US20090262641A1 - Signaling method in an ofdm multiple access system - Google Patents

Abstract

A method for reducing the peak-to-average ratio in an OFDM communication signal is provided. The method includes defining a constellation having a plurality of symbols, defining a symbol duration for the OFDM communication signal, and defining a plurality of time instants in the symbol duration. A plurality of tones are allocated to a particular communication device, and a discrete signal is constructed in the time domain by mapping symbols from the constellation to the time instants. A continuous signal is generated by applying an interpolation function to the discrete signal such that the continuous signal only includes sinusoids having frequencies which are equal to the allocated tones.

Description

CROSS-REFERENCE TO RELATED APPLICATIONS
• The present application is a continuation of U.S. patent application Ser. No. 11/186,490, entitled “SIGNALING METHOD IN AN OFDM MULTIPLE ACCESS SYSTEM”, filed on Jul. 20, 2005, now pending, which is a divisional of U.S. patent application Ser. No. 09/805,887, entitled “SIGNALING METHOD IN AN OFDM MULTIPLE ACCESS SYSTEM”, filed on Mar. 15, 2001, now U.S. Pat. No. 7,295,509, which claims priority to U.S. Provisional Application No. 60/230,937, entitled “SIGNALING METHOD IN AN OFDM MULTIPLE ACCESS WIRELESS SYSTEM”, filed on Sep. 13, 2000, all of which are assigned to the assignee hereof, and expressly incorporated herein 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 Δ 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_ccan 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_cis 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 aspect, a digital processing system is defined that is utilized in a transmitter that generates a transmission signal having a first circuit that receives a discrete signal of complex data symbols, calculates a frequency response of the discrete signal, and outputs a first vector of length M, a second circuit that receives the vector of length M and generates a second vector of length N, wherein the second vector of length N includes elements of the first vector at positions corresponding to tones allocated to the transmitter, and a third circuit that receives the second vector and generates a digital time domain vector therefrom for transmission by the transmitter.
• In one aspect, an apparatus that generates a transmission signal is defined which comprises means for receiving a discrete signal of complex data symbols, calculates a frequency response of the discrete signal, and outputs a first vector of length M, means for receiving the vector of length M and generates a second vector of length N, wherein the second vector of length N includes elements of the first vector at positions corresponding to tones allocated to a transmitter, and means for generating a digital time domain vector of the second vector for transmission by the transmitter.
• 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 ofFIG. 1.
• FIG. 2B is a block diagram of another interpolation system used by the OFDM system ofFIG. 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 ofFIG. 1.
• FIG. 3B is a graph showing the frequency domain response of the graph ofFIG. 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 ofFIGS. 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 ofFIGS. 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 ofFIG. 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 ofFIGS. 8A-8C.
• Like reference symbols in the various drawings indicate like elements.
DETAILED DESCRIPTION
• Referring toFIG. 1, an orthogonal frequency division multiplexing (OFDM)communication system10 is shown.OFDM communication system10 receives a first constellation of symbols {B_i}12 and provides the symbols to a symbol-to-symbol mapping circuit14, that produces a second constellation of complex symbols {C_i}16. Thecomplex symbols16 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 themapping circuit14 is an optional step performed by theOFDM communication system10.
