WO2017078561A1

WO2017078561A1 - Method and apparatus for synchronizing a receive sequence with a known transmit sequence

Info

Publication number: WO2017078561A1
Application number: PCT/RU2015/000734
Authority: WO
Inventors: Yanxing Zeng; Jianqiang Shen; Lev Borisovich Rapoport; Alexey Vyacheslavovich FEDOROV
Original assignee: Huawei Technologies Co., Ltd.
Priority date: 2015-11-02
Filing date: 2015-11-02
Publication date: 2017-05-11
Also published as: CN108351865A; WO2017078561A8

Abstract

A method (200) for synchronizing a receive sequence with a known transmit sequence includes: compressing (201) a replica of the known transmit sequence from a length N to a reduced length B, the length N being a multiple of the reduced length B, by applying a B-by-N compression matrix Ψ to the replica to obtain a compressed replica of the reduced length B, wherein the compression matrix Ψ comprises a plurality of shifting matrices each of the shifting matrices depending on a first compression parameter p that is an integer ratio of the length N and the reduced length В and a second compression parameter α that is an angle between 0 and 2π; compressing (202) the receive sequence from the length N to the reduced length В by applying the B-by-N compression matrix Ψ to the receive sequence to obtain a compressed receive sequence of the length B; and multiplying (203) the compressed replica with the compressed receive sequence in frequency domain to obtain a compressed product sequence, wherein a time-domain transform of the compressed product sequence indicates a correlation between the receive sequence and the replica of the known transmit sequence.

Description

Method and apparatus for synchronizing a receive sequence with a known transmit sequence

TECHNICAL FIELD

The present disclosure relates to a method and an apparatus for synchronizing a receive sequence with a known transmit sequence, in particular for use in a wireless communication system, especially for reducing complexity of a synchronization algorithm. The disclosure further relates to quick synchronization methods based on sparse FFT (Fast Fourier Transform).

BACKGROUND

The process of synchronization 100, e.g., as illustrated in Fig. 1 takes particular role in digital signal processing. It is very important to know when the transmitted signal has started. The receiver computes the correlation of the replica 102, 106 of the transmitted signal with the received signal 104, 108 for all possible shifts of the replica with respect to the signal. In Fig. 1 the reference 102 denotes the replica of the transmitted signal in time domain, the reference 106 denotes the replica of the transmitted signal in frequency domain, the reference 104 denotes the received signal in time domain and the reference 108 denotes the received signal in frequency domain. The maximum 1 14 of correlation function 1 12 determines the start point i.e. the correct shift of the useful signal. Usefulness is different from case to case, it depends on many factors, for example environment condition, equipment features etc. The correlation function can be calculated in several ways, for example by using linear convolution or by using circular convolution.

Thus, the output 1 12 of the inverse FFT 107 will spike at the correct shift 1 14 that synchronizes the replica 102 with the received signal 104. Since the output 1 12 of the synchronization process has a single major spike at the correct shift 1 14 the inverse FFT 107 is very sparse. The synchronization 100 may be mathematically described according to the following: Given a spreading signal (replica) r = r_x, r₂,... , r_n 102 of size n and a received signal s = s_l, s₂ , . . ., s_n 104, find the time shift t 1 14 that maximizes the correlation between r and s , i.e. compute: / = arg max(r_„ * s), where " * " is a circular convolution and r__n is time reversed replica; i.e. r__n = r_n, r_n_,, . .. , r_x . As it was mentioned above, circular convolution can be calculated more efficiently in frequency domain as follows: aig max(r _w * s) = arg max(F~^l (F(r) - F(s))} Here, F, F^~l are FFT 101 , 103 and

IFFT 107, respectively. The multiplication in frequency domain is realized by a multiplier 105. Accordingly, in this disclosure only the FFT-based synchronization algorithm is considered as a baseline for evaluating synchronization performance.

SUMMARY

It is the object of the invention to provide an improved synchronization technique for synchronizing a receive sequence with a replica of a transmit sequence.

This object is achieved by the features of the independent claims. Further implementation forms are apparent from the dependent claims, the description and the figures. The basic concept described in this disclosure can be described according to the following: The quick synchronization technique according to the disclosure exploits the sparse nature of the synchronization problem, where the correct alignment between the received signal and the replica causes their cross-correlation to spike. The main idea is to use this property to perform both the Fourier and inverse Fourier transforms in a time faster than 0(n \og n) , thereby reducing the complexity of synchronization. I.e., the complexity is reduced by using the concept of sparsity of the correlation vector.

This concept can be applied to different fields, for example to the signal synchronization of all wireless communication systems. Further, the concept can be applied in signal processing and pattern recognition

In order to describe the invention in detail, the following terms, abbreviations and notations will be used: FFT: Fast Fourier Transform.

IFFT: Inverse Fast Fourier Transform.

PRN: Pseudo Random Noise.

DFT: Discrete Fourier Transform.

FT: Frequency Transform. SFFT: Sparse FFT.

Systems, devices and methods according to the disclosure may be based on using convolution operations, in particular linear and circular convolution that may be implemented by FFTs and IFFTs. The convolution operation, for example used in Radio Engineering, can be described as follows:

s(t) = r(t) * h(t) = I r(r)/i(t - τ)άτ.

