Communication signal comprising a sequence

Abstract

Examples of the invention relate to first and second communication devices for the transmission and reception of a communication signal comprising a sequence in communication systems. The communication signal (510) comprises a sequence y [k], where k = 0,1,..., N − 1, N is an odd integer, and where the sequence y [k] is formed based on a superposition of a first Zadoff-Chu, ZC, sequence (formula I) with a first root index u 0, which is interleaved according to (formula II), and a second ZC sequence (formula III) with a second root index u 1 , which is interleaved according to (formula IV), and wherein u 0 , u 1 , g a , g b , g c , g d are integers fulfilling the conditions: (formula V)(formula VI) formula (VII) where gcd(m,n) is the greatest common divisor of integers m and n. Furthermore, examples of the invention also relate to corresponding methods and a computer program.

Description

[0001] COMMUNICATION SIGNAL COMPRISING A SEQUENCE

[0002] TECHNICAL FIELD

[0003] Examples of the invention relate to first and second communication devices for the transmission and reception of a communication signal comprising a sequence in communication systems. Furthermore, examples of the invention also relate to corresponding methods and a computer program.

[0004] BACKGROUND

[0005] When a transmitter and a receiver of radio signals move relatively to each other, so called Doppler effects will arise. That is, the bandwidth of the transmitted signal could become different in the receiver, e.g., it may be broader, or the transmitted signal appears shifted in frequency at the receiver. In satellite communications with low orbiting satellites, e.g., non-terrestrial networks (NTNs), the frequency shift of the received signal, i.e., the frequency offset, could be substantial and needs to be compensated for in order to provide reliable communications. These frequency offsets are typically much larger than what is experienced in the terrestrial cellular networks. Nevertheless, the transmitter and the receiver need to be synchronized, therefore it is required that the timing and frequency offset is estimated by the receiver. For this purpose, synchronization signals or random access preambles can be transmitted. These signals are predetermined and are selected from a set of signals which is known to the receiver and are used by the receiver to achieve time- and frequency synchronization.

[0006] SUMMARY

[0007] An objective of examples of the invention is to provide a solution which mitigates or solves the drawbacks and problems of conventional solutions.

[0008] Another objective of examples of the invention is to provide sequences from which time- and frequency synchronization can be obtained, while requiring small detection complexity, in communication systems with channels having large frequency offsets.

[0009] The above and further objectives are solved by the subject matter of the independent claims.

[0010] Further examples of the invention can be found in the dependent claims.

[0011] According to a first aspect of the invention, the above mentioned and other objectives are achieved with a first communication device configured to:

[0012] transmit a communication signal to a second communication device, the communication signal comprising a sequence y[fc], where k = 0,1,.... IV — 1, N is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first Zadoff-Chu, ZC, sequence xu,| / <| with a first root index u0, which is interleaved according to xu,\gak + gb\, and a second ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd\, and wherein itg.u-^g^ gb,gc, 9aareintegers fulfilling the conditions:

[0013] u0,u1,ga,gc6 {1,2. N - 1},

[0014] 9b.9d 6 {0,1. N - l},

[0015] gcd(

[0016]

[0017] gcd(u0,lV) = gcd(u1,lV) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n.

[0018] An advantage of the first communication device according to the first aspect is that a large set of sequences could be produced by using different sequence parameters u^u^g^ gb,gc, gd. Furthermore, the probability of detecting the sequence can be improved by proper selection of the sequence parameters, since the sequence parameters determine the shape of a periodic correlation function of the sequence y[fc].

[0019] In an implementation form of a first communication device according to the first aspect, the sequence y[fc] is expressed as y[fc] = xUo[gak + gb] + xUi[gck + gd]

[0020] where the first ZC sequence xu[fc] and the second ZC sequence xu[fc] are defined by

[0021]

[0022] where j=√−1.

[0023] Zadoff-Chu sequences have perfect periodic autocorrelation function and low mutual cross-correlation.

[0024] In an implementation form of a first communication device according to the first aspect, ga= gc= 1 when gb= gd= 0.

[0025] An advantage with this implementation form is to avoid selections of the sequence parameters which produce worse detection probability of the sequence y[fc].

[0026] In an implementation form of a first communication device according to the first aspect, u0,u1,ga, gb,gc, gdare selected to maximize an absolute value of a function y,-(d) determined by a periodic correlation between the sequence y[fc] and an interleaved ZC sequence with root index

[0027]

[0028] An advantage with this implementation form is that it improves the probability of detecting the sequence y[fc].

[0029] In an implementation form of a first communication device according to the first aspect, the function y;[ d ] is expressed for i = 0, 1 and d = 0, 1,..., N — 1 as

[0030] dlV)]

[0031] dlV)]

[0032]

[0033] where (.)* denotes complex conjugate and (mod IV) is the modulo-lV operator.

[0034] An advantage with this implementation form is that only two correlators are needed at the second communication device for jointly determining the timing synchronization and the frequency offset.

[0035] In an implementation form of a first communication device according to the first aspect, maximizing the absolute value of the function y;(d) comprises

[0036]

[0037] or

[0038]

[0039] where d0= arg max| y0(d) | and d, = arg max| yt(d) |, and where |. | denotes the absolute value.

[0040] An advantage with this implementation form is that it maximizes the probability of detecting the timing and the frequency offset of the received sequence. In an implementation form of a first communication device according to the first aspect,

[0041] I«o5a -?l = 1,

[0042] and one or both of

[0043]

[0044] or

[0045]

[0046] An advantage with this implementation form is that this selection of sequence parameters maximizes the probability of detecting the timing and the frequency offset.

[0047] In an implementation form of a first communication device according to the first aspect, the communication signal is an orthogonal frequency division multiplexing, OFDM, signal, a discrete Fourier transform spread OFDM, DFT-s-OFDM, signal or a single-carrier signal.

[0048] An advantage with this implementation form is that the sequence y[fc] can be contained in either an OFDM based multicarrier signal or a single-carrier signal.

[0049] In an implementation form of a first communication device according to the first aspect, the first communication device is configured to:

[0050] performing a DFT of size N on the sequence y[fc] to obtain N DFT outputs; and

[0051] allocate the N DFT outputs to a set of subcarriers in an OFDM signal to form the communication signal.

[0052] An advantage with this implementation form is that the sequence y [fc] can be transmitted on a dedicated set of the subcarriers of an OFDM signal.

[0053] In an implementation form of a first communication device according to the first aspect, the communication signal is a synchronization signal or a random access preamble.

[0054] An advantage with this implementation form is that it enables synchronization and random access in channels with large frequency offsets.

[0055] In an implementation form of a first communication device according to the first aspect, the first communication device is configured to:

[0056] transmit a control message to the second communication device, the control message indicating at least one of

[0057]

[0058] receive a control message from the second communication device, the control message indicating at least one of

[0059] An advantage with this implementation form is that either one sequence or a set of sequences can be transmitted. In the first case, the receiver will know the transmitted sequence. In the second case, the receiver will know the set of sequences from which a transmitted sequence is selected.

[0060] In an implementation form of a first communication device according to the first aspect, the first communication device is a satellite and the second communication device is a user equipment, UE, or vice versa. An advantage with this implementation form is that the issue with large frequency offsets in satellite communications can be mitigated.

[0061]

[0062] An advantage of the second communication device according to the second aspect is that a large set of sequences could be used by using different sequence parameters u^u^g^ gb,gc, gd. Furthermore, the probability of detecting the sequence can be improved by proper selection of the sequence parameters, since the sequence parameters determine the shape of a periodic correlation function of the sequence y[fc].

[0063] In an implementation form of a second communication device according to the second aspect, the first and second periodic correlations are expressed as

[0064]

[0065] where r [fc] is determined based on the received communication signal.

