illustrative embodiments and exemplary applications will now be described with reference to the accompanying drawings to disclose the advantageous teachings of the present invention . while the present invention is described herein with reference to illustrative embodiments for particular applications , it should be understood that the invention is not limited thereto . those having ordinary skill in the art and access to the teachings provided herein will recognize additional modifications , applications , and embodiments within the scope thereof and additional fields in which the present invention would be of significant utility . an illustrative embodiment communications systems according to the present invention is illustrated in fig1 . a plurality of terminal radio units 2 communicate through radio signals 4 with a radio repeater system 6 . the radio terminal units comprise a transceiver circuit , controller , and user interface ( not shown ). the radio repeater system 6 comprises a plurality of radio repeaters 8 that operate on discrete frequency allocations . each radio repeater 6 comprises a transceiver . the transmitted and received radio signals 4 are combined and coupled to one or more radiating structures with radio frequency combining circuits 10 . the operation of the radio repeater system is generally controlled by controller 12 , a computing device , the general function of which is understood by those of ordinary skill in the art . the controller 12 also comprises communications circuits . external networks , such as the public switched telephone network (“ pstn ”) 14 and public or private data networks 16 , are interfaced to the controller 12 , which is operable to switch communications signals between the repeaters 8 and between the repeaters 8 and the pstn 14 and networks 16 . more particularly , the illustrative embodiment in fig1 is implemented in a trunked land mobile radio system that employs fdm channelization of the allocated radio spectrum and tdma access via packetized data for channel trunking management , system control , data communications , and voice communications within each of the fdm sub - channels . such systems are deployed in the 800 mhz smr band , the 900 mhz band , and certain other vhf and uhf bands . however , such techniques are equally applicable in any reasonable frequency band , as are those of the present invention . the controller 12 may be any of a variety of computers , processors , microprocessor , or other suitable digital controlling devices as are understood by those of ordinary skill in the art . in fig1 , three repeaters 8 are shown , however , those skilled in the art will appreciate that the number of repeaters is dependent upon the radio spectrum allocated to the system and may range from one to twenty , or more . the controller 12 provides various kinds of control of the resources within the repeater systems 6 , including interconnecting radio and wireline communications resources , generating and interpreting communication protocols , encoding and decoding voice and data , and other control functions . in an illustrative embodiment , the channel allocations are 25 khz wide , and the aforementioned fdm channelization sub - divides each 25 khz channel into four sub - channels , which are spaced 4 . 8 khz apart . the tdma channelization further divides each sub - band into two communications time slots with multiplexed data packets that are arranged in time to accomplish simplex , half - duplex , and duplex communications in real time . the illustrative embodiment system operates through the transmission and reception of four bit symbols at a 4 khz symbol rate . the symbols are modulated and demodulated to the rf band according to a 16 - qam modulation scheme with a 16 - point constellation in phase and amplitude , as is understood by those of ordinary skill in the art . the present invention teaches novel approaches to the base - band