Patent Publication Number: US-11032035-B2

Title: Signaling method in an OFDM multiple access system

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application is a continuation of U.S. patent application Ser. No. 15/226,181, filed on Aug. 2, 2016, now allowed, which is a continuation of U.S. patent application Ser. No. 13/619,460, filed Sep. 14, 2012, now issued as U.S. Pat. No. 9,426,012 on Aug. 23, 2016, which is a continuation of U.S. patent application Ser. No. 13/158,170, filed Jun. 10, 2011, now issued as U.S. Pat. No. 8,295,154 on Oct. 23, 2012, which is a continuation of U.S. patent application Ser. No. 12/171,155, filed Jul. 10, 2008, now issued as U.S. Pat. No. 8,014,271 on Sep. 6, 2011, which is a continuation of U.S. patent application Ser. No. 09/805,887, filed on Mar. 15, 2001, now issued as U.S. Pat. No. 7,295,509 on Nov. 13, 2007 which is hereby expressly incorporated by reference and which claims the benefit of U.S. Provisional Patent Application Ser. No. 60/230,937 filed Sep. 13, 2000, and titled “SIGNALING METHOD IN AN OFDM MULTIPLE ACCESS WIRELESS SYSTEM,” which is also incorporated by reference. 
    
    
     TECHNICAL FIELD 
     This invention relates to an orthogonal frequency division multiplexing (OFDM) communication system, and more particularly to an OFDM communication system for a multiple access communication network. 
     BACKGROUND 
     Orthogonal frequency division multiplexing (OFDM) is a relatively well known multiplexing technique for communication systems. OFDM communication systems can be used to provide multiple access communication, where different users are allocated different orthogonal tones within a frequency bandwidth to transmit data at the same time. In an OFDM communication system, the entire bandwidth allocated to the system is divided into orthogonal tones. In particular, for a given symbol duration T available for user data transmission, and a given bandwidth W, the number of available orthogonal tones F is given by WT. The spacing between the orthogonal tones A is chosen to be 1/T, thereby making the tones orthogonal. In addition to the symbol duration T which is available for user data transmission, an additional period of time T c  can be used for transmission of a cyclic prefix. The cyclic prefix is prepended to each symbol duration T and is used to compensate for the dispersion introduced by the channel response and by the pulse shaping filter used at the transmitter. Thus, although a total symbol duration of T+T c  is employed for transmitting an OFDM symbol, only the symbol duration T is available for user data transmission and is therefore called an OFDM symbol duration. 
     In prior OFDM techniques, an OFDM signal is first constructed in the frequency domain by mapping symbols of a constellation to prescribed frequency tones. The signal constructed in the frequency domain is then transformed to the time domain by an inverse discrete Fourier transform (IDFT) or inverse fast Fourier transform (IFFT) to obtain the digital signal samples to be transmitted. In general, symbols of the constellation have a relatively low peak-to-average ratio property. For example, symbols of a QPSK constellation all have the same amplitude. However, after being transformed by the IDFT or IFFT, the resultant time domain signal samples are the weighted sum of all the symbols, and therefore generally do not preserve the desirable low peak-to-average ratio property. In particular, the resulting time domain signal typically has a high peak-to-average ratio. 
     Existing techniques for implementing OFDM communication systems can be highly inefficient due to the relatively high peak-to-average ratio when compared with other signaling schemes, such as single carrier modulation schemes. As a result, existing OFDM techniques are not well suited for a wireless multiple access communication network with highly mobile users because the high peak-to-average ratio of the transmitted signal requires a large amount of power at the base station and at the wireless device. The large power requirements result in short battery life and more expensive power amplifiers for handheld wireless communication devices or terminals. Accordingly, it is desirable to provide an OFDM technique which reduces the peak-to-average ratio of the signal to be transmitted, while simultaneously taking advantage of the larger communication bandwidth offered by an OFDM communication system. 
     SUMMARY 
     In one aspect of the communication system, power consumption associated with generating and transmitting OFDM signals is reduced as compared to the prior OFDM systems discussed above. The OFDM signaling method includes defining a constellation having a plurality of symbols, defining the symbol duration for the OFDM communication signal, and defining a plurality of time instants in the symbol duration. In a given symbol duration, a plurality of tones in the symbol duration are allocated to a particular transmitter and the signal to be transmitted is represented by a vector of data symbols from the symbol constellation. The symbols are first directly mapped to the prescribed time instants in the symbol duration. A continuous signal is then constructed by applying continuous interpolation functions to the mapped symbols such that the values of the continuous signal at the prescribed time instants are respectively equal to the mapped symbols and the frequency response of the continuous signal only contains sinusoids at the allocated tones. Finally the digital signal, which is to be transmitted, consists of samples of the continuous signal. Alternatively, the digital signal can be generated directly by applying discrete interpolation functions to the mapped symbols. As symbols from the constellation generally have good peak-to-average ratio property, proper choices of allocated frequency tones, prescribed time instants and interpolation functions can result in a minimized peak-to-average ratio of the continuous function and the digital signal samples. 
     In one implementation the method of directly generating the digital signal samples is to multiply the symbol vector consisting of symbols to be transmitted with a constant matrix, where the constant matrix is determined by the allocated frequency tones and the prescribed time instants. The matrix can be precomputed and stored in a memory. 
     In one aspect, a transmitter associated with the communication system is allocated a number of contiguous tones and the prescribed time instants are equally-spaced time instants over the entire OFDM symbol duration. 
     In another aspect, the transmitter is allocated a number of equally-spaced tones and the prescribed time instants are equally-spaced time instants over a fraction of the OFDM symbol duration. 
     In the above aspects, in addition to the general method, the digital signal samples can be constructed by expanding the mapped symbols to a prescribed set of time instants from minus infinity to plus infinity and interpolating the expanded set of the mapped symbols with a sinc function. Equivalently, the digital signal samples can also be generated by a series of operations including discrete Fourier transformation, zero insertion, and inverse discrete Fourier transformation. 
     To further reduce the peak-to-average ratio of the digital signal samples obtained through interpolation, when symbols of the constellation are mapped to the prescribed time instants, the constellations used by two adjacent time instants are offset by π/4. 
     In another aspect of the system, the real and the imaginary components of the resultant digital sample vector are cyclically offset before the cyclic prefix is added. In yet another aspect of the communication system, the intended transmitter is allocated more tones than the number of symbols to be transmitted. Symbols of the constellation are directly mapped to prescribed equally-spaced time instants. The digital signal samples are constructed by expanding the mapped symbols to a prescribed set of time instants from minus infinity to plus infinity and interpolating the expanded set of the mapped symbols with a function whose Fourier transformation satisfies the Nyquist zero intersymbol interference criterion, such as raised cosine functions. The digital signal samples can also be generated by a series of operations including discrete Fourier transformation, windowing, and inverse discrete Fourier transformation. 
     The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. 
    
