Instantaneous Frequency Estimation of Radio Frequency Signal Based on Rydberg Atomic Receiver

Rydberg atom receiver attracts much research interest for promising applications in communications and sensing. Generally, Rydberg atomic receiver is utilized with microwave frequency comb (MFC) to expand its detectable frequency range. MFC consists of a set of equally spaced discrete frequency lines, resembling a comb in frequency space. In such receives, the mixing of RF signal and its closest MFC component excites the atoms into the desired Rydberg state. Since the mixing rather than RF signal is detected, there exists inevitable ambiguity of frequency estimation. In this paper, we provide a novel frequency ambiguity resolution based on improved Chinese remainder theorem (I-CRT). It realizes the instantaneous frequency estimation of RF signal with the MFC-based Rydberg atomic receiver. The effectiveness of proposed resolution in this paper is verified by both simulation experiments and theoretical analysis.


Instantaneous Frequency Estimation of Radio Frequency Signal Based on Rydberg Atomic Receiver
Guanyu Chen , Cheng Wang , Bin Yang , and Tiantian Chen Abstract-Rydberg atom receiver attracts much research interest for promising applications in communications and sensing.Generally, Rydberg atomic receiver is utilized with microwave frequency comb (MFC) to expand its detectable frequency range.MFC consists of a set of equally spaced discrete frequency lines, resembling a comb in frequency space.In such receives, the mixing of RF signal and its closest MFC component excites the atoms into the desired Rydberg state.Since the mixing rather than RF signal is detected, there exists inevitable ambiguity of frequency estimation.In this paper, we provide a novel frequency ambiguity resolution based on improved Chinese remainder theorem (I-CRT).

It realizes the instantaneous frequency estimation of RF signal with the MFC-based Rydberg atomic receiver. The effectiveness of proposed resolution in this paper is verified by both simulation experiments and theoretical analysis.
Index Terms-Rydberg atomic receiver, microwave frequency comb, Chinese remainder theorem, instantaneous frequency estimation.

I. INTRODUCTION
T HE RF signal detection has a wide range of applica- tions in communication, navigation, radar, electromagnetic spectrum monitoring, and aerospace fields.Limited by Johnson-Nyquist noise and the influence of antenna size on radiation efficiency [1], [2], [3], it is difficult to achieve wideband signal detection for electronic system.Rydberg atoms, with one highly excited, nearly ionized electron, are highly sensitive to applied electromagnetic field [4], [5], [6].In recent years, significant progress has been made in electromagnetic perception based on Rydberg atomic receiver with a record sensitivity down to 55 nV • cm −1 • Hz −1/2 [7].The Rydberg atomic receiver can also detect frequency, phase, polarization, and direction of arrival of RF signals [8], [9], [10], [11].Therefore, it has the potential to become the next generation of radio receiver.
Limited by the relaxation time of electromagnetically induced transparency (EIT) phenomenon, the Rydberg atomic receiver has about 10 MHz instantaneous bandwidth [12] which is related to the natural lifetime of Rydberg atomic energy level.Compared with traditional receivers, limited instantaneous bandwidth is the main disadvantage of Rydberg atomic receiver.Based on superheterodyne technique, Zhang [13] used the microwave frequency comb (MFC) instead of the single-frequency local oscillator field to break through the limitation of EIT relaxation time and realize real-time frequency measurement with a range of 125 MHz.The MFC significantly expands the instantaneous bandwidth of Rydberg atomic receiver.
MFC has multiple equally spaced discrete frequency lines, resembling a comb in frequency space.Using MFC, the mixing of RF signal with its closest MFC component would be detected by Rydberg atoms.Hence, there exists inevitable ambiguity of instantaneous frequency estimation for MFC-based Rydberg atomic receivers.To solve the problem, additional MFC is introduced in [13] to provide frequency measurement without ambiguity.However, this method requires RF signal lies in certain frequency range and has limited estimation accuracy.Moreover, the restrictions of MFC lines and detectable RF signal have not been well explained yet.
In this paper, the instantaneous frequency estimation is studied based on MFC-based Rydberg atomic receiver.Specifically, a frequency ambiguity resolution with improved Chinese remainder theorem (I-CRT) is proposed.CRT is to reconstruct a single integer by its remainders modulo several moduli.It has been widely studied and applied in frequency estimation, phase unwrapping and error code correcting codes [14], [15], [16], [17], [18], [19], [20], [21].According to MFC structure, first it requires to determine which side from a particular MFC line the mixing signal is located at.Hence, we improve traditional CRT and propose improved Chinese remainder theorem (I-CRT).The I-CRT obtains unambiguous instantaneous frequency estimation with limited computation burden.Both simulation experiments and theoretical analysis confirm its effectiveness.The upper bound of frequency that the RF frequency can be uniquely obtained from MFC-based receivers are provided in addition.
The remaining content is organized as follows.In Section II, we first introduce the measurement principle of the MFCbased Rydberg atomic receiver and give the formulation of frequency estimation.In Section III, we propose I-CRT and derive its upper bound of frequency estimation.In Section IV, we present simulation results to verify the performance of the proposed algorithm.Finally, in Section V, we conclude the study.