• Next, a timeinstant mapping circuit18 maps eachcomplex symbol16 to a prescribed time instant within a given OFDM symbol duration. The mapping operation is performed in the time domain such that themapping circuit18 generates a discrete signal of mapped symbols within the time domain symbol duration. The output of themapping circuit18 is provided to aninterpolation circuit20, that produces a series of digital signal samples {S_i}22. Thedigital signal samples22 are formed by sampling a continuous signal, which is constructed by applying one or more predetermined continuous interpolation functions to the mappedcomplex symbols19. Alternatively, thedigital signal samples22 are formed by directly applying one or more predetermined discrete interpolation functions to the mappedcomplex symbols19. 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 theinterpolation circuit20 is described in greater detail below. Acyclic prefix circuit24 receives the series ofdigital signal samples22 from theinterpolation circuit20 and prepends a cyclic prefix to thedigital signal samples22. Thecyclic prefix circuit24 operates to copy and prepend the last portion of the digital signalsample vector S22 to the beginning of the OFDM symbol duration. The resultingdigital signal samples22 with the prepended cyclic prefix are converted to an analog signal by a digital toanalog converter28. The resulting analog signal is further processed by apulse shaping filter30, the output of which is modulated to a carrier frequency, and amplified by apower amplifier unit32 for transmission through anantenna34.
• In one implementation of theOFDM communication system10, the symbol-to-symbol mapping circuit14, the timeinstant mapping circuit18, theinterpolation circuit20, and thecyclic prefix circuit24 are implemented in a digital signal processor (DSP)26, and may include a combination of hardware modules and/or software modules. Thesecircuits14,18,20, and24 can also be implemented as separate discrete circuits within theOFDM communication system10.
• The details of theinterpolation circuit20 are shown inFIG. 2A. Theinterpolation circuit20 includes aninterpolation function module21 that applies one or more continuous interpolation functions to the discrete signal of mappedsymbols19 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 aninterpolation function memory23 connected to theinterpolation function module21. A frequency tone and timeinstant allocation circuit27 is connected to theinterpolation function memory23 and defines an allocated tone set selected from frequency tones distributed over a predetermined bandwidth associated with theOFDM communication system10. The allocated tone set is then provided to theinterpolation function memory23. The frequency tone and timeinstant allocation circuit27 also defines the prescribed time instants distributed over the time domain symbol duration, which can also be stored in theinterpolation function memory23 for use by theinterpolation function module21 as well as other modules within theDSP26. Theinterpolation circuit20 also includes asampling circuit25 for receiving and sampling the continuous signal at discrete time instants distributed over the time domain symbol duration to generate the vector ofdigital signal samples22. Alternatively, inFIG. 2B theinterpolation function module21 applies one or more discrete interpolation functions to the discrete signal of mappedsymbols19 to directly generate the digitalsignal sample vector22, in which case the sampling circuit25 (ofFIG. 2A) is not needed. Through applying the discrete interpolation functions, theinterpolation function module21 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 theDSP26. More specifically,FIG. 3A shows the construction of the signal to be transmitted in a given OFDM timedomain symbol duration40. The timedomain symbol duration40 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 thesymbol duration40 is represented by complex symbols C₁, C₂, C₃, . . . ,C_M16 that are mapped to the prescribed time instants, where M denotes the number of symbols to be transmitted in thesymbol duration40.
• In one implementation, theOFDM communication system10 is a multiple access communication system where the entire bandwidth available to all transmitters within the system is divided into F orthogonal frequency tones, f₁, f₂, . . . , f_F. In the givensymbol duration40, 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 f₁, f₂. . . , 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 inFIG. 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 allowsOFDM communication system10 to operate as a multiple access communication system.
• The complex data symbols C₁, C₂, C₃, . . . ,C_M16 are first mapped to t₁, t₂, t₃, . . . , t_M, respectively, where t₁, t₂, t₃, . . . , t_Mare M prescribed time instants within the timedomain symbol duration40. 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 functions42 are then applied to the discrete signal of mappedsymbols16 to generate a continuous function CF(t) for t in the time interval from 0 to T.