    
     
       DESCRIPTION OF DRAWINGS 
         FIG. 1  is a block diagram of an OFDM system. 
         FIG. 2A  is a block diagram of an interpolation system used by the OFDM system of  FIG. 1 . 
         FIG. 2B  is a block diagram of another interpolation system used by the OFDM system of  FIG. 1 . 
         FIG. 3A  is a graph showing symbols mapped to prescribed time instants in the time domain according to the OFDM technique implemented by the system of  FIG. 1 . 
         FIG. 3B  is a graph showing the frequency domain response of the graph of  FIG. 3B . 
         FIG. 4A  shows an implementation technique for producing a digital signal sample vector using time domain symbol mapping in the case where the allocated tones are contiguous. 
         FIG. 4B  is a block diagram showing a communication system for producing a digital signal sample vector in the case where the allocated frequency tones are contiguous. 
         FIG. 4C  is a graph showing the mapping of the symbols to the prescribed time instants, the expansion of the mapped symbols, and the use of a sinc function to interpolate the expanded symbols. 
         FIG. 4D  is a graph showing the large peak-to-average ratio of the resulting digital signal sample vector when the symbols are mapped in the frequency domain in the prior OFDM systems. 
         FIG. 4E  is a graph showing the reduced peak-to-average ratio of the resulting digital signal sample vector when the symbols are mapped in the time domain using the technique of  FIGS. 4A-4C . 
         FIG. 5A  shows another implementation technique for producing the digital signal sample vector using time domain symbol mapping in the case where the allocated tones are equally spaced in frequency. 
         FIG. 5B  is a block diagram showing a communication system for producing a digital signal sample vector in the case where the allocated frequency tones are equally spaced. 
         FIG. 5C  is a graph showing the mapping of the symbols to the prescribed time instants, the expansion of the mapped symbols, and the use of a sinc function to interpolate the symbols. 
         FIG. 5D  is a graph showing the reduced peak-to-average ratio of the resulting digital signal sample vector when the symbols are mapped in the time domain using the technique of  FIGS. 5A-5C . 
         FIG. 6  is a graph showing π/4 symbol rotation. 
         FIG. 7  shows the use of a cyclic shift of the real and imaginary signal components. 
         FIG. 8A  is a graph showing application of a windowing function in the frequency domain to further reduce the peak-to-average ratio. 
         FIG. 8B  is a block diagram showing a technique using more tones than the number of symbols to be transmitted for producing a digital signal sample vector. 
         FIG. 8C  is a graph showing the use of an interpolation function corresponding to the window function of  FIG. 8B  to the symbols mapped to the prescribed time instants. 
         FIG. 8D  is a graph showing the reduced peak-to-average ratio of the resulting digital signal sample vector when the symbols are mapped in the time domain using the technique of  FIGS. 8A-8C . 
     
