A method for efficiently calculating the DFT of ZC sequences

Abstract

This invention belongs to the field of wireless communication and signal processing technology, specifically relating to an efficient method for calculating the DFT of a ZC sequence. Starting from the concept of micro-frequency hopping, and based on the fact that the essence of a ZC sequence is a linear micro-frequency hopping symbol, this invention transforms the classic method for calculating the DFT of ZC into calculating a linear micro-frequency hopping symbol with frequency shift and initial phase offset, according to the principle of phase equal to the frequency point accumulation of the micro-frequency hopping pattern and micro-frequency hopping cyclic frequency shift modulation. Therefore, the DFT result can be quickly calculated simply through frequency point accumulation. Specifically, this method initializes parameters such as the modulo P inverse iu of the root value u, Legendre symbol, quasi-phase offset value Po, quasi-frequency shift value Fs, and frequency point step size Fstep, and then iteratively updates the quasi-frequency value and quasi-phase value, thereby obtaining the final DFT result of the ZC sequence based on the phase. The entire iterative calculation process only requires 2(P-1) additions and P complex exponential operations, reducing the classical DFT computation by approximately 80%.

Description

Technical Field

[0001] This invention belongs to the field of wireless communication and signal processing technology, and specifically relates to a method for efficiently calculating the DFT of ZC sequences. Background Technology

[0002] The Zadoff-Chu (ZC) sequence is a complex sequence with excellent autocorrelation and cross-correlation properties, widely used in various fields. In 4G LTE and 5G NR systems, the ZC sequence is widely used for the generation and processing of various physical layer signals, such as synchronization signals, preamble sequences for random access channels, reference signals for uplink control channels, and demodulation and detection reference signals for uplink data channels. These applications utilize the low autocorrelation and low cross-correlation properties of the ZC sequence to improve signal synchronization performance and anti-interference capabilities. The constant envelope and zero autocorrelation properties of the ZC sequence also make it valuable in radar and Global Navigation Satellite Systems (GNSS). For example, in radar systems, the ZC sequence is used for pulse compression, improving the resolution and anti-interference capabilities of radar signals; in GNSS, the ZC sequence is used to generate navigation signals, supporting high-precision positioning and synchronization. In particular, the Fourier transform of the ZC sequence is still a ZC sequence, a characteristic that makes it very useful in Orthogonal Frequency Division Multiplexing (OFDM) systems, as it can maintain consistent signal characteristics in both the time and frequency domains, thereby simplifying signal processing and improving system performance.

[0003] In 2009, S. Beyme and C. Leung proposed an efficient method for computing the DFT of Zadoff-Chu sequences. Specifically, it can be expressed as: Fu(k) = conj(Zu(iu*k))*Fu(0); k = 0, 1, ..., P-1, P is a prime number; where Fu(k) is the DFT value of the ZC sequence with root value u, conj is the conjugate, Zu is the time-domain ZC signal with root value u, iu is the modulo P inverse of root value u, k is the index value, and Fu(0) is the cumulative sum of Zu, that is, Fu(0) equals sum(Zu), where sum is the summation.

[0004] This method reduces the complexity of the traditional Discrete Fourier Transform (DFT) from O(P*log(P)) to O(P). Specifically, it can be achieved by first calculating the time-domain ZC sequence Zu with root value u, then rearranging the Zu sequence according to mod(iu*k,P), and finally multiplying the conjugate of the sequence by the sum of Zu, sum(Zu). This method requires storing the time-domain Zu sequence, which places high demands on storage resources. Alternatively, it can be generated as follows: Fu(k) = exp(2i*pi*mod(iu*k,P)*(mod(iu*k,P)+1) / P / 2)*sum(Zu); thus, calculating Fu(k) includes:

[0005] 1. Calculate sum(Zu), including:

[0006] 1.1. Calculate u*i*(i+1) using 2*P multiplications;

[0007] 1.2. Calculate the Zu sequence based on the phase 2*pi*u*i*(i+1) / 2 / P, and calculate the Pth complex exponent;

[0008] 1.3. Calculate the cumulative sum of the Zu sequence, P-1 times of complex addition.