• The interpolation functions42 are constructed such that the values of the continuous function CF(t) at time instants t₁, t₂, t₃, . . . , t_Mare respectively equal to C₁, C₂, C₃, . . . , C_Mand the frequency response of the continuous function CF(t) contains only sinusoids at the allocated tones. Therefore, CF(t) is constructed as
• $\mathrm{CF}\left(t\right)=\sum _{k=1}^MA_k{}^{J2\pi f_{i\left(k\right)}t}$
• where J=√{square root over (−1)} and coefficients A_kare given by
• $\left[\begin{array}{c}{A}_1\\ ⋮\\ A_M\end{array}\right]={\left[\begin{array}{ccc}{}^{J2\pi f_{i\left(1\right)}{\mathrm{<figure-callout\; label="t"\; filenames="US20090262641A1-20091022-D00007.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">t</figure-callout>}}_1}& \dots & {}^{J2\pi f_{i\left(M\right)}{\mathrm{<figure-callout\; label="t"\; filenames="US20090262641A1-20091022-D00007.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">t</figure-callout>}}_1}\\ ⋮& & ⋮\\ {}^{J2\pi f_{i\left(1\right)}t_M}& \dots & {}^{J2\pi f_{i\left(M\right)}t_M}\end{array}\right]}^{-1}\left[\begin{array}{c}{\mathrm{<figure-callout\; label="C"\; filenames="US20090262641A1-20091022-D00007.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">C</figure-callout>}}_1\\ ⋮\\ C_M\end{array}\right]$
• Thus, each coefficient A_kis generated by multiplying a matrix of predetermined sinusoids with the single column of data symbols C₁, C₂, C₃, . . . ,C_M16.
• 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 theDSP26 is a vector of digital signal samples S22, which are the samples of the continuous function CF(t) atdiscrete time instants 0, T/N, 2 T/N, . . . , T(N−1)/N, that is, S₁=CF(0), S₂=CF(T/N), S₃=CF(2 T/N), . . . , S_N=CF(T(N−1)/N), where N is the number of discrete time instants in the vector ofdigital signal samples22. In a general form, t₁, . . . , t_Mmay not necessarily be equal to any of thetime instants 0, T/N, 2 T/N . . . , T(N−1)/N. Therefore, while the digital signal samples S22 may occur at the time instants t₁, . . . , t_M, theOFDM communication system10 does not require that thetime instants 0, T/N, 2 T/N . . . T(N−1)/N be equal to t₁, . . . , t_M.
• In another implementation ofOFDM communication system10, the digital signal samples S22 may be generated by theDSP26 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:
• $\begin{array}{c}S=\left[\begin{array}{c}{\mathrm{<figure-callout\; label="S"\; filenames="US20090262641A1-20091022-D00007.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">S</figure-callout>}}_1\\ ⋮\\ S_N\end{array}\right]\\ =\left[\begin{array}{ccc}{}^{J2\pi f_{i\left(1\right)}0}& \dots & {}^{J2\pi f_{i\left(M\right)}0}\\ ⋮& & ⋮\\ {}^{J2\pi f_{i\left(1\right)}T\frac{N-1}{N}}& \dots & {}^{J2\pi f_{i\left(M\right)}T\frac{N-1}{N}}\end{array}\right]\left[\begin{array}{c}{A}_1\\ ⋮\\ A_M\end{array}\right]\\ =\left[\begin{array}{ccc}{}^{J2\pi f_{i\left(1\right)}0}& \dots & {}^{J2\pi f_{i\left(M\right)}0}\\ ⋮& & ⋮\\ {}^{J2\pi f_{i\left(1\right)}T\frac{N-1}{N}}& \dots & {}^{J2\pi f_{i\left(M\right)}T\frac{N-1}{N}}\end{array}\right]\\ {\left[\begin{array}{ccc}{}^{J2\pi f_{i\left(1\right)}{\mathrm{<figure-callout\; label="t"\; filenames="US20090262641A1-20091022-D00007.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">t</figure-callout>}}_1}& \dots & {}^{J2\pi f_{i\left(M\right)}{\mathrm{<figure-callout\; label="t"\; filenames="US20090262641A1-20091022-D00007.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">t</figure-callout>}}_1}\\ ⋮& & ⋮\\ {}^{J2\pi f_{i\left(1\right)}t_M}& \dots & {}^{J2\pi f_{i\left(M\right)}t_M}\end{array}\right]}^{-1}\left[\begin{array}{c}{\mathrm{<figure-callout\; label="C"\; filenames="US20090262641A1-20091022-D00007.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">C</figure-callout>}}_1\\ ⋮\\ C_M\end{array}\right]\\ =\mathrm{ZC}\end{array}$
• 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 function42 of a corresponding symbol C_ito generate the digitalsignal samples S22. As such, the matrix Z can be pre-computed and stored in theinterpolation function memory23 of the interpolation circuit20 (FIG. 2B). Theinterpolation circuit20 then applies the discrete interpolation functions42 defined by the matrix Z to the discrete signal of mappedcomplex symbols C16 in order to satisfy the criteria of S=ZC and to generate the vector ofdigital signal samples22.
• The purpose of constructing the signal in the time domain is to directly map thesymbols16, which have a desirable low peak-to-average ratio property, to the prescribed time instants within thesymbol duration40. Appropriate interpolation functions42 are selected to obtain the continuous function CF(t) and thedigital signal samples22 such that the desirable low peak-to-average ratio property of thesymbols16 is substantially preserved for the continuous function and for thedigital signal samples22. The peak-to-average ratio property of the resulting (interpolated) continuous function CF(t) and thedigital signal samples22 is dependent upon the interpolation functions42, 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 t₁, . . . , t_M.