[0066] An advantage with this implementation form is that only two correlators are needed at the second communication device to jointly determine the timing and the frequency offset.

[0067]

[0068]

[0069] In an implementation form of a second communication device according to the second aspect, the sequence y[fc] is expressed as

[0070]

[0071] An advantage with this implementation form is to avoid selections of the sequence parameters which produce worse detection probability of the sequence y[fc].

[0072] In an implementation form of a second communication device according to the second aspect, the communication signal is an OFDM signal, a DFT-s-OFDM signal or a single-carrier signal.

[0073] An advantage with this implementation form is that the sequence y[fc] can be contained in either an OFDM based multicarrier signal or a single-carrier signal.

[0074] In an implementation form of a second communication device according to the second aspect, the communication signal is a synchronization signal or a random access preamble.

[0075] An advantage with this implementation form is that it enables synchronization and random access in channels with large frequency offsets.

[0076] In an implementation form of a second communication device according to the second aspect, the second communication device is configured to:

[0077] receive a control message from the first communication device, the control message indicating at least one of u0, u1:ga, 9b.9c, 9d,

[0078] transmit a control message to the first communication device, the control message indicating at least one of u^u^g^ 9b.9c, 9d-

[0079] An advantage with this implementation form is that either one sequence or a set of sequences can be transmitted. In the first case, the receiver will know the transmitted sequence. In the second case, the receiver will know the set of sequences from which a transmitted sequence is selected.

[0080] In an implementation form of a second communication device according to the second aspect, the first communication device is a satellite and the second communication device is a UE, or vice versa. An advantage with this implementation form is that the issue with large frequency offsets in satellite communications can be mitigated.

[0081] According to a third aspect of the invention, the above mentioned and other objectives are achieved with a method for a first communication device, the method comprises:

[0082] transmitting a communication signal to a second communication device, the communication signal comprising a sequence y[fc], where k = 0,1,, IV — 1, N is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first Zadoff-Chu, ZC, sequence xUo[fc] with a first root index u0, which is interleaved according to xUo[gak + gb\, and a second ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd\, and wherein u0,u1,5a, gb.9c, 9aareintegers fulfilling the conditions:

[0083]

[0084] gcd(

[0085]

[0086] gcd(u0,lV) = gcd(u1,lV) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n.

[0087] The method according to the third aspect can be extended into implementation forms corresponding to the implementation forms of the first communication device according to the first aspect. Hence, an implementation form of the method comprises the feature(s) of the corresponding implementation form of the first communication device.

[0088] The advantages of the methods according to the third aspect are the same as those for the corresponding implementation forms of the first communication device according to the first aspect.

[0089] According to a fourth aspect of the invention, the above mentioned and other objectives are achieved with a method for a second communication device, the method comprises:

[0090] receiving a communication signal from a first communication device, the communication signal comprising a sequence y[fc], where k = 0,1,.... IV — 1, N is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first ZC sequence xu,| / <| with a first root index u0, which is interleaved according to xu. \gak + gb\, and a second ZC sequence xu[fc] withasecond root index u1, which is interleaved according to xu[gck + gd\, and wherein Ug. Uj.g^ gb,gc, gdare integers fulfilling the conditions:

[0091] Uo. UL9a.gc G {1,2. N - 1},

[0092] gb,gd∈ {0,1,..., N − 1},

[0093] gcd(u05a - u^. N) = gcd(u0,lV) = gcd(u1,lV) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n:

[0094] performing a first p0and a second p±periodic correlation between the received communication signal and the first interleaved ZC sequence xu. \gak + gb\ and the second interleaved ZC sequence xUi[gck + gd\, respectively, to obtain correlation values; and

[0095] estimating at least one of a timing t and a frequency offset f of the received communication signal based on the correlation values.

[0096] The method according to the fourth aspect can be extended into implementation forms corresponding to the implementation forms of the second communication device according to the second aspect. Hence, an implementation form of the method comprises the feature(s) of the corresponding implementation form of the second communication device. The advantages of the methods according to the fourth aspect are the same as those for the corresponding implementation forms of the second communication device according to the second aspect.

[0097] Examples of the invention also relate to a computer program, characterized in program code, which when run by at least one processor causes the at least one processor to execute any method according to examples of the invention. Further, examples of the invention also relate to a computer program product comprising a computer readable medium and the mentioned computer program, wherein the computer program is included in the computer readable medium, and may comprises one or more from the group of: read-only memory (ROM), programmable ROM (PROM), erasable PROM (EPROM), flash memory, electrically erasable PROM (EEPROM), hard disk drive, etc.

[0098] Further applications and advantages of examples of the invention will be apparent from the following detailed description.

[0099] BRIEF DESCRIPTION OF THE DRAWINGS

[0100] The appended drawings are intended to clarify and explain different examples of the invention, in which:

[0101] - Fig. 1 shows a first communication device according to examples of the invention;

[0102] - Fig. 2 shows a flow chart of a method for a first communication device according to examples of the invention; - Fig. 3 shows a second communication device according to examples of the invention;

[0103] - Fig. 4 shows a flow chart of a method for a second communication device according to examples of the invention; - Fig. 5 shows a communication system according to examples of the invention;

[0104] - Fig. 6 shows a peak value as function of the peak-to-sidelobe-ratio;

[0105] - Fig. 7 shows a maximum peak-to-sidelobe-ratio for different sequence lengths;

[0106] - Fig. 8 shows detection probability as function of peak-to-sidelobe-ratio;

[0107] - Fig. 9 shows probability of timing and frequency offset detection without LPPs for different distance between root indices;

[0108] - Fig. 10 shows probability of timing and frequency offset detection when using LPP compared to without using LPP;

[0109] and

[0110] - Fig. 11 shows comparison of detection methods using the same sequence.

[0111] DETAILED DESCRIPTION

[0112] The subject of constructing suitable synchronization signals / preambles subject to high Doppler shifts in low-earth orbit (LEO) satellites was considered in 3GPP New Radio (NR) Release-17. However, it was decided that the ground receiver, i.e., the user equipment (UE) is equipped with a global navigation satellite system (GNSS) module such that it can determine its position and velocity. This information, combined with satellite ephemeris information, either broadcast by the network or provisioned via other means, allows the UE to determine the satellite position / velocity and then compute the Doppler shift and carry out frequency pre-compensation and timing adjustment. The pre-compensation could be applied when receiving synchronization signals from the satellite or when transmitting random access (RACH) preamble to the satellite. Hence, the Doppler shift is mostly eliminated, and no particular signals need to be provided for the purpose of synchronization under high Doppler shift. While this assumption alleviates the issue, it puts more requirements on the UE and its implementation, considering that the GNSS signal could be weak and that low-cost UEs might neither have the time nor the energy to determine its position and / or receive the ephemeris transmissions. A difficulty is that the GNSS signals may not be detected by the UE, e.g., if it is located indoors, or if there is severe interference. The GNSS signals may also be jammed by an adversary. Thereto, there needs to be a network that provides the satellite ephemeris to the UE. Thus, in the future 3GPP releases, this issue may be revisited, and a robust solution may not be based on the GNSS. In multi-carrier signals, the Doppler frequency can be divided into an integer frequency offset and a fractional frequency offset. The integer part characterizes how many subcarriers the signal is shifted in frequency. The fractional part characterizes how large fraction of a subcarrier spacing the signal is shifted. The fractional frequency offset could, e.g., be estimated from the phase shift of the signal, obtained by a correlation between repeated orthogonal frequency division multiplexing (OFDM) symbols. If the signals of the OFDM symbols are not pre-determined, autocorrelation can be used, otherwise cross-correlation between the known and received signals is a possible method. Alternatively, autocorrelation can be between the cyclic prefix and the corresponding repeated samples within the OFDM symbol.