modulation and demodulation of such multiple sub - channelized systems . fig2 is a functional block diagram of a radio transmitter 20 according to an illustrative embodiment of the present invention . a plurality of data signals 22 , which may be arranged as a plurality of data channels , are coupled to a data / voice conditioning circuit 24 . the data voice conditioning circuit may employ any of a variety of error correction , interleaving , or other coding techniques that may be applicable to the communications objective of the systems . such signal conditioning circuitry is understood by those having ordinary skill in the art . the voice data conditioning circuit 24 is operative to generate ‘ m ’ base - band signals 25 corresponding to ‘ m ’ sub - channels , that are sampled at a first base - band sampling rate . ‘ m ’ is an integer value that is greater than one . in one illustrative embodiment , ‘ m ’ is equal to four . the data / voice conditioning circuit 24 is coupled to a processing system 26 , which is a modulation engine , for accepting and processing the ‘ m ’ base - band signals . the processing system 26 comprises a conventional digital signal processor (“ dsp ”) 28 and a conventional memory 30 including ram for providing an input buffer 34 , and rom for storing pre - programmed parameters and software , such as filter coefficients 32 , a multiple channel polyphase filter program 36 , and an inverse discrete fourier transform (“ idft ”) program 38 . processed communication signals , in the form of frequency multiplexed composite signals , are output from the processing system 26 to a digital to analog converter (“ dac ”) 40 , which converts the digital base - band signals to analog base - band signals . the analog signals output from the dac 40 are coupled to an rf modulator 42 , for modulating a wireless carrier with a frequency - multiplexed composite signal , which is an rf signal , having ‘ m ’ sub - channels . the signals are then radiated by antenna 44 for radio electromagnetic coupling via radio waves to one or more receivers . the dsp 28 , memory 30 , and dac 40 can be any of the variety of such devices known to be suitable for telecommunications applications by those or ordinary skill in the art , or that may later become known . operation of the processing system 20 in accordance with the present invention will be described further herein after . an exemplary receiver 48 in an illustrative embodiment of the present invention appears as a functional block diagram in fig3 . the receiver 48 is operable to demodulate a plurality of ‘ m ’ sub - channels that are received from the transmitter 20 . the receiver 48 comprises an antenna 50 for intercepting radio signals . the antenna 50 is coupled to a conventional receiver front end rf demodulator 52 for receiving the radio signals and converting them to base - band analog signals . the rf demodulator 52 output produces a composite signal that is coupled to analog to digital converter (“ adc ”) 56 . adc 56 is operable to convert the analog signals into base - band digital signals , having the ‘ m ’ sub - channels of information , into a composite signal having a sampling rate and derived from the ‘ m ’ sub - channels . the adc 56 is coupled to a processing system 58 for processing the signals and for controlling the receiver 48 . the processing system 58 comprises a conventional digital signal processor ( dsp ) 60 and a conventional memory 62 . the dsp 60 , memory 62 , and adc 56 can be any of the variety of such devices known to be suitable for telecommunications applications by those or ordinary skill in the art . the memory 62 comprises software elements including executable code and variables storage for programming the processing system 58 in accordance with the present invention . the memory 62 includes filter coefficients 64 describing a filter impulse response in accordance with the present invention , as are