight Technologies UXR_25 oscilloscope as the RF receiver that has an analogue-to-digital converter (ADC) chip of 256 billion samples per second. In 1 $\mathsf{\mu }$s, then, 256 thousand voltage samples are taken. We want to determine if a signal is present using the 256 thousand voltage samples using random matrix theory (RMT). The answer for this particular ADC is that we can detect any signals with SNR >−20 dB, a thousand-fold increase from SNR > 10. This paper gives the physics bound of signal detection.

1. Introduction

To determine the actual physics bound of signal detection, we consider a modern digital radio receiver that uses an analogue-to-digital converter (ADC) chip, taking the output of a de-multiplexed, de-modulated radio-frequency receiver (RF). This means we are discretizing each channel’s baseband analogue signal (meaning the carrier wave frequency, phase and amplitude have been demodulated to the baseband). We demonstrate in the paper that these ADC voltage samples constitute statistical mechanics, with fluctuations and correlations. The mathematics employed is random matrix theory (RMT). Thus it should not be surprising that associated with this statistical mechanics of signals is the occurrence of critical phenomena. If no signal is present, the voltage samples are the famous Johnson–Nyquist noise [1,2] (Figure 1). We show, using RMT, that the presence of a signal is a liquid–liquid phase separation, ending in a liquid–liquid critical point. The RF receiver in the signal phase has broken ${\mathcal{Z}}_2$ symmetry, and in the noise phase, the ${\mathcal{Z}}_2$ symmetry is restored. In Equation (20), the time correlation function $F(\tau =k\mathrm{\Delta }t)$ is given. In Figure 4, the noise floor correlation average is zero ($<F\left(\tau \right)>=0$, where <......> denotes the average), whereas in the signal phase (Figure 5), $<F\left(\tau \right)>\ne 0$. Now $F\left(\tau \right)$ has an algebraic sign. The ${\mathcal{Z}}_2$ transformation $F\left(\tau \right)\to -F\left(\tau \right)$ is a symmetry of $<F\left(\tau \right)>=0$ and not a symmetry of $<F\left(\tau \right)>\ne 0$.

We show that physics of signals has an order parameter $\eta$ that delineates the coexistence curve for phase separation: signal above and noise below. Associated with this order parameter is a critical exponent $\beta$ that we calculate to be $\beta =0.67\pm 0.03$, in a linear regression log–log plot. Finally, we show that this discovery of a signal phase transition greatly improves the operational efficiency of radars.

Shannon established the modern engineering aspects of signals with the Shannon–Hartley theorem [3] giving the channel capacity of analog communications with assumed Gaussian noise and the Nyquist–Shannon sampling theorem [4], which specifies the requirements to avoid aliasing. However, the physics of signals was not discovered. Basically, what is a radio-frequency (RF) signal? Physics-wise, it can be defined in power terms whereby an RF receiver (sensor) has increased power during a time interval $\mathrm{\Delta }t$, when compared to a known no-signal receiver state. This increased power level is embodied in a signal-to-noise power ratio ($SNR$), and conventional signal processing (Kalman filter) requires $SNR>10$ (>10 dB), in order to identify the presence of a signal in noise. Signal detection theory has not changed much in years. An early review is [5] and a recent 21st-century review is [6]. The paucity of new radar patents reflects this.

Here, we take a different approach and look at the correlation of the voltage samples $\left\{V_i\right\}$ created by the ADC in the digital RF receiver baseband, within the time interval $\mathrm{\Delta }t$. We use Random Matrix Theory (RMT) [7] for analysis, because it determines the properties of the sample covariance matrix, the statistical tool for communications [8]. These voltage samples $V_i$ constitute statistical mechanics, which has both fluctuations and correlations.

The statistical mechanics comprises two independent networks whose external driving parameters are causality and randomness. Causality voltages are the signals and the random voltages are the noise. Looking at the voltage samples determined by the analogue-to-digital chip, we ask what is the relation of the voltage $V_{now}$ measured at time = now, compared to the previous measured voltage $V_{now-1}$ measured at previous = now − 1? If the voltages are white noise, then each voltage sample is drawn from some probability distribution (e.g., Gaussian) and so $V_{now}$ and $V_{now-1}$ are uncorrelated. Colored noise, in contradistinction to white noise, has a finite time correlation, meaning that the voltages $V_{now}$ and $V_{now-1}$ have a probability of being causally related. As the time sequence unfolds, colored noise voltages that are separated from each other by more than the colored time correlation length have reduced probability of being causally related. Synthetic colored noise simulations can be found in [9,10].

