Abstract:
The present invention discloses an optimal hardware implementation of the FFT/IFFT operation that minimizes the number of clock cycles required to compute the FFT/IFFT while at the same time minimizing the number of complex multipliers needed. For performing an N-point FFT/IFFT operation in N clock cycles, the optimal hardware implementation consists of several modules. An input module receives a plurality of inputs in parallel and combines the inputs after applying a multiplication factor to each of the inputs. At least one multiplicand generator is used to provide multiplicands to the system. At least two complex multiplier modules for performing complex multiplications are required with at least one of the complex multiplier modules receiving an output from the input module. Each of the complex multiplier modules receives multiplicands from the at least one multiplicand generator. Furthermore, at least one of the complex multiplier modules receives an output of another complex multiplier module. A map module is provided for receiving outputs of the at least two complex multiplier modules, the map module selecting and applying a multiplication factor to each of the outputs received to generate multiple outputs. Finally, an accumulation module receives and performs an accumulation task on each of the multiple outputs of the map module thereby generating a corresponding number of multiple outputs. In a preferred embodiment, the N-point FFT/IFFT operation is performed in N clock cycles using  
       (       N   32     +   1     )       
complex multipliers. In a specific implementation, a system comprising 3 complex multipliers is used to compute a 64-point FFT/IFFT operation in 64 clock cycles. Advantageously, the total number of clock cycles required to complete the FFT/IFFT operation is minimized while at the same time minimizing the number of complex multipliers needed.

Description:
FIELD OF INVENTION  
       [0001]     This invention relates to OFDM systems and, more particularly, to an optimized hardware implementation of the FFT/IFFT module which minimizes the number of clock cycles for computing the FFT/IFFT of a signal.  
       BACKGROUND OF THE INVENTION  
       [0002]     Wireless LAN (WLAN) technology is one of the most widely deployed and most rapidly expanding areas of radio communications. As demand for mobile data grows, networks will have to offer more bandwidth to support both a larger numbers of users as well as higher data transfer rates for individual users. Satisfying these demands involves the deployment of newer air interface technologies such as 3G cellular and the IEEE 802.11a standard.  
         [0003]     The IEEE 802.11a standard is based on a multicarrier modulation scheme called orthogonal frequency domain multiplexing (OFDM) in the 5 GHz band. In multicarrier modulation, data signals (bits) are modulated onto a number of carriers rather than on a single carrier as in traditional AM or FM systems. The result is an optimum usage of bandwidth. The basic principle of OFDM is to split a high rate data stream into a number of lower rate streams, which are then transmitted simultaneously over a number of sub-carriers (overlapping, orthogonal narrow band signals). The frequencies used in OFDM are orthogonal. Neighboring frequencies with overlapping spectrum can therefore be used. This results in a more efficient usage of bandwidth. OFDM is therefore able to provide higher data rates for the same bandwidth. It also offers several advantages over single carrier systems such as better multi-path effect immunity, simpler channel equalization and relaxed timing acquisition constraints. Accordingly, OFDM has become the modulation method of choice for many new systems.  
         [0004]     Each sub-carrier in OFDM has a fixed phase and amplitude for a certain time duration, during which a small portion of the information is carried. This unit of data is called a symbol and the time period during which the symbol is available is called the symbol duration. After that time period, the modulation is changed and the next symbol carries the next portion of information. A set of orthogonal sub-carriers together forms an OFDM symbol. To avoid inter symbol interference (ISI) due to multi-path propagation, successive OFDM symbols are separated by a guard band. This makes the OFDM system resistant to multi-path effects. Although OFDM has been in existence for a long time, recent developments in DSP and VLSI technologies have made it a feasible option. As a result, OFDM is fast gaining popularity in broadband standards and high-speed wireless LAN standards such as the IEEE 802.11a.  
         [0005]     In practice, the most efficient way to generate the sum of a large number of sub-carriers is by using the Inverse Fast Fourier Transform (IFFT). At the receiver side, a fast and efficient implementation of the well known discrete fourier transform (DFT) function called the Fast Fourier Transform (FFT) can be used to demodulate all the sub-carriers. All sub-carriers differ by an integer number of cycles within the FFT integration time, which ensures the orthogonality between different sub-carriers.  
         [0006]     Several choices are available for implementing an OFDM modem: digital signal processing (DSP) based implementation, DSP-based implementation with hardware accelerators or a complete ASIC implementation.  
         [0007]     High performance digital signal processors (DSPs) are widely available in the market today. The computation intensive and time critical functions that were traditionally implemented in hardware are nowadays being implemented in software running on these processors. However, a DSP-based implementation of an OFDM modem has the disadvantage of not being very optimum in terms of chip area occupied and power consumption.  
         [0008]     To overcome limitations incurred with a DSP-based implementation while still retaining the flexibility of a software implementation, some blocks of an OFDM transceiver can be implemented in hardware. Alternatively, the entire functionality may be implemented in hardware. Advantages of this ASIC-based approach include lower gate count and hence, lower cost and lower power consumption.  
         [0009]     When general purpose DSP chips do not meet the required performance parameters of an application, an ASIC (application specific integrated circuit) DSP may be developed. When a particular algorithm has to be implemented, for example the FFT/IFFT algorithm, an application specific DSP chip is generated with an architectural structure dependent upon the algorithm&#39;s computational structure. Alternatively, the algorithm can be restructured to better fit an available target architecture (for example, that of a parallel computational arrangement). Most current implementations of the FFT/IFFT engine for an OFDM modem are done using a DSP chip with software and concentrate on minimizing calls to the multiplier block.  
         [0010]     However, it would be advantageous to implement an FFT/IFFT engine entirely in ASIC technology so that each of the functional blocks of the FFT/IFFT engine be mapped onto dedicated, parallel hardware resources thereby avoiding the difficult programming and optimization challenges of scheduling time-critical operations through a single DSP core. An optimized hardware implementation which minimizes the total run time while at the same time minimizing the number of complex multiplier is, therefore, sought.  
       SUMMARY OF THE INVENTION  
       [0011]     The present invention pertains to symbolic or mathematical manipulation of the FFT/IFFT formula in order to derive an optimal hardware implementation. The invention involves restructuring the FFT/IFFT formula to minimize the number of clock cycles required to compute the FFT/IFFT while at the same time minimizing the number of complex multipliers required.  
         [0012]     According to one embodiment of the present invention, a system for performing an N-point FFT/IFFT operation is provided comprising an input module for receiving a plurality of inputs in parallel and for combining said inputs after applying a multiplication factor to each of said inputs, at least one multiplicand generator for providing multiplicands to said system, at least two multiplier modules for performing complex multiplications, at least one of said multiplier modules receiving an output of said input module, each of said multiplier modules receiving multiplicands from said at least one multiplicand generator, at least one of said multiplier modules receiving an output of another multiplier module, a map module for receiving outputs of all of said at least two multiplier modules, said map module selecting and applying a multiplication factor to each of said outputs of said at least two multiplier modules, said map module generating multiple outputs and an accumulation module for receiving and accumulating said multiple outputs of said map module.  
         [0013]     In accordance with an aspect of the present invention, an N-point FFT/IFFT operation, with N being the number of input samples, may be performed in N clock cycles using  
       (       N   16     +   1     )       
 