understood by those of ordinary skill in the art . in addition , the memory 62 includes an input buffer 66 used for storing input data samples . the memory 62 further comprises a multiple channel polyphase filter program 68 and a discrete fourier transform (“ dft ”) program 70 , in accordance with the present invention . operation of the processing system 58 in accordance with the present invention will be described further herein after . in several of the following illustrative examples , a multiple sub - carrier modulation system , developed by comspace corporation and known in the marketplace by the “ dcma ” trademark , is used to exemplify the prior art and the advantageous teaching of the present invention . the dcma systems modulation approach transmits and receives a single or multiple sub - carriers within a fcc allocated radio channels that are spaced at 25 khz centers . such channel allocations exist in the united states in the 800 mhz and 900 mhz bands . however , the present invention is in no way limited by these examples , and , those of ordinary skill in the art will appreciate that the teachings herein are applicable to a vast array of communications systems and encompass a vast range of communications bandwidths . the exemplary dcma systems modulates each sub - channel with a frequency offset that takes on one of four possible values with respect to the fcc allocated channel center frequency , and these values are ; − 7200 hz , − 2400 hz , + 2400 hz , and + 7200 hz . modulations is by 16 - qam comprised of 4 - bit symbols transmitted at a symbol rate of 4000 baud . according to the aforementioned frequency offsets , the sub - carrier channel spacing is 4800 hz . the output sampling rate of the combined sub - carriers ultimately transmitted over the allocated channel is 260 kilo samples per second (“ ksps ”). reference is directed to fig4 , which is an architectural diagram of the prior art modulation engine as used in the dcma modulation scheme . a plurality of ‘ m ’ sub - channel signals sm ( m ) 80 are input to upsample and pulse shaping filters 82 . the filters 82 are digitally implemented finite impulse response (“ fir ”) filters in one embodiment and output time domain signal pm ( n ) 84 to frequency shifting circuits 86 that shift the base - band signals according to the aforementioned frequency offsets . the frequency shifted signals are summed be a summation function 88 and the combined output signal y ( n ) 90 is coupled to rf transmitting circuitry ( not shown ). p l ⁡ ( n ) = ∑ m = - ∞ ∞ ⁢ h ⁡ ( nd - mi ) ⁢ s l ⁡ ( m ) ( 1 ) where d is the polyphase filter decimation rate , i is the polyphase filter interpolation rate , m is the input signal time at the input signal sampling rate , and n is the output time at the upsampled output rate . and , s l ( m ) characterizes one symbol of input as a function of time . given that the upsample rate , establishes a modulo ratio , the expression becomes : p l ⁡ ( n ) = ∑ m = - ∞ ∞ ⁢ h ⁡ ( i ⁡ ( ⌊ nd i ⌋ - m ) + nd ⁢ ⁢ mod ⁢ ⁢ i ) ⁢ s l ⁡ ( m ) ( 2 ) and since any realizable digital filter must be truncated in time , the summation is limited to the number of taps per phase of the filter , the expression becomes : p l ⁡ ( n ) = ∑ m = 0 n taps / phase - 1 ⁢ h ⁡ ( im + nd ⁢ ⁢ mod ⁢ ⁢ i ) ⁢ s l ⁢ ⁢ ( ⌊ nd i ⌋ - m ) ( 3 ) adding the frequency shift and summing the plural singles , the output y ( n ) 90 becomes : y ⁡ ( n ) = ∑ l = 1 m ⁢ p l ⁡ ( n ) ⁢ ⅇ j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ nlf δ ⁢ t s ( 4 ) one of the advantages of the present invention is the reduction in processor operation required to achieve the desired modulation and demodulation of the plural sub - channels . respecting the prior art modulation engine of fig1 , the number of processor multiplication operations ( a reasonable indication of processor demand ) required to implement the modulation engine is as follows . n mult =( 2 * m * n taps / phase + m ( 4 + p )) f so ( 5 ) where p is the equivalent number of multiplies required in order to generate the complex sinusoid , and f so is the final sampling rate . given the parameters of the dcma system , the number of multiplies in the modulation engine of the repeater is : fig5 is an architectural diagram of a modulation engine according to an illustrative embodiment of the present invention . the input signals sm ( m ) 100 are coupled to an inverse discrete fourier transform (“ idft ”) 102 . the output bins of the idft are coupled to a multiple channel polyphase filter 104 . the outputs of the filter 104 are combined by a commutator 106 to yield the combined signal output y ( n ) 108 . while the architecture in fig5 differs from the prior art architecture described with respect to fig4 , the function of the modulation engine in fig5 , in the general case , is mathematically the same . the structure in fig5 can be mathematically described , and the present invention further derived as follows . consider a composite expression for the modulation engine , which combines equations ( 1 ) and ( 4 ). y ⁡ ( n ) = ∑ l = 1 m ⁢ ∑ m = - ∞ ∞ ⁢ h ⁡ ( nd - mi ) ⁢ s l ⁡ ( m ) ⁢ ⅇ j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ nlf δ ⁢ t s ( 7 ) y ⁡ ( n ) = ∑ m = - ∞ ∞ ⁢ h ⁡ ( nd - mi ) ⁢ ∑ l = 1 m ⁢ s l ⁡ ( m ) ⁢ ⅇ j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ nlf δ ⁢ t s ( 8 ) use the substitution , n = r + sm , where r is an indicator of the commutator cycle position in time , and s is an indicator of overall time , to yield a modified summation . y ⁡ ( n ) = y ⁡ ( r + sm ) = ∑ m = - ∞ ∞ ⁢ h ⁡ ( nd - mi ) ⁢ ∑ l = 1 m ⁢ s l ⁡ ( m ) ⁢ ⅇ j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ ( r + sm ) ⁢ lf δ ⁢ t s ( 9 ) equation 9 will be readily understood by those possessing ordinary skill in the art . an important advancement in the art made by the present invention occurs when the constraint that f δ t s is equal to k / m , which equates f δ t s . to a ratio of integers , otherwise stated as a rational number . it is not required in general , but for illustration purposes , k is set to 1 and equation ( 9 ) rewritten as : y ⁡ ( n ) = ⁢ ∑ m = - ∞ ∞ ⁢ h ⁡ ( nd - mi ) ⁢ ∑ l = 1 m ⁢ s l ⁡ ( m ) ⁢ ⅇ j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ ( r + sm ) ⁢ l / m = ⁢ ∑ m = - ∞ ∞ ⁢ h ⁡ ( nd - mi ) ⁢ ∑ l = 1 m ⁢ s l ⁡ ( m ) ⁢ ⅇ j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ rl / m ( 10 ) which is algebraically manipulated in the same fashion as between equations ( 1 ) and ( 2 ) to yield : y ⁡ ( n ) = ⁢ ∑ m = - ∞ ∞ ⁢ h ⁡ ( nd - mi ) ⁢ s r ⁡ ( m ) = ⁢ ∑ m = 0 n taps / phase - 1 ⁢ h ⁡ ( im + nd ⁢ ⁢ mod ⁢ ⁢ i ) ⁢ s r ⁡ ( ⌊ nd i ⌋ - m ) ( 11 ) the commutator position s r ( m ) is interpreted as the rth idft output bin at time , m . in equation ( 10 ), the sm term in the phasor repeats every 2π radians and thus drops out of the equation . this filtering operation has the same number of processor multiplies as a single sub - channel rate change . the added processing cost for computing the multiple sub - carrier modulation bank is the cost of computing the idft , which performs the mixing operation . however , the idft is computed at the symbol rate and not the output sampling rate , and is therefore substantially less processor intensive . this demonstrates that the illustrative embodiment or the present invention not only achieves the desirable decoupling of the input frequency spacing from the filter parameters ( decimation rate , interpolation rate , and number of phases ) but , does so at a substantially reduced processor load . now , considering the forgoing advancement in view of the illustrative embodiment dcma system , which operation parameters were enumerated herein before , it will be appreciated that the foregoing modulation model can be applied to the modulation bank for the downlink . the downlink is the communications path from the repeater systems to the terminal