Abstract:
Provided is a recursive method and apparatus for processing a signal for determining a plurality of frequency components of the signal, the signal being a chirp-like polyphase sequence. In one embodiment, the method includes: (1) determining a first frequency component of the plurality of frequency components, (2) determining a component factor by accessing a factor table, (3) determining the second frequency component using the determined first frequency component and the determined component factor. If there is at least one further frequency component of the signal, the method further comprising for each of the further frequency components: (4) determining a respective further component factor by accessing the factor table, and (5) determining the further frequency component using a previously determined frequency component and the determined further component factor, wherein the previously determined frequency component is the frequency component determined most recently prior to determining each respective further frequency component.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 13/638,271 filed on Nov. 13, 2012, entitled “FOURIER TRANSFORM FOR A SIGNAL TO BE TRANSMITTED ON A RANDOM ACCESS CHANNEL,” which is the National Stage of, and therefore claims the benefit of, International Application No. PCT/EP2011/054954 filed on Mar. 30, 2011, entitled “FOURIER TRANSFORM FOR A SIGNAL TO BE TRANSMITTED ON A RANDOM ACCESS CHANNEL,” which was published in English under International Publication Number WO 2011/121044 on Oct. 6, 2011, and has a priority date of Mar. 30, 2010, based on GB application 1005319.7. All of the above applications are commonly assigned with this application and are incorporated herein by reference in their entirety. 
    
    
     TECHNICAL FIELD 
     This disclosure relates to processing a signal to be transmitted on a random access channel, in particular where the signal is for use as a random access channel preamble. 
     BACKGROUND 
     In a wireless cellular communication system, the procedure of establishing communication between a mobile terminal or User Equipment (UE) and a base station is called random access. Such random access can be implemented using a random access channel (RACH) in, for example, an orthogonal frequency division multiplexing (OFDM) communication system or a single carrier frequency division multiplexing (SC-FDMA) communication system. Random access enables the establishment of the uplink from a UE to a base station. Using the RACH, a UE can send a notification to the network indicating that the UE has data to transmit. Receipt of the notification at the base station allows the base station to estimate the UE timing, to thereby realize uplink synchronization between the UE and the base station. 
     The random access channel (RACH) typically consists of a ranging signal or a preamble. The preamble is designed to allow the base station to detect the random access attempt within target detection and false alarm probabilities, and to minimise the impact of collisions on the RACH, as is known in the art. Moreover, the base station should be able to detect several simultaneous random preambles sent from different UEs and correctly estimate the timing of each of the UEs. In order to achieve that goal, the RACH preambles should have i) good cross-correlation properties to allow for accurate timing estimation of different simultaneous and asynchronous RACH preambles, ii) good auto-correlation properties to allow for accurate timing estimation, iii) zero cross-correlation for synchronous and simultaneous RACH preambles. 
     Long Term Evolution (LTE) wireless networks, also known as Evolved Universal Terrestrial Radio Access Networks (E-UTRAN), are being standardized by the 3GPP working groups. The Orthogonal Frequency Division Multiple Access (OFDMA) access scheme and the Single Carrier Frequency Division Multiple Access (SC-FDMA) access scheme were chosen for the downlink (DL) and the uplink (UL) of E-UTRAN, respectively. Signals from different User Equipments (UEs) to a base station are time and frequency multiplexed on a physical uplink shared channel (PUSCH). In the case that the UE is not UL synchronized, the UE uses a non-synchronized Physical Random Access Channel (PRACH) to communicate with the base station, and in response the base station provides UL resources and timing advance information to allow the UE to transmit on the PUSCH. 
     The 3GPP RAN Working Group 1 (WG1) has agreed on the preamble based physical structure of the PRACH (as described in “3GPP TS 36.211 Evolved Universal Terrestrial Radio Access (E-UTRA); Physical channels and modulation”). The RAN WG1 also agreed on the number of available preambles that can be used concurrently to minimize the collision probability between UEs accessing the PRACH in a contention-based manner. The Zadoff-Chu (ZC) sequence has been selected for RACH preambles for LTE networks. 
     A Zadoff-Chu sequence is a complex-valued mathematical sequence which, when used for radio signals, gives rise to a signal, whereby cyclically shifted versions of the signal do not cross-correlate with each other when the signal is recovered, for example at the base station. A generated Zadoff-Chu sequence that has not been shifted is known as a “root sequence”. The Zadoff-Chu sequence exhibits the useful property that cyclically shifted versions of the sequence remain orthogonal to one another, provided that each cyclic shift, when viewed within the time domain of the signal, is greater than the combined propagation delay and multi-path delay-spread of the signal as it is transmitted between the UE and base station. 
     The complex value at each position (n) of each root (μ) of the Zadoff-Chu sequence (for odd N ZC , where N ZC  is the length of the Zadoff-Chu sequence) is given by: 
     
       
         
           
             
               
                 
                   x 
                   μ 
                 
                 ⁡ 
                 
                   ( 
                   n 
                   ) 
                 
               
               = 
               
                 e 
                 
                   
                     - 
                     j 
                   
                   ⁢ 
                   
                     
                       πμ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         n 
                         ⁡ 
                         
                           ( 
                           
                             n 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                     
                       N 
                       ZC 
                     
                   
                 
               
             
             , 
           
         
       
         
         
           
             where 0≦n≦N ZC −1. 
           
