A method for bit error rate analysis based on non-full load OCDM system

Abstract

The application discloses a bit error rate analysis method based on a non-full-load OCDM system, relates to the technical field of performance analysis of a wireless communication system, and gives a modulation and demodulation process of the non-full-load OCDM system, deduces and analyzes the BER performance of the system; a time-domain channel circulation matrix is diagonalized, general BER expression under a single-path and two-path Rayleigh channel is given, and the BER expression of the system under the single-path and two-path Rayleigh channel is deduced based on this; the upper limit and the lower limit of the BER under the second channel environment are deduced, and simulation results prove the effectiveness and accuracy of the BER theoretical value and the upper limit and the lower limit deduced, and also prove the effectiveness and accuracy of the method.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of performance analysis of wireless communication systems, and particularly relates to a bit error rate analysis method based on a non-full-load OCDM system. BACKGROUND

[0002] Orthogonal Chirp Division Multiplexing (OCDM) is a kind of multi-carrier modulation mode proposed in 2016. Each subcarrier is a chirp signal, and each subcarrier occupies the same bandwidth and signal period, so the OCDM system can achieve the maximum spectral efficiency under chirp spread spectrum technology. OCDM has better performance than Orthogonal Frequency Division Multiplexing (OFDM) technology in many scenarios, especially in time-frequency dual-selected channels, so many studies on OCDM systems have emerged, including the following points: 1. Designing an OCDM system based on a digital filter to reduce out-of-band spectrum leakage; 2. Carrier frequency offset estimation; 3. Reducing the peak-to-average power ratio (PAPR); 4. Channel estimation; 5. Channel equalization, etc.

[0003] However, existing research is basically based on full-load communication systems, that is, each subcarrier transmits a constellation point. However, in actual communication systems, empty subcarriers can be set, that is, a non-full-load communication system is designed. The non-full-load OCDM system means that some chirp subcarriers do not transmit symbols, and this structure is also applicable to OFDM systems. Compared with full-load systems, non-full-load systems can obtain signal-to-noise ratio (SNR) gain, but at the cost of sacrificing certain information rate, which can be compromised according to actual needs.

[0004] In underwater acoustic communication, especially in terms of bit error rate (BER) performance, non-full-load OCDM systems have been proven to be superior to non-full-load OFDM systems. Some scholars have also pointed out that frequency domain spatial modulation systems are actually a special kind of non-full-load MIMO (Multiple-Input Multiple-Output)-OFDM system, because they transmit symbols through multiple antennas, reducing the power demand of high-power amplifiers. However, existing technologies only analyze the BER performance of non-full-load OCDM / OFDM systems, but do not derive the BER expression of non-full-load OCDM systems, especially in fading channels. SUMMARY

[0005] To solve the above technical problems, the application provides a bit error rate analysis method based on a non-full-load OCDM system, comprising the following steps:

[0006] S1, confirming a non-full-load factor a at a transmitting end, that is, modulating one subcarrier in every a subcarriers;

[0007] S2, acquiring 4-order quadrature amplitude modulation 4-QAM sending symbols s=[s(0), s(a),... s(N-a+1)], wherein N is the number of OCDM subcarriers;

[0008] S3, setting a power normalization factor as P nor , multiplying the discrete Fresnel transform matrix Φ on the left of s, and then multiplying the coefficient so as to keep the transmitting power unchanged;

[0009] S4, processing a received signal by using a zero-forcing ZF equalization at a receiving end, setting a time-domain channel circulation matrix as H, a receiving end noise vector as n=[n(0), n(1),... n(N-1)], and an average power as obtaining a general expression of a receiving end instantaneous signal-to-noise ratio expression ρ by diagonalizing H through fast Fourier transform;

[0010] S5, calculating a probability density function PDF of ρ about an eigenvalue of H, combining a bit error rate BER expression of M-order quadrature amplitude modulation M-QAM constellation points under an additive white Gaussian noise AWGN channel, integrating ρ, and obtaining expressions of the bit error rate BER and upper and lower limits thereof under a single-path Rayleigh channel and a two-path Rayleigh channel.