• Referring toFIG. 4A, one implementation of theOFDM communication system10 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 f₁, f₂, . . . , f_F. Therefore, f_i(k)=f₀+(k−1)Δ, for k=1, . . . , M, where M is the number of symbols. If theOFDM communication system10 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 f₀=0. The construction for the other cases where f₀≠0 can be similarly obtained.
• Complex symbols C₁, . . . ,C_M16 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 t₁, . . . , t_Mare equally-spaced time instants uniformly distributed over the entireOFDM symbol duration40 as shown in the first time domain graph ofFIG. 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 inFIGS. 3A-3B, can be simplified to
• $Z=\frac{1}{M}\hspace{1em}[\begin{array}{ccc}{}^{J2\pi f_{i\left(1\right)}0}& \dots & {}^{J2\pi f_{i\left(M\right)}0}\\ ⋮& & ⋮\\ {}^{J2\pi f_{i\left(1\right)}T\frac{N-1}{N}}& \dots & {}^{J2\pi f_{i\left(M\right)}T\frac{N-1}{N}}\end{array}]\hspace{1em}\hspace{1em}[\begin{array}{ccc}{}^{-J2\pi f_{i\left(1\right)}{\mathrm{<figure-callout\; label="t"\; filenames="US20090262641A1-20091022-D00007.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">t</figure-callout>}}_1}& \dots & {}^{-J2\pi f_{i\left(1\right)}t_M}\\ ⋮& & ⋮\\ {}^{-J2\pi f_{i\left(M\right)}{\mathrm{<figure-callout\; label="t"\; filenames="US20090262641A1-20091022-D00007.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">t</figure-callout>}}_1}& \dots & {}^{-J2\pi f_{i\left(M\right)}t_M}\end{array}].$
• The second time domain graph ofFIG. 4A shows the resulting digital signalsample vector S22 after theinterpolation circuit20 applies the interpolation functions42 defined by the matrix Z to thecomplex symbols16 according to the expression S=ZC. As part of this implementation, thesampling module25 is not generally used as the digital signalsample vector S22 is directly generated from the discrete signal of mapped symbols using the transformation function S=ZC.
• Turning toFIG. 4B, a digital processing system50 provides another technique for obtaining the vector of digital signal samples S. ADFT circuit52 receives a discrete signal of complex data symbols C_iand calculates the frequency responses A₁, . . . , A_M, at tones f_i(1), f_i(2), . . . , f_i(M), through an M-point discrete Fourier transform (DFT). The vector [A₁, . . . , A_M]54 output by theDFT circuit52 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 atblock56. More specifically, this process involves putting the k^thsymbol A_kto the i(k)^thelement of the new vector, for k=1, . . . , M, where f_i(k)is the k tone allocated to the transmitter, and inserting zeros in all the remaining elements. Finally, an IDFT circuit58 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 toFIG. 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 atfrequency 0. The construction for the other cases where the allocated tones are not centered atfrequency 0 can be similarly obtained. As withFIG. 4A, the prescribed time instants t₁, . . . , t_Mare equally-spaced time instants uniformly distributed over the entireOFDM symbol duration40.
• The complex symbols C₁, . . . , C_Mare first mapped in the time domain to time instants t₁, . . . , t_Mrespectively. Next, the mapped symbols C₁, . . . , C_Mare leftwards and rightwards shifted and replicated to an expanded set of prescribed time instants, which is a superset of t₁, . . . , t_Mand 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 asinc interpolation function60. Mathematically, the above steps construct the continuous function CF(t) as
• $\mathrm{CF}\left(t\right)=\sum _{i=1}^M\left\{C_i\sum _{k=-\infty }^{\infty }\sin c\left(t-t_i-\mathrm{kT},\frac{T}{M}\right)\right\}.$
• where sinc(a,b)=sin(πa/b)/(πa/b). Thesinc interpolation function60 can also be precomputed and stored in theinterpolation function memory23. As discussed inFIG. 3A the digital signal samples S22 are the samples of the continuous function CF(t) attime instants 0, T/N, . . . , T(N−1)/N. InFIGS. 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 thesymbol duration40 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 signalsample vector S62 and its associated transmittedOFDM signal64 produced bysymbols16 where the signal is constructed in the frequency domain. As described above, this known technique of mapping thesymbols16 in the frequency domain produces a large signal variation in the transmittedOFDM signal64 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 signalsample vector S66 associated with the transmittedOFDM signal68. As will be appreciated by comparingFIGS. 4D and 4E, mapping the constellation ofcomplex symbols16 in the time domain produces anOFDM signal68 having a significantly reduced peak-to-average ratio.
