Patent Publication Number: US-9432079-B1

Title: Frequency hopped frequency modulation spread spectrum (FHFMSS) multiple accessing communication systems and methods

Description:
BACKGROUND 
     Broadband wireless systems are in a rapidly evolutionary phase in terms of the development of various technologies, development of various applications, deployment of various services and generation of many important standards in the field. Although there are many factors to be considered in the design of these systems, the key factors have been the bandwidth utilization efficiency due to the limited bandwidth allocation, flexibility in operation and robustness of the communication link in the presence of various disturbances while achieving the specified performance. At present, the OFDM and spread spectrum techniques have been adapted in many wireless communication standards, such as the World-wide Interoperability for Microwave ACCESS (Wimax), digital audio broadcasting (DAB), digital video broadcasting-terrestrial (DVB-T), Long Term Evolution (LTE), Universal Mobile Telecommunications Service (UMTS) or Code Division Multiple Accessing (CDMA) 2000, Wideband CDMA (W-CDMA), CDMA standard IS95, IEEE 802.11 Wireless Local Area Network (WLAN), IEEE 802.15 Wireless Private Area Network (WPAN), etc. 
     One of the advantages of the OFDM system is the mitigation of a major source of distortion present in high data rate wireless communication links, namely the inter symbol interference (ISI) achieved by increasing the symbol period by the use of multiple carrier transmission. However, the use of a large number of carriers based on the orthogonality property in the OFDM system makes the performance of the system very sensitive to any carrier frequency offsets introduced, for example, by the Doppler shifts encountered in the wireless channels. The proper operation of the OFDM system requires means for precise estimate of the Doppler that may be different for different carriers in the frequency selective fading channel, and means to mitigate such a Doppler effect from the received OFDM signal. Another important problem arising with the use of a relatively large number N of carriers used in the OFDM signal is a relatively high peak to average power ratio resulting in a much reduced radio frequency (RF) power amplifier efficiency. Various methods exist in the prior art to solve these problem. 
     In multiple accessing mode of the OFDM system wherein relatively narrow band OFDM channels are assigned to various users in a mobile communication network, the frequency selective fading of the communication channel may cause severe fading of some of the user signals resulting in frequent hand offs or call drop for such users in a mobile communication network. The solution to such a problem may be the transmission of the user signal over multiple OFDM channels for providing a diversity gain and/or transmission at a much reduced data rate. However, such a solution results in a reduced capacity of the OFDM system. Another problem with the OFDM system is that the narrow band signals of various users have no protection against any intended or unintended interference. Any significant narrowband interference in any of the OFDM channels may disrupt communication to users assigned such channels. 
     Spread spectrum systems provide protection against narrow band interference and possess various other desirable properties such as graceful degradation with increased number of active multiple access users, etc. There are mainly two types of spread spectrum systems in use in the mobile communication networks, viz., the direct sequence spread spectrum (DSSS) system and the frequency hopped spread spectrum (FHSS) system. In the DSSS system, the baseband modulated signal is modulated in a second stage by a pseudo random binary sequence resulting in a spreading of the signal bandwidth that results in the protection against narrow band interference. However, in the wireless networks, due to the increased self noise in the system resulting from multipath propagation, the DSSS system may be more suited for a relatively short range multiple accessing application or long range point to point communication by using separation of the multipath components using, or example, a rake receiver. In the FHSS system, the carrier frequency of the baseband modulated signal is varied according to some pseudo random (PN) pattern or sequence that varies at the information symbol rate or a integer multiple thereof in the fast FHSS system, and results in the spreading of the signal bandwidth that depends upon the number of frequencies in the pseudo random pattern. The FHSS system may offer more protection against multipath over relatively longer distances compared to the DSSS system. 