[0009] 2. Calculate the ZC conjugate sequence rooted at iu, including:

[0010] 2.1. Calculate the modular inverse iu of u;

[0011] 2.2. Calculate mod(iu*k,P), performing P modulo operations;

[0012] 2.3. Calculate u*mod(iu*k,P)*(mod(iu*k,P)+1), which involves 2*P multiplications;

[0013] 2.4. Calculate the Fu sequence based on the phase 2*pi*u*mod(iu*k,P)*(mod(iu*k,P)+1) / 2 / P, and calculate the P-th complex exponent.

[0014] 3. Multiplying the ZC conjugate sequence with root value iu by sum(Zu) requires P complex multiplications.

[0015] This method was probably the most efficient way to calculate the DFT of ZC sequences before the concept of micro-frequency hopping emerged. However, based on the patented technology and ideas of micro-frequency hopping, the efficiency of calculating the DFT of ZC sequences will be greatly improved from a completely new perspective. Summary of the Invention

[0016] The purpose of this invention is to provide an efficient method for calculating the DFT of a ZC sequence. This method is based on the cyclic phase recursion method and reduces the computational cost of the DFT of the current optimal ZC sequence by more than 80%, thereby solving the problems mentioned in the background art.

[0017] To achieve the above objectives, the present invention adopts the following technical solution: a method for efficiently calculating the DFT of a ZC sequence, wherein the method is for a ZC sequence Zu with length P being a prime number and root value u, Zu(k) = exp(-1i*pi*u*k*(k+1) / P), and the DFT calculation of the ZC sequence can be efficiently achieved only through cyclic phase recursion, including the following steps:

[0018] First, initialize: Calculate the modular inverse iu of the root value u modulo P, calculate the Legendre symbol corresponding to the root value u, calculate the quasi-phase deviation value Po, calculate the quasi-frequency shift value Fs, calculate the frequency point step size Fstep; reset the count value k, assign the initial value of the quasi-frequency value freq to the quasi-frequency shift value Fs, and assign the initial value of the quasi-phase value phase to the quasi-phase deviation value Po: k = 0; freq = Fs; phase = Po.

[0019] Then, calculate the DFT of Zu in a loop: Let Fu be the DFT result of Zu. When k < P, first calculate the complex exponential value of Fu(k) according to the quasi-phase value phase divided by P: Fu(k) = exp(-2i * pi * phase / P).

[0020] Then, recursively update the count value, quasi-frequency value, and quasi-phase value: k = k + 1; freq = freq + Fstep; phase = phase + freq.

[0021] Preferably, calculating the quasi-phase deviation value Po includes:

[0022] First, calculate the Legendre symbol corresponding to the root value u. Let: t = (mod(u, 2) * P + u) / 2; then the Legendre symbol of t, (t / P), is the Legendre symbol corresponding to u.

[0023] Then, calculate the quasi-phase deviation Po. The formula is as follows: Po = ((mod(P, 4) * t + 1 - (t / P) + (mod(P, 4) == 3)) * P - t) / 4.

[0024] Preferably, the formula for calculating the quasi-frequency shift value Fs is: Fs = (P + 1) / 2 * (iu - 1).

[0025] Preferably, the formula for calculating the frequency point step size Fstep is: Fstep = P - iu.

[0026] Preferably, directly obtain the accumulated sum of the ZC sequence using the quasi-phase deviation value Po. The formula is: Fu(0) = sqrt(P) * exp(-2i * pi * Po / P).

[0027] Preferably, calculating the complex exponential value of Fu(k) according to the quasi-phase value phase divided by P is implemented using Cordic to reduce complex exponential calculations, including:

[0028] By normalizing the pre-stored angle theta of Cordic and multiplying it by P, that is, theta = arctan(2^(-i)) / 2 / pi * P, Fu(k) can be directly calculated using the quasi-phase phase without the operation of dividing by P.

[0029] Preferably, the calculation of the modular inverse iu of the root value u and the Legendre symbol corresponding to the root value u includes:

[0030] Calculate the inverses iu of all root values ​​u in advance, as well as the Legendre symbols (t / P) corresponding to all u, and store them in a table to reduce the amount of calculation.

[0031] Preferably, the method further includes: reducing the data bit width by modifying the quasi-frequency value freq and quasi-phase value phase modulo P in each recursive calculation, i.e., freq = mod(freq + Fstep, P) and phase = mod(phase + freq, P).