• FIG. 5A shows a second implementation of theOFDM communication system10, and serves to further generalize the system shown inFIGS. 4A-4C. As part ofOFDM system10, 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 f₁, f₂, . . . f_F. Therefore, f_i(k)=f₀+(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 inFIGS. 4A-4C. For the sake of description, let f₀=0. The construction for the other cases where f₀≠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,2 T/L), and [(L−1)T/L, T/L). As part of this technique, symbols C₁, . . . ,C_M16 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 t₁, . . . , t_Mare equally-spaced time instants uniformly distributed over a fraction (1/L) of thesymbol duration70. 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 toFIG. 4A.
• The procedure for obtaining the digital signal samples S22 described inFIG. 4A can also be applied with respect toFIG. 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 thedigital signal samples22 from the discrete signal of mapped symbols, can be simplified to the same formula as inFIG. 4A with the only change in the definition of f_i(1), f_i(2), . . . , f_i(M)and t₁, . . . , t_M.
• InFIG. 5B, the procedure of obtaining the digital signalsample vector S22 described inFIG. 4B can also be applied to the case of allocated frequency tones that are equally spaced tones. More specifically, adigital processing system100 provides another technique for obtaining the vector of digital signal samples S. ADFT circuit102 receives a discrete signal of complex data symbols C_iand calculates the frequency responses A₁, . . . , A_M, at tones f_i(1), f_i(2), . . . , f_i(M), through an M-point discrete Fourier transform (DFT). The vector [A₁, . . . , A_M]104 output by theDFT circuit102 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 atblock106. More specifically, this process involves putting the k^thsymbol A_kto the i(k)th element of the new vector, for k=1, . . . , M, where f_i(k)is the k^thtone allocated to the transmitter, and inserting zeros in all the remaining elements. Finally, an IDFT circuit108 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 ofFIG. 4C, where symbols C₁, . . . , C_Mare first mapped to t₁, . . . , t_Mrespectively over a fraction (1/L) of thesymbol duration70. The symbol mapping is also performed in the time domain. Next the mapped symbols C₁, . . . , C_Mare 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 asinc interpolation function72. 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
• $\mathrm{CF}\left(t\right)=\sum _{i=1}^M\left\{C_i\sum _{k=-\infty }^{\infty }\sin c\left(t-t_i-\mathrm{kT}\frac{1}{L},\frac{T}{M}\frac{1}{L}\right)\right\}.$
• With continued reference toFIG. 5C, eachsinc interpolation function72 is narrower and therefore decays faster than thesinc interpolation function60 shown inFIG. 4C. Thesinc interpolation function72 can also be precomputed and stored in theinterpolation function memory23 for use by theinterpolation function module21. The digitalsample vector S22 can be obtained in the same technique shown inFIG. 4C. InFIGS. 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 S74 associated with the transmittedOFDM signal76. As will be appreciated by comparingFIGS. 4D and 5D, mapping the constellation ofcomplex symbols16 in the time domain produces anOFDM signal76 having a significantly lower peak-to-average ratio.
• Referring now toFIG. 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 B₁, . . . , B_Mof the constellation are to be transmitted, symbols B₁, . . . , B_Mare mapped to another block of complex symbols C₁, . . . , C_M, where each odd number symbol remains unchanged and each even number symbol is phase rotated by π/4. For example, if symbols B₁, . . . , B_Mbelong to a QPSK constellation {0, π/2, π, π3/2}, the odd number symbols C_kstill belong to the same QPSK constellation, while after being phase rotated the even number symbols C_kbelong to another QPSK constellation {π/4, π3/4, π5/4, π7/4}. Symbols C₁, . . . , C_Mare then used to construct thedigital signal samples22 in the time domain as described above with respect toFIGS. 3A-5C.