A. The Measurement Principle of Rydberg Atomic Receiver
In order to excite the Rydberg atoms, the commonly used alkali metal atoms are cesium (Cs) atoms and rubidium (Rb) atoms.This paper takes Cs atom as an example to introduce the measurement principle of Rydberg atomic receiver.The energy level diagram of Cs atom and the schematic diagram of the experimental equipment are depicted in Fig. 1(a) and (b), respectively.In Cs atom vapor cell, probe (λ p = 852 nm) and coupling lasers (λ c = 510 nm) are counterpropagating [22], [23], [24], [25].The 852 nm probe laser is frequency-stabilized to |6S 1/2 → |6P 3/2 transition.The Cs atoms transit from the ground state to an excited state.The 510 nm coupling laser is resonant with the transition |6P 3/2 → |47S 1/2 .The cesium atoms are excited to the Rydberg state, which forms a Rydberg EIT system.A strong MFC field E MFC is set as a local oscillator field, and a RF signal field E S couples the two Rydberg states |3 → |4 .The probe light is detected by photodetector, and analyzed via an oscilloscope and a spectrum analyzer.E MFC and E S are combined through a resistance power divider, and transmitted to the vapor cell by a horn antenna.
In [9], the single-frequency local oscillator field denote the amplitude, frequency and phase of local field (signal field) respectively.Rydberg atoms have extreme sensitivity to RF fields due to their large dipole moments; RF signal field is small compared to the LO field E S <<E L .Total electric field E atom in the vapor cell can be expressed as (1). where As limited by the evolution time to reach the steady state, the instantaneous bandwidth of the Rydberg atomic receiver is less than 10 MHz [12], while the MFC method could expand instantaneous working bandwidth.As shown in Fig. 1(c), MFC field consists of multiple phase-stabilized frequency lines with equidistant frequency intervals.The frequency lines are different microwave fields with approximately equal power and different frequencies [26], [27].
In this paper, the single local oscillator field E L in ( 1) is replaced by an MFC field E MFC .In Fig. 1(c), f 0 is the MFC offset frequency and the f M is MFC frequency interval.E L i denotes the electric field strength of the i th MFC comb line, Then, the electric field strength of the MFC is denoted as E MFC = i E L cos(2πf i t + φ i ).Thus, the electric field strengths of the Rydberg atoms when they receive E MFC and E S is as follows.
where N C denotes the total number of MFC comb lines.The subscript j represents the MFC comb line number that generates the mixing frequency response with the RF signal field, and the frequency of this comb line is the closet to RF signal frequency.
The transmission coefficient T probe of the probe light passing through the atomic vapor cell is a function of |E atom |, T probe ∝ |E atom | 2 [9].From the detected probe optical spectrum varies with the mixing frequency Δf j , then the mixing frequency f R = Δf j can be measured.The f R is an absolute value, and the positive-negative of f R can't be directly judged.
The relationship between the frequency of RF signal f S and the MFC offset frequency f 0 and the MFC frequency interval f M is shown below where n is the mode-order number of RF signal field's closet MFC comb line, b takes the value of ±1.Since, n and b are unknown integers, f S can't be determined by only one MFC.We need more than two MFCs to determine the RF signal frequency.In Fig. 2, three MFCs are used to measure f S .