A (pure) signal, however, has its time correlation infinite because causality is creating the voltage samples by the presence of the RF wave; we do not need to know the exact signal waveform to determine its causality. This discretizing of the analogue RF signal and measuring the correlation of the samples mean that fluctuations and correlation lengths are statistical variables that are quantifiable in terms of critical phase transitions. In quantum field theory, a Wick rotation from Minkowski Space to Euclidean Space converts the generating functional of correlations into the partition function of statistical mechanics. That is, the Wick rotation on quantum fluctuations produces thermal flucuations from correlations. The set of voltages $\left\{V_i\right\}$ produced by the ADC creates statistical mechanics, where the voltages have some correlation with each other. A signal means the voltage samples are correlated with each other, but noise means the voltage samples are random fluctuations. We show by Monte Carlo simulation [11] that the RF receiver in the signal phase has broken ${\mathcal{Z}}_2$ symmetry (correlation function $F\left(\tau \right)\to -F\left(\tau \right)$) and in the noise phase, the ${\mathcal{Z}}_2$ symmetry is restored. This is a symmetry that has nothing to do with a spatial symmetry; instead, it is a multiplicative symmetry associated with the correlations themselves. The two competing networks form a liquid–liquid phase separation transition, which ends in a liquid–liquid critical point (LLCP), with a critical exponent which we measure. The LLCP is that $SNR$ where the correlation length of the voltages is the same correlation length of the noise, designated $SN{R}_{critical}$. It also is the $SNR$ where the order parameter changes value. The LLCP define a coexistence phase line separating signals above and noise below. The payoff of the LLCP discovery is a new radar processing algorithm.

2. Materials and Methods

In order to apply the mathematical analysis of random matrix theory, we need to list the various types of noise in a radio-frequency receiver. As mentioned above, a radio-frequency receiver, disconnected from the antenna (meaning no signal is present), and interrogated by an analogue-to-digital (ADC) hardware chip, will produce random positive and negative voltage samples $\left\{V_i\right\}$. If the noise has a correlation among these samples, we call it ’colored noise’. If the random voltages are not correlated, we call it ’white noise’. White noise means that each voltage sample $V_i$ is drawn from some probability distribution, which may be Gaussian or not, but it has a variance ${\sigma }^2$. How do we tell whether the noise is white noise or not?

2.1. White Noise

We form the sample covariant matrix $\mathcal{R}$ from the voltage samples and measure its condition number $\zeta$ (condition number ≡ maximum eigenvalue/minimum eigenvalue). By forming the sample covariance matrix, we use all the information available to determine if a signal is present. Random Matrix Theory (RMT) is the mathematics of the sample covariance matrix $\mathcal{R}$, whether the entries are all random or random + signal. A fundamental theorem of RMT is the famous Marchenko & Pastur eigenvalue distribution [12] which states that if $\mathcal{R}$ only has white noise (with K rows and N columns), then the eigenvalues $\lambda$ of $\mathcal{R}$ are given by the famous Marchenko & Pastur distribution. The largest ${\lambda }_{max}$ and smallest ${\lambda }_{min}$ of the eigenvalues from this distribution are

$$\begin{array}{ccc}\hfill {\lambda }_{max}& =& {\sigma }^2{(1+\sqrt{\alpha })}^2\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\lambda }_{min}& =& {\sigma }^2{(1-\sqrt{\alpha })}^2\hfill \end{array}$$

(2)

where ${\sigma }^2$ is the variance of the white noise random distribution and

$$\alpha =K/N$$

(3)

The condition number $\zeta$ is the ratio of largest to smallest eigenvalue and for white noise

$${\zeta }_{noise}={(1+\sqrt{\alpha })}^2/{(1-\sqrt{\alpha })}^2$$

(4)

If the noise floor of the radio-receiver has a condition number different from Equation (4), then that receiver has colored noise, not white noise. It turns out that different covariant metrics will give the same results in the asymptotic limit

$${\displaystyle \frac{K}{N}}<1$$

(5)

$$K,N\to \infty ,{\displaystyle \frac{K}{N}}\to \alpha$$

(6)

which is the physics limit of infinite volume for critical phenomena.