The convolution can calculate signal s (t) at the output of a linear filter with impulse response h(t) when the spreading signal is r(t). In the discrete case, there are two types of convolution: linear and circular. Circular convolution is often called cyclic or periodic. In the following circular convolution is considered with respect to two discrete periodic signals a n) and b(n) with the same period N. The equation for circular convolution may be written as:

N-l

c :{(nn)) == ^ a(m)b{n - m) , n = 0, . . , N - 1.

m=0

The result of circular convolution will have the same length N . Complexity of this operation takes N² multiplications. The circular convolution may be based on using the Fast Fourier Transform (FFT) which may be implemented by the Discrete Fourier Transform (DFT) according to:

W-l

C(fc) = ^ c(n)W ^k , k = 0, . . , N - 1,

n=0

The expression for the circular convolution may be substituted obtaining:

W-l N-l

C(fc) = ^ ^jT a{m)b(n - m) W$^k.

n=0 m=0

Interchanging the summation operations results in the following fundamental

C{k) =

C(k) = A(k)B(k). This fundamental result shows that the spectrum of circular convolution equals to the multiplication of the spectra of the convoluted signals. The prototype c(n) can be determined by applying the Inverse Discrete Fourier Transform (IDFT). That means that for calculating a circular convolution, in case when the period length N is a power of 2, it may be better to use an FFT. The complexity of this operation takes about 0(N^■ log₂ N) multiplications.

Systems, devices and methods according to the disclosure may be based on using Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT). The Fourier Transform is considered most common analysis tool in all fields of science and technology. Fast algorithms for Discrete Fourier Transform (DFT) are highly valuable, because current fastest algorithm, Fast Fourier Transform (FFT), has a complexity 0 (n log n) for n-dimensional signal. The idea to reduce complexity of FFT algorithms is one of the central questions in the theory of algorithms. In many applications most of the Fourier coefficients of a signal are small, miniscule or even equal to zero. This fact means that the output of the DFT is sparse. If a signal has a small number k of non-zero Fourier coefficients the output of the Fourier transform can be represented using only k coefficients. Hence, for such signals, DFT algorithms such as SFFT can be used whose runtime is sub-linear in the signal size n.

Systems, devices and methods according to the disclosure may be based on using sparse Fast Fourier Transforms (SFFT). The SFFT, e.g. as described by Ή. Hassanieh, P. Indyk, D. Katabi, and E. Price: Simple and practical algorithm for sparse fourier transform in SODA'12" is a new algorithm of FFT that works with sparse signals. For exactly fe-sparse case, when FT (Frequency Transform) of the signal includes k non-zeros frequencies, the complexity of algorithm SFFT is equal to 0 {k logn). For general case, when FT of the signal includes approximately k heavily weighted frequencies, complexity of SFFT is equal to 0(k logn log(n/fc )). The main idea of the SFFT is to apply the frequency transform to a small array instead of the big original signal. For this purpose random permutation of the spectrum of original signal may be used, and aliasing of the permuted signal into small number of elements may be applied. After these steps "heavy" coefficients of FT may be estimated. This method is probabilistic, and for a special class of signals the method has a good probability to correct reconstruction of Fourier coefficients. According to a first aspect, the invention relates to a method for synchronizing a receive sequence with a known transmit sequence, the method comprising: Compressing the known transmit sequence from a length N to a reduced length B, the length N being a multiple of the reduced length B, by applying a B-by-N compression matrix Ψ to the transmit sequence to obtain a compressed transmit sequence of the reduced length B, wherein the compression matrix Ψ comprises a plurality of shifting matrices each of the shifting matrices depending on a first compression parameter p that is an integer ratio of the length N and the reduced length B and a second compression parameter a that is an angle between 0 and 2π ; Compressing the receive sequence from the length N to the reduced length B by applying the B-by-N compression matrix Ψ to the receive sequence to obtain a compressed receive sequence of the length B; and Multiplying the compressed transmit sequence with the compressed receive sequence in frequency domain to obtain a compressed product sequence, wherein a time-domain transform of the compressed product sequence indicates a correlation between the receive sequence and the transmit sequence; computing a delay of the receive sequence with respect to the transmit sequence; and synchronizing the receive sequence with the transmit sequence according to the delay.

Such a method provides a quick synchronization technique by exploiting the sparse nature of the synchronization problem. The correct alignment between the received signal and the replica causes their cross-correlation to spike. By applying this property both the Fourier and inverse Fourier transforms can be applied in a time faster than 0(n log n) , thereby reducing the complexity of synchronization. In a first possible implementation form of the method according to the first aspect, the method further comprises: determining a maximum magnitude element E_max and a position N_max of the maximum magnitude element E_max in the time domain transform of the compressed product sequence; determining an argument φ of the maximum magnitude element; determining a floor value based on the argument φ of the maximum magnitude element E_max, the first compression parameter p and the second compression parameter a ; and if the floor value equals zero: use the position N_max of the maximum magnitude element E_max as the delay of the receive sequence with respect to the known transmit sequence; otherwise: determine the delay based on a threshold comparison of a function of the receive sequence, the transmit sequence, the floor value and the position N_max of the maximum magnitude element E_max.

Such method provides the advantage of finding the optimum delay of the receive sequence with respect to the transmit sequence and thus providing an optimum synchronization.

In a second possible implementation form of the method according to the first aspect as such or according to the first implementation form of the first aspect, the compression matrix Ψ is constructed by a number of p shifting matrices of dimension B-by-B, wherein p is the first compression parameter.

By constructing the compression matrix by a number of shifting matrices this operation provides a reduction of complexity of correlation function calculation via FFT.