[0113] If the fractional frequency offset is to be estimated before the integer frequency offset, there needs to be some form of time synchronization method, in order to identify the repeated signal parts over which correlation should be performed for the fractional frequency offset estimation. The time synchronization then needs to be performed subject to both the integer and fractional frequency offset. Alternatively, the fractional frequency offset could be estimated after the timing and the integer frequency offset has been estimated and compensated for. However, this requires a solution which provides robust timing synchronization and integer frequency offset estimation, subject to both the integer and fractional frequency offset, which is within the scope of examples of the invention.

[0114] An objective of examples of the invention is to provide a communication signal which can be detected subject to large frequency offset. The communication signal can serve the purpose of a synchronization signal or a random access preamble. From the received signal, the timing and frequency offset of the signal may be estimated by the receiver. The signal may further comprise an identity of the cell or of a UE. This solves the issue of maintaining synchronization between transmitter and receiver, or performing access to the network, subject to large Doppler shifts, which is especially relevant for NTN communications. A particular objective of examples of the invention is focused on estimating the integer frequency offset and the timing in the presence of an uncompensated fractional frequency offset.

[0115] Fig. 1 shows a first communication device 100 according to examples of the invention. In the example shown in Fig. 1, the first communication device 100 comprises a processor 102, a transceiver 104 and a memory 106. The processor 102 is coupled to the transceiver 104 and the memory 106 by communication means 108 known in the art. The first communication device 100 may be configured for wireless and / or wired communications in a communication system. The wireless communication capability may be provided with an antenna or antenna array 110 coupled to the transceiver 104, while the wired communication capability may be provided with a wired communication interface 112 e.g., coupled to the transceiver 104.

[0116] The processor 102 may be referred to as one or more general-purpose central processing units (CPUs), one or more digital signal processors (DSPs), one or more application-specific integrated circuits (ASICs), one or more field programmable gate arrays (FPGAs), one or more programmable logic devices, one or more discrete gates, one or more transistor logic devices, one or more discrete hardware components, or one or more chipsets. The memory 106 may be a read-only memory, a random access memory (RAM), or a non-volatile RAM (NVRAM). The transceiver 104 may be a transceiver circuit, a power controller, or an interface providing capability to communicate with other communication modules or communication devices, such as network nodes and network servers. The transceiver 104, memory 106 and / or processor 102 may be implemented in separate chipsets or may be implemented in a common chipset. That the first communication device 100 is configured to perform certain actions can in this disclosure be understood to mean that the first communication device 100 comprises suitable means and devices, such as e.g., the processor 102 and the transceiver 104, configured to perform the actions.

[0117] According to examples of the invention the first communication device 100 is configured to transmit a communication signal 510 to a second communication device 300, the communication signal 510 comprising a sequence y[fc], where k = 0,1,..., N — 1, N is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first Zadoff-Chu, ZC, sequencexu0[fc] withafirst root index u0, which is interleaved according to xUo[gak + gb], and a second ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd], and wherein ul),u-],ga, gh,gc, gdare integers fulfilling the conditions:

[0118] u0, u1,ga,gc∈ {1,2,..., N − 1},

[0119] gb,gd∈ {0,1,..., N − 1},

[0120] gcd(

[0121]

[0122] gcd(u0,! V) = gcd(u1,lV) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n.

[0123] The above conditions assure that the herein disclosed sequence y [fc] is interleaved by a linear permutation polynomial (LPP).

[0124] Furthermore, in an example of the invention, the first communication device 100 comprises a transceiver configured to transmit a communication signal 510 to a second communication device 300, the communication signal 510 comprising a sequence y[fc], where k = 0,1,.... IV — 1, N is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first Zadoff-Chu, ZC, sequence xu,| / <| with a first root index u0, which is interleaved according to xu. \gak + gb\, and a second ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd\, and wherein u0, ULga, 9b.9c. 9aareintegers fulfilling the conditions:

[0125] Uo. UL9a.gc G {1,2. N - 1},

[0126] gb,gd∈ {0,1,..., N − 1},

[0127] gcd(u05a - u^. N) = gcd(u0,! V) = gcd(u1,lV) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n.

[0128] Moreover, in yet another example of the invention, the first communication device 100 for a communication system 500 comprises a processor and a memory having computer readable instructions stored thereon which, when executed by the processor, cause the processor to transmit a communication signal 510 to a second communication device 300, the communication signal 510 comprising a sequence y[fc], where k = 0,1,...,1V — 1, IV is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first Zadoff-Chu, ZC, sequence xu,| / <| with a first root index u0, which is interleaved according to xUo[gak + gb], and a second ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd\, and wherein u0, ULga, gb,gc, gdare integers fulfilling the conditions:

[0129] Uo. UL9a.9c G {1,2. IV - 1},

[0130] gb,gd∈ {0,1,..., N − 1},

[0131] gcd(u05a - u^. N) = gcd(u0,! V) = gcd(u1,lV) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n.

[0132] Fig. 2 shows a flow chart of a corresponding method 200 which may be executed in a first communication device 100, such as the one shown in Fig. 1. The method 200 comprises transmitting 202 a communication signal 510 to a second communication device 300, the communication signal 510 comprising a sequence y[fc], where k = 0,1,...,1V — 1, IV is an odd integer, and where the sequence y [fc] is formed based on a superposition of a first Zadoff-Chu, ZC, sequence xUo[fc] with a first root index u0, which is interleaved according to xUo[gak + gb], and a second ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd\, and wherein u0, ULga, 9b, 9c. 9dareintegers fulfilling the conditions:

[0133] Uo. UL9a.9c G {1,2. IV - 1},

[0134] 9b.9d G {0,1. IV - 1},

[0135] gcd(u05a - u^. N) = gcd(u0,! V) = gcd(u1,lV) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n. Fig. 3 shows a second communication device 300 according to examples of the invention. In the example shown in Fig. 3, the second communication device 300 comprises a processor 302, a transceiver 304 and a memory 306. The processor 302 is coupled to the transceiver 304 and the memory 306 by communication means 308 known in the art. The second communication device 300 may be configured for wireless and / or wired communications in a communication system. The wireless communication capability may be provided with an antenna or antenna array 310 coupled to the transceiver 304, while the wired communication capability may be provided with a wired communication interface 312 e.g., coupled to the transceiver 304.

[0136] The processor 302 may be referred to as one or more general-purpose CPUs, one or more DSPs, one or more ASICs, one or more FPGAs, one or more programmable logic devices, one or more discrete gates, one or more transistor logic devices, one or more discrete hardware components, one or more chipsets. The memory 306 may be a read-only memory, a RAM, or a NVRAM. The transceiver 304 may be a transceiver circuit, a power controller, or an interface providing capability to communicate with other communication modules or communication devices. The transceiver 304, the memory 306 and / or the processor 302 may be implemented in separate chipsets or may be implemented in a common chipset. That the second communication device 300 is configured to perform certain actions can in this disclosure be understood to mean that the second communication device 300 comprises suitable means and devices, such as e.g., the processor 302 and the transceiver 304, configured to perform the actions.

[0137] According to examples of the invention the second communication device 300 is configured to:

[0138] receive a communication signal 510 from a first communication device 100, the communication signal 510 comprising a sequence y[fc], where k = 0,1,, IV — 1, N is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first ZC sequence xu,| / <| with a first root index u0, which is interleaved according to xu,\gak + gb\, and a second ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd\, and wherein u^.u^ga, gb,gc, gdare integers fulfilling the conditions:

[0139] u0, u1,ga,gc∈ {1,2,..., N − 1},

[0140] gb,gd∈ {0,1,..., N − 1},

[0141] gcd(

[0142]

[0143] gcd(u0,lV) = gcd(u1, / V) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n:

[0144] perform a first p0and a second p±periodic correlation between the received communication signal 510 and the first interleaved ZC sequence xu,\gak + gb\ and the second interleaved ZC sequence xUi[gck + gd\, respectively, to obtain correlation values; and

[0145] estimate at least one of a timing t and a frequency offset f of the received communication signal 510 based on the correlation values.

