Receiving and transmitting end IQ imbalance estimation method based on constant modulus characteristic

Abstract

The invention belongs to the technical field of wireless communication, and particularly relates to a receiving and transmitting end IQ imbalance estimation method based on constant modulus characteristics. The method is used for a terahertz CE-OFDM communication perception integrated system, and the IQ imbalance parameter is solved by establishing an optimization problem based on the constant modulus characteristic by using a signal received by a receiver, so that digital domain compensation in the receiver is facilitated, and a feedback loop and a training process of a traditional method are not needed.

Description

Technical Field

[0001] The present invention belongs to the technical field of wireless communication, and in particular relates to a method for estimating IQ imbalance between a transmitting end and a receiving end based on constant mode characteristics. Background Art

[0002] As a more advanced next-generation mobile communication technology, the sixth generation mobile communication technology (6th Generation Mobile Networks, 6G) has a higher frequency band, larger bandwidth, higher communication rate, and lower latency than the fifth generation mobile communication technology (5G). A single system can integrate wireless signal perception and communication capabilities. The significance of Integrated Sensing and Communication (ISAC) is that the communication system can be regarded as a sensor, which can understand the physical environment through the transmission, reflection and scattering of radio waves. At the same time, perception can realize the ability of imaging, high-precision speed measurement and positioning, grasp the precise information of the channel, and thus improve communication performance. As one of the leading trends of 6G technology and services, ISAC can realize communication and radar functions in a fusion system within the same spectrum, improve the utilization rate of spectrum resources, and avoid interference between communication and perception. The THz band has a larger bandwidth and a smaller wavelength than the low frequency band. Compared with millimeter waves and lower frequency electromagnetic waves, the THz band has a stronger transmission capability; from the perspective of perception, the THz bandwidth is large enough to achieve high-precision positioning and imaging at close range, and has broad application prospects.

[0003] Existing ISAC waveforms are mainly divided into two categories: 1) realizing communication functions on radar signals. 2) realizing radar perception on communication signals. Since 6G has very high requirements for communication rates, the former is not applicable. The technical route of realizing radar perception on communication signals can achieve higher communication rates, but the cost is that the receiver implementation is relatively complex. This technical route is to design ISAC waveforms based on the communication system. The performance indicators are mainly communication performance indicators such as bit error rate, spectrum efficiency, and channel capacity. The perception function is supported under the premise of maximizing communication performance. Communication-friendly waveforms can be used to meet the needs of high-speed data transmission. According to the modulation method, this type of waveform can be divided into single-carrier waveforms and multi-carrier waveforms. Single-carrier waveforms are simple to implement, have low power consumption, and have a low peak to average power ratio (PAPR), but the communication rate is not high, the spectrum efficiency is low, and the ability to resist interference and multipath propagation is weak.

[0004] Unlike single-carrier waveforms, multi-carrier waveforms have higher spectral efficiency and can communicate with high-speed moving targets. Multi-carrier waveforms are generally implemented using Orthogonal Frequency Division Multiplexing (OFDM). Cyclic Prefix OFDM (CP-OFDM) inserts a cyclic prefix on the basis of the OFDM waveform, achieving higher spectral efficiency and being able to resist multipath effects. It is widely used in many communication standards. With the increase of transmission bandwidth and modulation order, the PAPR of CP-OFDM waveform will also increase. The bandwidth of the THz band is very large, so the PAPR will be higher. A higher PAPR will cause the power amplifier to operate in a nonlinear region during transmission, resulting in nonlinear distortion of the signal, causing obvious spectrum spread interference and in-band signal distortion, and seriously degrading system performance. It is necessary to introduce power back-off (IBO) technology. Power back-off is to reduce the input power to make the power amplifier work away from the saturation region to improve the linearity of the power amplifier, but this method will significantly reduce the amplification efficiency of the power amplifier.

[0005] In order to prevent the power amplifier from operating in the nonlinear region, constant envelope waveforms have received considerable attention because such waveforms allow the power amplifier to operate close to its saturation point. Constant Envelope OFDM (CE-OFDM) was first proposed by Thompson et al. in 2004. This waveform uses the OFDM signal as a modulation waveform to drive the complex exponential phase to overcome this problem of continuous phase modulation. The PAPR of the terahertz communication and perception integrated waveform based on CE-OFDM is 0dB, which avoids signal distortion caused by the power amplifier operating in the nonlinear region. However, the IQ imbalance in the ISAC system will still affect the communication and perception performance.