In a third possible implementation form of the method according to the first aspect as such or according to any of the preceding implementation forms of the first aspect, the plurality of shifting matrices are arranged side-by-side in the compression matrix Ψ . When arranging the shifting matrices side-by-side in the compression matrix, the compression matrix can be efficiently computed.

In a fourth possible implementation form of the method according to the first aspect as such or according to any of the preceding implementation forms of the first aspect, the shifting matrices depend on a complex-valued shifting operator that is a function of the first compression parameter p and the second compression parameter a .

By such dependency, the shifting matrices can be efficiently constructed. In a fifth possible implementation form of the method according to the fourth implementation form of the first aspect, the complex-valued shifting operator is e ^p , wherein i is an imaginary unit, p is the first compression parameter and a is the second compression parameter. By using such a complex-valued shifting operator, FFT processing can be applied for efficiently implementing the method.

In a sixth possible implementation form of the method according to the first aspect as such or according to any of the preceding implementation forms of the first aspect, each of the shifting matrices depends on an index indicating a position of the respective shifting matrix in the compression matrix Ψ .

This provides the advantage that the index can be used to determine the correct shifting.

In a seventh possible implementation form of the method according to the sixth implementation form of the first aspect, the index ranges from 0 to p-1 in steps of 1 , wherein p is the first compression parameter.

This provides the advantage that equal ranges for the shifting can be applied.

In an eighth possible implementation form of the method according to the first aspect as such or according to any of the preceding implementation forms of the first aspect, the

wherein I_BXB is a B-by-B identity matrix, p is the first compression parameter, i is an imaginary unit and a is the second compression parameter.

This provides the advantage that the compression matrix can be implemented by using FFT processing.

In a ninth possible implementation form of the method according to the first aspect as such or according to any of the preceding implementation forms of the first aspect, the first compression parameter p and the second compression parameter a satisfy the following relation: \ ' ~ > Θ, wherein Θ is a predetermined phase error.

It is advantageous to choose parameters a and p to satisfy the ratio above, i.e. the parameters should be chosen based on information about Θ . In a tenth possible implementation form of the method according to the ninth

implementation form of the first aspect, the predetermined phase error satisfies the following relation: \ θ \ < sin^-1 _^'°^gA, _| wherein N is the length of the known transmit sequence and of the receive sequence.

It is advantageous to first estimate Θ and only after that to choose parameters.

In an eleventh possible implementation form of the method according to the ninth implementation form of the first aspect, the predetermined phase error satisfies the following relation: \ θ\ < 0.2.

This provides the advantage that the phase error is smaller than the threshold 0.2. This fact gives information that in case of noise with the same power as the signal Θ can be estimated less than 0.2. Also, this fact says that the algorithm according to the disclosure can be used, i.e. that appropriate parameters a and p can be found.

In a twelfth possible implementation form of the method according to the first aspect as such or according to any of the preceding implementation forms of the first aspect, the known transmit sequence comprises a RACH preamble of a random access channel according to a mobile communication standard.

This provides the advantage that the method can be applied to radio signals and mobile communications.

In a thirteenth possible implementation form of the method according to the first aspect as such or according to any of the preceding implementation forms of the first aspect, the first compression parameter p is a power of 2. This provides the advantage that the first compression parameter can be easily computed.

According to a second aspect, the invention relates to a synchronization apparatus for synchronizing a receive sequence with a known transmit sequence, the synchronization apparatus comprising: a compressing circuit configured to compress a replica of the known transmit sequence from a length N to a reduced length B, the length N being a multiple of the reduced length B, by applying a B-by-N compression matrix Ψ to the replica to obtain a compressed replica of the reduced length B, wherein the compression matrix Ψ comprises a plurality of shifting matrices each of the shifting matrices depending on a first compression parameter p that is an integer ratio of the length N and the reduced length B and a second compression parameter that is an angle between 0 and 2π , and to compress the receive sequence from the length N to the reduced length B by applying the B-by-N compression matrix Ψ to the receive sequence to obtain a compressed receive sequence of the length B; a frequency transform circuit configured to transform the compressed replica and the compressed receive sequence into frequency domain; a multiplication circuit configured to multiply the frequency-transformed compressed replica with the frequency-transformed compressed receive sequence to provide a compressed product sequence in frequency domain; an inverse frequency transform circuit configured to transform the compressed product sequence from frequency domain to time domain; and a threshold circuit configured to compare the time- domain transform of the compressed product sequence with a threshold to determine a correlation between the receive sequence and the replica of the known transmit sequence. Such an apparatus provides a quick synchronization by exploiting the sparse nature of the synchronization problem. The correct alignment between the received signal and the replica causes their cross-correlation to spike. By applying this property both the Fourier and inverse Fourier transforms can be applied in a time faster than 0(n \og ri) , thereby reducing the complexity of synchronization.

