Self-adaptive CPFSK multi-symbol incoherent demodulation method

Abstract

The invention discloses a self-adaptive CPFSK multi-symbol incoherent demodulation method, and belongs to the field of communication signal processing. The implementation method comprises the following steps: initializing a receiving signal and the length of an observation window; performing orthogonal down-conversion on the received signal to obtain a baseband received signal; constructing a local template sequence according to the length of the observation window, and performing baseband modulation to generate a local baseband template signal; according to the baseband receiving signal and a local baseband template signal, constructing an incoherent judgment statistic, and selecting a candidate sequence; classifying the candidate sequences into credible sequences and uncredible sequences by comparing the judgment of the candidate sequences before and after about the same symbol; taking an intermediate symbol of the credible sequence as an incoherent judgment result, and adding one to the position of the symbol to be judged; for an untrusted sequence, the length of an observation window is expanded, a candidate sequence is selected again, an intermediate symbol is used as a new incoherent judgment result, the position of a symbol to be judged is increased by one, iteration is repeated, and self-adaptive multi-symbol incoherent demodulation is achieved.

Description

Technical Field

[0001] This invention relates to an adaptive CPFSK multi-symbol incoherent demodulation method, specifically an improved method for demodulation based on the maximum likelihood criterion, belonging to the field of communication signal processing. Background Technology

[0002] Maritime communications face significant challenges due to severe multipath effects and poor channel stability, placing higher demands on the reliability and robustness of communication systems. To address this challenge, Continuous Phase Frequency Shift Keying (CPFSK) technology, with its constant envelope and phase continuity, demonstrates significant advantages in maritime communications.

[0003] In maritime communication scenarios, to avoid the impact of phase distortion due to multipath effects on signal demodulation and improve the reliability of CPFSK communication systems, high-performance incoherent demodulation methods are typically employed. Traditional CPFSK incoherent demodulation methods are divided into symbol-by-symbol differential demodulation and multi-symbol incoherent demodulation methods: the former demodulates by comparing the phase difference between adjacent symbols, which has a simple structure but limited performance; the latter combines the phase continuity of the signal and the correlation between multiple symbols, significantly improving demodulation performance, but with higher implementation complexity.

[0004] Existing research addresses the high computational complexity of CPFSK multi-symbol incoherent demodulation methods by proposing a series of optimization schemes, primarily including simplifying decision rules and reducing the number of local templates. The former approximates the decision rules under specific signal-to-noise ratio conditions, avoiding the computational overhead of exponential operations; the latter uses a decision feedback mechanism to filter templates based on already decided information, reducing redundant computation. While these schemes reduce complexity to some extent, the overall computational load remains significant due to the reliance on relatively long observation windows.

[0005] Furthermore, complex weather conditions lead to unstable maritime communication channels, and traditional CPFSK multi-symbol incoherent demodulation methods, due to their use of fixed observation windows, cannot respond to time-varying signal-to-noise ratio channel conditions. Taking all factors into consideration, this invention, based on the aforementioned optimization schemes, further investigates a more efficient computational complexity control strategy from the perspective of optimizing the observation window length, in order to achieve a better balance between performance and complexity. Summary of the Invention

[0006] To address the following technical shortcomings of existing CPFSK multi-symbol incoherent demodulation methods: (i) Traditional multi-symbol incoherent demodulation methods have high computational complexity when using a long observation window for symbol demodulation; (ii) Traditional multi-symbol incoherent demodulation methods use a fixed observation window length, which cannot adapt well to channels with time-varying signal-to-noise ratios. The main objective of this invention is to provide an adaptive CPFSK multi-symbol incoherent demodulation method. In scenarios with limited hardware resources, this method utilizes the decision results for the same symbol in consecutive decision sequences. When the decision results are inconsistent, the observation window for the current symbol is expanded and re-determined, thereby achieving adaptive adjustment of the observation window length and improving its demodulation performance. This invention has the following main advantages: (i) Using a shorter observation window as the basis for decision-making effectively reduces computational complexity. At the same time, observation window expansion is achieved through consistent decision-making, resulting in high-performance demodulation with relatively low computational complexity growth; (ii) Down-converting the received signal to baseband effectively reduces computational complexity by utilizing the low processing rate of baseband; (iii) Adaptive adjustment of the observation window length can better adapt to channels with time-varying signal-to-noise ratios.

[0007] The objective of this invention is achieved through the following technical solution.