[0032] Preferably, the method further includes:

[0033] Multiply the DFT result Fu of the ZC sequence by the square root of P, i.e., Fu = sqrt(P) * Fu; to make the result conform to the definition of nonnormalization.

[0034] Preferably, the method further includes:

[0035] When performing a DFT on the ZC sequence with a cyclic shift "S" used in current 4G and 5G communication systems, the method requires changing the quasi-frequency shift value Fs from Fs = (P+1) / 2*(iu-1) to Fs = (P+1) / 2*(iu-1)-S; where S is the cyclic shift value of the ZC sequence.

[0036] Technical effects and advantages of the present invention: The method for efficiently computing the DFT of ZC sequences proposed in this invention has the following advantages compared with the prior art:

[0037] This invention starts from the concept of micro-frequency hopping, taking the essence of the ZC sequence as a linear micro-frequency hopping symbol. Based on the principle of phase equal to the frequency point accumulation of the micro-frequency hopping pattern and the micro-frequency hopping cyclic frequency shift modulation, the classic method for calculating the DFT of ZC is transformed into calculating a linear micro-frequency hopping symbol with frequency shift and initial phase offset. Therefore, the DFT result can be quickly calculated by simply accumulating frequency points. Specifically, this method initializes parameters such as the modulo P inverse iu of the root value u, Legendre symbol, quasi-phase offset value Po, quasi-frequency shift value Fs, and frequency point step size Fstep, and then cyclically updates the quasi-frequency value and quasi-phase value. Finally, the DFT result of the ZC sequence is obtained based on the phase. The entire cyclic calculation process only requires 2(P-1) additions and P complex exponential operations, reducing the classical DFT calculation workload by about 80%. Attached Figure Description

[0038] Figure 1 This is a flowchart of the method for calculating the DFT of a ZC sequence according to the present invention. Detailed Implementation

[0039] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0040] This embodiment proposes an efficient method for computing the DFT of ZC sequences, which reduces the computational cost of the DFT of the current best ZC sequence by more than 80%, as detailed below:

[0041] First, micro-frequency hopping proves that the essence of the ZC sequence is a linear special case of the micro-frequency hopping symbol. Thus, based on the fact that phase is the accumulation of frequency, calculating a ZC sequence only requires a simple cyclic phase recursion, i.e., Zu = exp(-2i*pi*cumsum(u*(0:P-1)) / P), where cumsum is the cumulative sum. Furthermore, according to the principle of micro-frequency hopping cyclic frequency shift modulation, a ZC sequence with a frequency shift Fs can be expressed as Zu = exp(-2i*pi*cumsum(u*(0:P-1)+Fs) / P), where exp is a complex exponential function, i is the imaginary unit, pi is pi, and k = 0, 1, ..., P-1.

[0042] Furthermore, a ZC sequence with frequency shift Fs and phase offset Po can be expressed using micro-frequency hopping as Zu = exp(-2i*pi*(cumsum(u*(0:P-1)+Fs)+Po) / P).

[0043] On the other hand, the DFT of a ZC sequence can also be represented as the conjugate of a ZC sequence with root value iu, frequency shifted by Fs, and multiplied by the cumulative sum Fu(0) of the Zu sequence:

[0044] Fu = exp(1i*pi*iu*k*(k+1) / P)*exp(-2i*pi*Fs / P)*Fu(0); where the frequency shift Fs = (P+1) / 2*(iu-1), and the cumulative sum of the ZC sequence Fu(0) = sum(Zu).

[0045] According to the concept of micro-frequency hopping cyclic frequency shift modulation, the above formula can be rewritten as: Fu=exp(-2i*pi*cumsum(Fstep*(0:P-1)+Fs) / P)*Fu(0); where, the frequency step size Fstep=P-iu.

[0046] At this point, the first half of Fu can be calculated in the same way as the general ZC sequence is calculated using the cyclic phase recursion method. However, it is also necessary to calculate and multiply by Fu(0). In order to make full use of the concept of micro-frequency hopping, it is necessary to further represent the multiplication by Fu(0) using phase offset Po.