    
    
     Like reference symbols in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     Referring to  FIG. 1 , an orthogonal frequency division multiplexing (OFDM) communication system  10  is shown. OFDM communication system  10  receives a first constellation of symbols {B i }  12  and provides the symbols to a symbol-to-symbol mapping circuit  14 , that produces a second constellation of complex symbols {C i }  16 . The complex symbols  16  represent data or a stream of data to be transmitted by the OFDM communication system, and may be chosen from a variety of symbol constellations including, but not limited to phase shift keying (PSK) and quadrature amplitude modulation (QAM) symbol constellations. The symbol-to-symbol mapping performed by the mapping circuit  14  is an optional step performed by the OFDM communication system  10 . 
     Next, a time instant mapping circuit  18  maps each complex symbol  16  to a prescribed time instant within a given OFDM symbol duration. The mapping operation is performed in the time domain such that the mapping circuit  18  generates a discrete signal of mapped symbols within the time domain symbol duration. The output of the mapping circuit  18  is provided to an interpolation circuit  20 , that produces a series of digital signal samples {S i }  22 . The digital signal samples  22  are formed by sampling a continuous signal, which is constructed by applying one or more predetermined continuous interpolation functions to the mapped complex symbols  19 . Alternatively, the digital signal samples  22  are formed by directly applying one or more predetermined discrete interpolation functions to the mapped complex symbols  19 . When using the technique of applying discrete interpolation functions, no intermediate continuous signal is generated and the step of sampling the continuous signal is not necessary. The operation of the interpolation circuit  20  is described in greater detail below. A cyclic prefix circuit  24  receives the series of digital signal samples  22  from the interpolation circuit  20  and prepends a cyclic prefix to the digital signal samples  22 . The cyclic prefix circuit  24  operates to copy and prepend the last portion of the digital signal sample vector S  22  to the beginning of the OFDM symbol duration. The resulting digital signal samples  22  with the prepended cyclic prefix are converted to an analog signal by a digital to analog converter  28 . The resulting analog signal is further processed by a pulse shaping filter  30 , the output of which is modulated to a carrier frequency, and amplified by a power amplifier unit  32  for transmission through an antenna  34 . 
     In one implementation of the OFDM communication system  10 , the symbol-to-symbol mapping circuit  14 , the time instant mapping circuit  18 , the interpolation circuit  20 , and the cyclic prefix circuit  24  are implemented in a digital signal processor (DSP)  26 , and may include a combination of hardware modules and/or software modules. These circuits  14 ,  18 ,  20 , and  24  can also be implemented as separate discrete circuits within the OFDM communication system  10 . 
     The details of the interpolation circuit  20  are shown in  FIG. 2A . The interpolation circuit  20  includes an interpolation function module  21  that applies one or more continuous interpolation functions to the discrete signal of mapped symbols  19  to generate a continuous signal in which signal variation between adjacent symbols is minimized. Thus, the continuous signal has a low peak-to-average ratio. The interpolation functions may be precomputed and stored in an interpolation function memory  23  connected to the interpolation function module  21 . A frequency tone and time instant allocation circuit  27  is connected to the interpolation function memory  23  and defines an allocated tone set selected from frequency tones distributed over a predetermined bandwidth associated with the OFDM communication system  10 . The allocated tone set is then provided to the interpolation function memory  23 . The frequency tone and time instant allocation circuit  27  also defines the prescribed time instants distributed over the time domain symbol duration, which can also be stored in the interpolation function memory  23  for use by the interpolation function module  21  as well as other modules within the DSP  26 . The interpolation circuit  20  also includes a sampling circuit  25  for receiving and sampling the continuous signal at discrete time instants distributed over the time domain symbol duration to generate the vector of digital signal samples  22 . Alternatively, in  FIG. 2B  the interpolation function module  21  applies one or more discrete interpolation functions to the discrete signal of mapped symbols  19  to directly generate the digital signal sample vector  22 , in which case the sampling circuit  25  (of  FIG. 2A ) is not needed. Through applying the discrete interpolation functions, the interpolation function module  21  effectively combines the processing steps of applying the continuous interpolation functions and sampling the intermediate continuous signal. 
       FIG. 3A  graphically depicts the signal processing steps performed by the various circuits of the DSP  26 . More specifically,  FIG. 3A  shows the construction of the signal to be transmitted in a given OFDM time domain symbol duration  40 . The time domain symbol duration  40  is a time interval from 0 to T. For purposes of the following description, the OFDM symbol duration T does not include the cyclic prefix. The signal to be transmitted in the symbol duration  40  is represented by complex symbols C 1 , C 2 , C 3 , . . . , C M    16  that are mapped to the prescribed time instants, where M denotes the number of symbols to be transmitted in the symbol duration  40 . 
     In one implementation, the OFDM communication system  10  is a multiple access communication system where the entire bandwidth available to all transmitters within the system is divided into F orthogonal frequency tones, f 1 , f 2 , . . . , f F . In the given symbol duration  40 , a particular transmitter operating within a multiple access communication system is allocated M frequency tones f i(1) , f i(2) , . . . , f i(M) , which is a subset of f 1 , f 2 , . . . , f F , (the total number of frequency tones) in order to transmit the signal. As part of this implementation, the number of tones allocated to a particular transmitter is equal to the number of symbols to be transmitted by that transmitter. Later in  FIG. 8A , the number of allocated tones can be greater than the number of symbols to be transmitted. The remaining frequency tones can be used by other transmitters within the communication system. This technique allows OFDM communication system  10  to operate as a multiple access communication system. 
     The complex data symbols C 1 , C 2 , C 3 , . . . , C M    16  are first mapped to t 1 , t 2 , t 3 , . . . , t M , respectively, where t 1 , t 2 , t 3 , . . . , t M  are M prescribed time instants within the time domain symbol duration  40 . The mapping operation generates a discrete signal of mapped symbols. It should be noted that the number of prescribed time instants is equal to the number of symbols M to be transmitted. As described above, the symbol mapping occurs in the time domain. Continuous interpolation functions  42  are then applied to the discrete signal of mapped symbols  16  to generate a continuous function CF(t) for tin the time interval from 0 to T. 
     The interpolation functions  42  are constructed such that the values of the continuous function CF(t) at time instants t 1 , t 2 , t 3 , . . . , t M  are respectively equal to C 1 , C 2 , C 3 , . . . , C M  and the frequency response of the continuous function CF(t) contains only sinusoids at the allocated tones. Therefore, CF(t) is constructed as 
               CF   ⁡     (   t   )       =       ∑     k   =   1     M     ⁢       A   k     ⁢     e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   k   )         ⁢   t                 
where J=√{square root over (−1)} and coefficients A k  are given by
 