[0008] This invention discloses an adaptive CPFSK multi-symbol incoherent demodulation method, comprising the following steps:

[0009] Step 1: Initialize the received signal and observation window length. The time-domain expression for the received signal r(t) is:

[0010]

[0011] In the formula, f c Let t be the carrier frequency, t be the time, h be the modulation index, T be the symbol period, and b be the frequency. k b represents the k-th binary symbol, taking the value 1 or -1. i The i-th binary symbol is represented by 1 or -1, θ0 is the initial phase of the received signal, and n(t) is the noise.

[0012] To achieve multi-symbol incoherent demodulation, an observation window is set with a length of N = 2n + 1, where n is a positive integer. The m-th symbol is the symbol to be decided. When m < n + 1, the observation window centered on the symbol to be decided does not satisfy the above length, and the performance of multi-symbol incoherent demodulation degrades. Therefore, the symbol to be decided is defined as a symbol known to both parties. When m ≥ n + 1, the observation window satisfies:

[0013] (mn-1)T≤t<(m+n)T

[0014] Step 2: Perform orthogonal down-conversion on the received signal to obtain the baseband received signal. The orthogonal representation of the received signal is as follows:

[0015]

[0016] In the formula, n c (t) represents the in-phase component of the noise, n s (t) represents the orthogonal component of the noise.

[0017] The received signal is orthogonally down-converted to obtain the baseband received signal. The baseband received signal is represented as:

[0018] d(t)=LPF{r(t)e -j2πfct} = I r (t)+jQ r (t)kT≤t<(k+1)T

[0019] In the formula, LPF{·} represents low-pass filtering, and I r (t) represents the in-phase component of the baseband received signal, Q r (t) represents the quadrature components of the baseband received signal, which are expressed as follows:

[0020]

[0021] Step 3: Based on the length of the observation window, construct a local template sequence, and generate a local baseband template signal using baseband modulation. The local template sequence is represented as:

[0022] A = {a0, a1, ..., a} n-1 ,a n ,a n+1 ,...,a 2n-1 ,a 2n}

[0023] In the formula, a n Let n be the nth binary symbol, taking the values ​​1 and -1; where the set {a0, a1, ..., a...} is... n-1 Let {a} be a known sequence, and {a} be a set. n ,a n+1 ,...,a 2n-1 ,a 2n} represents an unknown sequence, generated by traversing the entire space.

[0024] CPFSK modulation is used to perform baseband modulation on the local template sequence to generate a local baseband template signal. The in-phase and quadrature components of the local baseband template signal are represented as follows:

[0025]

[0026] In the formula, h is the modulation index, and a l This represents the l-th binary symbol, taking the value 1 or -1, where t is time, T is the symbol period, and a iθ represents the i-th binary symbol, with a value of 1 or -1, and θ is the initial modulation phase, with a value of 0 or π / 2.

[0027] Step 4: Based on the baseband received signal from Step 2 and the local baseband template signal from Step 3, construct incoherent decision statistics and select candidate sequences.

[0028] The baseband received signal is correlated with the local baseband template signal to obtain the correlation result:

[0029]

[0030] In the formula, I s (t-mT+nT+T,A,θ) and Q s (t-mT+nT+T,A,θ) represent the time shifts (mn-1)T of the in-phase and quadrature components of the local baseband template signal, respectively.

[0031] The incoherent decision statistic is obtained by summing the squares of the correlation results for the same local template sequence with different modulation initial phases:

[0032]

[0033] In the formula, A i Let z be the i-th local template sequence. i 2 Let be the statistic for the i-th incoherent decision.

[0034] The local template sequence that maximizes the incoherent decision statistic is selected as the candidate sequence; the decision criterion for the candidate sequence is expressed as:

[0035]

[0036] In the formula, A c These are candidate sequences.

[0037] Step 5: By comparing the decisions of the preceding and following candidate sequences regarding the same symbol, the candidate sequences are classified into trustworthy sequences and untrustworthy sequences.

[0038] Compare the candidate sequence with the candidate sequence when the m-1th symbol is to be decided; when the two decisions about the mth symbol are consistent, the current candidate sequence is determined to be a reliable sequence, the intermediate symbol of the candidate sequence is taken as the incoherent decision result, m is incremented by one, and steps one to six are repeated; when the two decisions about the mth symbol are inconsistent, the current candidate sequence is determined to be an unreliable sequence, the observation window is expanded and a new decision is made, and step six is ​​entered.