B. Constructing System of Simultaneous Congruences for Frequency Estimation
It can be seen from the above that the frequency of RF signal field f S is measured by using K MFCs (K ≥ 2) with the same number of comb line, the same offset frequency and different frequency intervals.In the MFC measurement range, combined with Fig. 2 and (3), the mixing frequency f R1 , f R2 , . . ., f RK measured by K MFCs is related to RF signal frequency by the following system of simultaneous congruences where

III. FREQUENCY ESTIMATION METHOD BASED ON I-CRT ALGORITHM
CRT algorithm can be used to solve system of simultaneous congruences.Assume that the integers m 1 , m 2 . . ., m K are mutually prime, then for any integer a 1 , a 2 . . ., a K , the following system of equations exists as an integer solution, where Here is an example of traditional CRT, find an integer X that satisfies the conditions of dividing by 3 with a remainder of 2, dividing by 5 with a remainder of 3, and dividing by 7 with a remainder of 2, namely from ( 6), X = (70 × 2 + 21 × 3 + 15 × 2) mod 105 = 23.The traditional CRT is not robust, because X cannot be accurately reconstructed even when the remainder errors are small.And, m 1 , m 2 . . ., m K should be mutually prime.traditional CRT.The application of traditional CRT algorithm is limited.
With the development of CRT algorithm, CRT algorithm has now been applied in many fields.
In [14], [15], [16], [17], the problem of subsampling signal processing can also be expressed by system of simultaneous congruences, and solved by CRT algorithm.CRT algorithm is widely used in subsampling signal processing, phase unwrapping, pulsed doppler radar and other fields [18], [19], [20], [21].Only when ambiguity of f Rk be resolved in advance, can we use CRT algorithm to solve (4).
In order to solve (4), this paper proposes a frequency ambiguity resolution based on I-CRT to disambiguate f Rk and estimate RF signal frequency.Equation ( 4) can be transformed into the following

A. Theorem and Lemmas
Let M denote the greatest common divisor of all f Mk such that Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
All Γ k are mutually prime and the greatest common divisor of any two Γ k is 1. where the set of all the first element n1 of the pairs Then, several theorems and lemmas related to algorithm solving process are introduced.
Theorem 1: and the error τ of f Rk satisfies τ < M 4 .Then, there exists a unique element n 1 in the set S, S = {n 1 }, and Lemma 1: Assume that all the conditions in Theorem 1 hold, and let Lemma 2: Under the conditions of Theorem 1, let Then, for any element (n 1 , nk ) ∈ S k , there exists an integer m k , for example This lemma indicates that if we search (n 1 , nk ) within the set Ω k , at least one element belonging to set Ω k can be found.Lemma 2 is proved in Appendix C.