 complex multipliers. In accordance with a preferred aspect of the present invention, an N-point FFT/IFFT operation is performed in N clock cycles using  
       (       N   32     +   1     )       
 
 complex multipliers. Accordingly, in a preferred implementation of the present invention, an optimized hardware configuration comprising 3 complex multipliers is used to compute a 64-point FFT/IFFT operation in 64 clock cycles. Advantageously, the total number of clock cycles required to complete the FFT/IFFT operation is minimized while at the same time minimizing the number of complex multipliers needed. 
 
         [0016]     The advantage of implementing an FFT/IFFT engine with ASIC technology is that each of the functional blocks of the FFT/IFFT engine be mapped onto dedicated, parallel hardware resources thereby avoiding the difficult programming and optimization challenges of scheduling time-critical operations through a single DSP core.  
         [0017]     Other aspects and features of the present invention will become apparent to those ordinarily skilled in the art upon review of the following description of specific embodiments of the invention in conjunction with the accompanying drawings. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0018]     A better understanding of the invention will be obtained by considering the detailed description below, with reference to the following drawings in which:  
         [0019]      FIG. 1  depicts a brute force hardware implementation for the FFT/IFFT operation;  
         [0020]      FIG. 2  depicts a partially optimized hardware implementation for the FFT/IFFT operation;  
         [0021]      FIG. 3  depicts a fully optimized hardware implementation for the FFT/IFFT operation according to the present invention;  
         [0022]      FIG. 4  depicts an example of the logic flow undertaken by the MAP module of  FIG. 3  in accordance with one aspect of the present invention; and  
         [0023]      FIG. 5  depicts the general operation of the accumulation module of  FIG. 3  in accordance with one aspect of the present invention.  
     