[0039] Step 6: For untrusted sequences, extend the observation window length, reselect candidate sequences, and use their middle symbols as new incoherent decision results. Increment the position m of the symbol to be decided by one. Repeat steps one to six to achieve the adaptive CPFSK multi-symbol incoherent demodulation method.

[0040] Extend the observation window length to N = 2n + 3, and construct a local template sequence:

[0041] A′={a′0,a′1,...,a′ n ,a′ n+1 ,a′ n+2 ,...,a′ 2n+1 ,a′ 2n+2}

[0042] Among them, the set {a′0,...,a′ n Let {a'} be a known sequence, and let {a'} be a set. n+1 ,a′ n+2} have different values, set {a′ n+3 ,...,a′ 2n+2} represents an unknown sequence, generated through traversal of the entire space;

[0043] After A′ is subjected to CPFSK baseband modulation to form a new local baseband template signal, a correlation operation is performed with the baseband received signal from step two to obtain a new correlation result. At this time, the observation window satisfies:

[0044] (mn-2)T≤t<(m+n+1)T

[0045] The new relevant results are expressed as follows:

[0046]

[0047] In the formula, I s (t-mT+nT+2T,A,θ) and Q s (t-mT+nT+2T,A,θ) represent the time shifts (mn-2)T ​​of the in-phase and quadrature components of the local baseband template signal, respectively.

[0048] By summing the squares of the new correlation results with different initial phases using the same local template, we obtain the decision statistic:

[0049]

[0050] In the formula, A′ i For the i-th local template sequence, Let be the statistic for the i-th incoherent decision.

[0051] The local template sequence that maximizes the decision statistic is selected as the candidate sequence, and its intermediate symbol is used as the new incoherent decision result. The position m of the symbol to be decided is incremented by one. Steps one to six are repeated to achieve adaptive CPFSK multi-symbol incoherent demodulation.

[0052] Beneficial effects:

[0053] 1. The adaptive CPFSK multi-symbol incoherent demodulation method disclosed in this invention utilizes the previous and subsequent decision results for the same symbol to selectively expand the observation window and re-determine. Compared with the traditional CPFSK incoherent multi-symbol demodulation method, it significantly reduces the computational complexity of CPFSK multi-symbol incoherent demodulation with slight performance loss.

[0054] 2. The adaptive CPFSK multi-symbol incoherent demodulation method disclosed in this invention utilizes an orthogonal down-conversion structure to convert the received signal into a baseband received signal, which is then correlated with a local baseband template signal. Since baseband processing allows for a lower processing rate, the computational complexity of CPFSK multi-symbol incoherent demodulation is further reduced compared to traditional CPFSK incoherent demodulation methods.

[0055] 3. The adaptive CPFSK multi-symbol incoherent demodulation method disclosed in this invention can dynamically adjust the observation window length and its processing strategy based on the consistency of the decision results. This adaptive mechanism enables it to effectively cope with time-varying signal-to-noise ratio channel environments, maintain low-complexity operation when channel conditions are good, and automatically enhance processing capabilities to maintain communication reliability when channel conditions deteriorate. Attached Figure Description

[0056] Figure 1 This is a schematic diagram of the adaptive CPFSK multi-symbol incoherent demodulation method of the present invention;

[0057] Figure 2 This is a schematic diagram of the consistency decision strategy in this invention;

[0058] Figure 3 This is a comparison of the BER curves of the adaptive CPFSK multi-symbol incoherent demodulation method of the present invention and the traditional CPFSK incoherent demodulation method.

[0059] Figure 4 This is a statistical chart of the number of window expansions in the adaptive CPFSK multi-symbol incoherent demodulation method of the present invention. Detailed Implementation

[0060] The present invention will now be described in detail with reference to the accompanying drawings and embodiments. The technical problems solved by the present invention and its beneficial effects are also described. It should be noted that the described embodiments are only intended to facilitate understanding of the present invention and do not constitute any limitation thereof.