2.2. Colored Noise

Colored noise was discussed in reference [11], where synthetic colored noise was used. Here, we use actual experimental data using the Keysight Technologies UXR_25GHz oscilloscope, which has an ADC of 256 billion samples per second. In one micro-second ($\mathsf{\mu }$s), 256,000 voltage samples are captured, {$V_i^{raw}$}. In Figure 1, a snapshot of the noise floor is shown, across a 25 GHz passband.

This is the Johnson–Nyquist noise. There is a 170 $\mathsf{\mu }$V = $V_0$ DC offset, so the corrected noise floor voltage samples {$V_i$} are

$$V_i=V_i^{raw}-V_0$$

(7)

The standard deviation $\sigma$ of this Johnson–Nyquist noise is

$$\sigma =0.2929431\times {10}^{-3}\mathrm{Volts}$$

(8)

This experimental data has noise power $P_N$:

$$P_N={\displaystyle \frac{1}{256000\ast 50\mathrm{\Omega }}}{\displaystyle \sum _{i=1}^{256000}}{V}_i^2=0.1716313\times {10}^{-8}\mathrm{Watts}$$

(9)

where the 50 $\mathrm{\Omega }$ in Equation (9) is from the standard 50 $\mathrm{\Omega }$ input/output impedance of the oscilloscope.

2.3. Condition Number

To check whether the Keysight Technologies UXR_25GHz oscilloscope noise floor is colored noise or white noise, we form the sample covariance matrix from these 256,000 voltage samples, by dividing it into 16 rows, with 16,000 columns. From the set {$V_i$}, we form a column matrix Y as shown in Figure 2.

The sample covariance matrix is then

$$\mathcal{R}={\displaystyle \frac{1}{N}}\mathbf{Y}\times {\mathbf{Y}}^{\prime }$$

(10)

where ′ denotes transpose. For our case, this is a 16 × 16 symmetric matrix, which is a normal matrix. The condition number $\zeta$ for a normal matrix is

$$\zeta =\left|\right|\mathcal{R}\left|\right|•\left|\right|{\mathcal{R}}^{-1}\left|\right|$$

(11)

Using Equation (4) for the 16 × 16 sample covariant matrix, we get

$${\zeta }_{\mathrm{white}\mathrm{noise}}=1.134887$$

(12)

Using the experimental Keysight Technologies UXR_25GHz oscilloscope noise floor, we have

$${\zeta }_{\mathrm{oscilloscope}}=1.266554$$

(13)

which proves that the oscilloscope has a colored noise floor.

2.4. Signal Model

The physics has a simple mathematics model, as given in the following Figure 3. The voltages $V\left(i\right)$ have additive (incoherent) noise $n\left(i\right)$ (which may be either white noise or colored noise) and multiplicative (coherent) noise $\mathcal{H}$. Physicists call these terms statistical error and systematic error, while engineers call them incoherent noise and coherent noise associated with a signal $s\left(i\right)$

$$V\left(i\right)=\mathcal{H}s\left(i\right)+n\left(i\right)$$

(14)

For computational physics purposes, we use the Keysight Technologies noise floor and the 16 × 16,000 sample covariance matrix $\mathcal{R}$. Numbering the {$V_i$} with respect to the time interval of creation i$\mathrm{\Delta }t$, where here $\mathrm{\Delta }t$ = sec/(256 $\times {10}^9$) = 3.90625 pico-seconds and i goes from 1 to 256,000, we have the statistical ensemble, which has fluctuations and correlations in it. Added to this noise will be various 1 $\mathsf{\mu }$s signals giving a wide $SNR$ to explore. When the system noise is added to the signal voltage, the infinite signal correlation length is reduced for low signal-to-noise power ratios ($SNR$) and for small enough $SNR$, the differentiation of a signal voltage from a noise voltage becomes impossible, at the critical point, the LLCP $\equiv SN{R}_{\mathrm{critical}}$. In this paper, we quantify these statements and show it is associated with a liquid–liquid phase separation and the phase space is a coexistence curve of signal and noise.