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0024]     The basic principle of OFDM is to split a high rate data stream into a number of lower rate streams each of which are transmitted simultaneously over a number of sub-carriers. In the IEEE 802.11a standard OFDM modulation scheme, the binary serial signal is divided into groups (symbols) of one, two, four or six bits, depending on the data rate chosen, and the symbols are converted into complex numbers representing applicable constellation points. Each symbol, having a duration of 4 microseconds, is assigned to a particular sub-carrier. An Inverse Fast Fourier Transform (IFFT) combines the sub-carriers to form a composite time-domain signal for transmission. The IEEE 802.11a standard system uses 52 sub-carriers that are modulated using binary or quadrature phase shift keying (BPSK/QPSK), 16 Quadrature Amplitude Modulation (QAM) or 64 QAM. On the receiver side, the Fast Fourier Transform (FFT) can be used to demodulate all sub-carriers. All sub-carriers differ by an integer number of cycles within the FFT integration time, and this ensures the orthogonality between the different sub-carriers.  
         [0025]     The heart of an OFDM baseband processor is, therefore, the FFT/IFFT engine. It is well known that the FFT operation is designed to perform complex multiplications and additions, even though the input data may be real valued. The reason for this situation is that the phase factors are complex and, hence, after the first stage of the operation all variables are complex-valued. Thus, in terms of a hardware implementation, the FFT operation can be implemented using summation modules and multiplication modules (multipliers).  
         [0026]     Multiplication modules are the most widely used circuit in an OFDM modem. However, multipliers are costly resources both in terms of chip area and power consumption. A greater number of multipliers will require greater chip area resulting in bulkier devices not suitable for mobile applications. However, the total time it takes for an FFT/IFFT engine to operate on a given set of input samples (i.e. the total run time) is also critical as the less number of clock cycles it takes, the less the power consumption. With regard to the FFT/IFFT engine, it would therefore be desirable to reduce the number of multiplier modules required while at the same time minimizing the number of clock cycles required to compute the FFT/IFFT.  
         [0027]     The present invention pertains to symbolic or mathematical manipulation of the FFT formula in order to derive an optimal hardware implementation. The invention involves restructuring the FFT formula to minimize the number of clock cycles required while at the same time minimizing the number of complex multiplier modules. Since both the FFT and IFFT operations involve the same type of computations, only a discussion on the IFFT is presented. Those skilled in the art will appreciate that the formulation presented applies equally to an efficient implementation of the FFT operation.  
         [0028]     The computational problem for the IFFT is to compute the sequence Y(n) of N complex-valued numbers given another sequence of data X(k) according to the formula  
               Y   ⁡     (   n   )       =         1   N     ⁢       ∑     k   =   0       N   -   1       ⁢           ⁢       X   ⁡     (   k   )       ⁢     ⅇ       j2   ⁢           ⁢   kn   ⁢           ⁢   π     N       ⁢           ⁢   0         ≤   n   ≤     N   -   1               equation   ⁢           ⁢     (   1   )               
 
 In the above formulation, one can see that for each sample n, direct computation of Y(n) involves N complex multiplications (4N real multiplications). Consequently, to compute the IFFT of all N samples, the IFFT requires N 2  complex multiplications. 
 
         [0030]      FIG. 1  depicts one possible hardware implementation of equation (1). In  FIG. 1 , N complex-valued numbers defining the input sequence X(k) are fed into a multiplexer (MUX)  120 . The MUX  120  selects one of the N complex-valued inputs and delivers it to a complex multiplier  140 . The complex multiplier  140  is adapted to access to a look-up table (LUT)  150  which contains the values  
       ⅇ       j2   ⁢           ⁢   kn   ⁢           ⁢   π     N           
 for some value N, 0≦k≦N−1 and 0≦n≦N− 1 . The output of the complex multiplier  140  is fed to an accumulation module  180  which may comprise a register  160 . Using a single complex multiplier  140  as in  FIG. 1 , it is readily seen that the computation of each output sample requires N complex multiplications and, hence, the use of the complex multiplier  140  N times. In other words, to compute each output time sample, the results of N complex multiplications are added and accumulated in the register. This process will have to repeat itself for each of the N input samples to derive the N output time samples. Since N output samples in total need to be computed, this results in a total runtime of N 2  clock cycles (assuming one complex multiplication per clock cycle) to compute the IFFT for the entire input sequence Y(n). 
 
         [0032]     However, computation of the IFFT using the brute force hardware implementation of  FIG. 1  is inefficient primarily because it does not exploit the symmetry and periodicity properties of the phase factor,  
         ⅇ   jθ     ,       
 
 in equation (1). The present invention exploits these properties to minimize the total run time (number of clock cycles) for computing the IFFT/FFT of a given set of sample data. 
 
         [0034]     Those skilled in the art will appreciate that Equation (1) may be rewritten as the expansion equation (1a) or  
               Y   ⁡     (   n   )       =         ∑     k   =   0     3     ⁢           ⁢       X   ⁡     (       N   4     ⁢   k     )       ⁢     ⅇ       j   ⁢           ⁢   kn   ⁢           ⁢   π     2           +       ∑     k   =   0     3     ⁢           ⁢       X   ⁡     (         N   4     ⁢   k     +   1     )       ⁢     ⅇ       j   ⁡     (       k   2     +     2   N       )       ⁢   n   ⁢           ⁢   π           +   …   +       ∑     k   =   0     3     ⁢           ⁢       X   ⁡     (         N   4     ⁢   k     +     N   4     -   1     )       ⁢     ⅇ       j   ⁡     (       k   2     +       2   N     ⁢     (       N   4     -   1     )         )       ⁢   n   ⁢           ⁢   π                     equation   ⁢           ⁢     (     1   ⁢   a     )                   Y   ⁡     (   n   )       =         ⅇ         j2   ⁢           ⁢   n   ⁢           ⁢   π     N     ⁢     (   0   )         ⁢       ∑     k   =   0     3     ⁢           ⁢       X   ⁡     (         N   4     ⁢   k     +   0     )       ⁢     ⅇ       j   ⁢           ⁢   kn   ⁢           ⁢   π     2             +       ⅇ         j2   ⁢           ⁢   n   ⁢           ⁢   π     N     ⁢     (   1   )         ⁢       ∑     k   =   0     3     ⁢           ⁢       X   ⁡     (         N   4     ⁢   k     +   1     )       ⁢     ⅇ       j   ⁢           ⁢   kn   ⁢           ⁢   π     2             +   …   +       ⅇ         j2   ⁢           ⁢   n   ⁢           ⁢   π     N     ⁢     (       N   4     -   1     )         ⁢       ∑     k   =   0     3     ⁢           ⁢       X   ⁡     (         N   4     ⁢   k     +     N   4     -   1     )       ⁢     ⅇ       j   ⁢           ⁢   kn   ⁢           ⁢   π     2                       equation   ⁢           ⁢     (     1   ⁢   b     )               
 