BRIEF DESCRIPTION OF THE DRAWINGS

Further embodiments of the invention will be described with respect to the following figures, in which:

Fig. 1 shows a block diagram of a synchronization system 100 with circular convolution using the FFT; Fig. 2 shows a schematic diagram illustrating a method 200 for synchronizing a receive sequence with a known transmit sequence according to an implementation form;

Fig. 3 shows a block diagram of an apparatus 300 for synchronizing a receive sequence with a known transmit sequence according to an implementation form;

Fig. 4 shows an exemplary correlation function 400 in the complex plane after bucketization according to an implementation form; Fig. 5 shows a diagram 500 of a complex plane illustrating the definition of maximum element's argument;

Fig. 6 shows a diagram 600 of a complex plane illustrating an exemplary location of maximum elements of correlation functions for a first exemplary parameter set;

Fig. 7 shows a diagram 700 illustrating the correlation functions of Fig. 6 over time delay before bucketization;

Fig. 8 shows a diagram 800 of a complex plane illustrating an exemplary location of maximum elements of correlation functions for a second exemplary parameter set;

Fig. 9 shows a diagram 900 illustrating the correlation functions of Fig. 8 over time delay before bucketization; Fig. 10 shows a diagram 1000 of a complex plane illustrating an exemplary location of maximum elements of correlation functions for a third exemplary parameter set;

Fig. 11 shows a diagram 1100 illustrating the correlation functions of Fig. 10 over time delay before bucketization;

Fig. 12 shows a schematic diagram illustrating an algorithm 1200 according to an implementation form for implementing a bucketization stage in a synchronization apparatus 300 as described in Fig. 3; and Fig. 13 shows a schematic diagram illustrating an exemplary instruction sequence 1300 for finding the unique solution of the synchronization algorithm.

DETAILED DESCRIPTION OF EMBODIMENTS

In the following detailed description, reference is made to the accompanying drawings, which form a part thereof, and in which is shown by way of illustration specific aspects in which the disclosure may be practiced. It is understood that other aspects may be utilized and structural or logical changes may be made without departing from the scope of the present disclosure. The following detailed description, therefore, is not to be taken in a limiting sense, and the scope of the present disclosure is defined by the appended claims.

It is understood that comments made in connection with a described method may also hold true for a corresponding device or system configured to perform the method and vice versa. For example, if a specific method step is described, a corresponding device may include a unit to perform the described method step, even if such unit is not explicitly described or illustrated in the figures. Further, it is understood that the features of the various exemplary aspects described herein may be combined with each other, unless specifically noted otherwise.

Fig. 2 shows a schematic diagram illustrating a method 200 for synchronizing a receive sequence with a known transmit sequence according to an implementation form. The method 200 includes compressing 201 a replica of the known transmit sequence from a length N to a reduced length B, the length N being a multiple of the reduced length B, by applying a B-by-N compression matrix Ψ to the replica to obtain a compressed replica of the reduced length B, wherein the compression matrix Ψ comprises a plurality of shifting matrices each of the shifting matrices depending on a first compression parameter p that is an integer ratio of the length N and the reduced length B and a second compression parameter that is an angle between 0 and 2π .

The method 200 includes compressing 202 the receive sequence from the length N to the reduced length B by applying the B-by-N compression matrix Ψ to the receive sequence to obtain a compressed receive sequence of the length B. The method 200 includes multiplying 203 the compressed replica with the compressed receive sequence in frequency domain to obtain a compressed product sequence, wherein a time-domain transform of the compressed product sequence indicates a correlation between the receive sequence and the replica of the known transmit sequence.

The compression matrix Ψ may be constructed by a number of p shifting matrices of dimension B-by-B, wherein p is the first compression parameter. The plurality of shifting matrices may be arranged side-by-side in the compression matrix Ψ . Each of the shifting matrices may depend on a complex-valued shifting operator that is a function of the first compression parameter p and the second compression parameter a .

The complex-valued shifting operator may be e ^p , wherein i is an imaginary unit, p is the first compression parameter and a is the second compression parameter. Each of the shifting matrices may depend on an index indicating a position of the respective shifting matrix in the compression matrix Ψ . The index may range from 0 to p-1 in steps of 1 , wherein p is the first compression parameter. The compression matrix Ψ may be determined according to:

IBXB, e^J * IBXB. ^β * hxB. ^e~ * !BXB, - - e * * /_BxB J,

wherein I_BXB is a B-by-B identity matrix, p is the first compression parameter, i is an imaginary unit and a is the second compression parameter. The first compression parameter p and the second compression parameter a may satisfy the following relation: > Θ, wherein Θ is a predetermined phase error.

The predetermined phase error may satisfy the following relation:

wherein N is the length of the known transmit sequence and of the receive sequence.

The predetermined phase error may satisfy the following relation: |0 | < 0.2. The method may include: comparing the time-domain transform of the compressed product sequence with a threshold to determine an element with maximum magnitude in the time-domain transform of the compressed product sequence, wherein a time index of the element indicates a delay of the receive sequence with respect to the known transmit sequence.

The known transmit sequence may include a RACH preamble of a random access channel according to a mobile communication standard, e.g. LTE. The first compression parameter p may be a power of 2.

The complex weights may be used to compress the received signal, e.g. a received pseudo-random signal, allowing to reduce the complexity of calculation while performing the synchronization with locally generated replica, yet not losing the information necessary for synchronization. The method 200 may uses parameters a e [ ,2π] and p taking values as the power of 2 (2, 4, 8,... ) which may be chosen according to conditions (C1 ), (C2), (C3) described below with respect to Fig. 3.

The method 200 may be applied by the apparatus 300 described below with respect to Fig. 3 and may be used in the algorithm 1200, 1300 described below with respect to Figs. 12 and 13.

Fig. 3 shows a block diagram of a synchronization apparatus 300 for synchronizing a receive sequence with a known transmit sequence according to an implementation form.