By taking enough data samples, the incoherent noise goes to zero ($SNR\to \infty$, but in practical terms $\ge 10$ is sufficient), but the coherent noise never drops out. However, we will see that the order parameter ($\eta$) involves the condition number ($\zeta$) of a normal matrix, so from Equation (11), systemic error does not play a role as long as $\mathcal{H}$ is a constant.

The statistical ensemble has a mathematical symmetry transformation $F\left(\tau \right)\to -F\left(\tau \right)$ [where $F\left(\tau \right)$ is the correlation function Equation (20)] called a ${\mathcal{Z}}_2$ transformation. If no signal is present, the average correlation $<F\left(\tau \right)>$ is unchanged by the ${\mathcal{Z}}_2$ transformation and the symmetry is intact. If a signal is present, the coherence length of the voltages is bigger than the noise floor coherence length and the average correlation $<F\left(\tau \right)>\ne 0$: the ${\mathcal{Z}}_2$ symmetry is broken. At $SN{R}_{\mathrm{critical}}$, the statistical ensemble {$V_i$} has the noise coherence length and the two networks, signal and noise, become indistinguishable.

2.5. Condition Number for a Signal

If a signal is present, $\mathcal{R}$ has voltages which have some embedded coherence among the statistical ensemble {$V_i$}. If a signal $\left\{s_i\right\}$ is present in the voltage fluctuations, then RMT predicts [11,13] that the condition number for the covariant matrix $\mathcal{R}$ in white noise becomes

$$\begin{array}{ccc}\hfill \rho & =& K\times SNR\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill {\zeta }_s& =& {\displaystyle \frac{(1+\rho )(1+\frac{\alpha }{\rho })}{{(1-\sqrt{\alpha })}^2}}\hfill \end{array}$$

(16)

where the signal-to noise power ratio $SNR$

$$SNR={\sigma }_s^2/{\sigma }^2$$

(17)

where ${\sigma }_s^2$ is the variance of the signal and ${\sigma }^2$ is the variance of the white noise. Examination of Equations (4) and (16) shows a ‘jump’ (discontinuity) in $\zeta$ for a signal in white noise, since $SNR\to 0$ is not continuous. The RMT solution Equation (16) is only true for white noise; if colored noise is present, no RMT analytical solution is possible and the condition number must be calculated numerically.

2.6. Order Parameter for Radio Receivers

Let us pretend that the RF receiver is a network of K-number of cell phone towers, each cell tower getting a representative set of N number of voltage samples from that cell phone tower’s passband [8]. From the set of voltage samples {$V_i$}, we form the RF sample covariance matrix $\mathcal{R}$ using the pretended cell phone tower network. We thus divide up the 256,000 voltage samples we measured above for the UXR_25GHz oscilloscope (for 1 $\mathsf{\mu }$s duration) into K vectors (say $K=16$), each having $N=256,000/16=16,000$ voltages. The question is whether the pretended cell phone tower voltage samples are coherent with each other (and therefore a signal is present from a cell phone customer). If this were a real cell phone tower network and a real mobile phone was communicating with each tower with a variable strength and time delay, the network of cell phone tower voltage samples would be different from each other. However, the central question is whether this network is servicing a caller by having a signal present, even though the individual cell phone tower voltages are different from each other. The signal presence (minimum coherence of the voltages) or signal absence (random coherence of the voltages) is determined by the sample covariance matrix properties of the cell-phone network (K towers having N samples each). If the voltages were noise (no caller signal present), $\mathcal{R}$ then just has random number entries for voltages.

If an RF receiver has a signal during a $\mathrm{\Delta }t$ interrogation of an ADC, then that creates coherency among the voltages of its bandpass $\left\{V_i\right\}$, and the condition number of the associated sample covariance matrix $\mathcal{R}$ will be different than its noise condition number. In our computational results, we actually compute the condition number numerically and do not rely on asymptotic RMT.