B. Solution Process of I-CRT Algorithm
Based on the above theorem and lemma, ( 4) is solved as follows.First find an element Based on Lemma 2 we can find an element belonging to S k in set Ω k , so we only need to search over Ω k .
Search for all integers n1 from 0 to Γ k − 1.According to Lemma 3, when n1 is determined, its corresponding nk in S i is determined by the following equation, and the value of nk is taken as an integer within the interval range of ( 17), and denoted by nk = nk (n 1 ).In  I.
n1 takes values from 0 to Γ k − 1, nk take values from nka (n 1 ) to nkd (n 1 ), and from the following equation, search for the pair (n 1 , nk (n 1 )) that minimizes (18), and the corresponding function value is recorded as T 1 .
Next search for all integers nk from 0 to Γ 1 − 1.According to Lemma 3, when nk is determined, its corresponding n1 in S i is determined by the following equation and denoted by n1 = n1 (n k ).
Find the minimum value of two minimums T 1 , T 2 .The (n 1 , nk ) corresponding to the minimum value is the (n 1,k , nk ) ∈ S k being searched for, and its corresponding b 1 , b k represents Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
positivity or negativity of f R1 , f Rk .The total number of searches is 4(Γ 1 + Γ i ).
Find one element (n 1,k , nk ) ∈ S k for each k with 2 ≤ k ≤ K. Next, determine the mode-order numbers n k (2 ≤ k ≤ K).By Lemma 1, n1,k and n 1 have the same remainder, remainder is denoted by ξ 1,k .
Thus, for each k (2 ≤ k ≤ K), from n1,k we get the remainder ξ 1,k of n 1 modulo Γ k .There is a total of K − 1 remainders of n 1 modulo Γ k .Thus n 1 can be determined by these remainders where d k is determined by the following equation When n 1 has been determined in the above way, other modeorder numbers n k can then be obtained.For each k with 2 ≤ k ≤ K, by Lemma 1, we have Furthermore, After all mode-order numbers n k (2 ≤ k ≤ K) are determined, the instantaneous frequency of RF signal field can be determined by the following equation.
fS is the solution of (4).
The pseudo-code of the solving algorithm process in this paper is as follows.And, the block-diagram is shown in Fig. 3.
CRT algorithm is to reconstruct a single integer by its remainders modulo several moduli.The single integer cannot be accurately reconstructed when the remainder errors are larger than remainder redundancy.In this paper, the mixing frequency f R measured by MFC is an absolute value, and the positive-negative of f R can't be directly judged.The uncertainty of the frequency remainder f R exceeds the remainder redundancy (M/4) of the CRT algorithm and reduces the accuracy of frequency estimation.The I-CRT algorithm proposed in this paper can determine the positive and negative of f R , that is, the value of b k in (4), and then realize the accurate estimation of signal frequency.

C. Upper Bound on Frequency Estimation for I-CRT Algorithm
Next, we will analyze the upper bound on frequency estimation for I-CRT algorithm.RF signal field is measured by K MFCs with frequency interval f M1 , . . ., f MK .The relationship between the frequency estimation upper f max of the I-CRT algorithm and f M1 , f M2 , . . ., f MK is as follows In the above equation, divides f M1 , . . ., f MK into two frequency sets Set1 and Set2, {f M1 , . . ., Only when the frequency of RF signal is lower than f max , can the Rydberg atomic receiver accurately estimate the frequency of the applied RF signal field.The proof of ( 25) is as follows.