The synchronization apparatus 300 includes a compressing circuit 332, 334, a frequency transform circuit 306, 308, a multiplication circuit 305, an inverse frequency transform circuit 310 and a threshold circuit 312. The compressing circuit 332, 334 is used to compress a replica 302 of the known transmit sequence from a length N to a reduced length B by applying a B-by-N compression matrix Ψ to the replica 302 to obtain a compressed replica 301 of the reduced length B. The length N is a multiple of the reduced length B. The compression matrix Ψ includes a plurality of shifting matrices. Each of the shifting matrices is depending on a first compression parameter p that is an integer ratio of the length N and the reduced length B and a second compression parameter a that is an angle between 0 and 2π . The compressing circuit 332, 334 is used to compress the receive sequence 304 from the length N to the reduced length B by applying the B-by-N compression matrix Ψ to the receive sequence 304 to obtain a compressed receive sequence 303 of the length B.

The frequency transform circuit (306, 308) is used to transform the compressed replica 301 and the compressed receive sequence 303 into frequency domain.

The multiplication circuit 305 is used to multiply the frequency-transformed compressed replica 309 with the frequency-transformed compressed receive sequence 311 to provide a compressed product sequence 313 in frequency domain.

The inverse frequency transform circuit 310 is used to transform the compressed product sequence from frequency domain 313 to time domain 307.

The threshold circuit 312 is used to compare the time-domain transform 307 of the compressed product sequence with a threshold to determine a correlation 314 between the receive sequence 304 and the replica of the known transmit sequence 302. In the following description, the compression operation may also be referred to as bucketization operation or simply as bucketization. This bucketization is described in the following.

In the following the special operation allowing to reduce the size of an array from N to B is considered. Here N must be multiple of B, i.e. - = p with p being an integer. The special operation is required to reduce complexity of circular convolution of two arrays. In our case, we talk about convolution of received signal and local signal in receiver, we call it replica. There are a lot of ways to reduce size of given array: cutting of the array with deletion of members, wrapping of the array into a small size array, wrapping of the array into a small size array with multiplication on complex weights. First way is not considerable, because of new array loses information. With loosing information sometimes there appear situations when we lose ability to achieve our aims. Wrapping of the array into a small size array is called 'bucketization' (like distribution quantities of a material between 'buckets') The B-by-N bucketization matrix Ψ can be defined as follows:

( ai 2ai 3ai (p-l)crt N

½xB, e P * /_BxB, e P * /_{B xB}, e P * /_BxB, ... , e P * I_BXB ) . where i is an imaginary unit, N-by-N matrix of shifting C,

and B-by-B matrix of cyclic shifting CC, cc = ( V'?B»-Ι-,β->Ι o ^υβ ¹ )·

Here I_XxX is the X-by-X identity matrix, where size X can be equal to B, N - 1 and B - 1. Matrix of shifting C will be used for setting spreading time delay of signal, matrix of cyclic shifting CC will be used for calculation correlation function. Two arrays are consider, a and b, both having size N x 1. Array a has PRN properties according to "B. M. Popovic, Generalized chirp-like polyphase sequences with optimum correlation properties, IEEE Transactions on Information Theory, vol. 38, no. 4, pp 1406- 1409, July 1992", and the b is K-ih shift of , i.e. b = C^Ka = [0,0, ... ,Ο, α^ a₂, ... a_N__K] , furthermore K = (Z - 1) * B + k, k e [Ο, Λί - e [i, p]. In this document b is called - received signal and K is called - signal propagation delay, array a is called - local code or replica. I denoted a "floor" of the received signal's start point, it means that the first element of vector a is located in the Z-th part of vector b after dividing b into p parts. The result of multiplication bucketization matrix and array will be next:

a_new = Ψ * a; b_new = Ψ * b.

It can be represented in the table form:

a_new takes following form:

b_new takes following form:

H , j e

[1, k] and b_new j = fy-_fc+me e ^ 0 - 1 + m) , ; e [ft + 1, B]. We notice that our new arrays are S-by-l vectors (columns).

The correlation function as depicted in Fig. 4 of two new arrays a_new and b_new can be calculated according:

Corr ) = (CC'^_1 * a_new)" * b_new.

Obviously, as it is shown in the table above, except just one position j^* - k + 1, all values of correlation function Corr(J), j≠ j^* do not contain members as a_t * a_t * exp - · rnj for some t and m. For j = j^* value Corr(j^*) consists N - K members as a_t * a_t * exp

This is shown in the following:

B-k

Corr(j^*) =

/ ^a

+^■■■ + α_;+(ρ__1)β exp I -£ - (p - 1)

B

^ ( _£ · exp (i ^ /) + ··· + a_t+(p_,__1)B exp

Due to PRN properties of array a (it is assumed that the array is normalized), finally the following result is obtained:

Corr(j^*) = (J? - k)(p - l + l) exp ^ (/ - 1) + fc(p - exp i ^ ή

exp ') + Δ = R exp(i<p) + Δ.

Here = (£ - fe)(p - / + 1), and = k(p - l), Λ, φ - length and argument of total vector, respectively. The result is shown in Figure 5. Obviously, number I can be estimated by using φ.

The value Δ has noise-like behavior, which expectation is E[A] = 0, and variance

V[A] = E[A * A^H]~ p²B = pN. In case, when p =

us take assumption: R_min = In accordance with the synchronization problem, in practice, this condition works almost always.

Here, the phase error is according to:

\θ\ < sin ¹ = sin ¹ „ ₂ = sin ¹ (C1 )

For N = 2048 the maximum phase error is |0| < 0,1-

As it shown above, the maximum element of the correlation function Corr_max = Corr(j^*) can be located in sector [- (/ - 1) - \θ\, - 1 + \θ\].