units . as the repeater systems must communicate simultaneously with a plurality of terminal units , it must do so on each of the sub - channels simultaneously as well . however , since the output sampling rate in dcma is fixed at 260 ksps , it is necessary to contemplate the more complex sampling rate change this implies ( 4 ksps to 260 ksps ). consider , generally , a process where the dft is greatly over sampled in order to accommodate a difficult rate change . such an operation can be mathematically constructed as follows . y ⁡ ( n ) = ∑ m = - ∞ ∞ ⁢ h ⁡ ( nd - mi ) ⁢ ∑ l = 1 m ⁢ s l ⁡ ( m ) ⁢ ⅇ j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ ( r + sm ) ⁢ lf δ ⁢ t s ( 12 ) the first summation generally defining the filtering operation and the second generally defining the transformation . again applying an advancement of the present invention , let f δ t s be a rational number , a / b . note that at this point in the design process . it is unknown how to select the idft resolution , m . using this substitution , equation ( 12 ) becomes : y ⁡ ( n ) = ∑ m = - ∞ ∞ ⁢ h ⁢ ⁢ ( nd - mi ) ⁢ ∑ l = 1 m ⁢ s l ⁡ ( m ) ⁢ ⅇ j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ ( r + sm ) ⁢ la / b ( 13 ) since f δ t s equals a / b and this is k / m , then b equals am / k , and k equals 1 , so b = am , therefore equation ( 13 ) becomes : y ⁡ ( n ) = ∑ m = - ∞ ∞ ⁢ h ⁢ ⁢ ( nd - mi ) ⁢ ∑ l = 1 m ⁢ s l ⁡ ( m ) ⁢ ⅇ j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ rla / m ( 14 ) again , applying the same practical application of a finite time filter , equation ( 14 ) becomes : y ⁡ ( n ) = ∑ m = 0 n tapsnyq - 1 ⁢ h ⁢ ⁢ ( mi + nd ⁢ ⁢ mod ⁢ ⁢ i ) ⁢ ∑ l = 1 m ⁢ s l ⁡ ( ⌊ nd i ⌋ - m ) ⁢ ⅇ j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ la / m ( 15 ) where n tapsnyq is the number of taps per phase of the nyquist filter , which is the filter chosen for use in the dcma implementation of the present invention . the inner summation is the sub - sampled idft . a key difference between this equation and the previous equation is that the filter bank output is fractionally sampled . this is an important advancement over the prior art . with this advancement , the baud rate can be related to the frequency by any rational number , and the output sampling frequency can be related to the frequency spacing by any rational number . this mathematical construct is very useful for any modulation engine or software radio . some of the tradeoffs involved with such a structure do exist . the resolution of the idft is given by the output sampling rate divided by the greatest common denominator of the output sampling frequency and the sub - channel frequency spacing . the decimation rate on the output commutator is given by the sub - channel frequency spacing divided by the greatest common divisor of the output sampling frequency and the sub - channel frequency spacing . however , this advancement allows the designer of a communications system great flexibility in selecting the input sampling rate , the sub - channel frequency spacing , and the output sampling rate to suit system requirements . the variable factors which are adjusted to achieve this result are the number of terms of the idft , ( the order of the idft ), and the relationship between phase of the filter , ( the sequence in which the phases are utilized ). respecting the question of processor overhead burden of the dcma illustrative embodiment of the present invention , in the worst case , the computational complexity of using this filter bank is as follows . n mult = 2 * n taps / phase f so + c dft f b ( 16 ) where c dft is the number of processor multiplies required in order to compute the idft . again , for the dcma illustrative embodiment , the following is an approximation of the complexity numbers . the question of how many processor multiplies will it take to implement the corresponding dft . in this case there are far fewer inputs than total output