         
       
    
     All of the RACH preambles are generated by cyclic shifts of a number of root sequences of the Zadoff-Chu sequence, which are configurable on a cell-basis. A RACH preamble is transmitted from a UE to the base station to allow the base station to estimate, and if needed, adjust the timing of the UE transmission. It has been agreed by the RAN WG1 that there are a total of 64 RACH preambles allocated for each cell of a base station. Specifically, a cell can use different cyclically shifted versions of the same ZC root sequence, or other ZC root sequences if needed, as RACH preambles. To maximize the number of available Zadoff-Chu sequences for a certain sequence length (N ZC ) it is preferred in one embodiment to choose the sequence length as a prime number, and therefore an odd number. Typically for LTE, the length of the Zadoff-Chu may be for example 839 or 139 depending on the format of the RACH preamble. 
     For the uplink in LTE wireless networks, SC-FDMA is used which is a single-carrier transmission based on Discrete Fourier Transform (DFT) spread OFDM. With reference to  FIG. 1 , the principle of SC-FDMA will now be described.  FIG. 1  shows a user equipment  101  and a base station  121 . The user equipment comprises a serial to parallel block  102  with a serial input for receiving a serial signal. The user equipment  101  further comprises an N-point Discrete Fourier Transform (DFT) block  104 , a subcarrier mapping block  106 , an M-point Inverse Discrete Fourier Transform (IDFT) block  108 , a parallel to serial block  110 , a Cyclic Prefixing (CP) and Pulse shaping (PS) block  112 , a Digital to Analogue converter (DAC) and Radio Frequency (RF) converter block  114  and an antenna  116  for transmitting a signal over a channel  118  of the communication system. The serial to parallel block  102  has a parallel output coupled to a parallel input of the N-point DFT block  104 . A parallel output of the N-point DFT block  104  is coupled to a parallel input of the subcarrier mapping block  106 . A parallel output of the subcarrier mapping block  106  is coupled to a parallel input of the M-point IDFT block  108 . A parallel output of the M-point IDFT block  108  is coupled to a parallel input of the parallel to serial block  110 . A serial output of the parallel to serial block  110  is coupled to an input of the CP and PS block  112 . An output of the CP and PS block  112  is coupled to an input of the DAC and RF converter block  114 . An output of the DAC and RF converter block  114  is coupled to the antenna  116 . 
     The base station  121  comprises an antenna  120  for receiving a signal over the channel  118  of the communication system. The base station further comprises a Radio Frequency (RF) converter and Analogue to Digital converter (ADC) block  122 , a remove Cyclic Prefixing (CP) block  124 , a serial to parallel block  126 , an M-point Discrete Fourier Transform (DFT) block  128 , a subcarrier demapping and equalization block  130 , an N-point Inverse Discrete Fourier Transform (IDFT) block  132 , a parallel to serial block  134  and a detection block  136  for detecting the signals. 
     A serial output of the antenna  120  is coupled to a serial input of the RF converter and ADC block  122 . A serial output of the RF converter and ADC block  122  is coupled to a serial input of the remove CP block  124 . A serial output of the remove CP block  124  is coupled to a serial input of the serial to parallel block  126 . A parallel output of the serial to parallel block is coupled to a parallel input of the M-point DFT block  128 . A parallel output of the M-point DFT block  128  is coupled to a parallel input of the subcarrier demapping and equalization block  130 . A parallel output of the subcarrier demapping and equalization block  130  is coupled to a parallel input of the N-point IDFT block  132 . A parallel output of the N-point IDFT block  132  is coupled to a parallel input of the parallel to serial block  134 . A serial output of the parallel to serial block  134  is coupled to a serial input of the detection block  136 . 
     In operation, for LTE uplink at the UE  101 , a block of N modulation symbols are received at the serial to parallel block  102  and are applied as a parallel input to the N-point DFT block  104 . The N-point DFT block  104  performs a discrete Fourier transform on the modulation symbols and then the output of the N-point DFT block  104  is applied to consecutive inputs of the M-point IFFT block  108  (where M&gt;N) via the subcarrier mapping block  106 . The output of the M-point IDFT block  108  is converted to a serial signal by the parallel to serial block  110  and a cyclic prefix is applied to each block of the serial signal in the CP and PS block  112 . The signal is converted to an analogue signal and modulated at radio frequency in the DAC and RF converter block  114  before being transmitted using the antenna  116  over the channel  118  to the antenna  120  of the base station  121 . 