[0011] The application is further defined by the technical solutions as follows:

[0012] Further, in the step S2, a time-domain continuous wave expression of the OCDM subcarrier is: wherein k is a subcarrier index value, ranges from 0 to N-1, N is an even number, T is a whole OCDM symbol duration, j is an imaginary unit, satisfies j 2 =-1, and t is a time independent variable;

[0013] Under a single sampling rate, one OCDM symbol is sampled at N points, and the sampling time points are Φ is expressed as:

[0014]

[0015] An OCDM waveform to be sent is expressed as:

[0016] x=Φ H s, #(2)

[0017] where s represents the transmitted OCDM waveform.

[0018] As the method for analyzing the bit error rate of the non-full OCDM system mentioned above, in step S3, assuming s is the power normalized 4-QAM signal, the power P of the transmitted signal x is x :

[0019]

[0020] Assuming , the power of the transmitted symbol is constant:

[0021]

[0022] where P nor represents the power normalization factor.

[0023] As the method for analyzing the bit error rate of the non-full OCDM system mentioned above, in step S4, after the transmitted signal passes through the channel, assuming the noise signal at the receiving end is n = [n(0), n(1),..., n(N-1)], the signal at the receiving end is:

[0024]

[0025] After zero-forcing (ZF) equalization, the estimated constellation point is represented as:

[0026]

[0027] Assuming x k is the kth element of x, since under the zero-forcing (ZF) equalization, the noise is uniformly distributed on each chirp subcarrier of the OCDM, the instantaneous signal-to-noise ratio ρ of the system at the receiving end is represented as:

[0028]

[0029] Assuming F is an N x N discrete Fourier transform (DFT) matrix, H and H -1 in equation (7) are represented as:

[0030] H = F H DF, H- 1 = F H DF, # (8)

[0031] where D is an N x N diagonal matrix, the elements are calculated as the N-point Fourier transform of the channel impulse response h = [h0, h1,..., h N-1 ], h n , n = 0, 1,..., N-1 is the nth path, and:

[0032]

[0033] where λ i , i = 0,..., N - 1 denote the N eigenvalues of H; define i.e. the general expression of instantaneous SNR is: i

[0034]

[0035] As described above, in step S5, the bit error rate BER expression of M-QAM under the white noise channel AWGN is:

[0036]

[0037] where Q denotes the Gaussian Q function, C M and ψ(i, k) are specific coefficients, i, k are corresponding indexes, and have

[0038]

[0039] Therefore, the bit error rate P b of 4-QAM can be expressed as Let the probability density function PDF of ρ be f ρ (ρ), and integrate ρ to obtain the general expression of P b under the zero-forcing ZF equalization, where the general expression of ρ is given by equation (10):

[0040]

[0041] As described above, in step S5, under the single-path Rayleigh channel, the channel impulse response CIR is expressed as |h0| 2 obeys the exponential distribution, and the average power is denoted as At this time, the N eigenvalues are all λ i = h0, i = 0, 1,..., N - 1, and the instantaneous SNR and its probability density function PDF are expressed as:

[0042]

[0043] Combined with the transformation formula of the Gaussian Q function where ψ is any integral variable, the bit error rate BER is further expressed as:

[0044]

[0045] Consider another integral equation:​

[0046]

[0047] Thus, the bit error rate BER is finally simplified as:

[0048]

[0049] As described above, in the method for analyzing the bit error rate based on the non-full OCDM system, in step S5, under the two-path Rayleigh channel, since H has two different eigenvalues, the channel impulse response CIR is expressed as and where θ0, θ1 are the phase angles corresponding to h0, h1; let the average powers of h0, h1 be that is, θ0, θ1 are uniformly distributed in (-π, π); at this time, the eigenvalues λ = [λ0, λ1,..., λ N-1 ] of H are:

[0050] λ0= λ2=... = λ N-2 = h0+ h1, λ1= λ3=... = λ N-1 = h0- h1, # (18)

[0051] and the instantaneous signal-to-noise ratio is expressed as:

[0052]

[0053] Let the random variable w = |λ0| 2 , v = |λ1| 2 , then X is simplified as:

[0054]

[0055] where the probability density function PDF of w, v is expressed as The probability density function PDF of w, v is Let the probability density function PDF of X be f X (x), and the probability cumulative distribution function CDF be F X (x) = P{X≤x}, then:

[0056]

[0057] As described above, in the method for analyzing the bit error rate based on the non-full OCDM system, in step S5,

[0058] When , it is concluded that the CDF of X at this time is:

[0059] ​

[0060] The PDF of X at this time is finally obtained as:

[0061]

[0062] When -1≤t(w,v,x)≤1, # (25), the integral domain about w, v, θ is obtained as:

[0063]

[0064]

[0065]

[0066] Let

[0067] The PDF of X at this time is finally obtained as:

[0068]

[0069] Combining and the expression of f X (x) is obtained as:

[0070]

[0071] At this time, the bit error rate BER is calculated by:

[0072]

[0073] As described above, in the bit error rate analysis method based on the non-full OCDM system, in step S5, the upper limit of the bit error rate BER is calculated by formula (20), in formula (20), if each pair (h0, h1) has coSθ=1, X takes the minimum value, ρ takes the minimum value, the bit error rate BER reaches the theoretical upper limit, at this time, the instantaneous signal-to-noise ratio is:

[0074]

[0075] The probability cumulative distribution function CDF of variable X l is expressed as:

[0076]

[0077] Let then When , the probability density function PDF is expressed as:

[0078] ​​

[0079] The upper bound of the bit error rate BER is represented as:

[0080]

[0081] wherein, represents the upper bound of the bit error rate BER.

[0082] As the bit error rate analysis method based on the non-full OCDM system as described previously, in step S5, the lower bound of the bit error rate BER is calculated by formula (20). In formula (20), if cos θ = -1 for each pair (h0, h1), X takes the maximum value, ρ takes the maximum value, and the bit error rate BER reaches the theoretical lower bound. At this time, the instantaneous signal-to-noise ratio is:

[0083]

[0084] At this time, the variable X u The probability cumulative distribution function CDF of the variable X

[0085]

[0086] When , the probability density function PDF of the variable X u is:

[0087]

[0088] The lower bound of the bit error rate BER is represented as:

[0089]

[0090] In combination with the transformation formula of the Gaussian Q function , formula (16) and formula (38) are further derived as:

[0091]

[0092] wherein

[0093] When , the variable X u obeys the two-parameter subexponential distribution with parameters (λ0, λ1), and has the probability density function PDF:

[0094]

[0095] At this time, the lower bound of the bit error rate BER is obtained as:

[0096]

[0097] The beneficial effects of the present application are:

[0098] The existing article only analyzes the BER performance of the non-full load OCDM / OFDM system through simulation results, but does not deduce the BER expression of the non-full load OCDM system, especially deduces and analyzes the BER performance in the fading channel, and the present application gives the modulation and demodulation process of the non-full load OCDM system, deduces and analyzes the BER performance of the system; diagonalizes the time domain channel circulation matrix, gives the general BER expression under the single-path and two-path Rayleigh channel, and deduces the BER expression of the system under the single-path and two-path Rayleigh channel based on this; deduces the upper and lower bounds of the BER in the second channel environment, and the simulation results also prove the correctness of the deduction. BRIEF DESCRIPTION OF DRAWINGS

[0099] Figure 1 It is the overall flowchart of the present application;

[0100] Figure 2 It is the measured BER and the theoretical BER comparison chart under the single-path Rayleigh channel when a=1, 2, 4 in the embodiment of the present application;

[0101] Figure 3 It is the measured BER and the theoretical BER comparison chart under the two-path Rayleigh channel when a=1, 2, 4 in the embodiment of the present application;

[0102] Figure 4 It is the measured BER and the theoretical BER upper and lower bounds comparison chart under the two-path Rayleigh channel when a=4 in the embodiment of the present application. DETAILED DESCRIPTION

[0103] The error rate analysis method based on the non-full load OCDM system provided by the embodiment, as shown in Figure 1 The method comprises the following steps:

[0104] S1, in the transmitting end, confirm the non-full load factor a, that is, modulate one subcarrier in every a subcarriers.

[0105] S2, obtain the 4-order quadrature amplitude modulation (QAM) sending symbol s=[s(0), s(a),... s(N-a+1)], wherein N is the OCDM subcarrier quantity; the time domain continuous wave expression of the OCM subcarrier is: Wherein, k is the subcarrier index value, the range is 0~N-1, assuming that N is even; T is a whole OCM symbol duration; j is an imaginary unit, and satisfies j 2 =-1; t is a time independent variable.