For a given $\mathrm{\Delta }t$ (a given ADC), the external control parameter is the $SNR$ (equivalent to magnetic field of traditional statistical mechanics of a metal–insulator transition) and an LLCP $\equiv SN{R}_{critical}$. $\mathrm{\Theta }$ is the reduced control parameter (thinking of $\mathrm{\Theta }$ as the reduced magnetic field, for example); the corresponding $\mathrm{\Theta }$ here is

$$\mathrm{\Theta }=SNR-SN{R}_{critical}$$

(18)

So the disordered phase (noise with intact ${\mathcal{Z}}_2$ symmetry) has $\mathrm{\Theta }<0$ and the ordered phase (signal present with broken ${\mathcal{Z}}_2$ symmetry) has $\mathrm{\Theta }>0$.

Thus there exists a universal signal order parameter $\eta$ which is positive for a signal and can be made zero whenever a signal cannot be measured:

$$\eta =max(0,\zeta -{\zeta }_{\mathrm{noise}})$$

(19)

where $\zeta$ is the measured condition number of the sample covariance matrix and ${\zeta }_{\mathrm{noise}}$ is the condition number of the noise floor. To determine $\eta$, we first must compute the noise sample covariance matrix (no signal present) ${\mathcal{R}}_{\mathrm{noise}}$ and determine its ${\zeta }_{\mathrm{noise}}$, before any signal measurement is made. This is the calibration of the RF detection apparatus.

3. Computational Results

We insert a 1 $\mathsf{\mu }$s signal using Equation (14) ($\mathcal{H}$ set to 1) and determine the condition number as a function of the inputted $SNR$, using the Keysight Technologies UXR_25GHz oscilloscope noise floor. The results are in

Table 1. Condition number as a function of $SNR$. Each unique ADC (here in this table, 256 billion samples per second) has an order parameter (Equation (19)) that ends in a liquid–liquid critical point, here close to $SNR=-20$ dB, because an $SNR$ = −20 dB gives a condition number close to noise (Equation (13)).

SNR in dB	Signal+Noise Condition Number (16 × 16)
1.494850 × 101	5.439931 × 102
8.927898 × 100	1.358643 × 102
2.907298 × 100	3.434518 × 101
−3.113302 × 100	9.217845 × 100
−9.133902 × 100	3.057018 × 100
−1.515450 × 101	1.575736 × 100
−2.003700 × 101	1.278857 × 100

.

From Table 1, we see that beyond around $SNR$ = −20 dB (see below critical exponent), a signal is indistinguishable from noise because $\eta$ is approximately zero, and thus an ADC of 256 billion samples per second has a liquid–liquid critical point (LLCP) around $SNR$ = −20 dB (see below for a more exact $SNR$). Different ADC values give different end point liquid–liquid critical points and the joining of these LLCPs is the signal and noise phase diagram or more precisely the coexistence boundary.

To fully grasp intuitively why signals have this liquid–liquid phase separation, we need the correlation of the voltages, since it is these voltage correlations that determine a signal from noise. We use the time correlation function $F(\tau =k\mathrm{\Delta }t)$

$$F(\tau =k\mathrm{\Delta }t)={\displaystyle \frac{{\sum }_{i=1}^{N-k}(V_i-\overline{V})(V_{i+k}-\overline{V})}{{\sum }_{i=1}^N{(V_i-\overline{V})}^2}}$$

(20)

where

$$\overline{V}={\displaystyle \sum _{i=1}^N}{\displaystyle \frac{V_i}{N}}$$

(21)

There are 256,000 voltage samples, so we plot $\tau$ in units of 500 time-samples. In Figure 4, we have the noise floor correlation function.

Perusal of Figure 4 shows no important correlation in the noise floor as expected. The noise ‘fluid’ has no significant correlation and the symmetry transformation $F\left(\tau \right)\to -F\left(\tau \right)$ has no effect on the correlation average $<F\left(\tau \right)>$; the ${\mathcal{Z}}_2$ symmetry is present. We now plot the correlation function for the $SNR$ = 14.948 signal in Figure 5.

