Bit error rate analysis method based on non-full-load OCDM system

Abstract

The invention 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, provides modulation and demodulation processes of the non-full-load OCDM system, and deduces and analyzes BER performance of the system. The method comprises the following steps: diagonalizing a time domain channel cyclic matrix, giving a general BER expression under single-path and two-path Rayleigh channels, and deducing a BER expression of a system under the single-path and two-path Rayleigh channels based on the general BER expression; the simulation result proves the effectiveness and accuracy of the BER theoretical value and the derivation of the upper and lower bounds of the BER theoretical value, and also proves the effectiveness and accuracy of the method disclosed by the invention.

Description

Technical Field

[0001] The present invention relates to the technical field of performance analysis of wireless communication systems, and in particular to a bit error rate analysis method based on a non-fully loaded OCDM system. Background Art

[0002] Orthogonal Chirp Division Multiplexing (OCDM) is a multi-carrier modulation method proposed in 2016. All subcarriers are chirp signals with linear frequency modulation, and each subcarrier occupies the same bandwidth and signal period. Therefore, the OCDM system can achieve the maximum spectrum efficiency under the chirp spread spectrum technology. OCDM performs better than Orthogonal Frequency Division Multiplexing (OFDM) technology in many scenarios, especially in time-frequency dual-selective channels. Therefore, many studies on OCDM systems have emerged, including the following: 1. Design of OCDM system based on digital filter to reduce out-of-band spectrum leakage; 2. Carrier frequency offset estimation; 3. Reducing Peak-to-average Power Ratio (PAPR); 4. Channel estimation; 5. Channel equalization, etc.

[0003] However, existing research is basically based on a fully loaded communication system, that is, each subcarrier sends a constellation point. But in an actual communication system, empty subcarriers can be set, that is, a non-fully loaded communication system can be designed. A non-fully loaded OCDM system means that some chirp subcarriers do not send symbols, and this structure is also applicable to OFDM systems. Compared with a fully loaded system, a non-fully loaded system can obtain a signal-to-noise ratio (SNR) gain, but at the cost of sacrificing a 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-fully loaded OCDM systems have been proven to be superior to non-fully loaded OFDM systems. Some scholars also pointed out that the frequency domain spatial modulation system is actually a special non-fully loaded MIMO (Multiple-Input Multiple-Output)-OFDM system, because it transmits symbols through multiple antennas, reducing the power requirements of high-power amplifiers. The existing technology only analyzes the BER performance of non-fully loaded OCDM / OFDM systems, but does not derive the BER expression of non-fully loaded OCDM systems, especially the derivation and analysis of BER performance under fading channels. Summary of the invention

[0005] In order to solve the above technical problems, the present invention provides a bit error rate analysis method based on a non-fully loaded OCDM system, comprising the following steps:

[0006] S1. At the transmitting end, confirm the non-full load factor a, that is, modulate one subcarrier in every a subcarriers;

[0007] S2. Obtain 4-QAM transmission symbol s=[s(0), s(a), ...s(N-a+1)], where N is the number of OCDM subcarriers;

[0008] S3. Let the power normalization factor be P nor , multiply s by the discrete Fresnel transformation matrix Φ on the left, and then multiply by the coefficient Keep the transmission power unchanged;

[0009] S4. Use zero-forcing ZF equalization to process the received signal at the receiving end. Suppose the time domain channel circulant matrix is ​​H, the receiving end noise vector is n = [n(0), n(1), ... n(N-1)], and the average power is After diagonalizing H through fast Fourier transform, the general expression of the instantaneous signal-to-noise ratio expression ρ at the receiving end is obtained;

[0010] S5. Calculate the probability density function PDF of ρ with respect to the eigenvalue of H, combine it with the bit error rate BER expression of the M-order orthogonal amplitude modulation M-QAM constellation point under the white noise channel AWGN, integrate ρ, and obtain the expressions of the bit error rate BER and its upper and lower bounds under single-path Rayleigh channel and two-path Rayleigh channel.