[0106] At a single sampling rate, one OCDM symbol is sampled at N points, and the sampling time points are Then, Φ is expressed as:

[0107]

[0108] The transmitted OCDM waveform is expressed as:

[0109] x = Φ H s, # (2)

[0110] Wherein, s represents the transmitted OCDM waveform.

[0111] S3, let the power normalization factor be P nor , left multiply s by a Discrete Fresnel Transformation Matrix (DFnT) Φ, and then multiply it by a coefficient to ensure that the transmission power is constant; let s be a 4-QAM signal after power normalization, then the power P x of the transmitted signal x is:

[0112]

[0113] Let Then, the transmitted symbol power is constant:

[0114]

[0115] Wherein, P nor represents a power normalization factor.

[0116] S4, at the receiving end, the received signal is processed by Zero-forcing (ZF) equalization, let the time-domain channel circulation matrix be H, the receiving end noise vector be n = [n(0), n(1),..., n(N-1)], and the average power be After diagonalization of H by fast Fourier transform, the general expression of the receiving end instantaneous signal-to-noise ratio expression ρ is obtained.

[0117] After the transmitted signal passes through the channel, let the receiving end noise signal be n = [n(0), n(1),..., n(N-1)], then the receiving end signal is:

[0118]

[0119] After zero-forcing ZF equalization, the estimated constellation point is expressed as:

[0120]

[0121] Let x kLet x be the k-th element. Since, under zero-forcing ZF equalization, the noise is uniformly distributed across the chirp subcarriers of the OCDM, the instantaneous signal-to-noise ratio ρ at the receiver is expressed as:

[0122]

[0123] Let F be an N×N Discrete Fourier Transform (DFT) matrix, then H and H in equation (7) -1 Represented as:

[0124] H = F H DF, H- 1 =F H DF, #(8)

[0125] Where D is an N×N diagonal matrix, and its elements are calculated as the Channel Impulse Response (CIR) h = [h0, h1, ... h2]. N-1 The N-point Fourier transform of h n If n = 0, 1, ..., N-1 is the nth path, then:

[0126]

[0127] Where, λ i Let i = 0, ..., N-1 represent the N eigenvalues ​​of H; define λ can be used i The general expression for instantaneous signal-to-noise ratio:

[0128]

[0129] S5. Calculate the probability density function (PDF) of ρ with respect to the eigenvalues ​​of H. Combine this with the expression for the bit error rate (BER) of M-QAM constellation points in an Additive Gaussian Qhite Noise (AWGN) channel. Integrate with ρ to obtain the expressions for the BER and its upper and lower bounds in single-path Rayleigh channels and two-path Rayleigh channels.

[0130] The bit error rate (BER) expression for M-QAM in a white noise (AWGN) channel is:

[0131]

[0132] in, Represents the Gaussian Q-function, C Mand ψ(i, k) is a specific coefficient, i, k are corresponding indexes, and

[0133]

[0134] Therefore, the bit error rate P b of 4-QAM can be expressed as Let the probability density function PDF of p be f ρ (ρ), and the integral of p is obtained to calculate the bit error rate P b under the zero-forcing ZF equalization, where the general expression of p is given by equation (10):

[0135]

[0136] Under the single-path Rayleigh channel, the channel impulse response (CIR) is expressed as |h0| 2 obeys an exponential distribution, and the average power is denoted as At this time, the N eigenvalues are all λ i =h0, i=0, 1,..., N-1, and the instantaneous signal-to-noise ratio and its probability density function PDF are expressed as:

[0137]

[0138] Combined with the transformation formula of the Gaussian Q function where ψ is any integral variable, and the bit error rate BER is further expressed as:

[0139]

[0140] Consider another integral equation:

[0141]

[0142] Therefore, the bit error rate BER is finally simplified as:

[0143]

[0144] Under the two-path Rayleigh channel, since H has two different eigenvalues, the channel impulse response CIR is expressed as and where θ0, θ1 are the corresponding phase angles of h0, h1; and the average powers of h0, h1 are denoted as , i.e., θ0, θ1 are uniformly distributed in (-π, π); at this time, the eigenvalues λ=[λ0, λ1,..., λ N-1 ] of H are:

[0145] λ0=λ2=…=λ N-2= h0+ h1, λ1= λ3=... = λ N-1 = h0- h1, # (18)

[0146] and the instantaneous signal-to-noise ratio is expressed as:

[0147]

[0148] Let the random variable w = | λ0| 2 , v = | λ1| 2 , then the random variable X is simplified as:

[0149]

[0150] where the probability density function (PDF) of w, v is expressed as The probability density function (PDF) of w, v is expressed as Let the probability density function (PDF) of X be f X (x), and the cumulative distribution function (CDF) be F X (x) = P{X≤x}, then we have:

[0151]

[0152] We discuss the following cases:

[0153] When , we derive the CDF of X in the first case:

[0154] =:

[0155]

[0156] We discuss the following cases: and Finally, we obtain the PDF of X in the first case: =:

[0157]

[0158] When -1≤t(w, v, x)≤1, # (25), we obtain the following integral domain about w, v, θ:

[0159]

[0160] and

[0161]

[0162] where, for the convenience of expression, let

[0163] The probability density function PDF of X at this time is calculated as:

[0164]

[0165] Combining and the expression of f X (x) is obtained:

[0166]

[0167] At this time, the bit error rate BER is calculated by:

[0168]

[0169] The upper bound of the bit error rate BER is calculated by formula (20), in which if CoSθ=1 for each pair (h0, h1), X takes the minimum value, p takes the minimum value, and the bit error rate BER reaches the theoretical upper bound, at which time the instantaneous signal-to-noise ratio is:

[0170]

[0171] The probability cumulative distribution function CDF of the variable Xl is represented as:

[0172]

[0173] Let then When , the probability density function PDF is represented as:

[0174]

[0175] At this time, the upper bound of the bit error rate BER is represented as:

[0176]

[0177] wherein represents the upper bound of the bit error rate BER.

[0178] The lower bound of the bit error rate BER is calculated by formula (20), in which if CoSθ=-1 for each pair (h0, h1), X takes the maximum value, p takes the maximum value, and the bit error rate BER reaches the theoretical lower bound, at which time the instantaneous signal-to-noise ratio is:

[0179]

[0180] At this time, the variable X uThe probability cumulative distribution function CDF of X is expressed as:

[0181]

[0182] When , X u has a probability density function PDF:

[0183]

[0184] The bit error rate BER lower bound is expressed as:

[0185]

[0186] The transformation formula combined with the Gaussian Q function is: The formula (16) and the formula (38) are further derived as:

[0187]

[0188] Wherein

[0189] When , X u obeys a two-parameter subexponential distribution with parameters (λ0, λ1), and has a probability density function PDF:

[0190]

[0191] At this time, the bit error rate BER lower bound is:

[0192]

[0193] As shown in Figures 2 to 3 , the non-full load OCDM system is described through a single-ray Rayleigh channel and a two-ray Rayleigh channel, and the measured BER after ZF equalization is compared with the theoretical BER, wherein the time-domain channel matrix H of the two-ray Rayleigh channel includes two different eigenvalues; it can be seen from the figure that when the non-full load factor takes a=1, 2, 4, the theoretical BER is closely matched with the measured BER; in addition, since the symbol power is constant, under the Rayleigh channel, the BER has a theoretical gain of 3dB when a is increased by one time, and the measured BER and the theoretical BER of the two channel models verify this point; Figure 2 And Figure 3 The independent variable γ of each curve in the figure covers a range of 16dB, 18dB, which proves that the BER analysis method is correct under the conditions of high symbol signal-to-noise ratio and low symbol signal-to-noise ratio at the sending end; the above points prove the effectiveness and accuracy of the BER analysis method proposed in the embodiment.