points to be computed . assuming the input is complex , the number of computations becomes : it is important to note in this case that the number of operations required for the mixing operation can be increased above the required amount for the prior art implementation . this situation can arise if the number of sub - carriers is much smaller than the dft size . however , in such situations , the utilized bandwidth in ratio to the output sampling rate will be small . if that is the case , the modulation bank can be implemented at a lower output sampling rate followed by an upsampler . the computational complexity of such an implementation in this case is as follows . where n tap is the number of taps per phase to implement an up by 5 rate change . this second filter could be readily implemented in 40 taps per phase . this yields a final complexity of 16 . 22 mips , a savings of 20 mips over the prior art implementation . respecting an implementation of the present invention modulation engine illustrated in fig5 to the dcma illustrative embodiment , there are four input sub - channel sub - carriers sm ( m ) 100 , each of which is sampled at 4 ksps . a total a 65 idft 102 bins coupled to 65 nyquist filter phases 104 . the filter 104 interpolation rate is 65 and the commutator 106 rate is 65 ksps . an upsample of the commutator output by 5 brings the final output sample , y ( n ), 108 to 260 ksps . the filtering and commutation operation of this bank can be completely defined for all time in terms of three parameters , the filter phase , the data pointer position , and the commutator position . in the case of a dcma illustrative embodiment 4 ksps baud system with frequency spacing of 4800 hz and an output sampling rate of 52 ksps , these parameters are defined as follows . h φ = h ( φ + 65 k ), k = 0 : n tapsnyq − 1 ( 21 ) δ ⁢ ⁢ ( n ) = ⌊ n 65 ⌋ ( 22 ) the foregoing equation set does not specify the implementation of the idft . the idft has four inputs and 65 outputs . there are four active sub - carriers . the sub - carriers should are located at − 7200 hz , − 2400 hz , + 2400 hz and + 7200 hz . in order to accommodate these frequencies , the basic idft is modified as follows . s r ⁡ ( m ) = ∑ l = 1 4 ⁢ s l ⁡ ( m ) ⁢ ⅇ j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ 6 ⁢ r ⁡ ( l - 2 . 5 ) / 65 = ⅇ - j ⁢ ⁢ 18 ⁢ ⁢ π ⁢ ⁢ r / 65 ⁢ ∑ l = 1 4 ⁢ s l ⁡ ( m ) ⁢ ⅇ j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ r6l / 65 ( 24 ) this additional shift is not expensive in terms of processor overhead because a direct implementation of the idft is used . the composite operation of the filter bank is summarized by the following steps : step 1 : compute the idft of one time - slice of symbols according to equation ( 24 ). this will account for 65 separate filter histories , each at the symbol rate , ( the original symbols will never be needed again .). step 2 : identify the filter phase according to equation ( 20 ). step 3 : identify the data pointer position according to equation ( 22 ). step 4 ; identify the commutator position according to equation ( 23 ). step 5 : compute the dot product of the filter phase and the data vector according to the following equation : y ⁡ ( n ) = ∑ k = 0 n tapsnyq - 1 ⁢ h ⁡ ( ϕ ⁡ ( n ) + 65 ⁢ k ) · s χ ⁡ ( n ) ⁡ ( δ ⁡ ( n ) - k ) ( 25 ) this operation is followed by a direct interpolation by 5 to achieve the design output sampling rate of 260 ksps , as discussed above . the interpolation by 5 operation is specified as follows . h φ = h ( φ + 5 k ), k = 0 : n tapsnyq − 1 ( 27 ) δ ⁡ ( n ) = ⌊ n 5 ⌋ ( 28 ) y ⁡ ( n ) = ∑ λ = 0 n tapsfill - 1 ⁢ h ⁡ ( ϕ ⁡ ( n ) + 5 ⁢ k ) · y ⁡ ( δ ⁡ ( n ) - k ) ( 29 ) the teachings of the present invention are also applicable to the demodulation of multiple sub - carrier signals in a similar fashion to the previously discussed modulation aspects . for a more detailed understanding of the demodulation aspects of the teachings of the present