[0011] The technical solution further defined in the present invention is:

[0012] Furthermore, in step S2, the time domain continuous waveform expression of the OCDM subcarrier is: Where k is the subcarrier index value, ranging from 0 to N-1, assuming N is an even number; T is the duration of a whole OCDM symbol; j is an imaginary unit, satisfying j 2 =-1; t is the time independent variable;

[0013] At a single sampling rate, an OCDM symbol is sampled at N points, and the sampling time point is Then Φ is expressed as:

[0014]

[0015] The transmitted OCDM waveform is represented as:

[0016] x=Φ H s, #(2)

[0017] Wherein, s represents the transmitted OCDM waveform.

[0018] As described above, in a bit error rate analysis method based on a non-fully loaded OCDM system, in step S3, assuming that s is a power-normalized 4-QAM signal, the power P of the transmitted signal x is x for:

[0019]

[0020] set up Then the transmitted symbol power is constant:

[0021]

[0022] Among them, P nor Represents the power normalization factor.

[0023] As described above, in the bit error rate analysis method based on a non-fully loaded OCDM system, in step S4, after the transmitted signal passes through the channel, assuming that the noise signal at the receiving end is n=[n(0), n(1), ..., n(N-1)], the receiving end signal is:

[0024]

[0025] After forcing the ZF equalizer to zero, the constellation point is estimated. It is expressed as:

[0026]

[0027] Let x k is the kth element of x. Since the noise is evenly distributed on each chirp subcarrier of the OCDM chirp signal under zero-forcing ZF equalization, the instantaneous signal-to-noise ratio ρ of the system at the receiving end is expressed as:

[0028]

[0029] Assume that F is an N×N discrete Fourier transform DFT matrix, then H and H in equation (7) are -1 It is expressed as:

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

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

[0032]

[0033] Among them, λ i , i = 0, ..., N-1 represents the N eigenvalues ​​of H; definition You can use λ i The general expression for the instantaneous signal-to-noise ratio is:

[0034]

[0035] As described above, in the bit error rate analysis method based on a non-fully loaded OCDM system, in step S5, the bit error rate BER of M-QAM under the white noise channel AWGN is expressed as:

[0036]

[0037] in, represents the Gaussian Q function, C M and ψ(i, k) are specific coefficients, i, k are corresponding indices, and

[0038]

[0039] Therefore, the bit error rate P of 4-QAM is b It can be expressed as Let the probability density function PDF of ρ be f ρ (ρ), integrating ρ, we can get P under zero-forcing ZF equilibrium b The general expression of , where the general expression of ρ is given by equation (10):

[0040]

[0041] As described above, in a bit error rate analysis method based on a non-fully loaded OCDM system, in step S5, under a single-path Rayleigh channel, the channel impulse response CIR is expressed as |h0| 2 It obeys exponential distribution, and 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 expressed as:

[0042]

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

[0044]

[0045] Consider another integral equation:

[0046]

[0047] Therefore, the bit error rate BER is finally simplified to:

[0048]

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

[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] Random variables w=|λ0| 2 , v = |λ1| 2 , then X is simplified to:

[0054]

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

[0056]

[0057] As described above, in the bit error rate analysis method based on a non-fully loaded OCDM system, in step S5,

[0058] when , we can deduce the CDF of X at this time: for:

[0059]

[0060] Finally, we get the PDF of X at this time: for:

[0061]

[0062] When -1≤t(w, v, x)≤1, #(25), the following integral domain with respect to w, v, θ is obtained:

[0063]

[0064] and

[0065]

[0066] set up

[0067] After calculation, the probability density function PDF of X at this time is:

[0068]

[0069] Combination and Get f X The expression of (x) is:

[0070]

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

[0072]

[0073] As described above, in a bit error rate analysis method based on a non-fully loaded OCDM system, in 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), then X takes a minimum value, ρ takes a minimum value, and the bit error rate BER reaches a theoretical upper bound. At this time, the instantaneous signal-to-noise ratio is:

[0074]

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

[0076]

[0077] set up Then there is when When , the probability density function PDF is expressed as:

[0078]

[0079] At this time, the upper bound of the bit error rate BER is expressed as:

[0080]

[0081] in, Indicates the upper bound of the bit error rate BER.