• With reference toFIG. 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 S22, which have been generated using the technique ofFIGS. 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 S₁, S₂, . . . , S_Nhave been obtained using the method as described inFIGS. 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′_kis equal to that of digital signal sample S_jwhere 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
• $\frac{\mathrm{<figure-callout\; label="N"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">N</figure-callout>}}{2\mathrm{LM}},$
• where L is discussed inFIG. 5A. In one aspect of this technique, d is chosen to be close to
• $\frac{\mathrm{<figure-callout\; label="N"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">N</figure-callout>}}{2\mathrm{LM}}.$
• For example, d can be the integer closest to
• $\frac{\mathrm{<figure-callout\; label="N"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">N</figure-callout>}}{2\mathrm{LM}},$
• the largest integer not greater than
• $\frac{\mathrm{<figure-callout\; label="N"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">N</figure-callout>}}{2\mathrm{LM}},$
• or the smallest integer not smaller than
• $\frac{\mathrm{<figure-callout\; label="N"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">N</figure-callout>}}{2\mathrm{LM}}.$
• In one example, d is chosen to be the largest integer not greater than
• $\frac{\mathrm{<figure-callout\; label="N"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">N</figure-callout>}}{2\mathrm{LM}}.$
• This example can be easily extended for other choices of d.
• The digital signal sample vector S′ is then passed to the cyclicprefix prepender circuit24, as shown inFIG. 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 thecyclic prefix circuit24 ofFIG. 1.
• Not specifically shown inFIG. 7, when or after the sample vector S′ and the cyclic prefix are outputted to the digital toanalog converter28, the imaginary components are further delayed by an amount of
• $\left(\frac{\mathrm{<figure-callout\; label="N"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">N</figure-callout>}}{2\mathrm{LM}}-d\right)\frac{T}{N},$
• which is a fraction of a sample period T/N.
• As a variation of the technique shown inFIG. 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 S22 are generated separately as described by the following.
• A first series ofdigital signal samples22 are generated using the technique ofFIGS. 3A-5C. The real components of the desireddigital signal samples22 are equal to those of the first series of samples. A second series ofdigital signal samples22 are generated using the technique ofFIGS. 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 inFIGS. 3,4A, and5A, the matrix
• $[\hspace{1em}\begin{array}{ccc}{}^{J2\pi f_{i\left(1\right)}0}& \dots & {}^{J2\pi f_{i\left(M\right)}0}\\ ⋮& & ⋮\\ {}^{J2\pi f_{i\left(1\right)}T\frac{N-1}{N}}& \dots & {}^{J2\pi f_{i\left(M\right)}T\frac{N-1}{N}}\end{array}]$
• is changed to
• $[\hspace{1em}\begin{array}{ccc}{}^{J2\pi f_{i\left(1\right)}\left(0-\frac{\mathrm{<figure-callout\; label="T"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">T</figure-callout>}}{2\mathrm{LM}}\right)}& \dots & {}^{J2\pi f_{i\left(M\right)}\left(0-\frac{\mathrm{<figure-callout\; label="T"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">T</figure-callout>}}{2\mathrm{LM}}\right)}\\ ⋮& & ⋮\\ {}^{J2\pi f_{i\left(1\right)}\left(T\frac{N-1}{N}-\frac{\mathrm{<figure-callout\; label="T"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">T</figure-callout>}}{2\mathrm{LM}}\right)}& \dots & {}^{J2\pi f_{i\left(M\right)}\left(T\frac{N-1}{N}-\frac{\mathrm{<figure-callout\; label="T"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">T</figure-callout>}}{2\mathrm{LM}}\right)}\end{array}].$
• In the block diagram method described with regard toFIG. 4B, an additional operation is required after zero insertion (block56) and before N-point IDFT (block58), where each element k in the expanded length N vector is phase rotated by
• ${}^{-J2\pi f_k\frac{\mathrm{<figure-callout\; label="T"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">T</figure-callout>}}{2\mathrm{LM}}}.$
• Referring toFIGS. 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 asymbol duration40. InFIGS. 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 ofFIGS. 8A-8D requires additional overhead of bandwidth to transmit the same number of complex symbols.
• For example, if thecommunication system10 is allocated M+M_excontiguous frequency tones, f_i(1), f_i(2), . . . , f_i(M+Mex), and M symbols C₁, . . . , C_Mof the constellation are to be transmitted at an OFDM symbol duration, from the comparison ofFIGS. 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 theOFDM communication system10, M_exis a positive number representing the number of excess tones to be used and is assumed to be an even number. Therefore, the allocated tone