invention , it is useful to consider the prior art , and especially the prior art dcma systems . by way of review , dcma modulation consists of a single ( in the case of a terminal unit operating on a single sub - channel ) or multiple sub - carriers with a frequency offset that takes on four possible values , − 7200 , − 2400 , + 2400 , or + 7200 hz from the allocated carrier frequency . the basic parameters of the system are a symbol rate is 4000 baud ; the sub - carrier spacing is 4800 hz , and an output sampling rate is 260 ksps . the prior art demodulation approach is diagrammed in fig6 . a multiple sub - carrier signal y ( n ) 110 is received from another circuit , which may be an analog to digital converter further coupled from an rf receiver demodulator . the signal is comprised of a digitally encoded signal at a relatively high sampling rate . a commutator 112 sequences the input signal through a number of positions equal to the number of sub - carriers represented in signal y ( n ), thus demultiplexing the received signals into their sub - carrier parts . each sub - carrier part is frequency shifted by a suitable amount in frequency shift circuits 114 . the frequency shifted circuits are output from the frequency shift circuits 114 as base - band sub - carrier signals pm ( n ) 116 . each of these signals is sampled at the relatively high input 110 sampling rate . down sampling to the sub - channel sampling rate and pulse shaping occur at down sampler / pulse shape circuits 118 for each sub carrier . the base - band signals , at the relatively lower output sampling rate are output at 120 as xm ( m ). note that ‘ n ’ is a representation of time at the relatively high input sampling rate , and ‘ m ’ is a representation of time at the relatively lower output sampling rate . the total number of sub - carriers , or sub - channels is represented by the integer ‘ m ’. mathematically , the processes illustrated in fig6 are expressed as follows . x l ⁡ ( m ) = ∑ n = - ∞ ∞ ⁢ h ⁡ ( md - ni ) ⁢ y ⁡ ( n ) ⁢ ⅇ - j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ n ⁡ ( f c + l ⁢ ⁢ δ ⁢ ⁢ f ) ⁢ t s ⁢ ⁢ x l ⁡ ( m ) = ⁢ ∑ s = - ∞ ∞ ⁢ ∑ r = 0 m - 1 ⁢ h ⁡ ( md - ( r + sm ) ⁢ i ) ⁢ y ⁢ ( r + sm ) ⁢ ⅇ - j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ ( r + sm ) ⁢ ( f c + l ⁢ ⁢ δ ⁢ ⁢ f ) ⁢ t s ( 31 ) where x l ( m ) is the output signal for each l th sub - channel , d is the filter decimation rate , i is the filter interpolation rate , y ( n ) is the multiplexed input signal , f c is the carrier center frequency , δf is the sub - carrier offset frequency , and ts is the sample time period . then , using the substitution , n = r + sm , as was applied in novel aspects of the modulation engine calculations , and the summation for the novel aspects of the illustrative embodiment demodulation engine becomes : x l ⁡ ( m ) = ⁢ ∑ s = - ∞ ∞ ⁢ ∑ r = 0 m - 1 ⁢ h ⁡ ( md - ( r + sm ) ⁢ i ) ⁢ y ⁢ ( r + sm ) ⁢ ⅇ - j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ ( r + sm ) ⁢ ( f c + l ⁢ ⁢ δ ⁢ ⁢ f ) ⁢ t s ( 32 ) x l ⁡ ( m ) = ∑ r = 0 m - 1 ⁢ ⅇ - j ⁢ ⁢ 2 ⁢ π ⁢ ⁢ r ⁡ ( f c + l ⁢ ⁢ δ ⁢ ⁢ f ) ⁢ t s ⁢ ∑ s = - ∞ ∞ ⁢ h ⁢ ⁢ ( md - ( r + sm ) ⁢ i ) ⁢ y ⁢ ⁢ ( r + sm ) ( 33 ) in the flowing analysis , the inner summation , indicating the filter response , is separated out , and , the modulo count remainder of the decimation / interpolation process incorporated for thoroughness . y ^ ⁡ ( m ) = ⁢ ∑ s = - ∞ ∝ ⁢ h ⁢ ⁢ ( mi ⁢ ⁢ ( ⌊ md - ri mi ⌋ - s ) + ⁢ ( md - ri ) ⁢ ⁢ mod ⁢ ⁢ mi ) ⁢ y ⁡ ( r + sm ) ( 34 ) s ~ = ⌊ md - ri mi ⌋ - s , and terminating the filter response time according to the number of taps employed in the digital filter , the expression becomes : y ^ r ⁡ ( m ) = ⁢ ∑ s ~ = 0 ⌈ n taps mi ⌉ - 1 ⁢ h ⁢ ⁢ ( m ⁢ i ~ ⁢ s + ( md - ri ) ⁢ ⁢ mod ⁢ ⁢ mi ) ⁢ ⁢ y ⁢ ( r + ( ⌊ md - ri mi ⌋ - s ~ ) ⁢ ⁢ m ) ( 35 ) the operation of commutation is expressed by , y r ( s )= y ( r + sm ), and is applied to equation ( 35 ) to