The signal ‘fluid’ has a dramatic difference in its correlation function, compared to the noise ‘fluid’. As the signal $SNR$ becomes smaller, the noise voltage samples become more important in the correlation function. For $SNR=-0.1515450\times {10}^2$ (Figure 6), the correlation among the voltage samples is falling apart.

At the liquid–liquid critical point, the signal power is so small that the correlation function is almost the same as the noise function (Figure 7), and the two ‘fluids’ are indistinguishable. The liquid–liquid critical point (LLCP) is a unique value of the ADC used.

3.1. Signal Phase Separation

In the radio receiver statistical mechanics, there are no temperature–pressure external variables that cause a voltage ensemble $\left\{V_i\right\}$ to be differentiated into a signal or noise. Instead, the external variables are the $SNR$ and the time interval $\mathrm{\Delta }t$ (that comes from the ADC discretizing the RF radio receiver passband). For a given ADC, the $SNR$ is decreased to a small level ($SN{R}_{\mathrm{critical}}$) where the statistical ensemble $\left\{V_i\right\}$ has the correlations of the receiver noise floor. The two ‘liquids’, noise and signal, then become totally undifferentiable. This physics is due to the presence of two competing networks, causality and random, whose phases are determined by a ${\mathcal{Z}}_2$ intrinsic symmetry. The order parameter ends in a liquid–liquid critical point (LLCP). The difference between the two networks, noise (random voltage) and signal (causal voltage), is in the coherence property: signal voltage samples are totally coherent with each other, while the ubiquitous noise is not. It is this grouping that makes voltage samples behave like a liquid. When the $SNR\to SN{R}_{\mathrm{critical}}\equiv$ LLCP, the signal coherence length becomes the noise coherence length, the order parameter vanishes and signal and noise merge. At that critical $SNR$, if we make $\mathrm{\Delta }t$ smaller (a new $\mathrm{\Delta }t$), we are again above a new order parameter and can say a signal is present. Going down to a smaller $SNR$ for the same new $\mathrm{\Delta }t$, we again hit a new liquid–liquid critical point where we no longer have a positive value order parameter. In this manner, we delineate the phase transition curve of an RF radio receiver by mapping the individual LLCP for each ADC. As for the ${\mathcal{Z}}_2$ symmetry, the transformation $F\left(\tau \right)\to -F\left(\tau \right)$ does not effect the coherence length in the random phase, but destroys the coherence length in the signal phase (broken). In summary, for a given ADC we have a line of positive order parameters that end in an LLCP. The phase diagram (noise on one side, signal on the other) for an RF radio receiver is the line of different ADCs, each of which has a unique order parameter LLCP: the phase transition line (coexistence line) is this line of LLCPs.

3.2. Phase Coexistence Line

We map the phase line by taking different ADC values. If we use a smaller ADC giving 16 billion samples per second, then in 1 $\mu$-second there will be 16,000 samples. Using a 16 × 16 sample covariant matrix gave an LLCP at a smaller $SNR$ (

Table 2. LLCP as a function of the signal external variables. The phase line is the line of LLCPs and this table gives three points on the line.

Noise Condition Number	SNRdB	$\mathrm{\Delta }\mathit{t}$ in Pico-Seconds
0.1036167 × 101	−0.3023894 × 102	0.390625
0.1266554 × 101	−2.003700 × 101	3.90625
0.2824512 × 101	−0.1484210 × 102	62.5

). If we use a very fast ADC of 2560 billion samples per second, the $SNR$ goes to the very low value of −30 dB (Table 2). The actual phase line separating noise from signal is the line of LLCPs determined at each ($SNR$, $\mathrm{\Delta }t$) pair. Each point on the phase line has a unique ${\zeta }_{noise}$. In other words, a given ADC has a unique LLCP and the phase line is the function of ADC. Figure 8 gives the RF receiver phase diagram.

3.3. Critical Exponent

As the signal gets weaker and approaches the LLCP on the signal side, there exists the $\beta$ critical exponent which gives the power law behavior of the order parameter $\eta$ near the critical point $\mathrm{\Theta }\to 0$:

$$\eta =c{\left(\mathrm{\Theta }\right)}^{\beta }$$

(22)

where c is a possible constant. In physics, the exponent $\beta$ is the slope of the log–log plot of the signal order parameter $\eta$ (Equation (19)) versus the reduced control parameter $\mathrm{\Theta }$ (Equation (18)).