[0194] As​Figure 4 As shown, the non-full load OCDM system is described in the comparison of the measured BER and the theoretical BER after ZF equalization through two-path Rayleigh channels, and the upper and lower bounds of the theoretical BER, wherein the time domain channel matrix H of the two-path Rayleigh channel contains two different eigenvalues; it can be seen from the figure that when a = 4, the theoretical BER closely matches the measured BER; the variable γ covers a range of 18 dB, proving that the BER analysis method is valid under the conditions of high symbol signal-to-noise ratio and low symbol signal-to-noise ratio at the sending end, and in this range, the upper and lower bounds of the theoretical BER can wrap the measured BER, proving the effectiveness and accuracy of the derivation of the BER theoretical value and its upper and lower bounds.

[0195] The existing articles only analyze the BER performance of the non-full load OCDM / OFDM system through simulation results, but do not derive the BER expression of the non-full load OCDM system, especially the BER performance under fading channels, while the modulation and demodulation process of the non-full load OCDM system is given, and the BER performance of the system is derived and analyzed; the time domain channel circulation matrix is diagonalized, and the general BER expression under single-path and two-path Rayleigh channels is given, and the BER expression of the system under single-path and two-path Rayleigh channels is derived; the upper and lower bounds of the BER under the second channel environment are derived, and the simulation results prove the effectiveness and accuracy of the derivation of the BER theoretical value and its upper and lower bounds, and also prove the effectiveness and accuracy of the method.

[0196] In addition to the above embodiments, the present application can have other embodiments. Any technical solution formed by equivalent substitution or equivalent transformation falls within the protection scope required by the present application.

Claims

1. A method for bit error rate analysis of a non-full load OCDM system, characterized by: The method comprises the following steps: S1, at the transmitting end, confirming a non-full load factor a, that is, modulating one subcarrier in every a subcarriers; S2, obtaining a 4-order quadrature amplitude modulation 4-QAM sending symbol s=[s(0), s(a),... s(N-a+1)], wherein N is the number of OCDM subcarriers; S3, set the power normalization factor as P nor , left multiplication of the discrete Fresnel transform matrix Φ, and then multiplied by the coefficient The transmitting power is unchanged; S4, the received signal is processed by zero forcing (ZF) equalization at the receiving end, let the time domain channel circulation matrix be H, the receiving end noise vector be n = [n(0), n(1),..., n(N-1)], and the average power be After diagonalization of H by fast Fourier transform, a general expression of the receiving end instantaneous signal-to-noise ratio expression ρ is obtained. S5, calculating a probability density function PDF of ρ about H characteristic value, combining a bit error rate BER expression of M-order quadrature amplitude modulation M-QAM constellation points under an additive white Gaussian noise AWGN channel, performing integration on ρ, and obtaining expressions of the bit error rate BER and upper and lower bounds thereof under a single-path Rayleigh channel and a two-path Rayleigh channel; In step S5, under the single-path Rayleigh channel, the channel impulse response CIR is represented as h0= h0 2 Subject to the exponential distribution, the average power is recorded as At this time, N eigenvalues are all λ i h0, i = 0, 1,..., N-1, the instantaneous signal-to-noise ratio and its probability density function PDF are represented as: Transformed version of the Gaussian Q function where ψ is any integration variable, and the bit error rate BER is further expressed as: Consider another integral equation: Therefore, the bit error rate BER is finally simplified as:

2. The method for bit error rate analysis based on non-full load OCDM system according to claim 1, characterized in that: The time-domain continuous wave expression of the OCDM sub-carrier in the step S2 is: wherein k is the sub-carrier index value, ranging from 0 to N-1, and N is an even number; T is a whole OCDM symbol duration; j is an imaginary unit, satisfying j 2 = -1; t is the time independent variable. At a single sampling rate, one OCDM symbol is sampled at N points, and the sampling time points are Φ is expressed as: The transmitted OCDM waveform is represented as: x = Φ H s (6) Wherein s represents the transmitted OCDM waveform.

3. The method for bit error rate analysis based on non-full load OCDM system according to claim 1, characterized in that: In the step S3, let s be the power normalized 4-QAM signal, then the power P of the transmitted signal x is x P = E[|s|2] = 1 Let The transmitted symbol power is constant at: where P nor represents a power normalization factor.