     Since typically for a RACH preamble, a DFT of size N=839 or 139 needs to be taken (depending on the format of the preamble), the operation performed in the N-point DFT block  104  is demanding in terms of computational complexity and memory. One method for implementing a DFT where the size of the DFT is a prime number is the Bluestein algorithm (Leo I. Bluestein, “A linear filtering approach to the computation of the discrete Fourier transform,” Northeast Electronics Research and Engineering Meeting Record 10, 218-219 (1968)). In the Bluestein algorithm the DFT is re-expressed as a convolution which provides a way to compute prime-size DFTs with a computational complexity of the order O(N log N). 
     This disclosure relates to reducing the computational complexity required to perform a prime number DFT for use in processing a signal to be transmitted on a Random Access Channel. 
     SUMMARY 
     According to a first aspect of the disclosure there is provided a recursive method for determining a plurality of frequency components of a signal, the signal being a chirp-like polyphase sequence. In one embodiment, the method includes: (1) determining a first frequency component of the plurality of frequency components, (2) determining a component factor by accessing a factor table, (3) determining the second frequency component using the determined first frequency component and the determined component factor. If there is at least one further frequency component of the signal, the method further comprising for each of the further frequency components: (4) determining a respective further component factor by accessing the factor table, and (5) determining the further frequency component using a previously determined frequency component and the determined further component factor, wherein the previously determined frequency component is the frequency component determined most recently prior to determining each respective further frequency component. 
     According to a second aspect of the disclosure there is provided an apparatus for processing a signal to be transmitted on a random access channel using a recursive method for determining a plurality of frequency components of the signal, the signal being a chirp-like polyphase sequence. In one embodiment, the apparatus includes a processor configured to: (1) determine a frequency component X(0) of the plurality of frequency components, (2) set a counter i to 1, (3) obtain a component factor for a frequency component X(i) of the signal from a factor table by indexing the factor table with an index corresponding to the component factor for the frequency component X(i), wherein the factor table stores a plurality of component factors in an indexed manner for use in determining the plurality of frequency components of the signal, (4) determine the frequency component X(i) using the determined frequency component X(i−1) and the obtained component factor for the frequency component X(i), wherein the frequency component X(i−1) is the frequency component determined most recently prior to determining the frequency component X(i), and (5) increment the counter i by 1 and continue to obtain the component factor for the frequency component X(i) of the signal and determine the frequency component X(i) until each frequency component of the plurality of frequency components is determined. 
     According to a third aspect of the disclosure there is provided a computer program product comprising computer readable instructions for execution on a computer, the instructions being for processing a signal to be transmitted on a random access channel, the signal being a chirp-like polyphase sequence, the instructions comprising instructions for executing a recursive method for determining a plurality of frequency components of the signal. In one embodiment, the recursive method includes the steps of: (1) determining a first frequency component of the plurality of frequency components, (2) determining a component factor by accessing a factor table for use in determining a second frequency component of the plurality of frequency components, (3) determining the second frequency component using the determined first frequency component and the determined component factor. If there is at least one further frequency component of the signal, the method further comprising for each of the further frequency components: (4) determining a respective further component factor by accessing the factor table for use in determining the further frequency component and (5) determining the further frequency component using a previously determined frequency component and the determined further component factor, wherein the previously determined frequency component is the frequency component determined most recently prior to determining each respective further frequency component. 
    