In order to calculate $\beta$ we need an accurate $SN{R}_{\mathrm{critical}}$ from which we can use Equation (22). This is given in

Table 3. Determination of $SN{R}_{\mathrm{critical}}$ for the $\mathrm{\Delta }t=3.90625$ pico-s LLCP.

Quantity	Value
signal power	1.51328 × 10−11 Watt
$SNR$ in dB	−20.54677
noise + signal $\zeta$	1.266722

.

Using the values in

Table 4. Determination of the critical exponent $\beta$ in Equation (22) using three $SNR$ values near $SN{R}_{\mathrm{critical}}$.

Quantity	Value
$SNR$ in dB	condition number
−20.52837	1.267117
−20.53756	1.266919
−20.51918	1.267315

, in a log–log plot with a linear regression analysis,

$$\beta =0.67\pm 0.03$$

(23)

3.4. Signal Statistical Mechanics Criticality

There are four possibilities governing the critical phenomena of signals:

• It is a liquid–liquid phase transition.
  The liquid–liquid phase transition is ruled out because the $\beta$ value derived here does not correspond to the Ising model $\beta$ exponent [14]. Furthermore, the correlation length does not go singular at the critical point.
• It is a percolation phase transition.
  Once the critical density is reached, the percolation extends across the community [15], which is not the case here, since the correlation length does not become infinite. Percolation is not correct.
• It is a liquid–glass transition.
  It is not a liquid–glass transition since this type of phase change is not abrupt, but extends over a range of external drivers (conventionally temperature, which here would be SNR) [16].
• It is a liquid–liquid phase separation.
  This is the critical phenomenon. The phase separation critical exponent $\beta$ is not universal: reference [17] in the experiment ‘Binary-liquid phase separation of lens protein solutions’ finds $\beta =0.325$, while reference [18] in the experiment ‘Binary liquid phase separation and critical phenomena in a protein/water solution’ measures $\beta =0.5$. In reference [19] for different kinds of hemoglobins, $\beta$ ranged from 0.708 to 0.4253. Also, the existence of the two phases coexisting together for a given external value of the driving parameter is not a singular correlation length.

In

Table 5. Table of RF statistical mechanics.

External Variables	Control parameters {$SNR,\mathrm{\Delta }t$}
Order parameter value	$\eta$ = 0, noise; $\eta >0$, signal
Critical Point $SN{R}_{\mathrm{critical}}$	signal coherence length = noise coherence
Intrinsic symmetry	broken signal phase, restored random phase
Critical phenomena	liquid–liquid phase separation

, we list the RF statistical mechanics, with ${\mathcal{Z}}_2$: ($F\left(\tau \right)\to -F\left(\tau \right)$).

4. Radar Processing

The physics result that RF signals being detected by an RF receiver (sensor) is a liquid–liquid phase separation transition has an immediate effect in radar. United States Patent 10,175,341 [20] shows that an enormous improvement in radar detection is achievable using the radar’s ADC with detection near the LLCP of the apparatus. The patent takes a Russian P37 Bar Lock search radar [21] illuminating a 10 m² third-generation fighter plane (centimetric radiation). The nominal detection range for $SNR=10$ dB is 220 km. With a special ADC of 160 giga-samples per second, the patent shows that detection near the LLCP is at 1200 km. Radar processing now depends on the available hardware chips.

5. Conclusions

A single voltage measurement carries little information; a collection of voltages can reveal the presence of a signal, which is a collective coherence created by causality. The RF receiver is an example of statistical mechanics having a phase transition, due to fluctuations and coherence properties. There is an order parameter that delineates which phase the RF receiver is in. The ‘liquids’ are the voltage samples’ coherent properties or lack thereof. For a given ADC, there exists an LLCP (whose order parameter vanishes) where noise and signal are indistinguishable. The line of ADC LLCP gives the coexistence phase line between noise and signals in an RF receiver. There exist numerous other examples in Nature where the fluctuations and coherence properties constitute statistical mechanics having deep physics implications. For the radio receiver, causality and randomness are the underlying statistical mechanical network drivers.