4. The method for bit error rate analysis based on non-full load OCDM system according to claim 1, characterized in that: In the step S4, after the transmitting signal passes through the channel, assuming that a receiving end noise signal is n=[n(0), n(1),..., n(N-1)], then a receiving end signal is: After zero-forcing (ZF) equalization, the constellation point is estimated is expressed as: Let x k The instantaneous signal-to-noise ratio ρ at the receiving end of the system is represented as follows, since the noise is uniformly distributed over each chirp subcarrier of the OCDM under zero-forcing (ZF) equalization: Let F be an N x N Discrete Fourier Transform (DFT) matrix, then H and H in equation (11) are given by -1 are given by: H = F H DF, H -1 = F H DF (12) where D is an N x N diagonal matrix with elements calculated as the N-point Fourier transform of the channel impulse response h = [h0, h1,... h N-1 N-1], h n n = 0, 1,..., N - 1 is the nth path, then: Where, λ i Let i = 0, ..., N-1 represent the N eigenvalues ​​of H; define That is, using λ i The general expression for instantaneous signal-to-noise ratio:

5. The method for bit error rate analysis of non-full load OCDM system according to claim 1, wherein: In the step S5, a bit error rate BER expression of M-QAM under an additive white Gaussian noise AWGN channel is: wherein, denotes the Gaussian Q-function, C M and ψ(i, k) are specific coefficients, i, k are corresponding indices, and have Thus, the bit error rate P b is given by Let the probability density function PDF of p be f ρ ( p ). Integrating over p, one obtains the general expression for P b under zero-forcing ZF equalization, where the general expression for p is given by equation ( 14 ) :

6. The method for bit error rate analysis of non-full load OCDM system according to claim 1, wherein: In the step S5, under the two-path Rayleigh channel, because H has two different eigenvalues, the channel impulse response CIR is expressed as and where θ0, θ1are the phase angles corresponding to h0, h1; and the average powers of h0, h1are respectively denoted as that is, θ0, θ1are uniformly distributed in (-π, π); at this time, the eigenvalues λ of H = [λ0, λ1, …, λ N-1 ] are: λ0= λ2=... = λ N-2 = h0+ h1, λ1= λ3=... = λ N-1 = h0- h1 (18) And an instantaneous signal-to-noise ratio is represented as: Random variables w = | λ0| 2 v = | λ1| 2 X is reduced to: where the probability density function PDF of w, v is denoted as with the probability density function PDF of w, v denoted as Let the probability density function PDF of X be f X (x), and the probability cumulative distribution function CDF be denoted as F X (x) = P{X≤x), then we have:

7. The method for bit error rate analysis of a non-full load OCDM system according to claim 6, wherein: In the step S5, When the CDF of X at this time is pushed out: is: The PDF of X at this time is finally obtained as: is: When -1≤t(w, v, x)≤1, #(25), the following integral domain about w, v, θ is obtained: And Set After calculation, the probability density function PDF of X at this time is obtained as: in combination and gives f X the expression for (x): At this time, the bit error rate BER is calculated by the following formula:

8. The method for bit error rate analysis of a non-full load OCDM system according to claim 6, wherein: In the step S5, the upper bound of the bit error rate BER is calculated by formula (20), in formula (20), if cosθ=1 for each pair (h0, h1), X takes the minimum value, ρ takes the minimum value, and the bit error rate BER reaches a theoretical upper bound, at this time, the instantaneous signal-to-noise ratio is: Variable X l The probability cumulative distribution function CDF of X is expressed as: Let Then, the probability density function PDF is given by When the probability density function PDF is given by At this time, the upper bound of the bit error rate BER is represented as: wherein denotes the upper bound on the bit error rate, BER.

9. The method for bit error rate analysis of a non-full load OCDM system according to claim 6, wherein: In the step S5, the lower bound of the bit error rate BER is calculated by formula (20), in formula (20), if cosθ=-1 for each pair (h0, h1), X takes the maximum value, ρ takes the maximum value, and the bit error rate BER reaches a theoretical lower bound, at this time, the instantaneous signal-to-noise ratio is: At this time the variable X u The probability cumulative distribution function CDF of X is represented as: When there is X u the probability density function PDF: Bit error rate (BER) lower bound is represented as: Transformations involving the Gaussian Q function Equations (3) and (38) are further derived as follows: wherein When X u The bivariate subexponential distribution with parameters (λ0, λ1) has the probability density function PDF: At this time, the lower bound of the bit error rate BER is obtained as:

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