     In the FHSS system, different users are assigned different pseudo random frequency sequences. A disadvantage of the FHSS system may be that whenever, the frequencies of more than one PN sequence coincide, that results in a collision whereby the symbols during such a collision period are erased placing a floor on the probability of symbol error irrespective of the signal to noise ratio or other parameters. Similarly the presence of narrow band interference around any one of the hop frequencies may result in the erasure of the symbol being transmitted during that hop. For example, if 2 out of 100 hop frequencies are interfered with, the probability of symbol error will be lower bonded by 0.02. This is different compared to the DSSS system wherein the narrow band interference is spread out in a wide band in the despreader at the receiver causing a relatively small degradation in the probability of symbol error. Moreover, the FHSS system requires a frequency synthesizer for the generation of pseudo random frequency sequence capable of generating a relatively large number of frequencies with relatively high switching speeds equal to the information symbol rate, or a integer multiple thereof with relatively low noise and a relatively high precision frequencies present in the pseudo random frequency sequence. 
     Kumar teaches an orthogonal frequency chirp multiple accessing (OFCM) spread spectrum system in Orthogonal Frequency Chirp Multiple Accessing Systems and Methods, U.S. patent application Ser. No., 14/244,774, 2014, wherein the baseband modulated signals are modulated by a time varying frequency waveforms such that an orthogonality is maintained among the various multiple accessing signals. The OFCM system taught by Kumar is a spectrally efficient and provides protection against both the interference and frequency selective fading due to multipath propagation. The OFCM system, however, may not provide any privacy protection to the multiple accessing users. As the time varying frequency waveforms are delayed versions of a common waveform whose frequency varies linearly with time, an unauthorized user can easily receive the baseband information symbols intended for the other multiple accessing users. 
     There is a strong motivation to come up with systems and methods that achieve the various advantages of the prior art spread spectrum systems while overcoming various possible weaknesses therein. The frequency hopped frequency modulation spread spectrum (FHFMSS) system of the invention provides protection against deep fades in some segments of the spectrum and against narrowband interference similar to that of the DSSS system in that the impact of any narrow band interference is distributed throughout the spread bandwidth after the despreading at the receiver rather than being concentrated in a few symbols as in the FHSS system of prior art placing a floor on the probability of symbol error. In the FHFMSS system, the wideband Frequency Hopped Frequency Modulation (FHFM) spreading waveform is generated by Frequency Modulation (FM) modulation of a multiplicity of the periodic waveforms with time varying frequencies determined by pseudo random sequences. The FHFM waveform in turn modulates the baseband modulated signal. A deterministic short sequence of frequency pairs has been previously used in the construction of a command destruct code in the space lift range system. 
     Unlike the FHSS system of the prior art, the spectrum of the FHFMSS signal occupies a wide bandwidth during any symbol period. Deep fading in a few segments of the wide band spectrum may not result in any significant performance degradation to any of the users. In the prior art FHSS system, the number of hop frequencies and the period of the pseudo random (PN) sequences may be both equal to the bandwidth spreading factor that is the ratio of the FHSS signal bandwidth and the baseband modulated signal bandwidth with the switching speed of the frequency synthesizer equal to or an integer multiple of the baseband information symbol rate. In the FHFMSS system of the invention, the various parameters such as the number of hop frequencies, the period of the PN code, the bandwidth spreading factor , and the switching speed of the frequency synthesizer may be selected independently so as to minimize the transmitter complexity while achieving the desired spread spectrum signal bandwidth. The FHFMSS has the advantage of not requiring a relatively very large number of hop frequencies and high switching speeds of the frequency synthesizer in the FHSS system of the prior art. 
     The FHFMSS system of the invention also inherits various advantages of the prior FHSS system in that it provides robustness against multipath propagation interference and even a better privacy protection compared to both the DSSS and FHSS system. In the FHFMSS system of the invention, individual user signals with modulations such as MPSK (Multi Frequency Shift Keying), are constant envelope signals inheriting the property from the frequency modulated (FM) signals with an advantage in terms of requiring relatively low amplifier back off in the user to base station transmission in a mobile communication network. Unlike the prior art DS spread spectrum systems, the power spectral density of the FHFMSS system of the invention does not have any spectral side lobes. These and other advantages of the FHFMSS system will be evident from the following specifications. 