 Simplifying the above yields,  
               Y   ⁡     (   n   )       =       ∑     l   =   0         N   4     -   1       ⁢           ⁢       ⅇ     j2lnπ   N       ⁢       ∑     k   =   0     3     ⁢           ⁢       X   ⁡     (         N   4     ⁢   k     +   l     )       ⁢     ⅇ       j   ⁢           ⁢   kn   ⁢           ⁢   π     2                       equation   ⁢           ⁢     (   2   )                   if   ⁢           ⁢   we   ⁢           ⁢   let   ⁢           ⁢       P   l     ⁡     (   n   )         =     ⅇ     j2lnπ   N               equation   ⁢           ⁢     (   3   )                   and   ⁢           ⁢       G   l     ⁡     (   n   )         =       ∑     k   =   0     3     ⁢           ⁢       X   ⁡     (         N   4     ⁢   k     +   l     )       ⁢     ⅇ       j   ⁢           ⁢   kn   ⁢           ⁢   π     2                   equation   ⁢             ⁢             (   4   )             
 
 the set of output samples may be rewritten as  
               Y   ⁡     (   n   )       =       ∑     l   =   0         N   4     -   1       ⁢           ⁢         P   l     ⁡     (   n   )       ⁢       G   l     ⁡     (   n   )                   equation   ⁢           ⁢     (   5   )               
 
 Letting 
 
 R   l ( n )= P   l ( n ) G   l ( n )  equation (6) 
 
 equation (5) may be rewritten  
               Y   ⁡     (   n   )       =       ∑     l   =   0         N   4     -   1       ⁢           ⁢       R   l     ⁡     (   n   )                 equation   ⁢           ⁢     (   7   )               
 
         [0039]      FIG. 2  depicts a hardware implementation for equation (5) above. Incoming complex numbers  216  arrive in groups of four at input ports (K 0 , K 1 , K 2  and K 3 ) of a G l (n) module  220 . Assuming N input samples, the four incoming complex samples of each group will have indices  
       N   4         
 apart. For example, the first group of incoming samples would be X( 0 ), X( 0 +N/4), X( 0 +2N/4) and X( 0 +3N/4). 
 
         [0041]     Similarly, the second group of incoming samples would be X( 1 ), X( 1 +N/4), X( 1 +2N/4) and X( 1 +3N/4). The output of the G l (n) module  220  is delivered to a complex multiplier  240  which is adapted to access a look-up table (LUT)  230  containing complex-valued constants P l (n) as defined by equation (3). The output R l (n) of the complex multiplier  240  is the product G l (n) with P l (n) as defined by equation (6). This product is sent to the accumulation module  250  which may comprise a register  260  as shown.  
         [0042]     Examining equation (4), it may be shown that the function of the G l (n) module  220  is to simply take the four incoming complex numbers  216  (with indices N/4 apart), multiply each one by a constant  
       ⅇ       j   ⁢           ⁢   k   ⁢           ⁢   n   ⁢           ⁢   π     2         
 
 and add them all. It may be shown that the value of the constant  
       ⅇ       j   ⁢           ⁢   k   ⁢           ⁢   n   ⁢           ⁢   π     2         
 
 in equation (4) reduces to +1, −1, +j or −j depending on the values of k and sample number n. Therefore, no complex multiplications are conducted in this module. 
 
         [0045]     Considering the implementation in  FIG. 2  and keeping equation (5) in mind, those skilled in the art will appreciate that  
       N   4       
 
 complex multiplications of P l (n)×G l (n) are required for the computation of each output sample. The results of these multiplications may then be added together in the accumulation module  250  to obtain each output sample. Therefore, a single output is generated every  
       N   4       
 
 clock cycles. Since N outputs need to be computed, the total run time required to compute the FFT/IFFT for a set of N input samples using the implementation in  FIG. 2  with one complex multiplier has been reduced from N 2  clock cycles to  
         N   2     4       
 
 clock cycles. Although the reduction in total run time from N 2  clock cycles in  FIG. 1  to  
         N   2     4       
 
 clock cycles in  FIG. 2  is an improvement, further optimization may be made by exploiting the periodicity of the phase factor, e jθ , in functions P l (n) and G l (n). 
 