               [           A   1             ⋮             A   M           ]     =         [           e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   1   )         ⁢     t   1             …         e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   M   )         ⁢     t   1                 ⋮                   ⋮             e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   1   )         ⁢     t   M             …         e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   M   )         ⁢     t   M               ]       -   1       ⁡     [           C   1             ⋮             C   M           ]             
Thus, each coefficient A k  is generated by multiplying a matrix of predetermined sinusoids with the single column of data symbols C 1 , C 2 , C 3 , . . . , C M    16 .
 
       FIG. 3B  shows the frequency response of the continuous function CF(t). 
     More specifically,  FIG. 3B  shows that the frequency response of the continuous function is non-zero only at the allocated frequency tones f i(1) , f i(2) , . . . , f i(M) , and is zero at all other frequency tones. 
     The output of the DSP  26  is a vector of digital signal samples S  22 , which are the samples of the continuous function CF(t) at discrete time instants 0, T/N, 2T/N, . . . , T(N−1)/N, that is, S 1 =CF(0), S 2 =CF(T/N), S 3 =CF(2T/N), . . . , S N =CF(T(N−1)/N), where N is the number of discrete time instants in the vector of digital signal samples  22 . In a general form, t 1 , . . . , t M  may not necessarily be equal to any of the time instants 0, T/N, 2T/N . . . , T(N−1)/N. Therefore, while the digital signal samples S  22  may occur at the time instants t 1 , . . . , t M , the OFDM communication system  10  does not require that the time instants 0, T/N, 2T/N . . . , T(N−1)/N be equal to t 1 , . . . , t M . 
     In another implementation of OFDM communication system  10 , the digital signal samples S  22  may be generated by the DSP  26  by directly multiplying a matrix of precomputed sinusoidal waveforms Z, operating as discrete interpolation functions, with the discrete signal of mapped symbols C in order to satisfy the transformation function S=ZC according to the following: 
                   S   =       [           S   1             ⋮             S   N           ]     =         [           e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   1   )         ⁢   0           …         e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   M   )         ⁢   0               ⋮                   ⋮             e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   1   )         ⁢   T   ⁢       N   -   1     N             …         e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   M   )         ⁢   T   ⁢       N   -   1     N               ]     ⁡     [           A   1             ⋮             A   M           ]       =         [           e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   1   )         ⁢   0           …         e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   M   )         ⁢   0               ⋮                   ⋮             e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   1   )         ⁢   T   ⁢       N   -   1     N             …         e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   M   )         ⁢   T   ⁢       N   -   1     N               ]     ⁡     [           e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   1   )         ⁢     t   1             …         e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   M   )         ⁢     t   1                 ⋮                   ⋮             e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   1   )         ⁢     t   M             …         e     J   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     f     i   ⁡     (   M   )         ⁢     t   M               ]         -   1                   ⁡     [           C   1             ⋮             C   M           ]       =   ZC         
where C represents the symbol vector, and the matrix Z represents the product of the two matrices in the second line of the above equation. Each column (i) of matrix Z represents the interpolation function  42  of a corresponding symbol C i  to generate the digital signal samples S  22 . As such, the matrix Z can be pre-computed and stored in the interpolation function memory  23  of the interpolation circuit  20  ( FIG. 2B ). The interpolation circuit  20  then applies the discrete interpolation functions  42  defined by the matrix Z to the discrete signal of mapped complex symbols C  16  in order to satisfy the criteria of S=ZC and to generate the vector of digital signal samples  22 .
 