Assume that there exist two RF signal fields with frequency f S and f S (f 0 < f s < f s < f max ).{f R1 , f R2 , . . ., f RK } and {f R1 , f R2 , . . ., f RK } are mixing frequencies of the two signal fields measured by same MFCs, respectively.The distance between mixing frequencies of f S and f S is defined as end if 15: b k corresponding to the minimum of T 1 and T 2 determines the positive-negative of f RK 16: end for 17: according to (n 1,k , nk ), ξ 1,k is calculated by (20) 18: calculate RF signal frequency f S is uniquely determined from {f R1 , f R2 , . . ., f RK }, which needs to satisfy any In order to satisfy this condition, the distance between mixing frequencies of f S and f S is greater than zero, min(D(f S , f S )) = D min > 0. The mixing frequency has a problem of frequency ambiguity, b k take the value of ±1.Therefore, there are four combinations between f Rk and f Rk .These combinations can be divided into two groups.The first group, f Rk and f Rk have different positive-negative sign.The second group, f Rk and f Rk have same positive-negative sign.
When f Rk and f Rk have same positive-negative sign From ( 27) and (28) we can divide the mixing frequency difference f Rk − f Rk , k ∈ [1, K] into two groups.The first group contains I frequencies whose frequency difference is equal to f + f +Λ k f Mk .The second group contains J frequencies whose frequency difference is equal to Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
In order to maximize the minimum distance D(f, f ), the frequency interval f Mk should be selected in a way that D(f, f ) should be maximized as much as possible.In order to find a closed-form solution of (29), two variables The values of f and f , f , that minimize the distance D(f, f ) can thus be obtained as follows By substitution (31) in ( 29) and after some manipulation (32) is obtained.After maximizing the distance between Λ i f Set1 Mi and Δ j f Set2 Mj , the minimum distance D min can be obtained by searching for different I and J values in (32).
We should find the minimum value of f , where min(D(f, f )) = D min = 0, which implies that f is incorrectly determined as f ; for this purpose, first, we find Λ i f Set1 Mi and Δ j f Set2 Mj that satisfies the condition D min = 0, and then, by substituting the values obtained in (31), we find the minimum value of f .
To determine the minimum values of Λ i f Set1 Mi and Δ j f Set2 Mj that satisfy D min = 0, we introduce the following definition: Definition 1: The Least Common Multiple (LCM) of a 1 , a 2 , . . ., a K is defined as LCM (a 1 , a 2 , . . ., a K ) then To complete the proof of ( 25), it is necessary to prove this point D min = 0.According to (32), when all terms in (32) are zero, it follows that Using the obtained minimum values of Λ i f Set1 Mi and Δ j f Set2 Mj , we should find the minimum value of f .Based on different values of I and J, we consider three different cases: When I = 0 and J = K. since we assume f < f , substituting (33) into (31), the resulting f (f min ) may be incorrectly determined as f (f min ).Therefore, the minimum frequency f min and its ambiguous frequency f min can be obtained by the following equation.
When I = 0 and J = K.According to (33) and ( 30), we have When I = K and J = 0.Then, according to (33) and (30), we have M1 , . . ., f Set1 MI ) − f .Sincef ≤ f , then the least possible value satisfying f = LCM(f Set1 M1 , . . ., f Set1 MI ) −f is realizable when f = f min .In this case: In summary, from (34)-(36), the frequency f min in (34) is the minimum of the frequency f min derived from (34)-(36).Therefore, in order to estimate RF signal frequency f S from f R1 , f R2 , . . ., f RK , f S should be less than the upper frequency limit f max , f max = f min + f 0 , as shown in (25).
The proposed algorithm greatly expands the instantaneous bandwidth of the Rydberg atomic receiver.For example, using 3 MFCs, set the frequency interval of MFCs:f M1 = 2.78 MHz, f M2 = 3.24 MHz, f M3 = 4.12 MHz.And, f max = 168.25 MHz calculated by (25), which is much larger than the instantaneous bandwidth of 10 MHz, breaking the limit of atomic relaxation time.From (25), it can be seen that the upper limit of RF signal frequency estimation can be improved by setting the frequency interval of MFC reasonably.