Lp p J

In the following section, an exemplary choice of parameters is described. It is

understandable, angle a and bucketization parameter p should satisfy the condition:

> (C2)

2 v

In case with noise phase \θ\ will be increased, because the radius of the circle 501 in Figure 5 will increase σ = σ_Α + a_noise. Let's calculate value of a_noise . Consider received vector b' - b + ξ = C^Ka + ξ, where ξ - Gaussian white noise with variance σξ. Correlation after bucketization will be as follows:

Corr ) = ( C^¹ * a_newf * Ψή' = ( C^¹ * a_new)^H * (b_new + Ψξ)

= {CCi-¹ * a_new)^H * b_new + {CCi-¹ * a_newf * Ψξ = R expG<p) + Δ + η.

Here 7 is a new noise-like process with zero mean and variance ν[η] = N^■ p · .

In case intensity of noise is equal to intensity of signal, value of square root of variance is

Jvfo] = a_noise = JNp. Renewed phase error |0| = sin ¹ = sin ¹ 2 ^ogW. For N = 2048 maximum phase error is obtained:

\θ\ < 0.2 . (C3) Again, in noise case angle a and bucketization parameter p should satisfy the following condition: -^■- > 0.

2 V

Based on array size and intensity of noise preferred parameters a and p can be chosen.

In the following section, "Floor" and Correct Shift estimation is described. The

bucketization operation provides a way to reduce complexity of correlation function calculation via to FFT. We now consider output array of correlation function Corr which may be calculated by using bucketization with parameters , p. Let CorrQ") be the maximum element of considered array, position j^* e [1, B] . The angle of maximum element is φ. Now, "floor" can be estimate as floor = round

Assume, that the received signal's start point is located in "floor" / - 1. As shown in Figure 5, floor can be estimated as I - 1 or I. Two options can be identified: check correlation between received signal b and local code a at positions (B · (floor - 1) + j^*) and (B - floor + j^*).

Va = b" (B ^■ (floor - 1) + ^* : B^■ floor + j^* - 1) · a(l: B); Val₂ = b^H(B^■ floor + j^*: B^■ (floor + 1) + j^* - 1) · a(l: S).

Then, one may choose that "floor" which has the greatest value. If floor estimated as /, then Val_r will be bigger than Val₂; if floor estimated as / - 1, then Val₂ will be bigger than Val_t.

Another way is described in the following: The threshold (threshold) can be pre-calculated for correlation value between received signal and replica. Then just one correlation value must be compared with threshold:

Val = b^H (B ^■ floor + ^*: B^■ (floor + 1) + f - 1)^■ a(l: B). In case when correlation value Val > threshold one can choose (floor = floor), otherwise one can choose (floor - floor - 1). Finally one may calculate the correct Shift CS as CS = B ^■ floor + j^*. The correct Shift CS gives an estimation of signal propagation delay.

Fig. 6 shows a diagram 600 of a complex plane illustrating an exemplary location of maximum elements of correlation functions for a first exemplary parameter set = π , N=1024, p=2 and B=512. Maximum elements are located at floor / = 1 (601 ) and floor I = 2 (602). Fig. 7 shows a diagram 700 illustrating the correlation functions of Fig. 6 over time delay before bucketization with the parameters Nfft = 1024, p = 2, B = 512.

Fig. 8 shows a diagram 800 of a complex plane illustrating an exemplary location of maximum elements of correlation functions for a second exemplary parameter set a = π , N=1024, p=4 and B=256. Maximum elements are located at floor / = 1 (801 ), floor I - 2 (802), floor I = 2 (803) and floor I = 3 (804). Fig. 9 shows a diagram 900 illustrating the correlation functions of Fig. 8 over time delay before bucketization with the parameters Nfft = 1024, p = A, B = 256.

Fig. 10 shows a diagram 1000 of a complex plane illustrating an exemplary location of maximum elements of correlation functions for a third exemplary parameter set α - π , N= 024, p=8 and B=128. Maximum elements are located at floor { = 1 (1001 ), floor I = 2 (1002), floor / = 2 (1003), floor I = 3 (1004), floor I = 4 (1005), floor / = 5 (1006), floor / = 6 (1007), floor i = 7 (1008) etc. Fig. 11 shows a diagram 1100 illustrating the correlation functions of Fig. 10 over time delay before bucketization with the parameters Nfft - 1024, = 8, B = 128.

Fig. 12 shows a schematic diagram illustrating a bucketization algorithm 1200 according to an implementation form for implementing the bucketization stage in a synchronization apparatus 300 as described in Fig. 3. The bucketization algorithm 1200 may be an exemplary implementation of a bucketization stage for a method 200 as described above with respect to Fig. 2 or for an apparatus 300 as described above with respect to Fig. 3. The bucketization algorithm 1200 is part of a synchronization algorithm according to Figures 2 and 3, also called "full algorithm" as described in the following. Fig. 13 shows a schematic diagram illustrating an exemplary instruction sequence 1300 for finding the unique solution of the synchronization algorithm.

The inputs of the synchronization algorithm are as follows:

• A discrete time signal s of length N.

• Replica r of length N.

• Bucketization parameters , p. The outputs of the synchronization algorithm are as follows:

• An estimate of the Correct Shift (CS) - N_cs

The synchronization algorithm performs the following blocks: 1. Bucketization (according to bucketization algorithm 1200):

Input: s, r, , p

• Based on parameter , p generate bucketization matrix Ψ

• Calculation of new vectors s', r' of length B: s' = Ψ · s, r' = Ψ · r. Output: s', r'

2. Small size FFT (see Figures 2 and 3):

Input: s', r'

• Perform an FFT of size B on the vectors s', r' S = FFT(s'), R = FFT(r').