[0146] Furthermore, in an example of the invention, the second communication device 300 comprises a transceiver configured to receive a communication signal 510 from a first communication device 100, the communication signal 510 comprising a sequence y[fc], where k = 0,1,..., IV — 1, N is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first ZC sequence xu,| / <| with a first root index u0, which is interleaved according to xu,\gak + gb\, and a second ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd\, and wherein u^.u^ga, gb,gc, gdare integers fulfilling the conditions:

[0147] Uo. UL9a.gc G {1,2. N - 1},

[0148] gb,gd∈ {0,1,..., N − 1},

[0149] gcd(u05a - u^. N) = gcd(u0,lV) = gcd(u1, / V) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n. The second communication device 300 comprises a processor configured to: perform a first ρ0and a second ρ1periodic correlation between the received communication signal 510 and the first interleaved ZC sequence xu. \gak + gb\ and the second interleaved ZC sequence xu[gck + gd\, respectively, to obtain correlation values; and estimate at least one of a timing t and a frequency offset f of the received communication signal 510 based on the correlation values.

[0150] Moreover, in yet another example of the invention, the second communication device 300 for a communication system 500 comprises a processor and a memory having computer readable instructions stored thereon which, when executed by the processor, cause the processor to:

[0151] receive a communication signal 510 from a first communication device 100, the communication signal 510 comprising a sequence y[fc], where k = 0,1,, IV — 1, N is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first ZC sequence xu,| / <| with a first root index u0, which is interleaved according to xu. \gak + gb\, and a second ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd\, and wherein u^.u^ga, gb,gc, gdare integers fulfilling the conditions:

[0152] u0, u1,ga,gc∈ {1,2,..., N − 1},

[0153] gb,gd∈ {0,1,..., N − 1},

[0154] gcd(

[0155]

[0156] gcd(u0,lV) = gcd(u1,lV) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n:

[0157] perform a first ρ0and a second ρ1periodic correlation between the received communication signal 510 and the first interleaved ZC sequence xu[gak + gb] and the second interleaved ZC sequence xu[gck + gd], respectively, to obtain correlation values; and

[0158] estimate at least one of a timing t and a frequency offset f of the received communication signal 510 based on the correlation values.

[0159] Fig. 4 shows a flow chart of a corresponding method 400 which may be executed in a second communication device 300, such as the one shown in Fig. 3. The method 400 comprises:

[0160] receiving 402 a communication signal 510 from a first communication device 100, the communication signal 510 comprising a sequence y[fc], where k = 0,1,.... IV — 1, N is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first ZC sequence xu[fc] with a first root index u0, which is interleaved according to xUo[gak + gb], and a second ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd\, and wherein u0,u1,5a, 9b.9c.9aareintegers fulfilling the conditions:

[0161] Uo. UL9a.gc G {1,2. N - 1},

[0162] gb,gd∈ {0,1,..., N − 1},

[0163] gcd(u05a - u^. N) = gcd(u0,lV) = gcd(u1,lV) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n:

[0164] performing 404 a first ρ0and a second ρ1periodic correlation between the received communication signal 510 and the first interleaved ZC sequence xUo[gak + gb\ and the second interleaved ZC sequence xu[gck + gd\, respectively, to obtain correlation values; and

[0165] estimating 406 at least one of a timing t and a frequency offset f of the received communication signal 510 based on the correlation values.

[0166] Fig. 5 shows a communication system or a communication network 500 according to an example of the invention. The communication system 500 in the disclosed example comprises a first communication device 100 and a second communication device 300 configured to communicate and operate in the communication system 500. In this particular example, the first communication device 100 is configured as a satellite, while the second communication device 300 is configured as a client device such as a HE. However, the reverse case is also possible, i.e., that the first communication device 100 is configured as a client device, while the second communication device 300 is configured as a satellite.

[0167] In Fig. 5 it is shown how the first communication device 100 transmits a communication signal 510 to the second communication device 300. The communication signal 510 comprises at least one sequence according to examples of the invention.

[0168] Control signaling according to examples of the invention is also illustrated in Fig. 5. The mentioned control signaling involves the transmission and reception of a control message 520 indicating at least one of u0,u1,ga, gb,gc, gd. The transmission direction of the control message 520 may either be from the first communication device 100 to the second communication device 300 or from the second communication device 300 to the first communication device 100. Hence, from the perspective of the first communication device transmit a control message 520 to the second communication device 300, the control message 520 indicating at least one of u0,u1,ga, gb,gc, gd', or receive a control message 520 from the second communication device 300, the control message 520 indicating at least one of u0,u1,ga, gb,gc, gd, where the control message may be contained in medium access control (MAC) signaling or radio resource control (RRC) signaling, or where the control message is obtained from other configurations of control signals in the system. The indication of the sequence parameters may be explicit or implicit in the control message 520.

[0169] Furthermore, in examples of the invention, the communication signal 510 is any of an OFDM signal, a discrete Fourier transform spread OFDM (DFT-s-OFDM) signal or a single-carrier signal.

[0170] The communication signal 510 may be used for synchronization or in random access procedures and thus the communication signal 510 is a synchronization signal or a random access preamble in such applications.

[0171] Further details related to examples of the invention will fully or partially be described in a 3GPP context to provide deeper understanding of the herein disclosed solution. Thus, 3GPP terminology, definitions, expressions and system architecture may be used. It may however be noted that examples of the invention are not limited thereto and that the disclosed solution may be set in communication systems and networks different to the ones standardized by 3GPP.

[0172] In general terms, examples of the invention relate to transmitting and receiving a sequence in a communication system 500. The sequence y[fc] herein used may be expressed as

[0173] y[k] = xUo[gak + gb] + xUi[gck + gd] (1) where k = 0,1,...,1V — 1, N is an odd integer, and where the first Zadoff-Chu sequence xu,| / <| and the second Zadoff-Chu sequence xu[fc] are defined by

[0174]

[0175] where j=√−1.

[0176] In examples of the invention, u^u^g^ gb,gc, gd, also denoted parameters of the sequence y[fc], are selected to maximize an absolute value of a function y,- (d) determined by a periodic correlation between the sequence y[fc] and an interleaved Zadoff-Chu sequence with root index u,-.

[0177] More specifically, in examples of the invention, the sequence parameters are selected according to

[0178]

[0179] or )l,,

[0180] 7377,a at(4)

[0181]

[0182] for i = 0 or i = 1, where

[0183] a(k -d) +gb(mod N)],d = 0,1. N - 1

[0184] (5) c(fc - <0 + 9d (modlV)],d = 0,1,..., N - 1

[0185]

[0186] with (mod IV) denoting the modulo-N operator and where

[0187] d0= argmax|y0(d)|, dt= argmaxly^d)!. (6) a av 7to maximizing the absolute value of the function y,-(d).

[0188] In a specific example of maximizing the absolute value of the function y,-(d). the sequence parameters u0, u1, ga. gb, gcand gdare selected such that |u05a— ui9c I=1al'4 either one or both of:

[0189]

[0190] Furthermore, in further examples of the invention, a receiver of the second communication device 300 is configured to perform the first and second periodic correlations expressed as

[0191]

[0192] between r [fc] and the first Zadoff-Chu sequence xUo[fc] and the second Zadoff-Chu sequence xu[fc], where r [fc] is determined based on the received communication signal 510, and where * denotes the complex conjugate.