[0082] As described above, in a bit error rate analysis method based on a non-fully loaded OCDM system, in step S5, the lower bound of the bit error rate BER is calculated by formula (20). In formula (20), if each pair (h0, h1) has cosθ=-1, then 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 is expressed as:

[0085]

[0086] when When there is X u The probability density function PDF is:

[0087]

[0088] Bit Error Rate BER Lower Bound It is expressed as:

[0089]

[0090] Combining the transformation formula of Gaussian Q function Formula (16) and formula (38) are further deduced to obtain:

[0091]

[0092] in

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

[0094]

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

[0096]

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

[0098] Existing articles only analyze the BER performance of non-fully loaded OCDM / OFDM systems through simulation results, but do not derive the BER expression of the non-fully loaded OCDM system, especially the derivation and analysis of the BER performance under a fading channel. The present invention provides the modulation and demodulation process of the non-fully loaded OCDM system, derives and analyzes the BER performance of the system; diagonalizes the time domain channel circulant matrix, provides a universal BER expression under a single-path and two-path Rayleigh channel, and derives the BER expression of the system under a single-path and two-path Rayleigh channel based on this; derives the BER upper and lower bounds under the second channel environment, and the simulation results also confirm the correctness of the derivation. BRIEF DESCRIPTION OF THE DRAWINGS

[0099] Figure 1 It is a schematic diagram of the overall process of the present invention;

[0100] Figure 2 This is a comparison diagram between the measured BER and the theoretical BER when a=1, 2, and 4 in a single-path Rayleigh channel according to an embodiment of the present invention;

[0101] Figure 3 This is a comparison diagram between the measured BER and the theoretical BER when a=1, 2, and 4 in a two-path Rayleigh channel according to an embodiment of the present invention;

[0102] Figure 4 1 is a comparison chart of the measured BER and the upper and lower bounds of the theoretical BER when a=4 in a two-path Rayleigh channel in an embodiment of the present invention. DETAILED DESCRIPTION

[0103] This embodiment provides a bit error rate analysis method based on a non-fully loaded OCDM system, such as Figure 1 As shown, the following steps are included:

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

[0105] S2. Obtain the 4th-order Quadrature Amplitude Modulation (QAM) transmission symbol s=[s(0), s(a), ...s(N-a+1)], where N is the number of OCDM subcarriers; the time domain continuous waveform expression of the OCDM subcarrier is: Where k is the subcarrier index value, ranging from 0 to N-1, assuming N is an even number; T is the duration of a whole OCDM symbol; j is an imaginary unit, satisfying j 2 =-1; t is the time independent variable.

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

[0107]

[0108] The transmitted OCDM waveform is represented 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 , multiply s by the Discrete Fresnel Transformation Matrix (DFnT)Φ on the left, and then multiply by the coefficient To ensure that the transmission power remains unchanged; let s be the power normalized 4-QAM signal, then the power P of the transmitted signal x x for:

[0112]

[0113] set up Then the transmitted symbol power is constant:

[0114]

[0115] Among them, P nor Represents the power normalization factor.

[0116] S4. Use zero-forcing (ZF) equalization to process the received signal at the receiving end. Suppose the time domain channel circulant matrix is ​​H, the receiving end noise vector is n = [n(0), n(1), ... n(N-1)], and the average power is After diagonalizing H through fast Fourier transform, the general expression of the instantaneous signal-to-noise ratio expression ρ at the receiving end is obtained.