IV. RESULTS AND ANALYSIS
In this section, the I-CRT algorithm is verified by numerical simulation experiments.The experiment is divided into three parts.The first part is the performance analysis of the I-CRT algorithm.The second part is to compare the I-CRT algorithm with other algorithms.The third part is the comparison of upper bound on frequency estimation for I-CRT algorithm and other algorithms.
In order to verify that the I-CRT algorithm proposed in this paper can accurately estimate the instantaneous frequency of the received RF signal from atomic receivers, the following root mean square error (RMSE) and the probability of detection P d are defined to illustrate the estimation accuracy.

RMSE(f
where f S denotes RF signal frequency, fSq is the estimated RF signal frequency of qth Monte Carlo simulation, and Q is the total number of Monte Carlo simulations.The P d probability of correct RF signal frequency estimation is as follows.The probability,P d , that RF signal frequency is estimated correctly is defined as follows In the experiment, if the absolute value of the difference between the estimated value fS and the true value f S is less than T , the estimation is correct, and T is the frequency error threshold.

A. Performance of I-CRT Algorithm
We set up three groups of experiments, each group uses three MFCs, frequency group 1: f M1 = 50 kHz, f M2 = 70 kHz, f M3 = 90 kHz; frequency group 2: In the experiment, the measurement noises of mixing frequency f Rk are independent and identically distributed and follow a normal distribution N (0, σ 2 ).The measurement noise ΔR = −10 lg(σ 2 ) [28] and the variation range of ΔR is set to −10∼5 dB.The Monte Carlo simulation is repeated for 1000 times at each value of ΔR.When the difference between the fS and f S is in the range of 0.1 kHz, the frequency estimation is considered to be correct.In the low frequency (LF) band (30 kHz∼300 MHz) and above, the frequency estimation error of 0.1 kHz is acceptable for frequency measurement in applications such as astronomy, radar detection, wireless communication, and navigation.
In Fig. 4, as the measurement noise decreases, the experiments with different frequency groups both can obtain P d = 100%.When the range of ΔR is −7 dB to −1 dB, it can be seen that at the same noise level, the frequency interval used in frequency group 3 is the largest and the estimation result is the best, while the frequency interval used in frequency group 1 is the smallest and the estimation result is the worst.The greatest common divisor of f M1 , f M2 , and f M3 is M.For CRT algorithm, M/4 is the upper limit of residual noise [28], ΔR must be less than M/4 in order to estimate f S correctly.Therefore, the greater the M, the better the noise-robust performance of the algorithm.
Set the frequency interval of MFCs:f M1 = 50 kHz, f M2 = 70 kHz, f M3 = 90 kHz; the frequency interval in the simula-  Under the condition of three measurement noises (ΔR = −10 lg(σ 2 )=− 5.2 dB, −4.7 dB, −3.2 dB), the Monte Carlo simulation is repeated 1000 times for each MN value.The experimental RF signal frequencies under different noise conditions for the same MN value are taken to be the same, and the RMSE of the estimated frequency calculated by (37) is shown in Fig. 5.
In Fig. 5, the lower the measurement noise, the better the convergence results obtained for the same MN.As MN increases, the maximum common divisor M of frequency group also increases, which increases the noise robustness of frequency estimation and reduces the influence of noise on estimation accuracy.Therefore, RMSE decreases with the increase of MN.

B. Multipart Figures Algorithm Performance Comparison
The I-CRT algorithm proposed in this paper is compared with the Closed Chinese remainder theorem (C-CRT), Frequency Band Division (FBD) algorithm [29] and the algorithm in Reference [13].The MFC frequency interval of I-CRT algorithm, C-CRT algorithm and FBD algorithm is set to f M1 = 3 MHz, f M2 = 4 MHz, f M3 = 5 MHz.The MFC frequency interval of the algorithm in Reference [13] is f M1 = 3MHz, f M2 = 2.9MHz.The measurement noise ΔR ranges from-10 dB to 15 dB, and Monte Carlo simulation is repeated 1000 times at Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.each ΔR.When the difference between the fS and f S is in the range of 0.1 kHz, the frequency estimation is considered to be correct.The P d , RMSE, and algorithm time consumption of four algorithms are as follows.
In Figs. 6 and 7, with the decrease of measurement noise, the I-CRT algorithm in this paper and the FBD algorithm can achieve P d = 100% and RMSE converge to the lowest value.The proposed method achieves P d = 100% and RMSE curve convergence, and the corresponding noise is about 11 dB higher than that of the FBD algorithm, indicating that I-CRT algorithm has better noise robust performance.And in the experimental environment of this paper, the FBD algorithm has the largest time consumption.
As for FBD algorithm, the relationship between error bound τ and noise variance σ 2 is σ 2 = τ 2 /3.In this paper, the abscissa of Figs. 6 and 7 is ΔR = −10 lg(σ 2 ), so the error bound of the FBD algorithm is τ = √ 3 • 10 − X 20 .For I-CRT algorithm, the maximum error bound is τ = M/4, so the noise robust performance is better than that of the FBD algorithm.
The C-CRT algorithm is also often used in the problem of deriving the original integer from the remainders, this algorithm also requires that the error of the remainder is not greater than M/4, but the problem solved in this paper needs to determine the positive-negative of the mixing frequency first.Therefore, the ambiguity of the mixing frequency causes the P d of the C-CRT algorithm to reach only about 35%.
The method in [13] uses two MFCs with a small difference in frequency intervals.The two MFCs have the same offset frequency.The mode-order number N in (3) is inferred by varying the size of the staggered comb lines.This method is simple, but it lacks noise robustness.It also needs to meet the constraint condition N C (f M1 − f M2 ) = f M1 /2 to estimate RF signal frequency.
From Fig. 8, it can be seen that the FBD algorithm has the largest time consumption, the I-CRT algorithm and the closed C-CRT algorithm have similar time consumption, and the algorithm in [13] has the smallest time consumption.The complexity of the I-CRT algorithm is o(4((K − 1) , where Γ k is obtained from (9) and K represents the number of MFCs.The complexity of the C-CRT algorithm is o(4K).The complexity of the FBD algorithm is o( , where N b is the total number of frequency bands.The complexity of method in [13] is o (1).In this experiment, I-CRT algorithm, closed C-CRT algorithm, and FBD algorithm have K = 3. N b reached 10 3 , making the FBD algorithm has the largest time consumption.