[0193] In yet further examples of the invention, the timing t and the frequency offset f are given by

[0194] t̂ = (u0ga2- u1gc2)-1(u0ga2d0- u1gc2d1)(mod N) (8) and

[0195] f̂ = u0ga2t̂ - u0ga2d0(mod N). (9) where

[0196]

[0197] In the following disclosure further aspects of the disclosed solution will be presented and described.

[0198] Synchronization sequence and signal

[0199] Let k = 0,1,..., IV — 1 where N is an odd integer, j = — 1 and define the Zadoff-Chu sequence

[0200] xu[k] = e-jπ / N·uk(k+1)(11) where the root index u is a positive integer being relatively prime to N, i.e., the greatest common divisor fulfills gcd(u, IV) = 1. A skilled person will be able to perform the examples of the invention using the complex-conjugate version of Eq. (11). A skilled reader will also be able to use the general definition of an odd-length Zadoff-Chu sequence

[0201]

[0202] where q is an integer and proceed with the same steps as will be given below.

[0203] Consider the linear permutation polynomial (LPP)

[0204] g[k] = g1k + g0(mod N) (12) where g0∈ {0,1,...,N − 1}, g1∈ {1,2,...,N − 1}. It can be shown that a requirement for Eq. (12) to be a permutation polynomial, i.e., that it permutes the integers in the set {0,1,..., IV — 1}, is that gcd(g1, N) = 1.

[0205] The following sequence is disclosed to be used for time- and frequency synchronization:

[0206] y[fc] = xUo[gak + gb] + xUi[gck + gd] (13) A skilled person will be able to preclude the combination ga= gc= 1 together with gb= gd= 0. Thus, in examples of the invention when gb= gd= 0.

[0207] Transmission of the sequence y[fc] by the first communication device 100 can as aforementioned be carried out in several ways.

[0208] In a first example, the sequence according to Eq. (13) is used as the Fourier coefficients of an OFDM signal, or as the input symbols of a discrete Fourier transform (DFT) spread OFDM (DFT-s-OFDM) signal. A cyclic prefix may be appended to the OFDM or DFT-s-OFDM signal. Thus, in the example when the communication signal 510 is an OFDM signal, the first communication device performs a DFT of size N on the sequence y[fc] to obtain N DFT outputs; and allocates the N DFT outputs to a set of subcarriers in an OFDM signal to form the communication signal 510.

[0209] In a second example, the sequence according to Eq. (13) is transmitted by a single-carrier waveform wherein each element of Eq. (13) modulates a pulse, p(t), of finite duration T, to provide the low-pass equivalent signal

[0210]

[0211] A cyclic prefix may be appended to the single carrier signal in Eq. (14).

[0212] The average power of Eq. (13) is given by

[0213]

[0214] and a normalized version can be defined by

[0215] 1

[0216] y[fc] = ^(.xUo[gak + gb] + xUi[gck + gd]). (16) It is noted that Eq. (13) and (16) do not necessarily have constant magnitude, as opposed to Eq. (11).

[0217] For a multicarrier signal with a sampling rate of fs= N f, where N is the number of subcarriers and A is the subcarrier spacing, and a Doppler shift ofD, the normalized frequency offset is defined by f = fD / A. The Doppler effect can be modeled by the multiplication with a sampled complex sinusoid ej(2π / N)fk

[0218]

[0219] such that for an additive white Gaussian (AWGN) channel with noise n[fc], the low-pass equivalent time-discrete received signal is given by

[0220]

[0221] where t is an integer capturing the time delay and f = fD / A is a normalized frequency offset. The representation in Eq. (17) is applicable if y[fc] is transmitted by a DFT-s-OFDM signal or according to Eq. (14). Both t and f are unknown to the receiver. The frequency offset comprises an integer part, i.e.,mtG {0,1,..., H — 1}, where H is an odd integer that should result in an integer frequency offset in the interval { — G [—0.5,0.5], The frequency offset can be expressed by

[0222]

[0223] and for an OFDM signal,mtdescribes the frequency shift in terms of number of subcarriers. If the signal contains a cyclic prefix, the communication signal 510 has a repetitive structure andfraccan be estimated, e.g., by auto-correlation based methods. The objective of the receiver is to estimate t and flntsubject to the uncompensated fractional frequency offsetfrac.

[0224] Estimation of timing and frequency offset

[0225] The receiver of the second communication device 300 determines two periodic correlation functions according to,

[0226] )

[0227]

[0228] where * denotes the complex conjugate. It should be noted that Eq. (19) and (19b) can equivalently be performed as the inverse DFT (IDFT) of the element-wise product between the DFT of r[fc] and the conjugate of the DFT of xu. \gak + gb\ or xU1[gck + gd].

[0229] Utilizing Eq. (13) and (17), the correlation function becomes

[0230]

[0231] where

[0232]

[0233] and

[0234]

[0235] Moreover, the correlation function becomes

[0236]

[0237] where

[0238]

[0239] and

[0240]

[0241] The noise ñ[k] in Eq. (20) and (23) is AWGN. The terms I01(d) and I11(d) represent the autocorrelation values of the first and second Zadoff-Chu sequences while I02(d) and I12(d) are cross-correlation values between the first and second Zadoff-Chu sequences. Depending on the magnitude and the phase of Ii1(d) relative to the phase of Ii2(d), the magnitude of ρi(d) could increase or decrease.

[0242] The first geometric sum / 01(d) can be expanded as

[0243] If the frequency offs

[0244]

[0245] moves the maximum peak value of | / 01(d)| away from the true delay t. If the frequency offset also contains the fractionalpart, the effect of is that it also reduces the peak value of | / 01(d)|, due to the sine functions in Eq. (26).

[0246] Suppose f = fint−

[0247]

[0248] then it follows that I01(d) in Eq. (23) achieves its maximum value a0N, since / is an integer, and we obtain by expanding the geometric sum

[0249] (mod IV) (27)

[0250]

[0251] Furthermore, let

[0252]

[0253] then due to Eq. (26), the correlation peaks at delays d0and d1will appear when the following conditions hold:

[0254] (mod IV)

[0255] (29)(mod N)

[0256]

[0257] Sincemtcan assume values from 0 to IV — 1 and is determined from Eq. (29), it is required that H < N. The solution to Eq. (29) can equivalently be obtained by either first eliminating / or t. In the first case, the solution is expressed by:

[0258]

[0259] In the second case, the solution is expressed by:

[0260]

[0261] It is known that the modular multiplicative inverse (a) fulfilling (a) 'a = 1 (mod N) exists if gcd(a,lV) = 1. Thus, the sequence parameters have to be chosen such that

[0262] gcd(u0ga2- u1gc2, N) = 1 (34) or, equivalently

[0263] gcd((u0ga2)-1- (u1gc2)-1, N) = 1. (35)

[0264] The complexity of solving Eq. (29) does not depend on H, i.e., the number of integer frequency offsets. Hence, only two correlators are needed at the second communication device 300 for any range of the frequency offset. Moreover, since the timing and integer frequency offset is determined from Eq. (29), the detection performance will not depend on H. If the maximum frequency offset is known to be H, a constraint can be applied such that the estimated value is

[0265]

[0266] Selection of sequence parameters

[0267] In order to determine suitable sequence parameters u0, u1, ga, gb, gcand gd, define the periodic correlation function of the sequence as follows:

[0268]

[0269] The probability of correct detection of t̂ = t and f̂ = fintdepends on the shapes of |γ0(d, t, f)| and |γ1(d, t, f)|. Suppose that the maximum correlation peaks are located according to