    
     
       BRIEF DESCRIPTION 
       Reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which: 
         FIG. 1  is a schematic representation of a SC-FDMA communication system; and 
         FIG. 2  is a flow chart of an embodiment of a process of processing a signal according to the principles of the disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     An efficient implementation of the DFT of a Zadoff-Chu sequence (or any other chirp-like polyphase sequence) is provided without needing to perform a Fourier transform. The method uses a recursive relation with reduced complexity. The Zadoff-Chu sequence has been chosen to be used for RACH preambles in LTE wireless networks, so the ability to implement a Fourier transform with reduced complexity on Zadoff-Chu sequences is particularly beneficial. However, it is noted that the method works with any signal that is a chirp-like polyphase sequence. 
     The Zadoff Chu sequence is just one example of a chirp-like polyphase sequence. As would be apparent to a skilled person, chirp-like polyphase sequences have ideal periodic autocorrelation functions. Details on chirp-like polyphase sequences can be found in “Generalized Chirp-Like Polyphase Sequences with optimum Correlation Properties” by Branislav M. Popović, IEEE Transactions on Information Theory, vol. 38, No. 4, July 1992, pages 1406 to 1409. It is described in that reference that as well as Zadoff-Chu sequences, Frank sequences and also Ipatov sequences are chirp-like polyphase sequences. 
     The complexity of implementing the Fourier Transform is reduced by using a lookup table with a simple index computation. Such indexing requires less processing power than performing a conventional DFT. The table may be stored at the UE. Component factors in the table may be calculated by the UE. Alternatively, the component factors stored in the table may be calculated by an entity other than the UE and passed to the UE for storage thereon. 
     Before describing an embodiment of the disclosure, there is provided a derivation of equations that are used in the embodiment to facilitate the understanding of the disclosure. 
     As described above, the Zadoff-Chu sequence (for odd N ZC ) is defined by 
               x   ⁡     (   n   )       =     e       -   j     ⁢       πμ   ⁢           ⁢     n   ⁡     (     n   +   1     )           N   zc                         x   ⁡     (   n   )       =     e     (     -       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢   μ   ⁢       ∑     i   =   1     n     ⁢   i         N   zc         )                     x   ⁡     (   n   )       =       ∏     i   =   1     n     ⁢           ⁢     e     (     -       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢   μ   ⁢           ⁢   i       N   zc         )               
where the fact that
 
                 ∑     i   =   1     n     ⁢   i     =       n   ⁡     (     n   +   1     )       2           
has been used.
 
     This can be rewritten as a recursive equation, such that: 
     
       
         
           
             
               x 
               ⁡ 
               
                 ( 
                 n 
                 ) 
               
             
             = 
             
               
                 x 
                 ⁡ 
                 
                   ( 
                   
                     n 
                     - 
                     1 
                   
                   ) 
                 
               
               ⁢ 
               
                 
                   e 
                   
                     
                       2 
                       ⁢ 
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       πμ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                     
                       N 
                       zc 
                     
                   
                 
                 . 
               
             
           
         
       
     
     Taking the Discrete Fourier transform of the above relation and using the DFT properties, one gets: 
                       X   ⁡     (   k   )       =       X   ⁡     (     k   +   μ     )       ⁢     e       2   ⁢   j   ⁢           ⁢     π   ⁡     (     μ   +   k     )           N   zc             ,           (   1   )               
where X(k) is the discrete Fourier transform of x(n).
 
     Based on equation (1), recursively one can write using the shift properties of the DFT: 
     
       
         
           
             
               
                 
                   
                     X 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       X 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           + 
                           μ 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       e 
                       
                         ( 
                         
                           - 
                           
                             
                               2 
                               ⁢ 
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     μ 
                                     + 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             
                               N 
                               zc 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                     ⁢ 
                     
                       
                         X 
                         ⁡ 
                         
                           ( 
                           
                             k 
                             + 
                             
                               2 
                               ⁢ 
                               μ 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         e 
                         
                           ( 
                           
                             - 
                             
                               
                                 2 
                                 ⁢ 
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   π 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       μ 
                                       + 
                                       k 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 N 
                                 zc 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           ( 
                           
                             - 
                             
                               
                                 2 
                                 ⁢ 
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   π 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         μ 
                                       
                                       + 
                                       k 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 N 
                                 zc 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   ⋮ 
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       X 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           + 
                           
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             μ 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         ∏ 
                         
                           i 
                           = 
                           1 
                         
                         m 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         e 
                         
                           ( 
                           
                             - 
                             
                               
                                 2 
                                 ⁢ 
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   μ 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         l 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         μ 
                                       
                                       + 
                                       k 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 N 
                                 zc 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       X 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           + 
                           
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             μ 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       e 
                       
                         ( 
                         
                           - 
                           
                             
                               2 
                               ⁢ 
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   m 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     
                                       l 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       μ 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               N 
                               zc 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       X 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           + 
                           
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             μ 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       e 
                       
                         - 
                         
                           
                             2 
                             ⁢ 
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             mk 
                           
                           
                             N 
                             zc 
                           
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         
                           - 
                           j 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         πμ 
                         ⁢ 
                         
                           
                             m 
                             ⁡ 
                             
                               ( 
                               
                                 m 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           
                             N 
                             zc 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Let us introduce the following notation. Two integers a and b are said to be congruent modulo n, if their difference a−b is an integer multiple of n. An equivalent definition is that both numbers have the same remainder when divided by n. If this is the case, it is expressed as:
 
 a=b  mod  n.  
 