[0061] To make the above-mentioned objects, features, and advantages of the present invention more readily understood, they will be further described in detail below with reference to the accompanying drawings and specific embodiments. This embodiment is an adaptive CPFSK multi-symbol incoherent demodulation method for hardware resource constraints, which realizes the function of correctly recovering the transmitted sequence at the receiving end. The parameters in this embodiment are shown in Table 1:

[0062] Table 1 Parameters of the Example

[0063]

[0064]

[0065] like Figure 1 As shown in the figure, the adaptive CPFSK multi-symbol incoherent demodulation method disclosed in this embodiment has the following specific implementation steps:

[0066] Step 1: Initialize the received signal and observation window length. The time-domain expression for the received signal r(t) is:

[0067]

[0068] In the formula, the carrier frequency f c =250kHz, t is time, modulation index h = 0.5, symbol period T = 0.1ms, b k b represents the k-th binary symbol, taking the value 1 or -1. i The i-th binary symbol is represented by 1 or -1, θ0 is the initial phase of the received signal, and n(t) is the noise.

[0069] If the initial observation window length N = 3, and the symbol to be decided is the 4th symbol, then the observation window satisfies:

[0070] 2T≤t<5T

[0071] Step 2: Perform orthogonal down-conversion on the received signal to obtain the baseband received signal. The orthogonal representation of the received signal is as follows:

[0072]

[0073] In the formula, n c (t) represents the in-phase component of the noise, n s (t) represents the orthogonal component of the noise.

[0074] The received signal is orthogonally down-converted to obtain the baseband received signal. The baseband received signal is represented as:

[0075]

[0076] In the formula, LPF{·} represents low-pass filtering, and I r (t) represents the in-phase component of the baseband received signal, Q r (t) represents the quadrature components of the baseband received signal, which are expressed as follows:

[0077]

[0078] Step 3: Based on the length of the observation window, construct a local template sequence, and generate a local baseband template signal using baseband modulation. The local template sequence is represented as:

[0079] A = {a0, a1, a2}

[0080] In the formula, a n Let represent the nth binary symbol, taking the values ​​1 and -1. The set {a0} represents the known sequence, and the set {a1,a2} represents the unknown sequence, generated through a full space traversal. Therefore, the local template sequences are {1,1,-1}, {1,1,1}, {1,-1,1}, and {1,-1,-1}.

[0081] CPFSK modulation is used to perform baseband modulation on the local template sequence to generate a local baseband template signal. The in-phase and quadrature components of the local baseband template signal are represented as follows:

[0082]

[0083] In the formula, h is the modulation index, and a l This represents the l-th binary symbol, taking the value 1 or -1, where t is time, T is the symbol period, and a i θ represents the i-th binary symbol, with a value of 1 or -1, and θ is the initial modulation phase, with a value of 0 or π / 2.

[0084] Step 4: Based on the baseband received signal from Step 2 and the local baseband template signal from Step 3, construct incoherent decision statistics and select candidate sequences.

[0085] The baseband received signal is correlated with the local baseband template signal to obtain the correlation result:

[0086]

[0087] In the formula, I s (t-2T,A,θ) and Q s(t-2T, A, θ) represent the results of shifting the in-phase and quadrature components of the local baseband template signal by 2T, respectively. Since the spectral components of the baseband received signal and the local baseband template signal are low, decimation can effectively reduce the number of sampling points involved in the correlation operation.

[0088] The incoherent decision statistic is obtained by summing the squares of the correlation results for the same local template sequence with different modulation initial phases:

[0089]

[0090] In the formula, A i Let z be the i-th local template sequence. i 2 Let be the statistic for the i-th incoherent decision.

[0091] The local template sequence that maximizes the incoherent decision statistic is selected as the candidate sequence. The decision criterion for the candidate sequence is expressed as:

[0092]

[0093] In the formula, A c These are candidate sequences.

[0094] Step 5: By comparing the decisions of the preceding and following candidate sequences regarding the same symbol, classify the candidate sequences into trustworthy and untrustworthy sequences. The classification strategy is as follows: Figure 2 As shown.

[0095] Compare the candidate sequence with the candidate sequence when the third symbol is to be decided. If the two decisions regarding the fourth symbol are consistent, the current candidate sequence is determined to be a reliable sequence. The middle symbol of the candidate sequence is used as the decision result, the position of the symbol to be decided is incremented by one, and steps one through five are repeated. If the two decisions regarding the fourth symbol are inconsistent, the current candidate sequence is determined to be an unreliable sequence. The observation window is expanded and a new decision is made, proceeding to step six.

[0096] Step 6: For unreliable sequences, extend the observation window length, reselect candidate sequences, use their middle symbols as new incoherent decision results, increment the position of the symbol to be decided by one, and repeat steps one to six.