         [0050]     For example, substituting (n+4) for n in equation (4) for G l (n) yields,  
                 G   l     ⁡     (     n   +   4     )       =         ∑     k   =   0     3     ⁢           ⁢       X   ⁡     (         N   4     ⁢   k     +   l     )       ⁢     ⅇ     j   ⁢           ⁢     k   ⁡     (     n   +   4     )       ⁢     π   2             =       ⅇ     j   ⁢           ⁢   k   ⁢           ⁢   2   ⁢   π       ⁢       ∑     k   =   0     3     ⁢           ⁢       (         N   4     ⁢   k     +   l     )     ⁢   ⅇ   ⁢       j   ⁢           ⁢   kn   ⁢           ⁢   π     2                     equation   ⁢           ⁢     (   8   )               
 
 or 
 
 G   l ( n+ 4)= e   jk2π   G   l ( n )= G   l ( n )  equation (9) 
 
 Similarly, substituting (n+4) for n in equation (3) for P l (n) yields,  
                 P   l     ⁡     (     n   +   4     )       =       ⅇ       j   ⁢           ⁢   2   ⁢     l   ⁡     (     n   +   4     )       ⁢   π     N       =       ⅇ         j   ⁢           ⁢   2   ⁢   l   ⁢           ⁢   n   ⁢           ⁢   π     +     j   ⁢           ⁢   8   ⁢   l   ⁢           ⁢   π       N       =       ⅇ       j   ⁢           ⁢   2   ⁢   l   ⁢           ⁢   n   ⁢           ⁢   π     N       ·     ⅇ       j   ⁢           ⁢   8   ⁢           ⁢   l   ⁢           ⁢   π     N                     equation   ⁢           ⁢     (   10   )               or                             P   l     ⁡     (     n   +   4     )       =       ⅇ       j   ⁢           ⁢   8   ⁢           ⁢   l   ⁢           ⁢   π     N       ⁢       P   l     ⁡     (   n   )                 equation   ⁢           ⁢     (   11   )               
 
 Substituting equation (11) and equation (9) into equation (6), it may be shown that  
                 R   l     ⁡     (     n   +   4     )       =           P   l     ⁡     (     n   +   4     )       ⁢       G   l     ⁡     (     n   +   4     )         =         ⅇ       j   ⁢           ⁢   8   ⁢           ⁢   l   ⁢           ⁢   π     N       ⁢       P   l     ⁡     (   n   )       ⁢       G   l     ⁡     (   n   )         =       ⅇ       j   ⁢           ⁢   8   ⁢           ⁢   l   ⁢           ⁢   π     N       ⁢       R   l     ⁡     (   n   )                     equation   ⁢           ⁢     (   12   )               
 
         [0054]     The relationship defined by equation (12) states that, for a given value of l, the function R l (n) for any given output sample is a phase rotation of the computed functions&#39; value four output samples before. Previously, each output sample required  
       N   4       
 
 computations of R l (n) which were then summed to arrive at a given output sample. With R l (n) displaying the recursive relationship defined by equation (12), each output sample still requires  
       N   4       
 
 computations of R l (n). However, once the first four output samples i.e. Y( 0 ), Y( 1 ), Y( 2 ), ( 3 ) are computed in N clock cycles (i.e. N computations), the values of R l (n) required to compute all other output samples are simply phase rotations of the previously calculated R l (n) values. In other words, the number of clock cycles required for the entire FFT/IFFT operation is reduced to N. 
 
         [0057]     In terms of simplifying a hardware implementation, a variable β can be defined with β being a multiple of 4. Then, the following relation can be shown to hold  
                 R   l     ⁡     (     n   +   β     )       =       ⅇ       j   ⁢           ⁢   2   ⁢   β   ⁢           ⁢   l   ⁢           ⁢   π     N       ⁢       R   l     ⁡     (   n   )                 equation   ⁢           ⁢     (   13   )               
 
 Accordingly, equation (7) may be rewritten as  
               Y   ⁡     (     n   +   β     )       =         1   N     ⁢       ∑     l   =   0         N   4     -   1       ⁢       R   l     ⁡     (     n   +   β     )           =       1   N     ⁢       ∑     l   =   0         N   4     -   1       ⁢         P   l     ⁡     (     n   +   β     )       ⁢       G   l     ⁡     (     n   +   β     )       ⁢           ⁢   or                   equation   ⁢           ⁢     (   14   )                   Y   ⁡     (     n   +   β     )       =       1   N     ⁢       ∑     l   =   0         N   4     -   1       ⁢         R   l     ⁡     (   n   )       ⁢     ⅇ       j   ⁢           ⁢   2   ⁢   β   ⁢           ⁢   l   ⁢           ⁢   π     N                     equation   ⁢           ⁢     (   15   )               
 
         [0059]     Those skilled in the art will appreciate that only the first  
       N   16       
 
 products in equation (15) require complex multiplications to be performed. For all other values of l, the product in equation (15) can be found by a trivial multiplication of one of the first  
       N   16       
 
 products. Accordingly,  
       (       N   16     +   1     )       
 
 complex multipliers are now required to perform the FFT/IFFT operation in N clock cycles. Although the number of clock cycles to perform the FFT/IFFT operation has been reduced by an order of magnitude from N 2  to N, this has been at the expense of adding  
       N   16       
 
 more complex multipliers. However, the number of complex multipliers required may be further reduced using a very useful property as described below. 
 
         [0064]     In the general case, let us define a first complex number A=x+jy with real part x and imaginary part y and a second complex number B=y+ix where B is the reflection of A about the 45 degree line in the unit circle. For complex numbers A and B each multiplied by a third complex number Z=R+jM, the following products are obtained: 
 
 A×Z =( xR−My )+ j ( Ry+Mx )  equation (16) 
 
 B×Z =( Ry−Mx )+ j ( My+Rx )  equation (17) 
 
 Examining equations (16) and (17) it is observed that all inner products (real multiplications) for both complex multiplications may be obtained by only carrying out one of the original complex multiplications. In other words, by computing A×Z, no new multiplications are required to compute B×Z. Computing B×Z is simply a matter of rearranging the way the different inner products from A×Z are added or subtracted. This useful property, called Image Multiplication (since B is a mirror image of A about the 45-degree line in the unit circle), may be exploited to halve the number of complex multipliers determined previously. 
 