     Let us choose m such that mμ=1 mod N ZC . m always exists since N ZC  and μ are relatively prime numbers (i.e. they share no common positive factors, or divisors, except 1) by construction of the Zadoff-Chu sequence. Then, from the periodicity property of the DFT:
 
 X ( k+m μ)= X ( k+ 1),
 
one obtains the final result as:
 
                     X   ⁡     (     k   +   1     )       =       X   ⁡     (   k   )       ⁢     e       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢   mk       N   zc         ⁢     e     j   ⁢           ⁢   πμ   ⁢       m   ⁡     (     m   +   1     )         N   zc                     (   2   )               
with
 
     
       
         
           
             
               X 
               ⁡ 
               
                 ( 
                 0 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   n 
                   = 
                   0 
                 
                 
                   
                     N 
                     ZC 
                   
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   e 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     πμ 
                     ⁢ 
                     
                       
                         n 
                         ⁡ 
                         
                           ( 
                           
                             n 
                             + 
                             1 
                           
                           ) 
                         
                       
                       
                         N 
                         zc 
                       
                     
                   
                 
                 . 
               
             
           
         
       
     
     From equation (2), one can get an expression for X(k) as: 
               X   ⁡     (   k   )       =       X   ⁡     (     k   -   1     )       ⁢     e       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢     m   ⁡     (     k   -   1     )           N   zc         ⁢     ⅇ     j   ⁢           ⁢   πμ   ⁢       m   ⁡     (     m   +   1     )         N   zc                   
By recursion one gets:
 
               X   ⁡     (   k   )       =       X   ⁡     (     k   -   2     )       ⁢     e       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢     m   ⁡     (     k   -   2     )           N   zc         ⁢     e     j   ⁢           ⁢   πμ   ⁢       m   ⁡     (     m   +   1     )         N   zc           ⁢       e   ⁢                 2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢     m   ⁡     (     k   -   1     )           N   zc         ⁢     e     j   ⁢           ⁢   πμ   ⁢       m   ⁡     (     m   +   1     )         N   zc                   
which leads to:
 
               X   ⁡     (   k   )       =       X   ⁡     (   0   )       ⁢       ∏     l   =   0       k   -   1       ⁢           ⁢       e       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢   ml       N   zc         ⁢     e     j   ⁢           ⁢   πμ   ⁢       m   ⁡     (     m   +   1     )         N   zc                       
and finally to the result that:
 
     
       
         
           
             
               
                 
                   
                     X 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       X 
                       ⁡ 
                       
                         ( 
                         0 
                         ) 
                       
                     
                     ⁢ 
                     
                       e 
                       
                         
                           2 
                           ⁢ 
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             mk 
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           N 
                           zc 
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         πμ 
                         ⁢ 
                         
                           
                             km 
                             ⁡ 
                             
                               ( 
                               
                                 m 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           
                             N 
                             zc 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     In the case where N ZC  is even, the Zadoff-Chu sequence is given by: 
               x   ⁡     (   n   )       =       e       -   j     ⁢           ⁢   πμ   ⁢       n   2       N   zc           .           
One can show by inductive proof that:
 
               x   ⁡     (   n   )       =       x   ⁡     (     n   -   1     )       ⁢     e     (     -       2   ⁢   j   ⁢           ⁢   πμ   ⁢           ⁢   n       N   zc         )       ⁢       e     (     -       j   ⁢           ⁢   πμ       N   zc         )       .             
Similarly to the derivations above, one obtains:
 
                     X   ⁡     (   k   )       =       X   ⁡     (     k   +   μ     )       ⁢     e       2   ⁢   j   ⁢           ⁢     π   ⁡     (     μ   +   k     )           N   zc         ⁢     e     -       j   ⁢           ⁢   πμ       N   zc                     (   4   )               
which leads to:
 
               X   ⁡     (   k   )       =       X   ⁡     (     k   +     m   ⁢           ⁢   μ       )       ⁢       ∏     l   =   1     m     ⁢       e     -       2   ⁢     jπ   ⁡     (       l   ⁢           ⁢   μ     +   k     )           N   zc           ⁢     e     -       j   ⁢           ⁢   πμ       N   zc                                     ⁢       X   ⁡     (   k   )       =       X   ⁡     (     k   +     m   ⁢           ⁢   μ       )       ⁢     e     -       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢   mk       N   zc           ⁢     e       -   j     ⁢           ⁢   πμ   ⁢       m   ⁡     (     m   +   1     )         N   zc           ⁢     e     -       j   ⁢           ⁢   π   ⁢           ⁢   m   ⁢           ⁢   μ       N   zc                     
If m is such that mμ=1 mod N ZC  exists, then one can rewrite the above equation as:
 
                     X   ⁡     (   k   )       =       X   ⁡     (     k   +   1     )       ⁢     e       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢   mk       N   ZC         ⁢     e       -   j     ⁢           ⁢   πμ   ⁢       m   ⁡     (     m   +   1     )         N   ZC           ⁢     e       j   ⁢           ⁢   π   ⁢           ⁢   m   ⁢           ⁢   μ       N   ZC                   (   5   )               
X(k) can be expressed as (if m is such that mμ=1 mod N ZC  exists):
 