[0047] For a ZC sequence with root value u, its cumulative sum Fu(0) can be expressed as: Fu(0)=sum(exp(-1i*pi*u*k*(k+1) / P)).

[0048] further,

[0049] Fu(0)=sum(exp(-2i*pi*u*(k+1 / 2)^2 / P / 2)*exp(2i*pi*u*1 / 4 / P / 2)).

[0050] further,

[0051] Fu(0)=exp(2i*pi*u*1 / 4 / P / 2)*sum(exp(-2i*pi*u*(k+1 / 2)^2 / P / 2)).

[0052] When u is even, let t = u / 2.

[0053] When u is odd, let t = (P + u) / 2.

[0054] Then Fu(0)=exp(-2i*pi*t*1 / 4 / P)*sum(exp(-2i*pi*t*(k+1 / 2)^2 / P)).

[0055] Where, sum(exp(-2i*pi*t*(k+1 / 2)^2 / P)) is the quadratic Gaussian sum G(1 / 2,t,P) with a fraction offset of (1 / 2), that is, Fu(0)=exp(-2i*pi*t*1 / 4 / P)*G(1 / 2,t,P); when P modulo 4 is 1, G(1 / 2,t,P)=(t / P)*sqrt(P)*exp(2i*pi*t / 4 / P).

[0056] When P modulus 4 is 3

[0057] G(1 / 2,t,P)=(t / P)*exp(2i*pi*1 / 4 / P)*sqrt(P)*exp(2i*pi*3*t / 4 / P);

[0058] Where (t / P) is the Legendre symbol for t. When t is a quadratic residue modulo P, (t / P) = 1, and when t is not a quadratic residue modulo P, (t / P) = -1.

[0059] Furthermore, Fu(0) can be expressed as: Fu(0)=sqrt(P)*exp(-2i*pi*Po / P); where, Po=((mod(P,4)*t+1-(t / P)+(mod(P,4)==3))*Pt) / 4, t=(mod(u,2)*P+u) / 2.

[0060] In summary, Fu can be represented as:

[0061] Fu = sqrt(P)*exp(-2i*pi*(cumsum((P-iu)*(0:P-1)+Fs)+Po) / P); ​​where Fs is the quasi-frequency shift value and Po is the quasi-phase offset value.

[0062] The Fu in the above formula is already a complete expression of micro-frequency hopping, so it can be efficiently implemented simply through cyclic phase recursion.

[0063] The above method also includes: when performing a DFT on the ZC sequence with cyclic shift "S" used in the current 4G and 5G communication systems, Zus(k) = exp(-1i*pi*u*(k+S)*((k+S)+1) / P), the method requires changing the quasi-frequency shift value Fs from Fs = (P+1) / 2*(iu-1) to Fs = (P+1) / 2*(iu-1)-S; where S is the cyclic shift value of the ZC sequence.

[0064] The following combination Figure 1 The flowchart illustrates how to quickly and efficiently calculate the DFT of a ZC sequence using the cyclic phase recursion method.

[0065] First, initialization is performed, including: calculating the modular inverse of u, iu, intermediate variable t, Legendre symbol of t (t / P), quasi-phase offset value Po, calculating the quasi-frequency shift value Fs, and calculating the frequency step size Fstep; then, the count value k is reset, the quasi-frequency value freq is initialized to the quasi-frequency shift value Fs, and the quasi-phase is initialized to the quasi-phase offset value Po: k = 0; freq = Fs; phase = Po; then, it is determined whether k is greater than P. If so, Fu(k) is calculated in a loop; otherwise, the calculation of Fu is completed and the process exits.

[0066] The iterative calculation of Fu(k) includes: first, dividing the quasi-phase value by P to calculate the current Fu(k), as follows: Fu(k) = exp(-2i*pi*phase / P).

[0067] Then recursively update the count value, quasi-frequency value, and quasi-phase value as follows: k = k + 1; freq = freq + Fstep;

[0068] phase = phase + freq;

[0069] Where Po = ((mod(P,4)*t+1-(t / P)+(mod(P,4)==3))*Pt) / 4, t = (mod(u,2)*P+u) / 2, (t / P) is the Legendre symbol for t, Fs = (P+1) / 2*(iu-1), Fstep = P-iu, iu is the modulo P inverse of the root value u, and mod is modulo.