• $f_{i\left(k\right)}={\mathrm{<figure-callout\; label="f"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00009.png"\; state="\{\{state\}\}">f</figure-callout>}}_0+\left(k-\frac{M_{\mathrm{ex}}}{2}-1\right)\Delta ,$
• for k=1, . . . , M+M_ex. For purposes of description, let f₀=0. The construction for the other cases where f₀≠0 can be similarly obtained.
• As with the technique described with respect toFIG. 4A, the prescribed time instants are t_k=(k−1)T/M, for k=1, . . . , M, that is, the prescribed time instants t₁, . . . , t_Mare equally-spaced time instants in thesymbol duration40.
• As part of this technique shown inFIG. 8A, P(f) is asmooth windowing function90 in the frequency domain, which is non-zero only over interval [f_i(1), f_i(M+Me)] In addition, P(f)90 also satisfies the Nyquist zero intersymbol interference criterion, i.e.,
• $\sum _{k=-\infty }^{\infty }P\left(f-\mathrm{KM}\Delta \right)=1$
• for any frequency f, where Δ 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 C₁, . . . , C_M,16. The frequency responses A₁, . . . , A_M84 are calculated through an M-point discrete Fourier transform (DFT) of thecomplex symbols16 atblock82. Atblock86, vector [A₁, . . . , A_M]84 is cyclically expanded to a new vector A′ of length N and windowed with awindowing function90 as follows:
• 
  A′k=A_g(k)*P((k−1)Δ+f₁)
• where index g(k)=mod(k−i(1)−M_ex/2, M)+1, for k=1, . . . , N.
• Atblock88, 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 bycyclic prefix circuit24 as described above with regard toFIG. 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 atfrequency 0. InFIG. 8C (as withFIG. 4C), symbols C₁, . . . , C_Mare first mapped to equally-spaced time instants in thesymbol duration40, and are then leftwards and rightwards shifted and replicated from −∞ to +∞. What is different fromFIG. 4C is that adifferent interpolation function92, which is determined by thewindowing function90, is used to generate the continuous function,
• 
  CF(t)=E_i=1^MC_iΣ_k=−∞^∞p(t−t_i−kT)
• where p(t)92 is the time domain response of P(f)90. As withFIG. 4C, the digital signal samples are obtained by letting t=0, T/N, . . . , T(N−1)/N.
• In one exemplary aspect of this technique, if a raised cosine windowing function is used, i.e.,
• $P\left(f\right)=\{\begin{array}{cc}\frac{T}{M}& \mathrm{if}f<\left(1-\beta \right)\frac{\mathrm{<figure-callout\; label="M"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">M</figure-callout>}}{2\mathrm{<figure-callout\; label="T\; T"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">T</figure-callout>}}& \\ \frac{T}{}\frac{}{2M}\left\{1+\cos \left[\begin{array}{c}\frac{\pi T}{\beta M}\\ \left(f-\frac{\left(1-\beta \right)\mathrm{<figure-callout\; label="M"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">M</figure-callout>}}{2T}\right)\end{array}\right]\right\}& \mathrm{if}\left(1-\beta \right)\frac{\mathrm{<figure-callout\; label="M"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">M</figure-callout>}}{2T}\le f\le \left(1+\beta \right)\le \frac{\mathrm{<figure-callout\; label="M"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">M</figure-callout>}}{2\mathrm{<figure-callout\; label="T"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00009.png"\; state="\{\{state\}\}">T</figure-callout>}}\\ 0& \mathrm{if}f>\left(1+\beta \right)\frac{\mathrm{<figure-callout\; label="M"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">M</figure-callout>}}{2T}\end{array}$
• where β=(M_ex+2)/M represents the percentage of excess tone overhead, then, the interpolation function p(t)92 is given by
• $p\left(t\right)=\frac{\sin \left(\pi \mathrm{tM}/T\right)}{\pi \mathrm{tM}/T}\frac{\cos \left(\pi \beta \mathrm{tM}/T\right)}{1-4{\mathrm{<figure-callout\; label="\beta "\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">\beta </figure-callout>}}^2{\mathrm{<figure-callout\; label="t"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">t</figure-callout>}}^2{\mathrm{<figure-callout\; label="M"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">M</figure-callout>}}^2/{\mathrm{<figure-callout\; label="T"\; filenames="US20090262641A1-20091022-D00004.png,US20090262641A1-20091022-D00008.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}.$
• 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 signalsample vector S94 associated with the transmitted OFDM signal96. As will be appreciated, mapping theconstellation symbols16 in the time domain produces an OFDM signal96 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.