[0006] Commonly used IQ imbalance parameter estimation methods are divided into training sequence method and blind estimation method, among which blind estimation method is further divided into statistical characteristic method and blind source separation method (adaptive method). The training sequence method has a "training" process, which requires the transmitter to first send a training sequence known to the receiver, which has a large overhead. The existing blind estimation method using signal cyclic symmetry will greatly reduce the compensation performance when estimating the IQ imbalance at the transmitting and receiving ends at the same time. The blind source separation method usually uses frequency domain least squares and frequency domain adaptive least mean squares algorithm for compensation, but it is necessary to add a feedback loop for adaptive correction. Summary of the invention

[0007] To solve the above problems, the present invention proposes a method for estimating IQ imbalance at the transmitting and receiving ends based on the constant mode characteristic, which utilizes the constant mode characteristic of the CE-OFDM waveform, has higher estimation accuracy than the training sequence method, and does not require the addition of an adaptive correction feedback loop.

[0008] The technical solution of the present invention is:

[0009] A method for estimating IQ imbalance between a transmitting end and a receiving end based on constant mode characteristics, for use in a CE-OFDM communication perception integrated system, comprises the following steps:

[0010] S1. Define the baseband signal when the IQ path is completely balanced as z I (t) and z Q (t), when the IQ paths are unbalanced, the orthogonal decomposition signal of the equivalent baseband signal x(t) transmitted by the transmitter is:

[0011]

[0012] Among them, a TX represents the transmitter amplitude imbalance parameter, Represents the transmitter phase imbalance parameter; the matrix form of the orthogonal decomposition signal is:

[0013]

[0014] The transmitter converts the equivalent baseband signal x(t)=x I (t)+jx Q (t) send;

[0015] S2. Define the receiver's received signal as:

[0016] c(t)=x(t)+H

[0017] Where H is the Gaussian white noise channel.

[0018] The received signal when the IQ paths are completely balanced is defined as c(t) = c I (t)+jc Q (t), when the IQ paths are unbalanced, the orthogonal decomposition signal of the equivalent baseband signal y(t) received by the receiver is:

[0019] y I (t) = c I (t)

[0020]

[0021] Among them, a RX represents the receiver amplitude imbalance parameter, Represents the receiver phase imbalance parameter; the matrix form of the orthogonal decomposition signal is:

[0022]

[0023] S3. Perform IQ imbalance compensation on y(t). The specific method is as follows:

[0024] Estimate the IQ imbalance parameters of the transmitter and receiver, and construct A in the receiver digital domain -1 and B -1 Correction is performed and the corrected equivalent baseband signal y'(t) = y' I (t)+jy' Q (t), the matrix form of the compensation process of the IQ imbalance between the transmitter and the receiver is:

[0025]

[0026] in, is a TX The estimated value of yes The estimated value of is a RX The estimated value of yes The estimated value of ; thereby correcting the baseband signal Z to be transmitted by the transmitter;

[0027] Since the CE-OFDM signal is a constant modulus signal, the corrected signal must meet the following requirements:

[0028]

[0029] Where y'(n) = y' I (n)+jy' Q (n), N represents the length of the digital signal y'(n), and P represents the power of the transmitted signal Z; let the power of the compensated signal y'(n) be constant, and establish the following problem so that the receiver can compensate the CE-OFDM signal affected by IQ imbalance in the digital domain:

[0030]

[0031] Solve the problem to get the IQ imbalance parameter To achieve IQ imbalance compensation, is a TX ,a RX A simplified representation of ; yes A simplified representation of .

[0032] Specifically, the problem The solution is:

[0033] Calculate var|y'(n)| for each The partial derivatives of are then solved using the Adam method.

[0034] Specifically, the problem Another way to solve is:

[0035] Use function Representation problem And define hour, Get the minimum value 0, expressed as:

[0036]

[0037] where a * and is the actual transmitter and receiver IQ imbalance parameter, is the constant modulus random sequence at the transmitting end, then For function The only global minimum point of ;

[0038] Let the domain of the IQ imbalance compensation parameter be And when hour, along with The change decreases first and then increases or increases / decreases monotonically. hour, along with The change of decreases first and then increases or increases / decreases monotonically; for the continuous single extreme value optimization problem with the above characteristics The alternating iterative method of the three-part method is used to solve the problem. The iterative process of the alternating iterative method of the three-part method is described as follows:

[0039] Set the iteration interval Threshold ε=10 -6 , the number of iterations M;

[0040] For the mth iteration:

[0041] The iterative parameter update is expressed as:

[0042]

[0043]

[0044] for Updates to the Rule of Thirds:

[0045] when a midr -a midl >ε, if Update a l =amidl , otherwise a r =a midr ;

[0046] for Updates to the Rule of Thirds:

[0047] when When renew otherwise

[0048] The output is

[0049] The beneficial effects of the present invention are:

[0050] 1) In the terahertz CE-OFDM interaceptive integrated system, the signal received by the receiver is used to establish an optimization problem based on the constant mode characteristic to solve the IQ imbalance parameters so as to compensate in the digital domain of the receiver without the feedback loop and "training" process of the traditional method;

[0051] 2) The present invention also proposes an alternating three-part method, which has good estimation performance and reduces the bit error rate by more than 50% compared with the training sequence method based on the minimum channel change energy. This paper also proposes an estimation strategy, in which the receiver first uses the Adam method for rough estimation. If the estimated IQ imbalance parameter is small, it can switch to the alternating three-part method for fine estimation to improve the estimation accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is the block diagram of the CE-OFDM communication perception integrated system;

[0053] Figure 2 yes Schematic diagram of the function graph of ;

[0054] Figure 3 It is a performance comparison chart of various compensation schemes;

[0055] Figure 4 It is a comparison chart of the local magnification of the performance of each compensation scheme. DETAILED DESCRIPTION

[0056] The present invention will be further described in detail below in conjunction with the accompanying drawings.

[0057] The present invention is used for CE-OFDM communication perception integrated system, such as Figure 1As shown in the figure, the system model communication transmitter completes channel coding, constellation mapping, serial-to-parallel conversion, IFFT, phase modulation, CP addition, digital-to-analog conversion, and transmission through a power amplifier for serial binary data. Since the CE-OFDM signal is realized by phase modulation of the OFDM signal, the generation method of the OFDM signal is first introduced.

[0058] In communication systems, multipath fading and frequency selective fading have a significant impact on single-carrier waveforms. Multi-carrier waveforms distribute data to N sub-channels for parallel transmission. When the frequency band is divided sufficiently, each sub-channel can be approximated as flat fading, greatly reducing the impact of frequency selective fading. As the basis of modern mobile communication technology, this communication scheme in which sub-carriers overlap but remain orthogonal is called OFDM.

[0059] Considering a series of subcarriers with a subcarrier spacing of △f, the complex envelope of the OFDM transmission waveform is:

[0060]

[0061] Where N represents the number of subcarriers; X OFDM (n) represents the data to be carried on the n+1th subcarrier; T represents the OFDM symbol length. In order to ensure orthogonality between OFDM subcarriers and maximize spectrum utilization, △f = 1 / T is generally selected. OFDM (t) performs Nyquist sampling (f s =B = N△f), that is k=0,1,…,N-1, we have:

[0062]

[0063] In the actual implementation of OFDM transmitters, FFT / IFFT is generally used instead of DFT / IDFT because its efficiency is much higher than DFT / IDFT. The butterfly operation of IFFT requires the number of FFT points to be an integer power of 2, so the number of subcarriers N of OFDM signals is usually an integer power of 2.

[0064] Next, the modulated OFDM signal is phase modulated to form a CE-OFDM signal. This process maps the traditional OFDM signal to a unit circle. The CE-OFDM baseband signal can be expressed as:

[0065]

[0066] Among them, A is the amplitude of the constant envelope signal, and h is the modulation index. This paper considers the OFDM signal x OFDMWhen (t) is a real number, the input of IFFT needs to be conjugate symmetric to ensure that the OFDM symbols formed after IFFT are all real numbers. Since the phase needs to be a real number, if the OFDM signal is a complex number, take its real part Re{m(t)}.

[0067] The signal transmitted by the transmitter is then modulated to a 220GHz (0.22THz) carrier frequency and transmitted to the receiver through the LOS channel. The transmitter link output power is 8dBm, and the power amplifier saturation output power is 20dBm. Considering the LOS channel, the path loss model of the terahertz electromagnetic wave transmission process is:

[0068] L comm =92.4+20lg(f)+20lg(d)

[0069] Where, f is in GHz and d is in km. Due to the high frequency of terahertz, we introduced directional antennas in both the transmitter and the receiver to alleviate the path loss caused by the high frequency of terahertz.