     SUMMARY OF THE INVENTION 
     Various embodiments of the invention are directed to methods and systems for frequency hopped frequency modulation spread spectrum (FHFMSS) transmitters and receivers for multiple accessing communication over wireless fading channels. For example, various embodiments of the transmitter may utilize an architecture comprised of a baseband modulation subsystem for receiving and modulating the input data providing the, in general complex valued, information baseband symbols, a code generation subsystem for generating a multiplicity M code vector sequences, a multi frequency synthesizer for the generation of the multiplicity M sequences of periodic ψ-waveforms, a complex baseband frequency modulator (FM) for providing a frequency hopped frequency modulation (FHFM) waveform, a spread spectrum modulator for modulating the information baseband symbols sequence by the FHFM waveform providing the baseband FHFMSS (frequency hopped frequency modulation spread spectrum) signal. 
     Various embodiments of the FHFMSS transmitter of the invention may be further comprised of a baseband to IF converter modulating the baseband FHFMSS signal by the IF local oscillator in phase and quadrature signals for providing the IF band pass FHFMSS signal, and an RF stages unit comprised of an up converter, an RF band pass filter and power amplifier for providing the RF band pass FHFMSS signal. 
     In various embodiments of the invention, the code generation subsystem for generating a multiplicity M code vector sequences may be comprised of a multiplicity M feedback shift registers of length N B  with K B  of the N B  stages&#39; outputs of the feedback shift registers comprising the elements of the multiplicity M code vectors. In various embodiments of the invention, different feedback connections in the feedback registers may correspond to distinct minimal polynomials of degree N B  resulting in a period of the code vector sequences equal to 2 N     B   −1. 
     In various embodiments of the invention, a multi frequency synthesizer for the generation of the multiplicity M sequences of periodic ψ-waveforms may comprise of a multiplexer for time multiplexing the multiplicity M code vector sequences, a code conversion unit for generating a sequence of periodic ψ-waveforms indices, and a time multiplexed direct digital frequency synthesizer (TMDDFS) for generation of the periodic ψ-waveforms corresponding to the indices provided by the code conversion unit. In various alternative embodiments of the invention, the multi frequency synthesizer unit may generate an integral version ψ I (t) of the ψ(t) waveforms. In various embodiments of the invention, the ψ(t) waveforms may be sinusoidal waveforms of some specified frequencies that may be integer multiples of a fundamental frequency f 0 . In various alternative embodiments of the invention, the frequency synthesizer may be an indirect frequency synthesizer based on phase lock loops, a direct analog frequency synthesizers, and the like. 
     In various digital embodiments of the invention, the complex baseband frequency modulator unit may be comprised of a multiplicity M scalar multipliers for scaling the discrete time waveforms ψ I (n), n denotes time index, generated by the frequency synthesizer unit by a set of modulation indices, an adder for adding the scaled waveforms, a rate converter for an up conversion of the sampling rate of the resulting sum, and an exponentiation unit for providing a frequency hopped frequency modulation (FHFM) waveform to the spread spectrum modulator. 
     In various embodiments of the invention, the time multiplexed direct digital frequency synthesizer (TMDDFS) may be comprised of a multiplicity M frequency registers for storing the ψ I -waveform indices that may be equal to the normalized frequencies of the of the sinusoidal ψ I -waveforms, a multiplicity M phase accumulators for modulo integer N s  accumulating the corresponding normalized frequencies, and a ROM memory wherein the sampled ψ I -waveforms are stored. In various embodiments of the invention wherein the ψ I -waveforms are sinusoidal waveforms with frequencies that are multiples of a fundamental frequency f 0 , the ROM may store the sampled version of only the sinusoidal waveform of frequency f 0  sampled at an appropriate sampling rate. The integer N s  may be equal to the number of samples in the ROM. 
     In various alternative embodiments of the invention, the code generation subsystem may be comprised of a feedback shift registers of length N B  with K B  of the N B  stages&#39; outputs of the feedback shift registers comprising the elements of the first of the multiplicity M code vectors with the other (M−1) code vectors obtained by feed forward circuits operating on the N B  stages&#39; outputs of the feedback shift register. 
     In various embodiments the baseband modulator may be one out of the group comprised of the Multiple Quadrature Amplitude Modulation (MQAM) modulator, the Multiple Phase Shift Keying (MPSK) modulator, the Multiple Amplitude Shift Keying (MASK) modulator, or any more general modulator architecture. The baseband modulator may also be comprised of error correction encoders and interleavers. 