C. Comparison of Upper Bound on Frequency Estimation for Different Algorithms
Comparing the theoretical upper bounds of each method, the theoretical upper bound of the four algorithms are related to the frequency interval of MFC.The theoretical upper bound of the I-CRT algorithm is given in Section II-I-C.Considering the frequency estimation problem in this paper, the theoretical upper bound of the C-CRT algorithm is Set the frequency:f M1 = 3 MHz, f M2 = 4 MHz, f M3 = 5 MHz, the MFC frequency interval of I-CRT algorithm, C-CRT algorithm and FBD algorithm is set to {f M1 , f M2 , f M3 } × MN, MNࢠ [1,20].The f max of the method in [13] is greatly affected by the frequency difference δf between f M1 and f M2 .Therefore, we set f M1 = 3 × MNMHz, δf = 100×MN kHz, MNࢠ [1,20].From Fig. 9, it can be seen that in the case of same values of f M1 , f M2 , and f M3 , the f max of the I-CRT algorithm is not superior to other methods.I-CRT algorithm achieves better estimation accuracy at the cost of reducing the maximum estimable frequency.Therefore, in practical applications, it is necessary to consider the frequency estimation range and reasonably set up the frequency interval of the MFC.The f max of I-CRT algorithm can be improved by adjusting the values to increase the least common multiplier of the frequency interval of the MFC.
Based on the above experiments, the performance comparison of the four algorithms is shown in the following table .Table II shows that only the P d of I-CRT and FBD algorithms can reach 100%.When P d = 100%, the measurement noise ΔR MAXP d of I-CRT algorithm is higher than that of FBD algorithm, indicating that the I-CRT algorithm has better robustness.In terms of algorithm complexity, the FBD algorithm has a higher complexity than I-CRT algorithm, as shown in Fig. 8, the FBD algorithm has the highest time consumption.As for the upper bound on frequency estimation of the algorithms, the I-CRT algorithm does not have an advantage.But, in practical application, in order to meet the measurement requirement on frequency upper bound, we could adjust frequency interval of the MFC to improve f max .In summary, the I-CRT algorithm is superior to other algorithms.

V. CONCLUSION
In this paper, on the basis of MFC-based Rydberg atomic measurements, the I-CRT algorithm is proposed to estimate instantaneous frequency of RF signals.This algorithm solves the frequency ambiguity of the mixing signal generated by mixing RF signal field and its closest MFC comb line via Rydberg atoms.And, the interference of the mirror frequency is avoided.By using the mixing frequencies measured by multiple MFCs, the I-CRT algorithm realize the accurate estimation of the RF signal frequency.And the proposed algorithm is compared with the C-CRT algorithm, FBD algorithm and the method in [13] verifying the superiority of proposed algorithm.
Limited by the relaxation time of Rydberg atomic electromagnetic induced transparency phenomenon, the maximum bandwidth of the real-time detection of Rydberg atomic receiver is about 10 MHz.Using MFC can break through the limitation of the instantaneous bandwidth.And, through a group of MFCs with different frequency intervals, a more reliable frequency response can be obtained.Therefore, MFC-based Rydberg atomic receiver has promising application prospects, and this study provides theoretical support for the design and application of Rydberg atomic receiver.