     The purpose of constructing the signal in the time domain is to directly map the symbols  16 , which have a desirable low peak-to-average ratio property, to the prescribed time instants within the symbol duration  40 . Appropriate interpolation functions  42  are selected to obtain the continuous function CF(t) and the digital signal samples  22  such that the desirable low peak-to-average ratio property of the symbols  16  is substantially preserved for the continuous function and for the digital signal samples  22 . The peak-to-average ratio property of the resulting (interpolated) continuous function CF(t) and the digital signal samples  22  is dependent upon the interpolation functions  42 , the choice of allocated frequency tones f i(1) , f i(2) , . . . , f i(M)  from the set of tones, and the prescribed time instants t 1 , . . . , t M . 
     Referring to  FIG. 4A , one implementation of the OFDM communication system  10  allocates tones f i(1) , f i(2) , . . . , f i(M)  to the transmitter associated with the communication system that are a subset of contiguous tones in the tone set f 1 , f 2 , . . . , f F . Therefore, f i(k) =f 0 +(k−1)Δ, for k=1, . . . , M, where M is the number of symbols. If the OFDM communication system  10  is a multiple access system, each transmitter associated with the communication system is allocated a non-overlapping subset of frequency tones. For purposes of description, let f 0 =0. The construction for the other cases where f 0 ≠0 can be similarly obtained. 
     Complex symbols C 1 , . . . , C M    16  are mapped in the time domain to the following time instants t k =(k−1)T/M, for k=1, . . . , M. As part of this implementation, the prescribed time instants t 1 , . . . , t M  are equally-spaced time instants uniformly distributed over the entire OFDM symbol duration  40  as shown in the first time domain graph of  FIG. 4A . Given the choice of the allocated frequency tones and prescribed time instants, the matrix Z, which is used to generate the digital signal samples S as discussed in  FIGS. 3A-3B , can be simplified to 
     
       
         
           
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     The second time domain graph of  FIG. 4A  shows the resulting digital signal sample vector S  22  after the interpolation circuit  20  applies the interpolation functions  42  defined by the matrix Z to the complex symbols  16  according to the expression S=ZC. As part of this implementation, the sampling module  25  is not generally used as the digital signal sample vector S  22  is directly generated from the discrete signal of mapped symbols using the transformation function S=ZC. 
     Turning to  FIG. 4B , a digital processing system  50  provides another technique for obtaining the vector of digital signal samples S. A DFT circuit  52  receives a discrete signal of complex data symbols C i , and calculates the frequency responses A 1 , . . . , A M , at tones f i(1) , f i(2) , . . . , f i(M) , through an M-point discrete Fourier transform (DFT). The vector [A 1 , . . . , A M ]  54  output by the DFT circuit  52  is then expanded to a new vector of length N (the total number of time instants in the discrete signal vector S) by zero insertion at block  56 . More specifically, this process involves putting the k th  symbol A k  to the i(k) th  element of the new vector, for k=1, . . . , M, where f i(k)  is the k th  tone allocated to the transmitter, and inserting zeros in all the remaining elements. Finally, an IDFT circuit  58  performs an N-point inverse discrete Fourier transform on the resulting vector (after zero insertion) to obtain the digital signal sample vector S. The collective procedure of DFT, zero insertion and IDFT is one way of implementing the discrete interpolation functions. 
     Turning to  FIG. 4C , another technique for obtaining the digital signal samples S is shown. For simplicity of description, it is assumed that the allocated contiguous tones f i(1) , f i(2) , . . . , f i(M)  are centered at frequency 0. The construction for the other cases where the allocated tones are not centered at frequency 0 can be similarly obtained. As with  FIG. 4A , the prescribed time instants t 1 , . . . , t M  are equally-spaced time instants uniformly distributed over the entire OFDM symbol duration  40 . 
     The complex symbols C 1 , . . . , C M  are first mapped in the time domain to time instants t 1 , . . . t M  respectively. Next, the mapped symbols C 1 , . . . , C M  are leftwards and rightwards shifted and replicated to an expanded set of prescribed time instants, which is a superset of t 1 , . . . t M  and consists of an infinite number of equally-spaced time instants covering the time interval from −∞ to +∞. This technique creates an infinite series of mapped symbols C. The continuous function CF(t) is then constructed by interpolating the infinite series of mapped symbols using a sinc interpolation function  60 . Mathematically, the above steps construct the continuous function CF(t) as 
               CF   ⁡     (   t   )       =       ∑     i   =   1     M     ⁢       {       C   i     ⁢       ∑     k   =     -   ∞       ∞     ⁢     sin   ⁢           ⁢     c   (       t   -     t   i     -   kT     ,     T   M       )           }     .             
where sinc(a,b)=sin(πa/b)/(πa/b). The sine interpolation function  60  can also be precomputed and stored in the interpolation function memory  23 . As discussed in  FIG. 3A  the digital signal samples S  22  are the samples of the continuous function CF(t) at time instants 0, T/N, . . . , T(N−1)/N. In  FIGS. 4A-4C , if N is a multiple of M, then S 1+(k−1)N/M =C k , for k=1, . . . , M. It should be noted that the continuous function CF(t) only applies to the symbol duration  40  from 0 to T. The use of time interval from −∞ to +∞ is solely for the purpose of mathematically constructing CF(t). The discrete interpolation functions, which combine the continuous interpolation functions and the sampling function, can be derived easily from the above description.
 