     
       
         
           
             
               X 
               ⁡ 
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               
                 X 
                 ⁡ 
                 
                   ( 
                   0 
                   ) 
                 
               
               ⁢ 
               
                 
                   ∏ 
                   
                     l 
                     = 
                     0 
                   
                   
                     k 
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       e 
                       
                         
                           2 
                           ⁢ 
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ml 
                         
                         
                           N 
                           ZC 
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         πμ 
                         ⁢ 
                         
                           
                             m 
                             ⁡ 
                             
                               ( 
                               
                                 m 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           
                             N 
                             ZC 
                           
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           m 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           μ 
                         
                         
                           N 
                           ZC 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               X 
               ⁡ 
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               
                 X 
                 ⁡ 
                 
                   ( 
                   0 
                   ) 
                 
               
               ⁢ 
               
                 e 
                 
                   
                     2 
                     ⁢ 
                     j 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       mk 
                       ( 
                       
                         k 
                         - 
                         1 
                       
                       ) 
                     
                   
                   
                     N 
                     ZC 
                   
                 
               
               ⁢ 
               
                 e 
                 
                   j 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   πμ 
                   ⁢ 
                   
                     
                       km 
                       ⁡ 
                       
                         ( 
                         
                           m 
                           + 
                           1 
                         
                         ) 
                       
                     
                     
                       N 
                       ZC 
                     
                   
                 
               
               ⁢ 
               
                 e 
                 
                   
                     j 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     km 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     μ 
                   
                   
                     N 
                     ZC 
                   
                 
               
             
           
         
       
     
     If m is such that mμ=1 mod N ZC  does not exist, one can find the smallest integer β such that min{β|β&lt;μ and mμ=β mod N ZC } in order to minimize the delay, and Equation (3) becomes: 
     
       
         
           
             
               
                 
                   
                     X 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       X 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           + 
                           β 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       e 
                       
                         
                           2 
                           ⁢ 
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           mk 
                         
                         
                           N 
                           ZC 
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         
                           - 
                           j 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         πμ 
                         ⁢ 
                         
                           
                             m 
                             ⁡ 
                             
                               ( 
                               
                                 m 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           
                             N 
                             ZC 
                           
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           πμ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           m 
                         
                         
                           N 
                           ZC 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     From the μth root of the Zadoff-Chu sequence, random access preambles with zero correlation zones of length N CS −1 are defined by cyclic shifts according to: 
               x     μ   ,   v       =       x   μ     ⁡     (       (     n   +     C   v       )     ⁢   mod   ⁢           ⁢     N   ZC       )                     x   μ     =     e       -   j     ⁢           ⁢   πμ   ⁢       n   ⁡     (     n   +   1     )         N   ZC                 
for 0≦n≦N ZC , where
 
               C   v     =     {           vN   CS             v   =   0     ,   1   ,   …   ⁢           ,       ⌊       N   ZC     /     N   CS       ⌋     -   1     ,       N   CS     ≠   0             for   ⁢           ⁢   unrestricted   ⁢           ⁢   sets             0           N   CS     =   0           for   ⁢           ⁢   unrestricted   ⁢           ⁢   sets                   d   start     ⁢     ⌊     v   /     n   shift   RA       ⌋       +       (     v   ⁢           ⁢   mod   ⁢             ⁢             ⁢     n   shift   RA       )     ⁢     N   CS                 v   =   0     ,   1   ,   …   ⁢           ,         n   shift   RA     ⁢     n   group   RA       +       n   _     shift   RA     -   1             for   ⁢           ⁢   restricted   ⁢           ⁢   sets                   
and N CS  is signalled by high layers.
 
     The DFT for a Zadoff-Chu sequence of odd length is given by Equation (2) above: 
                 X   μ     ⁡     (     k   +   1     )       =         X   μ     ⁡     (   k   )       ⁢     e       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢   mk       N   ZC         ⁢     e     j   ⁢           ⁢   πμ   ⁢       m   ⁡     (     m   +   1     )         N   ZC                   
The DFT of the with cyclically shifted Zadoff-Chu sequence is given by:
 
                 X   μ     ⁡     (   k   )       =         X   μ     ⁡     (   k   )       ⁢       e       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢     C   v     ⁢   k       N   ZC         .             
Therefore by modifying the recursive equation (2) shown above, one obtains:
 
                       X     μ   ,   v       ⁡     (     k   +   1     )       =         X     μ   ,   v       ⁡     (   k   )       ⁢     e       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢   mk       N   ZC         ⁢     e     j   ⁢           ⁢   πμ   ⁢       m   ⁡     (     m   +   1     )         N   ZC           ⁢     e       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢     C   v     ⁢   k       N   ZC                   (   7   )               
The exponential part of equation 7 for different values of k may be stored in a table at the user equipment, for use as component factors in determining the frequency components of the signal, as described below. Obtaining the exponential part of equation (i.e. a component factor) can then be easily implemented by indexing into the table of component factors which for ease of notation is restricted to a size of N ZC , which corresponds to 2π with a resolution of 2π/N ZC . In alternative embodiments, a table of different resolution and length may be used.
 