[0070] Based on the above process, the computational complexity of the cyclic phase recursion method used in this invention is as follows:

[0071] 1: Initialization, calculate the following parameters all at once, including:

[0072] 1.1: Calculate the inverse element iu modulo P;

[0073] 1.2: Calculate t corresponding to u;

[0074] 1.3: Calculate the Legendre symbol (t / P) for t;

[0075] 1.4: Calculate the phase offset Po;

[0076] 1.5: Calculate the frequency shift Fs;

[0077] 1.6: Calculate the frequency step size Fstep.

[0078] 2: P iterations of calculation, including:

[0079] 2.1: Calculate Fu by dividing the quasi-phase value by P, and then calculate the Pth complex exponent;

[0080] 2.2: Recursively update the quasi-frequency value freq, with P-1 additions;

[0081] 2.3: Recursively update the quasi-phase value phase, P-1 additions.

[0082] Apart from the initial computation, the entire loop requires only 2*(P-1) additions and P complex exponentiation operations.

[0083] During initialization, calculating the inverse of u modulo P, iu, requires the use of Fermat's Little Theorem: iu = mod(u^(P-2), P), which requires P-3 multiplications and modulo operations. Even with the power reduction method, there is still a certain amount of computation involved. It is advisable to calculate this in advance and store it in a lookup table.

[0084] Calculating the Legendre symbol (t / P) of t requires the use of Euler's criterion: (t / P) = mod(t^((P-1) / 2),P), which requires (P-1) / 2-1 multiplications and modulo operations. Even with the power reduction method, there is still a certain amount of computation involved. It is advisable to calculate this in advance and store it in a lookup table.

[0085] The remaining calculations, such as the temporary variable t, frequency step size, quasi-frequency shift, and quasi-phase offset, require only 8 additions, 3 multiplications, and 2 modulo operations in total.

[0086] t = (mod(u,2)*P + u) / 2;

[0087] Fstep = P - iu;

[0088] Fs = mod((P+1) / 2*(iu-1),P);

[0089] Po=((mod(P,4)*t+1-(t / P)+(mod(P,4)==3))*Pt) / 4.

[0090] Compared with the methods of S. Beyme and C. Leung, the computational complexity is reduced by more than 80% by using the cyclic phase recursion method.

[0091] Furthermore, the P-fold complex exponentiation operation accounts for more than 80% of the total computation in this invention. Therefore, efficient complex exponentiation calculation is required. This invention recommends the use of the Cordic algorithm. In particular, the angle theta is stored in this invention using the following method: theta(i)=arctan(2^(-i)) / 2 / pi*P, where arctan is the arctangent function and ^ is the exponentiation operation.

[0092] By normalizing the pre-stored angle of Cordic and multiplying it by P, when calculating the complex exponent Fu(k) = exp(-2i*pi*phase / P), the quasi-phase phase can be directly used as the phase without dividing by P, thus eliminating P division operations.

[0093] The result calculated by the above method is a power normalized result, which can be multiplied by the square root of P to obtain a general DFT result.

[0094] In addition, as an intermediate calculation result, to calculate the cumulative sum Fu(0) = sum(Zu) of a ZC sequence with root value u, we can simply use Fu(0) = sqrt(P) * exp(-2i * pi * Po / P), where sqrt is the square root operation, without having to calculate the Zu sequence first and then calculate the cumulative sum of Zu.