[0270] d0= argmaxd|γ0(d, t, f)|, d1= argmaxd|γ1(d, t, f)|. (40) a av 7

[0271] The value of |γi(di, t, f)| should be large since that increases the probability of determining the correct location of the main peak considering background noise. Also, the sidelobes should be small, i.e., the values of |γi(d, t, f)|, d ≠ dishould be small. Since this property should be achieved for any time delay t and frequency offset f, the problem of determining sequence parameters could be considered without loss of generalization for Yi(d, 0,0). Thus, the following two criteria could be used for choosing the parameters, i.e., for any of i = 0,1, solve for the maximum peak

[0272] 0max |yi(d

[0273] ,u1,ga,gb,gc,gli;)| (41)

[0274] ’ or the maximum peak- to- sidelobe-ratio

[0275]

[0276] Both Eq. (41) and (42) can be classified as a non-linear integer programming problem, which is known to be NP-complete, i.e., exhaustive search for the optimal solutions is needed, or some form of heuristics in order to obtain good solutions. Typically, there is not only a single combination of parameters which solve Eq. (41 ) or (42). Hence, a set of sequences could be obtained. It is therefore possible to associate, e.g., a cell identity or a UE identity to a certain sequence. The receiver may upon detection of the sequence therefore obtain time- and frequency synchronization with the transmitter and additionally determine the identity of the transmitter. With t = f = 0, Eq. (38) and (39) can be simplified, e.g.,

[0277]

[0278] with

[0279]

[0280] Then

[0281]

[0282] When d = d0, it follows from Eq. (27) that −u0ga2d0≡ 0 (mod N), so Eq. (46) reduces to

[0283] N-l

[0284] |γ0(d0)| = |N + ejπ / N·u(g+g)e-jπ / N·u(g+g)Σejπ / N·u(gk+(g+2gg)k)e-jπ / N·u(gk+(g+2gg)k)| (47) k=0

[0285] which trivially implies that |y0(d0)| 27V. By using the Property in Appendix A, the magnitude of the sum equals

[0286]

[0287] which implies that

[0288] N − √N ≤ |γ0(d0)| ≤ N + √N. (49)

[0289] When d d0, it follows from Eq. (46) that

[0290]

[0291] By using the Property in Appendix A for Eq. (50), it follows that |γ0(d)| = √N when d ≠ d0. Therefore, the peak-to- sideloberatio is bounded within the range:

[0292]

[0293] Moreover, it implies that the peak-to-sidelobe-ratio d ≠ diis constant for any sequence parameters. Hence, the

[0294]

[0295] parameters which maximize the peak-to-sidelobe-ratio also maximize the peak |γi(di)|.

[0296] It can be observed from Eq. (47) that if there is no LPP, i.e., ga= gc= 1, gb= gd= 0. the correlation magnitude depends only on the difference |u0− u1|.

[0297] As an example, for N = 9 a complete exhaustive computation is done over all feasible combinations of the parameters {u1,u2,ga,gb,gc,gd}. Fig. 6 shows the possible values of |γi(di)| as function of the associated ratio

[0298]

[0299] and it can be

[0300]

[0301] observed that a large value of |γi(di)| / |γi(d)| implies a large value of |γi(di)|. This is expected, since we showed above that |γ0(d)| =

[0302]

[0303] √N for all d ≠ d0. Thus, any of Eq. (41) or (42) would be a suitable condition for sequence parameter selection. Maximizing Eq. (41) therefore implies maximizing Eq. (42), and vice versa. In total, there is a set of 189 unique sequences, where any sequence in the set is not a circularly shifted version of another sequence in the set and / or is not the same sequence multiplied with a constant. When only using different combinations of root indices u, and u2, i.e., ga= gc= 1 and gb= gd= 0, there are only 9 unique sequences. That is, a 21 times increase of number of sequences is produced by the LPPs. The circles in Fig.

[0304] 6 are for the values without LPPs and it can be observed that a lower maximum peak-to-sidelobe-ratio will be achieved, suggesting that with proper selection of the LPPs, the detection performance will be better.

[0305] Based on the exhaustive evaluation, Fig. 7 shows that the maximum value of among all sequence parameters for a given

[0306]

[0307] N, is an increasing function of the sequence length, which implies that the detection performance will improve with the sequence length. Notably, the curve in Fig. 7 matches N + 1 for almost all values of N. It can be found by inspection that it does not match exactly

[0308] yfor IV = 3 and N = 7, where small differences can be observed. Thus, the maximum value of ’ ’ l / iWI over all sequence parameters, even if Eq. (48) does not hold, in most cases achieves the upper bound of Eq. (51).

[0309] A case for sequence parameter selection which achieves the maximum peak-to-sidelobe-ratio can be found as follows. By using the formula of Appendix B with the assumption that

[0310] |u0ga2− u1gc2| = 1 (52) then Eq. (47) can be expressed as

[0311]

[0312] Therefore, if gb= gd= 0, it follows from Eq. (53) that |γ0(d0)| = N + √N, when

[0313] |u0ga− u1gc| = √N (54) since the numerator in the exponential function in Eq. (53) becomes zero. The condition in Eq. (54) can only be fulfilled if √N is an integer.

[0314] Assuming Eq. (52) holds, a condition for having |γi(di)| = N + √N can be identified from Eq. (53) such that

[0315]

[0316] There is, however, no guarantee that Eq. (55) could always be fulfilled, e.g., since as noted the cases of N = 3 and N = 7 will not achieve |γi(di)| = N + √N.

[0317] Through exhaustive evaluation, it appears that:

[0318] • If N = 4k + 1, where k is an integer, there will be sequence parameters such that |γi(di)| = N + √N.

[0319] • If N = 8k + 1, where k is an integer, there will be sequence parameters such that |γi(di)| = N + √N while fulfilling Eq. (52) and (55).

[0320] • If N ≡ 1 (mod 4) and √N is not an integer, there will be sequence parameters such that |γi(di)| = N − √N.

[0321] Table 1 shows how many unique sequences there are for which |γi(di)| = N + √N and the associated maximum peak-to-sidelobe-ratio, i.e., -

[0322]

[0323] Table 1. Number of unique sequences having maximum peak-to-sidelobe-ratio with LPPs.

[0324]

[0325]

[0326] The exhaustive search for sequences with maximum peak-to- sidelobe-ratio can be simplified by fixing either gb= 0 or gd= 0. With that restriction, the same numbers as in Table 1 are still obtained.

[0327] Table 2 shows how many unique sequences there are which have the maximum peak-to-sidelobe ratio without using LPPs. These ratios are lower than in Table 1 and the numbers of sequences with the maximum ratio is also much smaller.

[0328] Table 2. Number of unique sequences having maximum

[0329] peak-to-sidelobe ratio without using LPPs.

[0330]

[0331] Table 3 in Appendix C lists examples of sequence parameters where |γi(di)| = N + √N.

[0332] In contrast to Eq. ( 11 ), the sequence according to Eq. (13) does not have constant magnitude. Thus, it may become distorted, or require a transmit power backoff, when transmitted through a nonlinear power amplifier. The well-known Rapp model simulates a solid-state power amplifier where the output signal is given by

[0333]

[0334] where Asatis the amplitude saturation level and p is a parameter modeling the amplifier. The relation between the input power backoff (IBO) and the power Pinof the input signal yin[k] is given by

[0335] 2

[0336] Pin= Asat2 / IBO. (57)

[0337] In this example, we set IBO = 0 dB, p = 2, Asat= 1 and yin[k] = ȳ[k]. Thus, a sequence with unit amplitude, |yin[k]| = 1, would not experience any clipping while sequences with amplitude variations will experience clipping and non-linear amplitude compression for levels below Asat. Furthermore, the number of integer frequency offsets is set to H = N, flntis a discrete uniform random variable andfracis a uniform random variable. Fig. 8 shows the detection probability Pr[t̂ = t, f̂ = fint] at a signal-to-noise-ratio (SNR) of 0 dB, for every feasible sequence, plotted as function of its |γi(di)| / |γi(d)| value. The SNR is defined as

[0338]

[0339] SNR = Pin / Pnoisewhere Pnoiseis the power of the background noise. The results show that better detection probability is achieved with larger peak-to- sidelobe-ratio. The circles in Fig. 8 are for the values without LPPs and it can be observed that a lower maximum detection probability will be achieved than with LPPs.