     For comparison purposes,  FIG. 4D  illustrates the resulting peak-to-average ratio for a digital signal sample vector S  62  and its associated transmitted OFDM signal  64  produced by symbols  16  where the signal is constructed in the frequency domain. As described above, this known technique of mapping the symbols  16  in the frequency domain produces a large signal variation in the transmitted OFDM signal  64  and results in a large peak-to-average ratio. 
       FIG. 4E  illustrates the resulting small signal variation and low peak-to-average ratio of the digital signal sample vector S  66  associated with the transmitted OFDM signal  68 . As will be appreciated by comparing  FIGS. 4D and 4E , mapping the constellation of complex symbols  16  in the time domain produces an OFDM signal  68  having a significantly reduced peak-to-average ratio. 
       FIG. 5A  shows a second implementation of the OFDM communication system  10 , and serves to further generalize the system shown in  FIGS. 4A-4C . As part of OFDM system  10 , tones, f i(1) , f i(2) , . . . , f i(M) , allocated to the transmitter associated with the communication system, are a subset of equally-spaced tones in the tone set f 1 , f 2 , . . . , f F . Therefore, f i(k) =f 0 +(k−1)LΔ, for k=1, . . . , M, and L is a positive integer number representing the spacing between two adjacent allocated frequency tones. When L=1, this implementation is equivalent to the implementation technique described in  FIGS. 4A-4C . For the sake of description, let f 0 =0. The construction for the other cases where f 0 ≠0 can be similarly obtained. 
     In this case where the allocated tones are equally-spaced tones, the constructed continuous function CF(t) is identical in each of the L time intervals, [0,T/L), [T/L,2T/L), . . . , and [(L−1)T/L, T/L). As part of this technique, symbols C 1 , . . . , C M    16  are mapped to the following time instants t k =(k−1)T/M/L, for k=1, . . . , M. In this implementation, the prescribed time instants t 1 , . . . , t M  are equally-spaced time instants uniformly distributed over a fraction (1/L) of the symbol duration  70 . As a comparison, in the case of allocated contiguous tones ( FIG. 4A ), the prescribed time instants are equally-spaced and distributed over the entire symbol duration, as discussed with respect to  FIG. 4A . 
     The procedure for obtaining the digital signal samples S  22  described in  FIG. 4A  can also be applied with respect to  FIG. 5A . More specifically, the digital signal sample vector S is the product of matrix Z (defining the discrete interpolation functions) and the symbol vector C. Given the choice of the allocated frequency tones and prescribed time instants, the matrix Z, which is used to generate the digital signal samples  22  from the discrete signal of mapped symbols, can be simplified to the same formula as in  FIG. 4A  with the only change in the definition off f i(1) , f i(2) , . . . , f i(M)  and t 1 , . . . , t M . 
     In  FIG. 5B , the procedure of obtaining the digital signal sample vector S  22  described in  FIG. 4B  can also be applied to the case of allocated frequency tones that are equally spaced tones. More specifically, a digital processing system  100  provides another technique for obtaining the vector of digital signal samples S. A DFT circuit  102  receives a discrete signal of complex data symbols C i  and calculates the frequency responses A 1 , . . . , A M , at tones f i(1) , f i(2) , . . . , f i(M) , through an M-point discrete Fourier transform (DFT). The vector [A 1 , . . . A M ]  104  output by the DFT circuit  102  is then expanded to a new vector of length N (the total number of time instants in the digital signal sample vector S) by zero insertion at block  106 . More specifically, this process involves putting the k th  symbol A k  to the i(k)th element of the new vector, for k=1, . . . , M, where f i(k)  is the k th  tone allocated to the transmitter, and inserting zeros in all the remaining elements. Finally, an IDFT circuit  108  performs an N-point inverse discrete Fourier transform on the resulting vector (after zero insertion) to obtain the time domain digital signal sample vector S. The collective procedure of DFT, zero insertion and IDFT is one way of implementing the discrete interpolation functions. 
       FIG. 5C  is the counterpart of  FIG. 4C , where symbols C 1 , . . . , C M  are first mapped to t 1 , . . . , t M  respectively over a fraction (1/L) of the symbol duration  70 . The symbol mapping is also performed in the time domain. Next the mapped symbols C 1 , . . . , C M  are leftwards and rightwards shifted and replicated to an expanded set of prescribed time instants from −∞ to +∞ which creates an infinite series of symbols. The continuous function CF(t) is then constructed by interpolating the infinite series of mapped symbols with a sinc interpolation function  72 . Thus, the continuous function CF(t) includes the digital signal samples mapped to the prescribed time instants as well as digital sample points between the prescribed time instants. Mathematically, the above steps construct the continuous function as 
               CF   ⁡     (   t   )       =       ∑     i   =   1     M     ⁢       {       C   i     ⁢       ∑     k   =     -   ∞       ∞     ⁢     sin   ⁢           ⁢     c   ⁡     (       t   -     t   i     -     kT   ⁢     1   L         ,       T   M     ⁢     1   L         )             }     .             
With continued reference to  FIG. 5C , each sinc interpolation function  72  is narrower and therefore decays faster than the sinc interpolation function  60  shown in  FIG. 4C . The sinc interpolation function  72  can also be precomputed and stored in the interpolation function memory  23  for use by the interpolation function module  21 . The digital sample vector S  22  can be obtained in the same technique shown in  FIG. 4C . In  FIGS. 5A and 5C , if N is a multiple of ML, then S 1+(k−1)N/M/L+(j−1)N/L =C k , for k=1, . . . , M, and j=1, . . . , L. The discrete interpolation functions, which combine the continuous interpolation functions and the sampling function, can be derived easily from the above description.
 