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

[0118]

[0119] After forcing the ZF equalizer to zero, the constellation point is estimated. It is expressed as:

[0120]

[0121] Let x kis the kth element of x. Since the noise is evenly distributed on each chirp subcarrier of the OCDM chirp signal under zero-forcing ZF equalization, the instantaneous signal-to-noise ratio ρ of the system at the receiving end is expressed as:

[0122]

[0123] Assume that F is an N×N Discrete Fourier Transform (DFT) matrix, then H and H in equation (7) are -1 It is expressed as:

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

[0125] Where D is an N×N diagonal matrix, and the element calculation is the channel impulse response (CIR) h = [h0, h1, ... h N-1 ] is the N-point Fourier transform, h n , n=0, 1, ..., N-1 is the nth path, then:

[0126]

[0127] Among them, λ i , i = 0, ..., N-1 represents the N eigenvalues ​​of H; definition You can use λ i The general expression for the instantaneous signal-to-noise ratio is:

[0128]

[0129] S5. Calculate the probability density function (PDF) of ρ with respect to the eigenvalue of H, combine it with the bit error rate (BER) expression of the M-order quadrature amplitude modulation M-QAM constellation point in the white noise channel (Additive Gaussian Qhite Noise, AWGN), integrate ρ, and obtain the expressions of the bit error rate BER and its upper and lower bounds in the single-path Rayleigh channel and the two-path Rayleigh channel.

[0130] The bit error rate BER of M-QAM in white noise channel AWGN is expressed as:

[0131]

[0132] in, represents the Gaussian Q function, C Mand ψ(i, k) are specific coefficients, i, k are corresponding indices, and

[0133]

[0134] Therefore, the bit error rate P of 4-QAM is b It can be expressed as Let the probability density function PDF of ρ be f ρ (ρ), integrating ρ, we can get P under zero-forcing ZF equilibrium b The general expression of , where the general expression of ρ is given by equation (10):

[0135]

[0136] In a single-path Rayleigh channel, the channel impulse response (CIR) is expressed as |h0| 2 It obeys exponential distribution, and 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 expressed as:

[0137]

[0138] Combining the transformation formula of 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 to:

[0143]

[0144] In a two-path Rayleigh channel, since H has two different eigenvalues, the channel impulse response CIR is expressed as and Where θ0 and θ1 are the phase angles corresponding to h0 and h1 respectively; the average powers of h0 and h1 are That is, θ0, θ1 are uniformly distributed on (-π, π); at this time, the eigenvalue λ of H = [λ0, λ1, ..., λ N-1 ]for:

[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] Random variables w=|λ0| 2 , v = |λ1| 2 , then X is simplified to:

[0149]

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

[0151]

[0152] Discuss the following situations:

[0153] when When , we can deduce the CDF of X in the first case:

[0154] for:

[0155]

[0156] Discuss the situation and Finally, we get the PDF of X in the first case: for:

[0157]

[0158] When -1≤t(w, v, x)≤1, #(25), the following integral domain with respect to w, v, θ is obtained:

[0159]

[0160] and

[0161]

[0162] For the convenience of expression,

[0163] After calculation, the probability density function PDF of X at this time is:

[0164]

[0165] Combination and Get f X The expression of (x) is:

[0166]

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

[0168]

[0169] The upper bound of the bit error rate BER is calculated by equation (20). In equation (20), if CoSθ=1 for every pair (h0, h1), then X takes the minimum value, ρ takes the minimum value, and the bit error rate BER reaches the theoretical upper bound. At this time, the instantaneous signal-to-noise ratio is:

[0170]

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

[0172]

[0173] set up Then there is when When , the probability density function PDF is expressed as:

[0174]

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

[0176]

[0177] in, Indicates 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 formula (20), if CoSθ=-1 for every pair (h0, h1), then X reaches its maximum value, ρ reaches its maximum value, and the bit error rate BER reaches its theoretical lower bound. At this time, the instantaneous signal-to-noise ratio is:

[0179]

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

[0181]

[0182] when When there is X u The probability density function PDF is:

[0183]

[0184] Bit Error Rate BER Lower Bound It is expressed as:

[0185]

[0186] Combining the transformation formula of Gaussian Q function Formula (16) and formula (38) are further deduced to obtain:

[0187]

[0188] in

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

[0190]

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

[0192]

[0193] like Figures 2 to 3 As shown in the figure, the comparison between the measured BER and the theoretical BER of the non-fully loaded OCDM system after ZF equalization through a single-path Rayleigh channel and a two-path Rayleigh channel is described. 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 the non-fully loaded factor is a=1, 2, 4, the theoretical BER is closely matched with the measured BER. In addition, since the symbol power is constant, in the Rayleigh channel, every time a doubles, the BER theoretically has a 3dB gain, and the measured BER and theoretical BER of the two channel models have verified this point. Figure 2 and Figure 3 The independent variable γ of each curve covers the range of 16dB and 18dB respectively, which proves 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 transmitting end; the above points prove the effectiveness and accuracy of the BER analysis method proposed in this embodiment.