[0340] If no LPPs are used (ga= gc= 1 and gb= gd= 0), Fig. 9 shows that the detection probability is sensitive to the choice of Δu = |u0− u1|. The values of Δu are constrained by the conditions gcd(u0− u1, N) = gcd(u0, N) = gcd(u1, N) = 1.

[0341] The slightly better peak-to- sidelobe-ratio of using LPP translates into a better probability of detection in comparison to without LPP, as shown in Fig. 10. Here, LPP with ga= 4, gb= 0 and u0= 1 is used with gc= 2, gd= 4 and u1= 2. Comparison to alternative receivers

[0342] One solution for determining the timing under large frequency offset is to detect the signal with several frequency offset hypotheses. The received signal is pre-multiplied with a complex sinusoid with a frequency corresponding to the hypothesis and is then fed to a correlator. One correlator is used per hypothesis. The hypothesis frequencies may be positive or negative, which implies that some hypothesis will cancel most of the frequency offset of the received signal. The timing is determined from the maximum peak of the outputs of all correlators. A drawback of this solution is the complexity, since a correlator is needed for each hypothesis. Furthermore, there could be a performance tradeoff since more frequency hypotheses increase the chance of better cancellation of the frequency offset, on the other hand, it results in more false timing peaks, since the number of correlators increases.

[0343] Another solution is to divide the received signal into smaller parts and perform correlation on each part, and then non-coherently accumulate the correlation values from the different parts. Thereby, the effect of the frequency offset is reduced. A drawback of this method is the complexity, since a correlator is needed for each part. Moreover, since the length over which the correlation is performed becomes shorter, there is a loss in the noise averaging, i.e., effectively the SNR decreases.

[0344] Fig. 11 compares the probability of correct timing for N = 9 where the sequence is the superposition of two Zadoff-Chu sequences with root indices u0= 1 and u, = 2 and corresponding LPPs ga= 4,gb= 0,gc= 2, gd= 4. This sequence gave the best performance according to the results of Fig. 11. The same sequence is used for all detection methods. It can be seen that 9 frequency offset hypotheses are needed to outperform the disclosed solution, which uses only 2 matched filters. Moreover, partial correlation with 2 parts does not work as well as the other methods, which is due to the large frequency offset since the number integer offset values are H = N. Hence, the herein disclosed invention with LPPs and using two correlators, provides better performance and lower detection complexity than conventional solutions.

[0345] Alternative transmitter

[0346] As an alternative implementation of an LPP interleaved Zadoff-Chu sequence, it can be shown that

[0347]

[0348] That is, the LPP interleaved Zadoff-Chu sequence with root index u is equivalent to a Zadoff-Chu sequence with root index glu, which is multiplied with a complex constant, a modulation sequence and a DFT sequence.

[0349] The case of even sequence length

[0350] If a Zadoff-Chu sequence of even length is used, i.e., it is defined by xu[fc] =

[0351]

[0352] and performing the same steps as previously described, it follows that the equation system (29) will still apply. However, it cannot be solved since gcd(u0g2 — u1g^, N') 1, which is due to that uog^ — u gl becomes even. A solution for constructing a sequence of length N + 1, where N is odd, is to define the sequence as Eq. (11 ) for k = 0,1,...,1V. This results in

[0353] (mod IV) (59)

[0354]

[0355] and thus the equation system (29) is solvable if gcd(u0g2 — u1g^, N') = 1.

[0356] A satellite herein may also be denoted by any non-geostationary satellite, such as low earth orbit (LEO) or medium earth orbit (MEO) satellites.

[0357] A client device herein may be denoted as a user device, a user equipment (UE), a mobile station, an internet of things (loT) device, a sensor device, a wireless terminal and / or a mobile terminal, and is enabled to communicate wirelessly in a wireless communication system, sometimes also referred to as a cellular radio system. The UEs may further be referred to as mobile telephones, cellular telephones, computer tablets or laptops with wireless capability. The UEs in this context may be, for example, portable, pocket-storable, hand-held, computer-comprised, or vehicle-mounted mobile devices, enabled to communicate voice and / or data, via a RAN, with another communication entity, such as another receiver or a server. The UE may further be a station, which is any device that contains an IEEE 802.11 -conformant MAC and PHY interface to the WM. The UE may be configured for communication in 3GPP related LEE, LTE-advanced, 5G wireless systems, such as NR, and their evolutions such as mobile networks for the future, as well as in IEEE related Wi-Fi, WiMAX and their evolutions.

[0358] Furthermore, any method according to examples of the invention may be implemented in a computer program, having code means, which when run by processing means causes the processing means to execute the steps of the method. The computer program is included in a computer readable medium of a computer program product. The computer readable medium may comprise essentially any memory, such as previously mentioned a ROM, a PROM, an EPROM, a flash memory, an EEPROM, or a hard disk drive.

[0359] Moreover, it should be realized that the first communication device 100 and the second communication device 300 comprise any necessary communication capabilities in the form of e.g., functions, means, units, elements, etc., for performing or implementing examples of the invention. Examples of other such means, units, elements and functions are: processors, memory, buffers, control logic, encoders, decoders, rate matchers, de-rate matchers, mapping units, multipliers, decision units, selecting units, switches, interleavers, de-interleavers, modulators, demodulators, inputs, outputs, antennas, amplifiers, receiver units, transmitter units, DSPs, TCM encoder, TCM decoder, power supply units, power feeders, communication interfaces, communication protocols, etc. which are suitably arranged together for performing the solution.

[0360] Therefore, the processor(s) of the first communication device 100 and the second communication device 300 may comprise, e.g., one or more instances of a CPU, a processing unit, a processing circuit, a processor, an ASIC, a microprocessor, or other processing logic that may interpret and execute instructions. The expression “processor” may thus represent a processing circuitry comprising a plurality of processing circuits, such as e.g., any, some or all of the ones mentioned above. The processing circuitry may further perform data processing functions for inputting, outputting, and processing of data comprising data buffering and device control functions, such as call processing control, user interface control, or the like.

[0361] Finally, it should be understood that the invention is not limited to the examples described above, but also relates to and incorporates all examples within the scope of the appended independent claims.

[0362] Appendix A.

[0363] Property: If S =

[0364]

[0365] then |S|2= N.

[0366] Proof: It follows that

[0367]

[0368] where the substitution t = k — p is used. The substitution is possible if the following condition (uog^ — ig j'N +

[0369]

[0370] + 2gcgd) = 0 (mod 2) holds, since it implies that + ga+2ga3bWe- i(gck + 3c+2gcgdW -gpen()jjc m / ( w^h period N. The condition can equivalently be expressed ~ui9c = 0 (mod 2). This condition is always true because if uoga— u1gc= 0 (mod 2), then Furthermore, if uoga— u1gc= 1 (mod 2), then uog^ — u±g^ = 1 (mod 2). Therefore, (u05a—

[0371]

[0372] (mod 2).

[0373] The inner sum is equal to N when (u0g2 — ui 5c)l = 0 (mod IV) and is zero otherwise. By the assumption that the inverseuo9a ~ui9c)~1exists modulo-lV, it follows that gcd(u(lt / 2— u^gl. N) = 1, therefore, t = 0 is the only solution where the inner sum is non-zero. Hence, we have |S |2= N.