         [0066]     Specifically, to compute all possible products  
           R   l     ⁡     (   n   )       ·     ⅇ       j   ⁢           ⁢   2   ⁢   β   ⁢           ⁢   l   ⁢           ⁢   π     N           
 
 in equation (15), only  
       N   32       
 
 multipliers are required. Since one additional multiplier is required to compute R l (n) itself, the total number of complex multipliers required is  
         (       N   32     +   1     )     .       
 
 Therefore, in accordance with an aspect of the present invention, the total number of clock cycles required for computing an N-point FFT/IFFT can reduced by an order of magnitude from N 2  to N by using only  
       (       N   32     +   1     )       
 
 complex multipliers. 
 
         [0071]     According to an embodiment of the present invention, it is assumed that a 64-point FFT/IFFT operation is required i.e. N=64. In this case, equation (15) reduces to  
               Y   ⁡     (     n   +   β     )       =       1   64     ⁢       ∑     l   =   0     15     ⁢         R   l     ⁡     (   n   )       ⁢     ⅇ       j   ⁢           ⁢   β   ⁢           ⁢   l   ⁢           ⁢   π     32                     equation   ⁢           ⁢     (   18   )               
 
 where β is a multiple of 4 and 0&lt;β≦60. Examining equation (18) above and noting that β is a multiple of 4, it is clear that in order to compute the IFFT/FFT, the multiplication of R l (n) with only three complex numbers  
         ⅇ       j   ⁢           ⁢   4   ⁢           ⁢   π     32       ,       ⅇ       j8   ⁢           ⁢   π     32       ⁢           ⁢   and   ⁢           ⁢     ⅇ       j   ⁢           ⁢   12   ⁢           ⁢   π     32             
 
 is required. All other multiplications simply entail multiplying one of these products by 1, −1, j or −j. 
 
         [0074]      FIG. 3  depicts an optimized hardware implementation  300  of the 64-point IFFT/FFT operation defined by equation (18) in accordance with one aspect of the present invention. A first G module  310  having four input ports K 0 , K 1 , K 2 , and K 3  receives four complex-valued input samples and is adapted to access a first look up table (LUT)  316 . An output G l (n) of the G module  310  is delivered to a first complex multiplier module GX  320  which is adapted to access a second look-up table (LUT)  326 . The output R l (n) of the first complex multiplier module GX  320  is delivered to a MAP module  360 . An output R l (n) of the first complex multiplier module GX  320  is further routed to a second complex multiplier module GX 1   330  and to a third complex multiplier module GX 2   340 . The second multiplier module GX 1   330  is adapted to access a first storage unit  336  containing a predefined complex-valued constant and delivers its output to the MAP module  360 . Similarly, the third multiplier module GX 2   340  is adapted to access a second storage unit  346  containing a predefined complex-valued constant and delivers two outputs to the MAP module  360 . The MAP module  360  generates a set of sixteen outputs  370  which are subsequently delivered to an accumulation module  380 . The accumulation module  380  generates a set of sixteen outputs  390  corresponding to sixteen output time/frequency samples. In the implementation of  FIG. 3 , therefore, sixteen output samples are generated at any given time from sixty-four input samples.  
         [0075]     The G module  310  is the first module to receive incoming complex numbers. As in  FIG. 2 , the G module  310  has four input ports (K 0 , K 1 , K 2  and K 3 ) and simply takes four incoming complex-valued samples  302  with indices being sixteen (N/4) apart, multiplies each one by a constant (+j, −j, +1 or −1) and adds them to form the output G l (n).  
         [0076]     Specifically, four new complex numbers get latched into this module during each clock cycle. In order to load all 64 input samples for a 64-point IFFT/FFT computation, sixteen clock cycles are required. Since sixteen output samples  390  are generated at the output of the entire IFFT/FFT block, the entire process is repeated 4 times to result in output time samples. A counter state n can thus be defined where n=0, 1, 2, 3 corresponding to the computation of each set of sixteen output samples.  
         [0077]     Once four complex numbers are loaded, the G module  310  accesses the look up table (LUT)  316  to obtain appropriate multiplication factors for each complex number. The multiplication factor for each complex number may take on one of four possible values: +1, −1, +j or −j. For each complex number, the multiplication is performed on both real and imaginary parts. The results are then added to generate the output G l (n) which is pushed to the output port. This process must be repeated sixteen times (l ranging from 0 to 15) in order to generate the sixteen G l (n) values necessary for each output sample.  
         [0078]     In one implementation, the look-up table (LUT)  316  accessed by the G module  310  can have sixteen entrees. Two stimulus variables, namely the port number (0, 1, 2 or 3) and counter state n may then be used to define the value of the multiplication factor. A single local controller (not shown) may be used to select one set of multiplication factors from the LUT  316  and subsequently push them to the G module  310 . Since the multiplication factors selected from the LUT  316  are determined by the two stimulus variables, the LUT  316  may take the form of a truth table.  
         [0079]     The output G l (n) of the G module  310  is delivered to the GX Module  320 . The GX module  320  is a complex multiplier used to generate R l (n) from G l (n). The output of this module may be described by the complex product  
         [       cos   ⁡     (       l   ⁢           ⁢   n   ⁢           ⁢   π     32     )       +     j   ⁢           ⁢   sin   ⁢           ⁢     (       l   ⁢           ⁢   n   ⁢           ⁢   π     32     )         ]     ×   output   ⁢           ⁢   from   ⁢           ⁢   G   ⁢           ⁢   module       
 