[0070] In the CE-OFDM communication receiver, the baseband signal is first demodulated, and then the transmitted binary data is obtained through A / D conversion, cyclic prefix removal, phase demodulation, serial-to-parallel conversion, FFT, parallel-to-serial conversion, QAM demodulation, and channel decoding. Assume that the CE-OFDM baseband signal obtained at the receiving end is y CE-OFDM (t) = s CE-OFDM (t), first perform phase demodulation to obtain the OFDM baseband signal y(t), perform serial-to-parallel conversion and perform FFT operation on it, and we have:

[0071]

[0072] From the demodulation, it can be seen that due to the orthogonality between the subcarriers of the OFDM signal, the original data can be perfectly demodulated without being interfered by other subcarriers.

[0073] In order to achieve the compensation of IQ imbalance at the transmitting and receiving ends, the present invention first performs IQ imbalance modeling of the transmitter and the receiver:

[0074] Assume that the baseband signal of IQ is z when the two paths are fully matched (balanced). I (t) and z Q (t). Then when the transmitter has no IQ imbalance, the transmitted equivalent baseband signal z(t) = z I (t)+jz Q (t).

[0075] When the IQ channels are unbalanced, the carrier signals of the I and Q channels are different. TX Represents the transmitter amplitude imbalance parameter; represents the transmitter phase imbalance parameter. And since x(t)=x I (t)+jx Q (t), the orthogonal decomposition signal of the equivalent baseband signal x(t) when TX IQI exists can be obtained as:

[0076]

[0077] Write (1) in matrix form:

[0078]

[0079] The equivalent baseband signal x(t) = x when TX IQI exists I (t)+jx Q After (t) passes through the AWGN channel, we get:

[0080] c(t)=x(t)+H (3)

[0081] Where c(t) = c I (t)+jc Q (t) is the equivalent baseband signal transmitted when the receiver has no IQ imbalance. Similarly, when the receiver also has IQ imbalance, the orthogonal decomposition signal of the equivalent baseband signal y(t) in the presence of RX IQI is:

[0082]

[0083] Write (4) in matrix form:

[0084]

[0085] After the receiver receives y(t), it needs to perform IQ imbalance compensation. One approach is to estimate the IQ imbalance parameters of the transmitter and receiver and construct A in the receiver digital domain. -1 and B -1 Correction is performed. Let the corrected equivalent baseband signal y'(t) = y' I (t)+jy' Q (t), the compensation process of the IQ imbalance between the transmitter and the receiver is written in matrix form as:

[0086]

[0087] It can be seen from formula (6) that if A -1 B -1 In: Transmitter amplitude imbalance parameter a TX ,a RX and transmitter phase imbalance parameters can all be accurately estimated, and the AWGN channel signal-to-noise ratio is relatively high, then X≈Z′, and the baseband signal Z to be transmitted by the transmitter can be better corrected.

[0088] Since the CE-OFDM signal is a constant modulus (constant power), the corrected signal must meet the following requirements:

[0089]

[0090] where y'(n) = y' I (n)+jy' Q (n), N represents the length of the digital signal y'(n), and P represents the power of the transmitted signal Z, which is known at the transmitter but unknown at the receiver. And due to the existence of path loss, the signal power reaching the receiver will also change, so problem (7) cannot be processed in the digital domain of the receiver. From the unbalanced matrix of formula (6) As can be seen from the figure, the IQ imbalance between the transmitter and the receiver shifts the power from |z I (n)+jz Q (n)|=P (i.e. constant modulus) becomes (Non-constant modulus, the power value at each moment is no longer constant). If better compensation can be obtained (through A -1 B -1 ), the power of the compensated signal y'(n) should still be constant, that is, the variance of the signal power is 0. In other words, problem (8) can be solved in the receiver digital domain to compensate for the CE-OFDM signal affected by IQ imbalance:

[0091]

[0092] Where y'(n) = y' I (n)+jy' Q (n). From the compensation process It can be seen that the receiver can use the received signal Y to compensate in the digital domain using problem (8), thereby estimating the optimal IQ imbalance parameter (Since in most studies a TX =a RX , Here we use Instead of a TX ,a RX ; replace to simplify the analysis). However, problem (8) involves the process of finding the modulus and variance, and is a non-convex and non-concave function.