[0097] Extend the observation window length to N=5 and construct a local template sequence:

[0098] A′={a′0,a′1,a′2,a′3,a′4}

[0099] Here, the set {a′0,a′1} represents the known sequences -1 and 1, and the set {a′2,a′3,a′4} represents the unknown sequences, generated through full space traversal. Therefore, the local template sequences are {-1,1,1,1,1}, {-1,1,1,1,-1}, {-1,1,1,-1,1}, {-1,1,1,-1,1}, {-1,1,-1,-1,1}, {-1,1,-1,1,1}, {-1,1,-1,-1,1}, and {-1,1,-1,-1,-1}.

[0100] After A′ is subjected to CPFSK baseband modulation to form a new local baseband template signal, a correlation operation is performed with the baseband received signal from step two to obtain a new correlation result. At this time, the observation window satisfies:

[0101] T≤t<6T

[0102] The new relevant results are expressed as follows:

[0103]

[0104] In the formula, I s (tT,A,θ) and Q s (tT,A,θ) represent the time shift T of the in-phase and quadrature components of the local baseband template signal, respectively.

[0105] By summing the squares of the correlation results for the same local template but different initial phases, we obtain the incoherent decision statistic:

[0106]

[0107] In the formula, A′ i For the i-th local template sequence, This is the statistic for the i-th incoherent decision;

[0108] Select the local template sequence that maximizes the decision statistic as the candidate sequence, use its middle symbol as the new incoherent decision result, increment the position of the symbol to be decided by one, and repeat steps one to six.

[0109] Through the above steps, the adaptive CPFSK multi-symbol incoherent demodulation method of the present invention can correctly recover the transmitted sequence. In a preferred embodiment of the present invention, based on the calculated bit error rate results under different signal-to-noise ratios, the bit error rate curves are plotted as follows. Figure 3 As shown, this embodiment improves demodulation performance by at least 0.5 dB compared to traditional methods using shorter observation windows. Meanwhile, as... Figure 4As shown, the adaptive CPFSK multi-symbol incoherent demodulation method disclosed in this example employs fewer long observation windows, significantly reducing computational complexity compared to traditional methods using longer observation windows. Furthermore, by converting the received signal to a baseband received signal and using decimation, the number of sampling points involved in correlation operations is reduced, further lowering computational complexity. Moreover, the number of long windows decreases as the signal-to-noise ratio increases, thus maintaining low-complexity operation under favorable channel conditions and automatically enhancing processing capabilities to maintain communication reliability when channel conditions deteriorate.

[0110] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. An adaptive CPFSK multi-symbol incoherent demodulation method, characterized in that: Includes the following steps: Step 1: Initialize the received signal and observation window length; Step 2: Perform orthogonal downconversion on the received signal to obtain the baseband received signal; Step 3: Based on the length of the observation window, construct a local template sequence and generate a local baseband template signal using baseband modulation; Step 4: Based on the baseband received signal from Step 2 and the local baseband template signal from Step 3, construct incoherent decision statistics and select candidate sequences. Step 5: By comparing the decisions of the preceding and following candidate sequences regarding the same symbol, classify the candidate sequences into trustworthy sequences and untrustworthy sequences; For a reliable sequence, the intermediate symbol of the reliable sequence is used as the incoherent decision result, the position m of the symbol to be decided is incremented by one, and steps one to six are repeated. Step 6: For untrusted sequences, extend the observation window length, reselect candidate sequences, use their middle symbols as new incoherent decision results, increment the position m of the symbol to be decided by one, and repeat steps one to six to achieve adaptive CPFSK multi-symbol incoherent demodulation.

2. The method as described in claim 1, characterized in that: The time-domain expression for the received signal r(t) mentioned in step one is: In the formula, f c Let t be the carrier frequency, t be the time, h be the modulation index, T be the symbol period, and b be the frequency. k b represents the k-th binary symbol, taking the value 1 or -1. i The i-th binary symbol is represented by 1 or -1, θ0 is the initial phase of the received signal, and n(t) is the noise.

3. The method as described in claim 1, characterized in that: The observation window length mentioned in step one is obtained by the following method: To achieve multi-symbol incoherent demodulation, an observation window is set with a length of N = 2n + 1, where n is a positive integer. The m-th symbol is the symbol to be decided. When m < n + 1, the observation window centered on the symbol to be decided does not satisfy the above length, and the performance of multi-symbol incoherent demodulation degrades. Therefore, the symbol to be decided is defined as a symbol known to both parties. When m ≥ n + 1, the observation window satisfies: (mn-1)T≤t<(m+n)T.