 where two global input variables are defined as before with n ranging from 0 to 3 and l ranging from 0 to 15. The GX module  320  performs a complex number multiplication of its received input G l (n) by a complex-valued constant,  
             P   l     ⁡     (   n   )       =     ⅇ     jlnπ   32         ,       
 
 where values for P l (n) are stored in the corresponding look-up table (LUT)  326 . 
 
         [0082]     In a specific implementation, the LUT  326  may comprise eight predefined values i.e.  
         ⅇ       j   ⁢           ⁢   π     32       ,       ⅇ       j2   ⁢           ⁢   π     32       ⁢           ⁢   …   ⁢           ⁢     ⅇ       j   ⁢           ⁢   8   ⁢           ⁢   π     32             
 
 hard coded into the LUT block. Generating these eight constants is sufficient since all other constants can be easily derived based on these constants and the application of an appropriate multiplication factor. Based on the value of the product l×n, one of the eight values is selected. The next step is to determine the multiplication factor which can be one of four possible numbers: +1, −1, +j, or −j. In this manner, any constant  
           P   l     ⁡     (   n   )       =     ⅇ       j   ⁢           ⁢   l   ⁢           ⁢   n   ⁢           ⁢   π     32           
 
 may be derived by performing a simple multiplication of a selected one of the eight values in the LUT  326  by an appropriate multiplication factor. The output R l (n) of the GX module  320  is subsequently delivered to the MAP module  360 . The output R l (n) is also routed to the GX1 module  330  and to the GX2 module  340 . 
 
         [0085]     The GX1 module  330  is a complex multiplier used to perform the complex multiplication of its received input R l (n) by a fixed complex-valued constant,  
               ⁢       ⅇ       j   ⁢           ⁢   8   ⁢           ⁢   π     32       .         
 
 Mathematically, the output of the GX1 module  330  may be described by the following product:  
         [       cos   ⁡     (       8   ⁢           ⁢   π     32     )       +     j   ⁢           ⁢   sin   ⁢           ⁢     (       8   ⁢           ⁢   π     32     )         ]     ×   output   ⁢           ⁢   from   ⁢           ⁢   GX   ⁢           ⁢   module       
 
 The GX1 module is adapted to access the storage unit  336  to obtain the complex-valued constant,  
               ⁢       ⅇ       j   ⁢           ⁢   8   ⁢           ⁢   π     32       .         
 
 The output of the GX1 module  330  is delivered to the MAP module  360 . 
 
         [0089]     The GX2 module  340  is also a complex multiplier used to perform the complex multiplication of its received input R l (n) by two fixed complex valued constants,  
       ⅇ       j4   ⁢           ⁢   π     32         
 
 and  
         ⅇ       j   ⁢           ⁢   12   ⁢           ⁢   π     32       .       
 
 Mathematically, the function of this module may be described by the following products:  
         [       cos   ⁡     (       4   ⁢           ⁢   π     32     )       +     j   ⁢           ⁢     sin   ⁡     (       4   ⁢           ⁢   π     32     )           ]     ×   output   ⁢           ⁢   from   ⁢           ⁢   GX   ⁢           ⁢     module   ⁢          [       cos   ⁡     (       12   ⁢           ⁢   π     32     )       +     j   ⁢           ⁢     sin   ⁡     (       12   ⁢           ⁢   π     32     )           ]     ×   output   ⁢           ⁢   from   ⁢           ⁢   GX   ⁢           ⁢   module       
 
 The GX2 module  340  receives the output R l (n) of the GX module  320  and is also adapted to receive the fixed constant  
       ⅇ       j4   ⁢           ⁢   π     32         
 
 from the corresponding storage unit  346 . Those skilled in the art will appreciate that  
       ⅇ       j4   ⁢           ⁢   π     32         
 
 is the same as e  
       ⅇ       j12   ⁢           ⁢   π     32         
 
 with the real and imaginary components reversed. Therefore, by multiplying R l (n) with  
         ⅇ       j4   ⁢           ⁢   π     32       ,       
 
 the product of R l (n) by  
       ⅇ       j12   ⁢           ⁢   π     32         
 
 may be obtained by manipulating the result of the first product thereby eliminating the need to perform an extra multiplication. Specifically, the second product  
           R   l     ⁡     (   n   )       ×     ⅇ       j12   ⁢           ⁢   π     32           
 
 may be obtained simply by rearranging the manner in which the inner products resulting from the first product i.e.  
           R   l     ⁡     (   n   )       ×     ⅇ       j4   ⁢           ⁢   π     32           
 
 are added or subtracted. The results of these two products are then delivered to the MAP module  360 . Once the products of the three complex multiplications performed by the GX module  320 , the GX1 module  330  and the GX2 module  340  are generated, they are sent to the MAP module  360  where the product of of  
           R   l     ⁡     (   n   )       ⁢           ⁢   by   ⁢           ⁢     ⅇ       jβ   ⁢           ⁢   l   ⁢           ⁢   π     32           
 
 for any value of β (multiple of 4) and l (integer) can be predicted. 
 