[0093] For problem (8), the present invention proposes two solutions:

[0094] The first is a solution using the Adam algorithm, which includes:

[0095] For problem (8), we can calculate var|y'(n)| for The partial derivatives of , and then use the Adam method to solve this problem. Specifically:

[0096]

[0097] We put The (i-th row, j-th column) element of the matrix is ​​represented by A -1 B -1 (i,j) represents, then in the above formula,

[0098]

[0099] in,

[0100]

[0101] Similarly, we can get

[0102]

[0103] In the above formula,

[0104]

[0105] in,

[0106]

[0107] Adam is a combination of SGDM and RMSProp. It takes into account momentum (it will not get stuck at the current small gradient point) and adaptive learning rate, and basically solves a series of problems of gradient descent, such as random small samples, adaptive learning rate, and easy to get stuck at the small gradient point. The Adam algorithm process is shown in Table 1:

[0108] Table 1 Adam algorithm

[0109]

[0110] The second is to use the alternating third method, specifically:

[0111] It can be observed that when When, problem (8) (using (represented by) takes the minimum value 0 (ignoring the influence of the AWGN channel because it only introduces a fixed variance), that is:

[0112] In the general expression (a * and is the actual transmitter / receiver IQ imbalance parameter, determined and unknown):

[0113] (because is a constant modulus random sequence at the transmitting end. To meet the minimum variance of statistical characteristics, it must be multiplied in front. )

[0114] in other words, For function The only global minimum point of .

[0115] Observe expression (14) (here we assume The same applies to other situations)

[0116]

[0117] It can be seen that when Gradually increase or decrease (away from the optimal solution ), due to For the constant modulus random sequence at the transmitting end, a gradually increasing interference term is introduced. Gradually increase.

[0118] Similarly, observe expression (15) (here we assume a * +△a>0, The same applies to other situations)

[0119]

[0120] It can be seen that when Gradually increase or decrease (away from the optimal solution a * ), due to For the constant modulus random sequence at the transmitting end, a gradually increasing interference term is introduced. Gradually increase.

[0121] It is usually assumed that the actual IQ imbalance parameters a and > 0 (the same applies to other cases), the domain of IQ imbalance compensation parameters can be reasonably set as Therefore, when When fixed, The increase of First monotonically decreasing, then monotonically increasing; when When fixed, The increase of First it decreases monotonically, then it increases monotonically.

[0122] But when hour, along with The changes in cannot be analyzed through expressions;

[0123]

[0124] when hour, along with The change of cannot be analyzed by expression (16); we draw it in a visual way The function graph of Figure 2 As shown, the non-convex and non-concave function There is only one global minimum point, and when hour, along with The change decreases first and then increases or increases / decreases monotonically. hour, along with The change first decreases and then increases or increases / decreases monotonically.

[0125] For the continuous single extremum optimization problem with the above characteristics This paper proposes an alternating iterative scheme based on the trichotomy. With the above characteristics, when performing the three-part method iteration for each variable, it can be ensured that the iteration interval of each variable is gradually reduced. The proposed solution consists of the following steps:

[0126] 1) Initialization and Iteration interval and threshold.

[0127] 2) Update the iteration parameters.

[0128] 3) Update using the rule of thirds The value of .

[0129] 4) Update using the rule of thirds The value of .

[0130] 5) Repeat steps 2), 3), and 4) until the target accuracy or the set number of iterations M is reached.

[0131] The iterative process of steps 2), 3), and 4) will each time increase the amplitude imbalance factor and The optimal value interval of is reduced to 2 / 3 of the original value, so the algorithm can converge quickly when used to solve the joint estimation problem of IQ imbalance parameters at the transmitter and receiver of the above CE-OFDM waveform. Assuming that the number of iterations (related to the target accuracy) is set to M, and the number of sample points of the received discrete sequence y(n) is N, the computational complexity of the proposed algorithm is o(MN). The detailed algorithm flow is shown in Table 2. The IQ imbalance compensation performance of the proposed algorithm in the CE-OFDM system where both the transmitter and the receiver have IQ imbalance is shown in the simulation analysis section.

[0132] Table 2 Alternating Trichotomy

[0133]

[0134]

[0135] Assume that the length of the signal y(n) received by the receiver for IQ imbalance parameter estimation is N, the number of iterations required by the Adam method / alternating three-part method is M, and the complexity of the Adam method for calculating the function gradient is O(D), then:

[0136] Due to the FFT transformation required in the training process, the time complexity of the training sequence method is: O(NlogN).