     In this way the exponential part of equation (7) (referred to herein as the component factor) for different frequency components (k) is calculated and stored in the table. Each frequency component of the signal (X(k+1)) can be calculated using the previously calculated frequency component and a component factor obtained from the table. In other words X(k+1)=X(k)F k+1 , where F k+1  is the component factor for the frequency component X(k+1) and is given by 
                 F     k   +   1       =       e       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢   mk       N   ZC         ⁢     e     j   ⁢           ⁢   πμ   ⁢       m   ⁡     (     m   +   1     )         N   ZC           ⁢     e       2   ⁢   j   ⁢           ⁢   π   ⁢           ⁢     C   v         N   ZC             ,         
and the values of F k+1  can be determined by accessing the table in an indexed manner. The value of the exponent
 
             (     divided   ⁢             ⁢             ⁢   by   ⁢           ⁢   a   ⁢             ⁢             ⁢   factor   ⁢           ⁢   of   ⁢           ⁢       2   ⁢   π   ⁢           ⁢   j       N   ZC         )         
is used as the index for accessing the table, as described in more detail below.
 
     The values of F k  for the different frequency components (k) in the signal may be calculated at the user equipment  101  and stored in the table. Alternatively, the values of F k  for the different frequency components (k) in the signal may be calculated at an entity other than the user equipment  101  and stored in the table. The values of F k  for the different frequency components (k) in the signal may be calculated before they are needed and stored in the table before they are needed. In this way, when the factors F k  are needed they just need to be looked up from the table rather than calculated. The table is stored in memory of the user equipment. 
     An embodiment of a method according to the principles of this disclosure is now described with reference to the flow chart of  FIG. 2 . The method is carried out to implement a DFT, for use in a method of processing signals for transmission on a RACH, e.g. where the signals are RACH preambles. For example, the method may be implemented in the N-point DFT block  104  as shown in the system of  FIG. 1 . 
     In step S 202  the frequency component X(0) is determined. X(0) may be determined by loading the frequency component from a store. Alternatively, X(0) may be calculated from the signal. Once the frequency component X(0) of the signal has been determined, the other frequency components in the signal can be calculated in a recursive manner using equation (7) above. 
     In order to begin the recursive method, in step S 204 , a counter i is set to 1 initially. Then in step S 206  the component factor F i  for the ith frequency component of the signal is looked up by indexing the table. Therefore on the first run through of the recursive method the component factor F 1  is obtained from the table. In step S 208  the ith frequency component (X(i)) is determined using the previously determined frequency component (X(i−1)) and the component factor for the ith frequency component obtained in step S 206 . In the first run through of the recursive method the frequency component X(1) is determined by multiplying X(0) with F 1 . In this sense, the component factors F 1  stored in the table are multiplying factors. Alternatively, the component factors F 1  stored in the table may be used to obtain the ith frequency component in other ways than by multiplication with a previously determined frequency component. The component factors obtained from the table may be used in conjunction with a previously determined frequency component in any way, as would be apparent to the skilled person, leading to a determination of the ith frequency component of the signal. 
     In the embodiments described above the component factors F k  are stored in the table. This is a much simpler operation than calculating the DFT for each component factor. 
     In step S 210  the counter i is incremented by 1 and in step S 212  it is determined whether the counter i is greater than or equal to the length of the Zadoff-Chu sequence N ZC . If the counter is greater than or equal to N ZC  then all of the frequency components of the signal have been determined and the process ends in step S 214 . However, if the counter i is less than N ZC  then the method passes back to step S 206  and the next frequency component of the signal is determined. The process continues until all of the frequency components of the signal have been determined. 
     In this way, a running table index is obtained which is initialised to 
                   μ   ⁢           ⁢     m   ⁡     (     m   +   1     )         2     +     C   v       =       I   ⁢           ⁢   mod   ⁢           ⁢     N   ZC     ⁢           ⁢   for   ⁢           ⁢   k     =   0.           
Using this index when accessing the table will return the value of the component factor F 1 , given by
 
               F   1     =     e         2   ⁢   j   ⁢           ⁢   π       N   ZC       ⁢     (         μ   ⁢           ⁢     m   ⁡     (     m   +   1     )         2     +     C   v       )               
(see the equation above for F k+1 ). This component factor is then multiplied with X(0) to give X(1). The next pass in the recursion requires I to be updated by m as:
 
 m =γ mod  N   ZC  
 
 I   i   +γ=I   i+1  mod  N   ZC  
 
where I i  is the index at iteration i. Note that the modulo operation does not need a divide since (I+γ) can never exceed 2N ZC .
 