[0095] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for efficiently computing the DFT of a ZC sequence, characterized in that, The method is directed to a ZC sequence \(Z_u\) with a prime length \(P\) and a root value \(u\), where \(Z_u(k)=\exp(-1i\pi uk(k + 1) / P)\), where \(\exp\) is the complex exponential function, \(i\) is the imaginary unit, \(\pi\) is the pi, and \(k = 0,1,\cdots,P - 1\). Through cyclic phase recursion, the DFT calculation of the ZC sequence is realized, including the following steps: First, initialize: Calculate the modular inverse \(i_u\) of the root value \(u\) modulo \(P\), calculate the Legendre symbol corresponding to the root value \(u\), calculate the quasi-phase offset value \(P_0\), calculate the quasi-frequency shift value \(F_s\), and calculate the frequency point step \(F_{step}\); Reset the count value \(k\), assign the initial value of the quasi-frequency value \(freq\) to the quasi-frequency shift value \(F_s\), and assign the initial value of the quasi-phase value \(phase\) to the quasi-phase offset value \(P_0\): k＝0； freq = Fs; phase = Po; Then, calculate the DFT of \(Z_u\) in a loop: Let \(F_u\) be the DFT result of \(Z_u\). When \(k < P\), first calculate the complex exponential value of \(F_u(k)\) according to the quasi-phase value \(phase\) divided by \(P\): \(F_u(k)=\exp(-2i\pi phase / P)\); Then recursively update the count value \(k\), the quasi-frequency value \(freq\), and the quasi-phase value \(phase\): k = k + 1; freq = freq+Fstep; phase = phase+freq; Calculate the quasi-phase offset value \(P_0\), including: First, calculate the Legendre symbol corresponding to the root value \(u\). Let \(t=(mod(u,2)*P + u) / 2\), then the Legendre symbol \((t / P)\) of \(t\) is the Legendre symbol corresponding to \(u\), where \(mod\) is the modulo operation; Then, calculate the quasi-phase offset \(P_0\), and the formula is as follows: \(P_0 = ((mod(P,4)*t + 1-(t / P)+(mod(P,4)==3))*P - t) / 4\); where \((t / P)\) is the Legendre symbol of the integer \(t\) with respect to the prime \(P\), not the general mathematical parentheses; The formula for calculating the quasi-frequency shift value \(F_s\) is: \(F_s=(P + 1) / 2*(i_u - 1)\); The formula for calculating the frequency point step \(F_{step}\) is: \(F_{step}=P - i_u\).

2. The method for efficiently computing the DFT of a ZC sequence according to claim 1, characterized in that, The sum of the ZC sequence is directly obtained by using the quasi-phase offset value \(P_0\), and the formula is: \(F_u(0)=\sqrt{P}\exp(-2i\pi P_0 / P)\), where \(\sqrt{}\) is the square root operation.

3. The method for efficiently computing the DFT of a ZC sequence according to claim 1, characterized in that, According to the quasi-phase value \(phase\) divided by \(P\), calculate the complex exponential value of \(F_u(k)\), which is implemented by Cordic to reduce the complex exponential calculation, including: By normalizing the pre-stored angle \(\theta\) of Cordic and multiplying it by \(P\), that is, \(\theta=\arctan(2^{-i}) / 2 / \pi*P\), the quasi-phase \(phase\) can be directly used to calculate \(F_u(k)\) without the operation of dividing by \(P\), where \(\arctan\) is the arctangent function and \(\hat{}\) is the power operation.

4. The method for efficiently computing the DFT of a ZC sequence according to claim 1, characterized in that, Calculate the modular inverse \(i_u\) of the root value \(u\) modulo \(P\), and calculate the Legendre symbol corresponding to the root value \(u\), including: Calculate the inverse \(i_u\) of all root values \(u\) and the Legendre symbol \((t / P)\) corresponding to all \(u\) in advance, and store them in a table to reduce the calculation amount.

5. The method for efficiently computing the DFT of a ZC sequence according to claim 1, characterized in that, The method further includes: The quasi-frequency value freq and quasi-phase value phase modulo P are calculated in each iteration, i.e., freq = mod(freq + Fstep, P) and phase = mod(phase + freq, P), thus reducing the data bit width.

6. The method for efficiently computing the DFT of a ZC sequence according to claim 1, characterized in that, The method further includes: Multiply the DFT result Fu of the ZC sequence by the square root of P, i.e., Fu = sqrt(P) * Fu; to make the result conform to the definition of nonnormalization.

7. The method for efficiently computing the DFT of a ZC sequence according to claim 1, characterized in that, The method further includes: When performing a DFT on the ZC sequence with cyclic shift "S" used in current 4G and 5G communication systems, Zus(k) = exp(-1i*pi*u*(k+S)*((k+S)+1) / P), The method changes the quasi-frequency shift value Fs from Fs=(P+1) / 2*(iu-1) to Fs=(P+1) / 2*(iu-1)-S; where S is the cyclic shift value of the ZC sequence.

Images