[0137] Since the time complexity required for the trisection method is O(2log3N), and each iteration of the submitted trisection method requires two steps of trisection, calculating the objective function value requires O(N), and the time complexity is: O(MNlogN).

[0138] For the Adam method, M updates are required. The computational complexity of each gradient descent update is O(N), and calculating the function gradient requires O(D), so the computational complexity is O(MND).

[0139] In summary, when a large number of samples are used, N>D>M, and the complexity of the training sequence method is the lowest among the three. The time complexity of the other two algorithms is not easy to compare directly. It should be noted that the lower complexity of the training sequence method is at the expense of the bit error rate. The Adam algorithm used to solve the optimization problem (8) and the proposed three-step method take into account the constant mode characteristics of the CE-OFDM waveform, making the IQ imbalance parameter compensation performance more superior. For details, see Figure 3 and Figure 4 .

Claims

1. A method for estimating IQ imbalance between a transmitter and receiver based on constant mode characteristics, for use in a CE-OFDM communication perception integrated system, characterized in that: The following steps are involved: S1. Define the baseband signal when the IQ path is completely balanced as z I (t) and z Q (t), when the IQ paths are unbalanced, the orthogonal decomposition signal of the equivalent baseband signal x(t) transmitted by the transmitter is: Among them, a TX represents the transmitter amplitude imbalance parameter, Represents the transmitter phase imbalance parameter; the matrix form of the orthogonal decomposition signal is: The transmitter converts the equivalent baseband signal x(t)=x I (t)+jx Q (t) send; S2. Define the receiver's received signal as: c(t)=x(t)+H Where H is the Gaussian white noise channel; The received signal when the IQ paths are completely balanced is defined as c(t) = c I (t)+jc Q (t), when the IQ paths are unbalanced, the orthogonal decomposition signal of the equivalent baseband signal y(t) received by the receiver is: y I (t)=c I (t) Among them, a RX represents the receiver amplitude imbalance parameter, Represents the receiver phase imbalance parameter; the matrix form of the orthogonal decomposition signal is: S3. Perform IQ imbalance compensation on y(t). The specific method is as follows: Estimate the IQ imbalance parameters of the transmitter and receiver, and construct A in the receiver digital domain -1 and B -1 Correction is performed and the corrected equivalent baseband signal y'(t) = y' I (t)+jy' Q (t), the matrix form of the compensation process of the IQ imbalance between the transmitter and the receiver is: in, is a TX The estimated value of yes The estimated value of is a RX The estimated value of yes The estimated value of ; thereby correcting the baseband signal Z to be transmitted by the transmitter; Since the CE-OFDM signal is a constant modulus signal, the corrected signal satisfies: Where y'(n) = y' I (n)+jy' Q (n), N represents the length of the digital signal y'(n), and P represents the power of the transmitted signal Z; let the power of the compensated signal y'(n) be constant, and establish the following problem so that the receiver can compensate the CE-OFDM signal affected by IQ imbalance in the digital domain: Solve the problem to get the IQ imbalance parameter To achieve IQ imbalance compensation, is a TX ,a RX A simplified representation of ; yes A simplified representation of .

2. The method for estimating IQ imbalance at a transmitting and receiving end based on constant mode characteristics according to claim 1, characterized in that: question The solution is: Calculate var|y'(n)| for each The partial derivatives of are then solved using the Adam method.

3. The method for estimating IQ imbalance at a transmitting and receiving end based on constant mode characteristics according to claim 1, characterized in that: question The solution is: Use function Representation problem And define hour, Get the minimum value 0, expressed as: where a * and is the actual transmitter and receiver IQ imbalance parameter, is the constant modulus random sequence at the transmitting end, then For function The only global minimum point of ; Let the domain of the IQ imbalance compensation parameter be And when hour, along with The change decreases first and then increases or increases / decreases monotonically. hour, along with The change of decreases first and then increases or increases / decreases monotonically; for the continuous single extreme value optimization problem with the above characteristics The alternating iterative method of the three-part method is used to solve the problem. The iterative process of the alternating iterative method of the three-part method is described as follows: Set the iteration interval Threshold ε=10 -6 , the number of iterations M; For the mth iteration: The iterative parameter update is expressed as: for Updates to the Rule of Thirds: when a midr -a midl >ε, if Update a l =a midl , otherwise a r =a midr ; for Updates to the Rule of Thirds: when When renew otherwise The output is

Images