     Pseudo code which may be used to implement the above described method will now be described. The pseudo code may be implemented in a computer program product for execution on a computer or other suitable hardware for carrying out the method as described above. Alternatively, the method may be carried out in hardware, rather than in software, as would be apparent to a skilled person. 
     In the following the notation a=b mod n is equivalent to b=mod(a,n). 
     The pseudo code may be written as follows: 
     
       
         
               
               
             
           
               
                   
                   
               
             
             
               
                   
                 load X(0) from a pre computed table or compute X(0) 
               
               
                   
                 Compute γ = mod(m,Nzc) 
               
               
                   
                 initialise I to mod(μm(m+1)/2 +Cv,Nzc) 
               
               
                   
                 for i=1 to Nzc−1 
               
               
                   
                   e = load exp from table with index I 
               
               
                   
                   X(i)=X(i−1)*e 
               
               
                   
                   I=I+γ 
               
               
                   
                   If I&gt;=Nzc I=I−Nzc 
               
               
                   
                 endfor 
               
               
                   
                   
               
             
          
         
       
     
     It will be apparent to a person skilled in the art that using the method described above, as provided for by the pseudo code above, the frequency components X(i) can be determined in a recursive manner, thus requiring less computing power and complexity than performing a conventional Fourier transform to determine the frequency components X(i). 
     In an alternative embodiment, the computational complexity may be reduced even further. If the signal may be multiplied by a complex constant we can ensure that the frequency component X(0)=1. In this way the first frequency component is set to 1 so it does not need to be computed. Multiplying the signal by a complex constant is equivalent to a scaling introduced in the communication channel  118 , which changes neither the received timing nor RACH detection probability nor false alarm probability as determined by the base station  121 . Therefore multiplying the signal by a complex constant does not detrimentally affect the use of the signal as a RACH preamble. 
     Where the signal is multiplied by a complex constant to ensure that the frequency component X(0) is equal to 1, the multiplication of the component factor obtained from the factor table and the previously determined frequency component in the recursive method described above may be avoided altogether. In this case the algorithm may be modified to have the following pseudo code. 
     
       
         
               
               
             
           
               
                   
                   
               
             
             
               
                   
                 Set X(0) = 1 
               
               
                   
                 Compute γ= mod(m,Nzc) 
               
               
                   
                 Initialise I to mod(μm(m+1)/2 +Cv,Nzc) 
               
               
                   
                 J=0 
               
               
                   
                 for i=1 to Nzc−1 
               
               
                   
                   J=J+I 
               
               
                   
                   If J&gt;=Nzc, J=J−Nzc 
               
               
                   
                   e = load exp from table with index J 
               
               
                   
                   X(i)=e 
               
               
                   
                   I=I+γ 
               
               
                   
                   If I&gt;=Nzc I=I−Nzc 
               
               
                   
                 endfor 
               
               
                   
                   
               
             
          
         
       
     
     In this alternative embodiment, it can be seen from equation (7) that with X(0) equal to 1, all of the frequency components will equal e d     i    where d i  is different for each of the frequency components X(i). Since e a e b =e a+b , each frequency component can be computed by adding up the indices for all previous frequency components and using the sum of the indices as the index for accessing the table. The pseudo code given above for this alternative embodiment implements this by loading the component factors using the index J where J is the sum of all previously determined indices I. 
     As would be apparent to the skilled person, a similar implementation approach could be used for the case where N ZC  is even based on Equations 5 and 6. 
     There have been described above methods of implementing a Fourier transform for use in the signal processing of RACH preambles using the Zadoff-Chu sequence. The methods described above do not use a dedicated Fourier transform algorithm for calculating the frequency components of the signal. This results in reduced complexity and reduced memory requirements. The implementation of the DFT is simplified by using a table-lookup with index computation. 
     While the specific description is directed towards signal processing of signals using the Zadoff-Chu sequence, it would be apparent to a skilled person that the method could also be applied to any other chirp-like polyphase sequence. 
     While this invention has been particularly shown and described with reference to embodiments, it will be understood to those skilled in the art that various changes in form and detail may be made without departing from the scope of the invention as defined by the appendant claims. 
     Those skilled in the art to which this application relates will appreciate that other and further additions, deletions, substitutions and modifications may be made to the described embodiments.