Output: S, R

3. Multiplying with the replica (see Figures 2 and 3):

Input: S, R

• Vector R is multiplied element-wise with the vector S. Result of element-wise multiplication is vector M.

Note: Algorithm can precompute R and store it in the frequency domain. There are several variants of choosing parameters a, p, and vector R can be precomputed for all these variants.

Output: M

4. Small size IFFT (see Figures 2 and 3):

Input: M • Perform an IFFT of size B on the vector M: m = IFFT(M).

Output: m

5. Find the unique solution (see Figure 13):

Input: m

· Find element E_max with maximum magnitude in vector m and its position iV_max. Estimate argument φ (angle) of complex value E_max.

• Estimate "floor" of start point localization floor = round (^ )-

• In case floor = 0, take N_cs - N_max, another case go further

• Calculate correlation between s and r:

Val = r"(l: B)^■ s(B^■ floor + N_max: B^■ (floor + 1) + N_max - 1). · Compare with threshold:

if \Val\ > threshold then floor = floor;

else floor = floor— 1

end if

N_cs = B ^■ floor + N_max.

Output: N_cs.

The following notations are applicable for the algorithm parameters:

N - input vector size, actually it equals to FFT size i.e. power of 2;

p - bucketization parameter, B = ^ must be integer number;

a - bucketization parameter, angle can be chosen from [0, 2π].

- output vector size. s = - local code (replica).

(p-i)tti \

/ax,,, e * l_BxB, e * I_BXB, e P * l_BxB, ... , e P * I_BxBj - bucketization matrix.

For evaluating performance of the algorithm 1200 numerically tests for time

synchronization in LTE were performed. LTE uses the Random Access Channel (RACH) for initial network access. There are four formats of RACH preambles:

- format 0 (for small and medium cells, cell's radius up to ~ 14 km),

- format 1 (for large cells, cell's radius up to ~ 77 km), format 2 (for medium cells, cell's radius up to ~ 29 km, supporting low data rates), - format 3 (for very large cells, cell's radius up to ~ 100 km).

The LTE system was simulated in the following steps:

1. Transmitter generates the RACH preamble with format 0 and spreads it through channel.

2. Cannel has two general conditions: single path channel with AWGN (SNR from -15 dB to -10dB), multipath ETU 70 channel with AWGN (SNR from -8 dB to -3 dB).

3. Receiver receives the signal and calculates correlation between signal and replica. A receiver with one antenna was simulated. Requirements for this simulation are as follows: The probability of correct detection shall be equal or exceed 99% for SNR -11 dB for single path cannel with AWGN and for SNR -5dB for ETU 70 channel. The

requirements were reached. The results are listed in Tables 1 and 2 below.

Table 1. Percentage of RACH preamble's correct detection for single path channel Multipath ETU 70 channel with AWGN

Standard method using convolution

SNR -8 dB -7 dB -6 dB -5 dB - dB -3 dB

99.4% 99.5% 99.5% 99.6% 99.6% 99.6%

P Method using SFFT

2 99.1% 99.2% 99.3% 99.4% 99.5% 99.5%

4 98.7% 98.9% 99.0% 99.1 % 99.2% 99.3%

P Method according to disclosure

2 99.1 % 99.2% 99.3% 99.4% 99.5% 99.5%

4 98.7% 98.8% 99.0% 99.1 % 99.2% 99.3%

Table 2. Percentage of RACH preamble's correct detection for multipath channel

The present disclosure also supports a computer program product including computer executable code or computer executable instructions that, when executed, causes at least one computer to execute the performing and computing steps described herein, in particular the method 200 as described above with respect to Fig. 2 or the algorithm 1200 or the instruction sequence 300 described above with respect to Figures 12 and 13. Such a computer program product may include a readable storage medium storing program code thereon for use by a computer. The program code may perform the method 200 as described above with respect to Fig. 2 or the algorithm 200 or the instruction sequence 1300 described above with respect to Figures 12 and 13.

While a particular feature or aspect of the disclosure may have been disclosed with respect to only one of several implementations, such feature or aspect may be combined with one or more other features or aspects of the other implementations as may be desired and advantageous for any given or particular application. Furthermore, to the extent that the terms "include", "have", "with", or other variants thereof are used in either the detailed description or the claims, such terms are intended to be inclusive in a manner similar to the term "comprise". Also, the terms "exemplary", "for example" and "e.g." are merely meant as an example, rather than the best or optimal. The terms "coupled" and "connected", along with derivatives may have been used. It should be understood that these terms may have been used to indicate that two elements cooperate or interact with each other regardless whether they are in direct physical or electrical contact, or they are not in direct contact with each other.

Although specific aspects have been illustrated and described herein, it will be appreciated by those of ordinary skill in the art that a variety of alternate and/or equivalent implementations may be substituted for the specific aspects shown and described without departing from the scope of the present disclosure. This application is intended to cover any adaptations or variations of the specific aspects discussed herein.

Although the elements in the following claims are recited in a particular sequence with corresponding labeling, unless the claim recitations otherwise imply a particular sequence for implementing some or all of those elements, those elements are not necessarily intended to be limited to being implemented in that particular sequence.