Claims (20)

1. A digital processing system utilized in a transmitter that generates a transmission signal, the system comprising:

a first circuit that receives a discrete signal of complex data symbols, calculates a frequency response of the discrete signal, and outputs a first vector of length M;

a second circuit that receives the vector of length M and generates a second vector of length N, wherein the second vector of length N includes elements of the first vector at positions corresponding to tones allocated to the transmitter; and

a third circuit that receives the second vector and generates a digital time domain vector therefrom for transmission by the transmitter.

2. The digital processing system ofclaim 1, wherein the transmitter is configured to operate in a multiple access system, and wherein the tones allocated to the transmitter are a non-overlapping subset of frequency tones with respect to other transmitters.

3. The digital processing system ofclaim 1, wherein the second circuit is configured to insert zeros in remaining elements of the second vector not corresponding to the tones allocated to the transmitter.

4. The digital processing system ofclaim 2, wherein the second circuit is configured to insert zeros in remaining elements of the second vector not corresponding to the tones allocated to the transmitter.

5. The digital processing system of1, wherein the vector of length N is larger than the vector of length M.

6. The digital processing system of1, wherein the digital time domain vector comprises equally spaced tones.

7. The digital processing system of5, wherein the digital time domain vector comprises equally spaced tones.

8. The digital processing system of4, wherein the vector of length N is larger than the vector of length M.

9. The digital processing system of4, wherein the digital time domain vector comprises equally spaced tones.

10. The digital processing system of9, wherein the digital time domain vector comprises equally spaced tones.

11. An apparatus that generates a transmission signal, the apparatus comprising:

means for receiving a discrete signal of complex data symbols, calculates a frequency response of the discrete signal, and outputs a first vector of length M;

means for receiving the vector of length M and generates a second vector of length N, wherein the second vector of length N includes elements of the first vector at positions corresponding to tones allocated to a transmitter; and

means for generating a digital time domain vector of the second vector for transmission by the transmitter.

12. The apparatus ofclaim 11, wherein the transmitter is configured to operate in a multiple access system, and wherein the tones allocated to the transmitter are a non-overlapping subset of frequency tones with respect to other transmitters.

13. The apparatus ofclaim 11, wherein the second circuit is configured to insert zeros in remaining elements of the second vector not corresponding to the tones allocated to the transmitter.

14. The apparatus ofclaim 12, wherein the second circuit is configured to insert zeros in remaining elements of the second vector not corresponding to the tones allocated to the transmitter.

15. The apparatus of11, wherein the vector of length N is larger than the vector of length M.

16. The apparatus of11, wherein the digital time domain vector comprises equally spaced tones.

17. The apparatus of15, wherein the digital time domain vector comprises equally spaced tones.

18. The apparatus of14, wherein the vector of length N is larger than the vector of length M.

19. The apparatus of14, wherein the digital time domain vector comprises equally spaced tones.

20. The apparatus of19, wherein the digital time domain vector comprises equally spaced tones.

Images