yield : y ^ r ⁡ ( m ) = ⁢ ∑ s ~ = 0 ⌈ n taps mi ⌉ - 1 ⁢ h ⁢ ⁢ ( m ⁢ i ~ ⁢ s + ( md - ri ) ⁢ ⁢ mod ⁢ ⁢ mi ) ⁢ ⁢ y r ⁢ ( ⌊ md - ri mi ⌋ - s ~ ) ( 36 ) equation 36 is the form of an ordinary polyphase filtering operation , with the slight modification of the ri term . this can be readily implemented in a digital signal processor , as will be appreciated by those of ordinary skill in the art . finally , substitution back in to equation ( 33 ) yields the following complete expression of the output channels . x l ⁡ ( m ) = ⁢ ∑ r = 0 m - 1 ⁢ ⅇ - j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ r ⁡ ( f c + l ⁢ ⁢ δ ⁢ ⁢ f ) ⁢ t s ⁢ y ^ r ⁡ ( m ) = ⁢ ⅇ - j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ rf c ⁢ t s ⁢ ∑ r = 0 m - 1 ⁢ ⅇ - j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ ⁢ rl ⁢ ⁢ δ ⁢ ⁢ ft s ⁢ y ^ r ⁡ ( m ) ( 37 ) the discrete fourier transform (“ dft ”) is not necessarily an ordinary dft , although it can be in certain cases , depending on the relationship between δf and t s . in general , according to the advantageous teachings of the present invention , the relationship between δf and t s can be any rational number . in order to accommodate this possibility , the input to the dft may have to be decimated . the processing structure employed to achieve the decimation , as well as the filter and transform according to an illustrative embodiment of the present invention is depicted in fig7 . fig7 is an architectural diagram of a demodulation engine according to an illustrative embodiment of the present invention . the composite multi - sub - carrier signal y ( n ) 122 is input to a commutator 124 , which commutes the signal 122 to the plurality of filter input signals 126 . the filters 130 pulse shape the signals to ŷm ( r , s , m ) filter signals 132 which are input to discrete fourier transform (“ dft ”) 134 . the ‘ m ’ individual outputs xm ( m ) 136 are output from dft 134 , and are used for subsequent signal processing and utilization , as is understood by those skilled in the art . respecting the illustrative embodiment dcma communications system , the input sampling rate is 260 ksps with 4 sub - carriers spaced at 4800 hz having a baud rate of 4 ksps . the calculations yield values as follows ; m = 325 , f c t s =− 9 / 325 , δft s = 6 / 325 , i = 1 and d = 65 . consequently , the operation of the analysis bank can be expressed as follows . y r ( s )= y ( r + s 325 ) ( 38 ) y ^ r ⁡ ( m ) = ⁢ ∑ s ~ = 0 ⌈ n taps 325 ⌉ - 1 ⁢ h ( 325 ⁢ s ~ + ( m65 - r ) ⁢ mod ⁢ ⁢ 325 ) ⁢ y r ⁡ ( ⌊ m65 - r 325 ⌋ - s ~ ) ( 39 ) x l ⁡ ( m ) = ⁢ ⅇ j ⁢ ⁢ 18 ⁢ π ⁢ ⁢ r / 325 ⁢ ∑ r = 0 324 ⁢ ⅇ - j ⁢ ⁢ 12 ⁢ π ⁢ ⁢ rl / 325 ⁢ y ^ r ⁡ ( m ) = ⁢ ⅇ j ⁢ ⁢ 18 ⁢ π ⁢ ⁢ r / 325 ⁢ ∑ r = 0 324 ⁢ ⅇ - j ⁢ ⁢ 2 ⁢ π ⁢ ⁢ l ⁡ ( 6 ⁢ r ⁢ ⁢ mod ⁢ ⁢ 325 ) / 325 ⁢ y ^ r ⁡ ( m ) ( 40 ) respecting the processor demand for this illustrative embodiment application of the present invention , the number of processor multiplications required in order to implement such an operation is approximated in the worst case as follows . n mult =( 2 * n taps + n dft ) f b =( 2 * n taps + 4 n sc ) f b ( 41 ) the number of computations required for a 10 sub - carrier system assuming a 11 symbol delay is 11 mips . the same system with a conventional implementation would require 70 mips . on the receive side at least 3 taps of the filter must be sampled . a fine timing offset ( early or late ) may be realized by changing the phase of the filter . these additional outputs scale the complexity requirements . for 3 output taps , the complexity requirements triple , which is true of both the prior art implementation and the illustrative embodiment implementation . thus , the present invention has been described herein with reference to a particular embodiment for a particular application . it is therefore intended by the appended claims to cover any and all such applications , modifications and embodiments within the scope of the present invention .