Many alternatives, modifications, and variations will be apparent to those skilled in the art in light of the above teachings. Of course, those skilled in the art readily recognize that there are numerous applications of the invention beyond those described herein. While the present invention has been described with reference to one or more particular embodiments, those skilled in the art recognize that many changes may be made thereto without departing from the scope of the present invention. It is therefore to be understood that within the scope of the appended claims and their equivalents, the invention may be practiced otherwise than as specifically described herein.

Claims

CLAIMS:

1. A method (200) for synchronizing a receive sequence with a known transmit sequence, the method comprising: compressing (201 ) the known transmit sequence from a length N to a reduced length B, the length N being a multiple of the reduced length B, by applying a B-by-N compression matrix Ψ to the transmit sequence to obtain a compressed transmit sequence of the reduced length B, wherein the compression matrix Ψ comprises a plurality of shifting matrices each of the shifting matrices depending on a first compression parameter p that is an integer ratio of the length N and the reduced length B and a second compression parameter a that is an angle between 0 and 2π ; compressing (202) the receive sequence from the length N to the reduced length B by applying the B-by-N compression matrix Ψ to the receive sequence to obtain a compressed receive sequence of the length B; multiplying (203) the compressed transmit sequence with the compressed receive sequence in frequency domain to obtain a compressed product sequence, wherein a time- domain transform of the compressed product sequence indicates a correlation between the receive sequence and the replica of the known transmit sequence; computing a delay of the receive sequence with respect to the known transmit sequence; and synchronizing the receive sequence with the transmit sequence according to the delay.

2. The method (200) of claim , comprising: determining a maximum magnitude element E_max and a position N_max of the maximum magnitude element E_max in the time domain transform of the compressed product sequence; determining an argument φ of the maximum magnitude element; determining a floor value based on the argument φ of the maximum magnitude element E_max, the first compression parameter p and the second compression parameter a ; and if the floor value equals zero: use the position N_max of the maximum magnitude element E_max as the delay of the receive sequence with respect to the known transmit sequence; otherwise: determine the delay based on a threshold comparison of a function of the receive sequence, the transmit sequence, the floor value and the position N_max of the maximum magnitude element E_max.

3. The method (200) of one of the preceding claims, wherein the compression matrix Ψ is constructed by a number of p shifting matrices of dimension B-by-B, wherein p is the first compression parameter.

4. The method (200) of one of the preceding claims, wherein the plurality of shifting matrices are arranged side-by-side in the compression matrix Ψ .

5. The method (200) of one of the preceding claims, wherein each of the shifting matrices depends on a complex-valued shifting operator that is a function of the first compression parameter p and the second compression parameter a .

6. The method (200) of claim 5, wherein the complex-valued shifting operator is e ^p , wherein i is an imaginary unit, p is the first compression parameter and a is the second compression parameter.

7. The method (200) of one of the preceding claims, wherein each of the shifting matrices depends on an index indicating a position of the respective shifting matrix in the compression matrix Ψ .

8. The method (200) of claim 7, wherein the index ranges from 0 to p-1 in steps of 1 , wherein p is the first compression parameter.

9. The method (200) of one of the preceding claims, wherein the compression matrix Ψ is according to:

( ai 2ai 3ai (p-i)ai \

I_{B xB}, e v * I_BxBl e > * I_BxB, e p * I_BxBl ... , e v * I_BxBj, wherein I_BXB is a B-by-B identity matrix, p is the first compression parameter, i is an imaginary unit and is the second compression parameter.

10. The method (200) of one of the preceding claims, wherein the first compression parameter p and the second compression parameter a satisfy the following relation:

2 P wherein Θ is a predetermined phase error.

1 1. The method (200) of claim 10, wherein the predetermined phase error satisfies the following relation:

12. The method (200) of claim 10, wherein the predetermined phase error satisfies the following relation: |0 | < 0.2.

13. The method (200) of one of the preceding claims, wherein the known transmit sequence comprises a RACH preamble of a random access channel according to a mobile communication standard.

14. The method (200) of one of the preceding claims, wherein the first compression parameter p is a power of 2.

15. A synchronization apparatus (300) for synchronizing a receive sequence with a known transmit sequence, the synchronization apparatus (300) comprising: a compressing circuit (332, 334) configured to compress a replica (302) of the known transmit sequence from a length N to a reduced length B, the length N being a multiple of the reduced length B, by applying a B- by-N compression matrix Ψ to the replica (302) to obtain a compressed replica (301 ) of the reduced length B, wherein the compression matrix Ψ comprises a plurality of shifting matrices each of the shifting matrices depending on a first compression parameter p that is an integer ratio of the length N and the reduced length B and a second compression parameter a that is an angle between 0 and 2π , and to compress the receive sequence (304) from the length N to the reduced length B by applying the B-by-N compression matrix Ψ to the receive sequence (304) to obtain a compressed receive sequence (303) of the length B; a frequency transform circuit (306, 308) configured to transform the compressed replica (301 ) and the compressed receive sequence (303) into frequency domain; a multiplication circuit (305) configured to multiply the frequency-transformed compressed replica (309) with the frequency-transformed compressed receive sequence (311 ) to provide a compressed product sequence (313) in frequency domain; an inverse frequency transform circuit (310) configured to transform the compressed product sequence from frequency domain (313) to time domain (307); and a threshold circuit (312) configured to compare the time-domain transform (307) of the compressed product sequence with a threshold to determine a correlation (314) between the receive sequence (304) and the replica of the known transmit sequence (302).