APPENDIX D PROOF OF LEMMA 3
Proof: Due to (n 1 , nk ) ∈ S k , we have otherwise L(n 1 − 1, nk ) or L(n 1 + 1, nk ) would be smaller than L(n 1 , nk ), which means that (n 1 , nk ) does not reach the minimum value of the function L(., .),whichcontradicts with (n 1 , nk ) ∈ S k From (52), we have We first consider the case when n1 is fixed.In this case, from (53), we obtain the searching range for nk The length of this searching range for nk is it means that at most one possible integer value for nk within the range to be selected, i.e., nk is uniquely determined by (51).We next consider the case when nk is fixed.From (50), we can obtain the searching range for n1 The length of this range for n1 is According to Lemma 1, two distinct n1 in S k,1 differ by mΓ k > Γ 1 for some integer m, i.e., the difference absolute value between any two distinct n1 in S k,1 is larger than 1.This proves that in the searching range in (54), there is only one valid element n1 in S k,1 , i.e., n1 is uniquely determined and given in (53).

Fig. 1 .
Fig. 1.(a) Schematic diagram of four energy levels.(b) Schematic diagram of experimental equipment for a Rydberg atomic receiver.(c) RF signal f s measured by MFC.
(17), due to the ambiguity of mixing frequency, the values of b 1 and b k are unknown, b 1 and b k represent the positivenegative values of f R1 and f Rk , and take the value of ±1.b 1 and b k have a total of four different combinations of values.Thus, according to the values of b 1 and b k , nk (n 1 ) has four different situations as shown in Table

Algorithm 2 :
Mixing Frequency Disambiguating.Input: n1 Output: nk (n 1 ), b 1 , b k corresponding to minimum function value of (18) 1: for each b 1 ∈ {−1, 1} do 2: for each b k ∈ {−1, 1} do 3: find nk by (14) 4: Calculate the function value of (18) by n1 , nk , b 1 , b k 5: end for 6: end for 7: Find the minimum of four function values 8: return nk (n 1 ), b 1 , b k corresponding to the minimum function value of (18) Input: nk Output: minimum function value corresponding ton 1 (n k ), b 1 , b k Same process as above When f Rk and f Rk have different positive-negative sign

Fig. 4 .
Fig. 4. Comparison of the probability of detection in terms of different frequency groups.

Fig. 5 .
Fig. 5. Comparison of the RMSE in terms of different measurement noises.

Fig. 6 .Fig. 7 .
Fig. 6.Comparison of the probability of detection in terms of different algorithms.

Fig. 8 .
Fig. 8.Comparison of the time consumption in terms of different algorithms.
to the derivation process from (39) to (43) in Appendix A, it can be inferred thatμ k Γ k = μ 1 Γ 1 , k = 2, 3, . . ., K, Γ k and Γ 1 are co-prime, thus μ 1 = m k Γ k and μ k = m k Γ 1 i.e., n1 = n 1 + m k Γ k and nk = n k + m k Γ 1 (48) for some integers m k .Because (n 1 , nk ) ∈ S k , we have 0 ≤ nk ≤ γ k − 1 (2 ≤ k ≤ K).For the necessity, if n1 = n 1 + m k Γ k and nk = n k + m k Γ 1 for some integer, we can obtain |n k f Mk + fRk − n1 f M1 − fR1 | = |n k f Mk + fRk − n 1 f M1 − fR1 |.From Theorem 1, we know that (n 1 , n k ) ∈ S k which implies (n 1 , nk ) ∈ S k .Lemma 1 indicates that all elements (n 1 , nk ) ∈ S k share the attribute (48) without exception.APPENDIX C PROOF OF LEMMA 2Proof: Lemma 2 follows from Lemma 1, forcing 0 (4) is the mode-order number of RF signal field's closet comb line in kth microwave frequency comb MFC k , f Mk is the frequency interval of MFC k , f Rk is the mixing frequency measured by MFC k , and fSk is the estimated frequency of RF signal field measured by MFC k .The f Rk has the problem of frequency ambiguity, because, b k takes the value of ±1.Here, n k and b k are unknown integers, which need to be solved by(4), n k and b k affect the accuracy of frequency estimation of the Rydberg atomic receiver.Next, we introduce how to disambiguate f Rk and estimate RF signal frequency form several f Rk (k = 1, 2, . . ., K).

TABLE I FOUR
SITUATIONS OF nk (n 1 )

TABLE II COMPARISON
OF THE FOUR DIFFERENT ALGORITHMS Fig. 9. Comparison of the upper bound on frequency estimation in terms of different algorithms.