[0374] Appendix B.

[0375] Let a, b and c be integers such that ac 0 and ac + b is even, then it has been shown that:

[0376]

[0377] Appendix C.

[0378] Table 3. Examples of sequence parameters with maximum peak-to-sidelobe-ratio.

[0379]

[0380]

Claims

CLAIMS1. A first communication device (100) configured to:transmit a communication signal (510) to a second communication device (300), the communication signal (510) comprising a sequence y[fc], where k = 0,1,..., N — 1, N is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first Zadoff-Chu, ZC, sequence xUo[fc] with a first root index u0, which is interleaved according to xUo[gak + gb\, and a second ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd\, and wherein u0,u1,ga, gb,gc, gdare integers fulfilling the conditions:u0, u1,ga,gc∈ {1,2,..., N − 1},gb,gd∈ {0,1,..., N − 1},gcd(gcd(u0,lV) = gcd(u1,lV) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n.

2. The first communication device (100) according to claim 1, wherein the sequence y[fc] is expressed asy[fc] = xUo[gak + gb] + xUi[gck + gd]where the first ZC sequence xu[fc] and the second ZC sequence xu[fc] are defined bywhere j=√−13. The first communication device (100) according to claim 1 or 2, wherein ga= gc= 1 when gb= gd= 0.

4. The first communication device (100) according to any one of the preceding claims, wherein u0,u1,ga, gb,gc, gdare selected to maximize an absolute value of a function y,-(d) determined by a periodic correlation between the sequence y[fc] and an interleaved ZC sequence with root index u,-.

5. The first communication device (100) according to claim 4, wherein the function y,|d| is expressed for i = 0, 1 and d = 0, 1 IV - l asdlV)]dlV)]where (. )* denotes complex conjugate and (mod N) is the modulo- operator.

6. The first communication device (100) according to claim 5, wherein maximizing the absolute value of the function yf(d) comprisesmax |yi(di)|u0,u1,ga,gb,gc,gciorl / )lmax - — 7377,u0,u1,ga,gb>gc>g(i | / i(d)|where d0= arg max| y0(d) | and dt= arg max| yt(d) |, and where |. | denotes the absolute value.

7. The first communication device (100) according to claim 6, wherein|Wo5a = 1,and one or both of|Wo5a- Wi5cl = 1or8. The first communication device (100) according to any one of the preceding claims, wherein the communication signal (510) is an orthogonal frequency division multiplexing, OFDM, signal, a discrete Fourier transform spread OFDM, DFT-s-OFDM, signal or a single-carrier signal.

9. The first communication device (100) according to claim 8, configured to:performing a DFT of size N on the sequence y[fc] to obtain N DFT outputs; andallocate the N DFT outputs to a set of subcarriers in an OFDM signal to form the communication signal (510).

10. The first communication device (100) according to any one of the preceding claims, wherein the communication signal (510) is a synchronization signal or a random access preamble.

11. The first communication device (100) according to any one of the preceding claims, configured to:transmit a control message (520) to the second communication device (300), the control message (520) indicating at least one of Uo.u^g^ gb,gc, gd, orreceive a control message (520) from the second communication device (300), the control message (520) indicating at least one12. The first communication device (100) according to any one of the preceding claims, wherein the first communication device (100) is a satellite and the second communication device (300) is a user equipment, UE, or vice versa.

13. A second communication device (300) configured to:receive a communication signal (510) from a first communication device (100), the communication signal (510) comprising a sequence y[fc], where k = 0,1,..., N — 1, N is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first ZC sequence xUo[fc] with a first root index u0, which is interleaved according to xUo[gak + gb], and a second ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd\. and wherein u0,u1,5a,b,c- 9aareintegers fulfilling the conditions:gcd(gcd(u0, W) = gcd(u1,lV) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n:perform a first p0and a second p±periodic correlation between the received communication signal (510) and the first interleaved ZC sequence xu. \gak + gb\ and the second interleaved ZC sequence xUi[gck + gd\. respectively, to obtain correlation values; andestimate at least one of a timing t and a frequency offset f of the received communication signal (510) based on the correlation values.

14. The second communication device (300) according to claim 13, wherein the first and second periodic correlations are expressed aswhere r [fc] is determined based on the received communication signal (510).

15. The second communication device (300) according to claim 13 or 14, wherein the timing t and the frequency offset f are given by16. The second communication device (300) according to any one of claims 13 to 15, wherein the sequence y[k] is expressed asy[fc] = xUo[gak + gb] + xUi[gck + gd]where the first ZC sequence xUo[fc] and the second ZC sequence xu[fc] are defined bywhere j=√−1.

17. The second communication device (300) according to any one of claims 13 to 16, wherein ga= gc= 1 when gb= gd= 0.

18. The second communication device (300) according to any one of claims 13 to 17, wherein the communication signal (510) is an OFDM signal, a DFT-s-OFDM signal or a single-carrier signal.

19. The second communication device (300) according to any one of claims 13 to 18, wherein the communication signal (510) is a synchronization signal or a random access preamble.

20. The second communication device (300) according to any one of claims 13 to 19, configured to:receive a control message (520) from the first communication device (100), the control message (520) indicating at least one of Ug.^.g^ gb,gc, gd, ortransmit a control message (520) to the first communication device (100), the control message (520) indicating at least one ofu0,u1,5a, gb,gc, gd.

21. The second communication device (300) according to any one of claims 13 to 20, wherein the first communication device (100) is a satellite and the second communication device (300) is a UE, or vice versa.

22. A method (200) for a first communication device (100), the method (200) comprising:transmitting (202) a communication signal (510) to a second communication device (300), the communication signal (510) comprising a sequence y[fc], where k = 0,1,...,1V — 1, N is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first Zadoff-Chu, ZC, sequence xu,| / <| with a first root index u0, which is interleaved according toxu0\9i(+ 9b andasecond ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd\. and wherein u0,u1,ga, gb,gc, gdare integers fulfilling the conditions:u0, u1,ga,gc∈ {1,2,..., N − 1},gb,gd∈ {0,1,..., N − 1},gcd(gcd(u0,lV) = gcd(u1,! V) = gcd ga, N) = gcd gc, N) = 1 where gcd(m,n) is the greatest common divisor of integers m and n.

23. A method (400) for a second communication device (300), the method (400) comprising:receiving (402) a communication signal (510) from a first communication device (100), the communication signal (510) comprising a sequence y[fc], where k = 0,1,...,1V — 1, N is an odd integer, and where the sequence y[fc] is formed based on a superposition of a first ZC sequence xUo[fc] with a first root index u0, which is interleaved according to xUo[gak + gb\. and a second ZC sequence xu[fc] with a second root index u1, which is interleaved according to xu[gck + gd\. and wherein u0,u1,5a, gb,gc- 9aareintegers fulfilling the conditions:u0, u1,ga,gc∈ {1,2,..., N − 1},gb,gd∈ {0,1,..., N − 1},gcd(u05a - u^. N) = gcd(u0,lV) = gcd(u1,lV) = gcd ga, N) = gcd(^c,lV) = 1 where gcd(m,n) is the greatest common divisor of integers m and n:performing (404) a first p0and a second p±periodic correlation between the received communication signal (510) and the first interleaved ZC sequence xu. \gak + gb\ and the second interleaved ZC sequence xUi[gck + gd\. respectively, to obtain correlation values; andestimating (406) at least one of a timing t and a frequency offset f of the received communication signal (510) based on the correlation values.

24. A computer program with a program code for performing a method according to claim 22 or 23 when the computer program runs on a computer.

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