4. The method as described in claim 1, characterized in that: The implementation method for step two is as follows: The orthogonal representation of the received signal is: In the formula, n c (t) represents the in-phase component of the noise, n s (t) represents the orthogonal components of the noise; The received signal is orthogonally down-converted to obtain the baseband received signal; the baseband received signal is represented as: In the formula, LPF{·} represents low-pass filtering, and I r (t) represents the in-phase component of the baseband received signal, Q r (t) represents the quadrature components of the baseband received signal, which are expressed as follows:

5. The method as described in claim 1, characterized in that: The implementation method for step three is as follows: The local template sequence is represented as: A={a0,a1,...,a n-1 ,a n ,a n+1 ,...,a 2n-1 ,a 2n } In the formula, a n Let n be the nth binary symbol, taking the values ​​1 and -1; where the set {a0, a1, ..., a...} is... n-1 Let {a} be a known sequence, and set {a} be a set {a}. n ,a n+1 ,...,a 2n-1 ,a 2n } represents an unknown sequence, generated through traversal of the entire space; The local template sequence is baseband modulated using CPFSK modulation to generate a local baseband template signal; the in-phase and quadrature components of the local baseband template signal are represented as follows: In the formula, h is the modulation index, and a l This represents the l-th binary symbol, taking the value 1 or -1, where t is time, T is the symbol period, and a i θ represents the i-th binary symbol, with a value of 1 or -1, and θ is the initial modulation phase, with a value of 0 or π / 2.

6. The method as described in claim 1, characterized in that: The implementation method for step four is as follows: The baseband received signal is correlated with the local baseband template signal to obtain the correlation result: In the formula, I s (t-mT+nT+T,A,θ) and Q s (t-mT+nT+T,A,θ) represent the time shifts (mn-1)T of the in-phase and quadrature components of the local baseband template signal, respectively. The incoherent decision statistic is obtained by summing the squares of the correlation results for the same local template sequence with different modulation initial phases: In the formula, A i Let z be the i-th local template sequence. i 2 This is the statistic for the i-th incoherent decision; The local template sequence that maximizes the incoherent decision statistic is selected as the candidate sequence; The decision criterion for candidate sequences is expressed as follows: In the formula, A c These are candidate sequences.

7. The method as described in claim 1, characterized in that: The implementation method for step five is as follows: Compare the candidate sequence with the candidate sequence when the symbol to be decided is the (m-1)th symbol; When the two decisions about the m-th symbol are consistent, the current candidate sequence is determined to be a reliable sequence. The intermediate symbol of the candidate sequence is used as the incoherent decision result, m is incremented by one, and steps one to six are repeated. When the two decisions regarding the m-th symbol are inconsistent, the current candidate sequence is determined to be an untrusted sequence, the observation window is expanded and a new decision is made, and the process proceeds to step six.

8. The method as described in claim 1, characterized in that: The implementation method for step six is ​​as follows: Extend the observation window length to N = 2n + 3, and construct a local template sequence: A′={a′0,a′1,...,a′ n ,a′ n+1 ,a′ n+2 ,...,a′ 2n+1 ,a′ 2n+2 } Among them, the set {a′0,...,a′ n Let {a'} be a known sequence, and set {a'} be a set {a'}. n+1 ,a′ n+2 } have different values, set {a′ n+3 ,...,a′ 2n+2 } represents an unknown sequence, generated through traversal of the entire space; After A′ is subjected to CPFSK baseband modulation to form a new local baseband template signal, a correlation operation is performed with the baseband received signal from step two to obtain a new correlation result. At this time, the observation window satisfies: (mn-2)T≤t<(m+n+1)T The new relevant results are expressed as follows: In the formula, I s (t-mT+nT+2T,A,θ) and Q s (t-mT+nT+2T,A,θ) represent the time shifts (mn-2)T ​​of the in-phase and quadrature components of the local baseband template signal, respectively. By summing the squares of the new correlation results with different initial phases using the same local template, we obtain the decision statistic: In the formula, A i Let ' be the i-th local template sequence. This is the statistic for the i-th incoherent decision; Select the local template sequence that maximizes the decision statistic as the candidate sequence, use its intermediate symbol as the new incoherent decision result, increment the position m of the symbol to be decided by one, and repeat steps one to six.

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