       FIG. 5D  illustrates the resulting small signal variation and low peak-to-average ratio of the digital signal sample vector S  74  associated with the transmitted OFDM signal  76 . As will be appreciated by comparing  FIGS. 4D and 5D , mapping the constellation of complex symbols  16  in the time domain produces an OFDM signal  76  having a significantly lower peak-to-average ratio. 
     Referring now to  FIG. 6 , a π/4 symbol rotation technique is used to further reduce the peak-to-average ratio of the transmitted OFDM signal. At an OFDM symbol duration, if symbols B 1 , . . . , B M  of the constellation are to be transmitted, symbols B 1 , . . . , B M  are mapped to another block of complex symbols C 1 , . . . , C M , where each odd number symbol remains unchanged and each even number symbol is phase rotated by π/4. For example, if symbols B 1 , . . . , B M  belong to a QPSK constellation {0, π/2, π, π3/2}, the odd number symbols C k  still belong to the same QPSK constellation, while after being phase rotated the even number symbols C k  belong to another QPSK constellation {π/4, π3/4, π5/4, π7/4}. Symbols C 1 , . . . , C M  are then used to construct the digital signal samples  22  in the time domain as described above with respect to  FIGS. 3A-5C . 
     With reference to  FIG. 7 , another technique for reducing the peak-to-average ratio is shown, which introduces a cyclic offset of the real and imaginary signal components. This technique involves a first step of offsetting the imaginary components of the digital signal samples S  22 , which have been generated using the technique of  FIGS. 3A-5C , by an integer number of samples. If necessary, the technique then involves a second step of adjusting the timing by a fraction of a sample period between the real and the imaginary signal components in the transmit path. 
     At an OFDM symbol duration, if the digital signal samples S 1 , S 2 , . . . , S N  have been obtained using the method as described in  FIGS. 3A-5C , the digital signal sample vector S is then mapped to another vector S′ as follows. The real component of digital signal sample S′ k  is equal to that of digital signal sample S k . The imaginary component of digital signal sample S′ k  is equal to that of digital signal sample S j  where index j=(k+d−1)mod N+1, for k=1, . . . , N, with mod representing a module operation. The parameter d is an integer representing the cyclic offset, in terms of number of samples, between the real and imaginary components. 
     In one implementation, the value of d is determined by 
               N     2   ⁢           ⁢   LM       ,         
where L is discussed in  FIG. 5A . In one aspect of this technique, d is chosen to be close to
 
               N     2   ⁢           ⁢   LM       .         
For example, d can be the integer closest to
 
               N     2   ⁢           ⁢   LM       ,         
the largest integer not greater than
 
               N     2   ⁢           ⁢   LM       ,         
or the smallest integer not smaller than
 
               N     2   ⁢           ⁢   LM       .         
In one example, d is chosen to be the largest integer not greater than
 
               N     2   ⁢           ⁢   LM       .         
This example can be easily extended for other choices of d.
 
     The digital signal sample vector S′ is then passed to the cyclic prefix prepender circuit  24 , as shown in  FIG. 1 . Therefore, the operation of half symbol cyclic shifting is carried out before the operation of prepending the cyclic prefix, such as that performed by the cyclic prefix circuit  24  of  FIG. 1 . 
     Not specifically shown in  FIG. 7 , when or after the sample vector S′ and the cyclic prefix are outputted to the digital to analog converter  28 , the imaginary components are further delayed by an amount of 
                 (       N     2   ⁢           ⁢   LM       -   d     )     ⁢     T   N       ,         
which is a fraction of a sample period T/N.
 
     As a variation of the technique shown in  FIG. 7  (not specifically shown), another technique for achieving a similar result can be used to eliminate the second step of adjusting timing by a fraction of a sample period between the real and the imaginary signal components in the transmit path. As part of this technique, the real and the imaginary components of the desired digital signal samples S  22  are generated separately as described by the following. 
     A first series of digital signal samples  22  are generated using the technique of  FIGS. 3A-5C . The real components of the desired digital signal samples  22  are equal to those of the first series of samples. A second series of digital signal samples  22  are generated using the technique of  FIGS. 3A-5C  except for the following changes. The imaginary components of the desired digital signal samples are equal to those of the second series of samples. In the general method described in  FIGS. 3, 4A, and 5A , the matrix 
                   [           e     J   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢     f     i   ⁡     (   1   )         ⁢   0           …         e     J   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢     f     i   ⁡     (   M   )         ⁢   0               ⋮                   ⋮             e     J   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢     f     i   ⁡     (   1   )         ⁢   T   ⁢       N   -   1     N             …         e     J   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢     f     i   ⁡     (   M   )         ⁢   T   ⁢       N   -   1     N               ]           
is changed to
 
     
       
         
           
               
             
               
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     Referring to  FIGS. 8A-8D , another technique for further reducing the peak-to-average ratio is implemented by allocating more frequency tones than the number of complex symbols to be transmitted in a symbol duration  40 . In  FIGS. 3-7 , the number of tones allocated to the transmitter associated with the communication system is equal to the number of symbols to be transmitted in a given OFDM symbol duration. Compared with the other techniques described with respect to the previous figures, the technique of  FIGS. 8A-8D  requires additional overhead of bandwidth to transmit the same number of complex symbols. 
     For example, if the communication system  10  is allocated M+M ex  contiguous frequency tones, f i(1) , f i(2) , . . . , f i(M+Mex) , and M symbols C 1 , . . . , C M  of the constellation are to be transmitted at an OFDM symbol duration, from the comparison of  FIGS. 4A and 5A , the case of allocated contiguous tones can be easily extended to the case of allocated equally-spaced tones. As part of this implementation of the OFDM communication system  10 , M ex  is a positive number representing the number of excess tones to be used and is assumed to be an even number. Therefore, the allocated tone 
                 f     i   ⁡     (   k   )         =       f   0     +       (     k   -       M   ex     2     -   1     )     ⁢   Δ         ,         
for k=1, . . . M+M ex . For purposes of description, let f 0 =0. The construction for the other cases where f 0 ≠0 can be similarly obtained.
 