[0194] like Figure 4 As shown in the figure, the comparison between the measured BER and the theoretical BER of a non-fully loaded OCDM system after ZF equalization through a two-path Rayleigh channel, as well as the upper and lower bounds of the theoretical BER, are described. 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 fits the measured BER closely. The independent variable γ covers a range of 18dB, 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 transmitter, and within this range, the theoretical upper and lower bounds can cover the measured BER, proving the validity and accuracy of the BER theoretical value and its upper and lower bounds.

[0195] Existing articles only analyze the BER performance of non-fully loaded OCDM / OFDM systems through simulation results, but do not derive the BER expression of non-fully loaded OCDM systems, especially deriving and analyzing the BER performance under fading channels. The method of this embodiment gives the modulation and demodulation process of the non-fully loaded OCDM system, derives and analyzes the BER performance of the system; diagonalizes the time domain channel circulant matrix, gives the general BER expression under single-path and two-path Rayleigh channels, and derives the BER expression of the system under single-path and two-path Rayleigh channels based on this; derives the BER upper and lower bounds of the second channel environment, and the simulation results prove the validity and accuracy of the BER theoretical value and the derivation of its upper and lower bounds, and also proves the validity and accuracy of the method of the present invention.

[0196] In addition to the above embodiments, the present invention may also have other implementation modes. Any technical solution formed by equivalent replacement or equivalent transformation falls within the protection scope required by the present invention.

Claims

1. A bit error rate analysis method based on a non-fully loaded OCDM system, characterized by: The following steps are involved: S1. At the transmitting end, confirm the non-full load factor a, that is, modulate one subcarrier in every a subcarriers; S2. Obtain 4-QAM transmission symbol s=[s(0), s(a), ...s(N-a+1)], where N is the number of OCDM subcarriers; S3. Let the power normalization factor be P nor , multiply s by the discrete Fresnel transformation matrix Φ on the left, and then multiply by the coefficient Keep the transmission power unchanged; S4. Use zero-forcing ZF equalization to process the received signal at the receiving end. Suppose the time domain channel circulant matrix is ​​H, the receiving end noise vector is n = [n(0), n(1), ... n(N-1)], and the average power is After diagonalizing H through fast Fourier transform, the general expression of the instantaneous signal-to-noise ratio expression ρ at the receiving end is obtained; S5. Calculate the probability density function PDF of ρ with respect to the eigenvalue of H, combine it with the bit error rate BER expression of the M-order orthogonal amplitude modulation M-QAM constellation point under the white noise channel AWGN, integrate ρ, and obtain the expressions of the bit error rate BER and its upper and lower bounds under single-path Rayleigh channel and two-path Rayleigh channel.

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

3. The bit error rate analysis method based on a non-fully loaded OCDM system according to claim 1, characterized in that: In step S3, let s be the power-normalized 4-QAM signal, then the power P of the transmitted signal x is x for: set up Then the transmitted symbol power is constant: Among them, P nor Represents the power normalization factor.