         [0101]     The mathematical function performed by the MAP module  360  is to compute the sixteen component values,  
           R   l     ⁡     (   n   )       ·     ⅇ       j   ⁢           ⁢   l   ⁢           ⁢   β   ⁢           ⁢   π     32           
 
 with l ranging from 0 to 15, for each required output time sample. The MAP module  360  in  FIG. 3  is adapted to receive four inputs corresponding to the complex products computed by the complex multiplier modules  320 ,  330  and  340 . In the embodiment of  FIG. 3 , the MAP module  360  has sixteen outputs corresponding to sixteen distinct output samples. 
 
         [0103]     Each input port of the MAP module  360  receives a unique complex number, R l (n), multiplied by a certain constant  
         (     l   ,     ⅇ       j4   ⁢           ⁢   π     32       ,       ⅇ       j8   ⁢           ⁢   π     32       ⁢           ⁢   or   ⁢           ⁢     ⅇ       j12   ⁢           ⁢   π     32           )     .       
 
 For each value of l with l ranging from 0 to 15, sixteen component products defined by  
           R   l     ⁡     (   n   )       ·     ⅇ       jβ   ⁢           ⁢   `l   ⁢           ⁢   π     32           
 
 and corresponding to different output time samples (granularity of 4) need to be computed and delivered to the output ports. However, out of the sixteen component products which need computing, four have already been computed. These are simply the four complex-valued inputs to the MAP module  360 . From these four inputs, any of the required sixteen component products may be generated by a simple multiplication of one of the four inputs by +1, −1, +j or −j.  FIG. 4  depicts an example of the logic flow  400  which may be undertaken in the MAP module  360  to arrive at one of sixteen component products. As shown, a MUX stage  420  receives the 4 inputs from modules GX, GX 1  and GX 2 . Depending on the output port (β) of the MAP module  360  being considered and the value of e, a complex product from one of the four input ports is selected and forwarded to a Decision stage  440  where an appropriate multiplication factor is applied. This process is implemented for each output port of the MAP module  360 . The sixteen outputs  370  of the MAP module are subsequently delivered to the accumulation module  380  whose functionality is described below. 
 
         [0106]     The accumulation module  380  receives sixteen inputs from the MAP module  360 . For each given input port, sixteen incoming complex numbers (these are the component values corresponding to values of equation (18) for l ranging from 0 to 15) arrive every clock cycle to be summed together in a register in order to generate one single output sample. This process occurs for each of sixteen input ports resulting in sixteen output time samples being computed in parallel. After the sixteen component values are summed for each input, the registers are cleared and the process is repeated for computation of the next set of sixteen output time samples. A general depiction of the operation performed by the accumulation module  380  is shown in  FIG. 5 .  
         [0107]     According to the embodiment in  FIG. 3 , using three (3) complex multipliers allows for the generation of 16 output samples every 16 clock cycles. Therefore, the total run time for applying the IFFT/FFT operation on the 64 complex-valued input samples would be 64 clock cycles. With regard to the brute force implementation depicted in  FIG. 1 , at the expense of adding two complex multipliers, the total number of clock cycles required to compute the 64-point FFT/IFFT has been reduced by an order of magnitude from (64) 2  to 64.  
         [0108]     Due to the similarity between the forward and inverse FFT (the IFFT differs from the FFT only by the sign of the exponent), the same module or circuitry with trivial modifications can be used for both modulation and demodulation in an OFDM transceiver. Although not shown, it should also be noted that depending on if the FFT or IFFT is to be computed, the accumulation module  380  treats its addition results differently. If the IFFT operation is required, the final result of the addition for each output sample is divided by the total number of samples (i.e. N). In the embodiment of  FIG. 3  and assuming the IFFT operation is desired, for example, the results of each accumulation would be divided by 64. If the FFT operation is desired, there is no division.  
         [0109]     The advantage of an FFT/IFFT engine implemented with ASIC technology is that each of the functional blocks of the FFT/IFFT engine be mapped onto dedicated, parallel hardware resources thereby avoiding the difficult programming and optimization challenges of scheduling time-critical operations through a single DSP core.  
         [0110]     It should be noted that the LUTs and other modules which provide multiplicands to the complex multiplier modules can be termed as multiplicand generators as they provide multiplicands for the system.  
         [0111]     While preferred embodiments of the invention have been described and illustrated, it will be apparent to one skilled in the art that numerous modifications, variations and adaptations may be made without departing from the scope of the invention as defined in the claims appended hereto.  
         [0112]     Although various exemplary embodiments of the invention have been disclosed, it should be apparent to those skilled in the art that various changes and modifications can be made which will achieve some of the advantages of the invention without departing from the true scope of the invention.  
         [0113]     A person understanding this invention may now conceive of alternative structures and embodiments or variations of the above all of which are intended to fall within the scope of the invention as defined in the claims that follow.