     As with the technique described with respect to  FIG. 4A , the prescribed time instants are t k =(k−1)T/M, for k=1, . . . , M, that is, the prescribed time instants t 1 , . . . , t M  are equally-spaced time instants in the symbol duration  40 . 
     As part of this technique shown in  FIG. 8A , P(f) is a smooth windowing function  90  in the frequency domain, which is non-zero only over interval [f i(1) , f i(M+Mex) ]. In addition, P(f)  90  also satisfies the Nyquist zero intersymbol interference criterion, i.e., 
                 ∑     k   =     -   ∞       ∞     ⁢     P   ⁡     (     f   -     kM   ⁢           ⁢   Δ       )         =   1         
for any frequency f, where □ is the spacing between adjacent tones.
 
       FIG. 8B  shows the block diagram of the technique. As described above, a symbol-to-symbol mapping is optionally performed to generate a discrete signal of mapped complex symbols C 1 , . . . , C M ,  16 . The frequency responses A 1 , . . . , A M    84  are calculated through an M-point discrete Fourier transform (DFT) of the complex symbols  16  at block  82 . At block  86 , vector [A 1 , . . . , A M ]  84  is cyclically expanded to a new vector A′ of length N and windowed with a windowing function  90  as follows:
 
 A′   k   A   g(k)   *P (( k− 1)□+ f   1 )
 
where index g(k)=mod(k−i(1)−M ex /2,M)+1, for k=1, . . . , N.
 
     At block  88 , the digital signal sample vector S is obtained by taking an N-point inverse discrete Fourier transform (IDFT) of the new vector A′. Finally, the cyclic prefix is added by cyclic prefix circuit  24  as described above with regard to  FIG. 1 . 
     To provide additional insight to the above signal construction technique, assume that the allocated tones f i(1) , f i(2) , . . . , f i(M+Mex)  are centered at frequency 0. In  FIG. 8C  (as with  FIG. 4C ), symbols C 1 , . . . , C M  are first mapped to equally-spaced time instants in the symbol duration  40 , and are then leftwards and rightwards shifted and replicated from −∞ to +∞. What is different from  FIG. 4C  is that a different interpolation function  92 , which is determined by the windowing function  90 , is used to generate the continuous function,
 
 CF ( t )=Σ i=1   M   C   i Σ k=−∞   ∞   p ( t−t   i   −kT )
 
where p(t)  92  is the time domain response of P(f)  90 . As with  FIG. 4C , the digital signal samples are obtained by letting t=0, . . . , T(N−1)/N.
 
     In one exemplary aspect of this technique, if a raised cosine windowing function is used, i.e., 
               P   ⁡     (   f   )       =     {             T   M     ⁢                       i   ⁢   f     ⁢           ⁢        f          &lt;       (     1   -   β     )     ⁢     M     2   ⁢           ⁢   T                         T     2   ⁢           ⁢   M       ⁢     {     1   +     cos   ⁡     [         π   ⁢           ⁢   T       β   ⁢           ⁢   M       ⁢     (          f        -         (     1   -   β     )     ⁢   M       2   ⁢           ⁢   T         )       ]         }       ⁢                     if   ⁢           ⁢     (     1   -   β     )     ⁢     M     2   ⁢           ⁢   T         ≤        f        ≤       (     1   +   β     )     ⁢     M     2   ⁢   T                     0   ⁢                     if   ⁢           ⁢        f          &gt;       (     1   +   β     )     ⁢     M     2   ⁢           ⁢   T                         
where β=(M ex +2)/M represents the percentage of excess tone overhead, then, the interpolation function p(t)  92  is given by
 
               p   ⁡     (   t   )       =         sin   ⁡     (     π   ⁢           ⁢     tM   /   T       )         π   ⁢           ⁢     tM   /   T         ⁢         cos   ⁡     (     π   ⁢           ⁢   β   ⁢           ⁢     tM   /   T       )         1   -     4   ⁢           ⁢     β   2     ⁢     t   2     ⁢       M   2     /     T   2             .             
As β increases, the interpolation function p(t)  92  decays faster, thereby reducing the probability of having large peak at samples between t i .
 
       FIG. 8D  shows the resulting small signal variation and low peak-to-average ratio of the digital signal sample vector S  94  associated with the transmitted OFDM signal  96 . As will be appreciated, mapping the constellation symbols  16  in the time domain produces an OFDM signal  96  having a significantly lower peak-to-average signal ratio. 
     A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. Accordingly, other embodiments are within the scope of the following claims.