4. The bit error rate analysis method based on a non-fully loaded OCDM system according to claim 1, characterized in that: In step S4, after the transmitted signal passes through the channel, assuming that the noise signal at the receiving end is n=[n(0), n(1), ..., n(N-1)], the signal at the receiving end is: After forcing the ZF equalizer to zero, the constellation point is estimated. It is expressed as: Let x k is the kth element of x. Since the noise is evenly distributed on each chirp subcarrier of the OCDM chirp signal under zero-forcing ZF equalization, the instantaneous signal-to-noise ratio ρ of the system at the receiving end is expressed as: Assume that F is the N×N discrete Fourier transform DFT matrix, then H and H in equation (7) are -1 It is expressed as: H=F H DF,H- 1 =F H DF,#(8) Where D is an N×N diagonal matrix, and the elements are calculated as the channel impulse response h=[h0, h1, ...h N-1 ] is the N-point Fourier transform, h n , n=0, 1, ..., N-1 is the nth path, then: Among them, λ i , i = 0, ..., N-1 represents the N eigenvalues ​​of H; definition You can use λ i The general expression for the instantaneous signal-to-noise ratio is:

5. The bit error rate analysis method based on a non-fully loaded OCDM system according to claim 1, characterized in that: In step S5, the bit error rate BER of M-QAM under the white noise channel AWGN is expressed as: in, represents the Gaussian Q function, C M and ψ(i, k) are specific coefficients, i, k are corresponding indices, and Therefore, the bit error rate P of 4-QAM is b It can be expressed as Let the probability density function PDF of ρ be f ρ (ρ), integrate ρ, and get the calculation of P under zero-forcing ZF equilibrium b The general expression of , where the general expression of ρ is given by equation (10):

6. The bit error rate analysis method based on a non-fully loaded OCDM system according to claim 1, characterized in that: In step S5, under a single-path Rayleigh channel, the channel impulse response CIR is expressed as |h0| 2 It obeys exponential distribution, and 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 expressed as: Combining the transformation formula of Gaussian Q function Where ψ is any integral variable, and the bit error rate BER is further expressed as: Consider another integral equation: Therefore, the bit error rate BER is finally simplified to:

7. The bit error rate analysis method based on a non-fully loaded OCDM system according to claim 1, characterized in that: In step S5, in a two-path Rayleigh channel, since H has two different eigenvalues, the channel impulse response CIR is expressed as and Where θ0 and θ1 are the phase angles corresponding to h0 and h1 respectively; the average powers of h0 and h1 are That is, θ0, θ1 are uniformly distributed on (-π, π); at this time, the eigenvalue λ of H = [λ0, λ1, ..., λ N-1 ]for: λ0=λ2=…=λ N-2 =h0+h1,λ1=λ3=…=λ N-1 =h0-h1,#(18) And the instantaneous signal-to-noise ratio is expressed as: Random variables w=|λ0| 2 , v=|λ1| 2 , then X is simplified to: Among them, the probability density function PDF of w, v is expressed as Its probability density function PDF is Let the probability density function PDF of X be f X (x), and the probability cumulative distribution function CDF is expressed as F X (x)=P{X≤x}, then:

8. The bit error rate analysis method based on a non-fully loaded OCDM system according to claim 7, characterized in that: In the step S5, when #(22), the CDF of X at this time is: for: Finally, we get the PDF of X at this time: for: When -1≤t(w, v, x)≤1, #(25), the following integral domain with respect to w, v, θ is obtained: and set up After calculation, the probability density function PDF of X at this time is: Combination and Get f X The expression of (x) is: At this time, the bit error rate BER is calculated by the following formula:

9. The bit error rate analysis method based on a non-fully loaded OCDM system according to claim 7, characterized in that: In 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 a minimum value, ρ takes a minimum value, and the bit error rate BER reaches the theoretical upper bound. At this time, the instantaneous signal-to-noise ratio is: VARIABLE X l The probability cumulative distribution function CDF is expressed as: set up Then there is when When , the probability density function PDF is expressed as: At this time, the upper bound of the bit error rate BER is expressed as: in, Indicates the upper bound of the bit error rate BER.

10. The bit error rate analysis method based on a non-fully loaded OCDM system according to claim 7, characterized in that: In 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 the 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 is expressed as: when When there is X u The probability density function PDF is: Bit Error Rate BER Lower Bound It is expressed as: Combining the transformation formula of Gaussian Q function Formula (16) and formula (38) are further deduced to obtain: in when When X u It obeys a two-parameter subexponential distribution with parameters (λ0, λ1) and has a probability density function PDF: At this time, the lower bound of the bit error rate BER is:

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