     Various embodiments of the FHFMSS receiver of the invention may utilize an architecture comprised of a receive antenna for receiving the radio Frequency (RF) band pass FHFMSS signal, an RF front stage unit comprised of an RF filter, amplifier and down converter for providing the Intermediate Frequency (IF) band pass FHFMSS signal, an IF to complex baseband converter providing the baseband FHFMSS signal, a code generation subsystem for generation of a multiplicity M code vectors, a multi frequency synthesizer for the generation of the multiplicity M sequences of periodic ψ-waveforms, a complex baseband frequency modulator (FM) for providing a frequency hopped frequency modulation (FHFM) waveform, a spread spectrum demodulator for despreading the baseband FHFMSS signal by the FHFM waveform providing the despread signal, a symbol detector and a baseband demodulator. The symbol detector may be comprised of an integrator and a decision device wherein the integrator may average out an integer N m  samples of the despread signal inputting the result to the decision device. 
     The decision device in the symbol detector detects the information baseband symbols on the basis of the signal constellation diagram of the baseband modulator at the FHFMSS transmitter providing the detected symbol to the baseband demodulator unit. 
     The baseband demodulator may map the detected baseband symbols into groups of m digits, wherein m=log 2  (M) assumed to be an integer, using the inverse of the map from group of m binary digits into 1 out of M possible information baseband symbols used in the baseband modulator at the FHFMSS transmitter. In various embodiments of the invention, the baseband demodulator may also be comprised of an error correction decoder and a deinterleaver. 
    
    
     
       BRIEF DESCRIPTIONS OF THE DRAWINGS 
       Various embodiments of the present invention are described here by way of examples in conjunction with the following figures, wherein: 
         FIG. 1  shows a block diagram of one embodiment of FHFMSS (Frequency Hopped Frequency Modulation Spread Spectrum) transmitter. 
         FIG. 1A  shows a diagram of a cellular communication system. 
         FIG. 1B  shows a plot of normalized interference power versus frequency pair indices. 
         FIG. 1C  shows a plot of magnitude of the correlation coefficient versus frequency pair indices. 
         FIG. 1D  shows a plot of the probability distribution of magnitude of the correlation coefficient. 
         FIG. 1E  shows a plot of normalized interference power versus frequency triplet indices. 
         FIG. 1F  shows a plot of the probability distribution of magnitude of the correlation coefficient. 
         FIG. 1G  shows a plot of the power spectral density of one embodiment of FHFM (Frequency Hopped Frequency Modulation) waveform. 
         FIG. 1H  shows a plot of the power spectral density in dBW of one embodiment of FHFM waveform. 
         FIG. 1J  shows a plot of the power spectral density in dBW of one embodiment of FHFM waveform. 
         FIG. 1K  shows a plot of the power spectral density in dBW of one embodiment of FHFM waveform. 
         FIG. 2  shows a block diagram of one embodiment of FHFMSS transmitter. 
         FIG. 3  shows a block diagram of one embodiment of time multiplexed direct digital frequency synthesizer (TMDDFS). 
         FIG. 4  shows a block diagram of one embodiment of code generation subsystem. 
         FIG. 4A  shows a block diagram of one embodiment of code generation subsystem. 
         FIG. 5  shows a plot of correlation function of a pseudo random frequency sequence. 
         FIG. 5A  shows a plot of the power spectral density in dBW of one embodiment of FHFM waveform. 
         FIG. 5B  shows a plot of the power spectral density in dBW of one embodiment of FHFM waveform. 
         FIG. 6  shows a block diagram of one embodiment of FHFMSS receiver. 
         FIG. 7  shows one embodiment of an example computer device. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The following description is provided to enable any person skilled in the art to make and use the invention and sets forth the best modes contemplated by the inventor of carrying out his invention. Various modifications, however, will remain readily apparent to those skilled in the art, since the generic principles of the present invention have been defined herein specifically to provide systems and methods for frequency hopped frequency modulation spread spectrum (FHFMSS) multiple accessing communication systems. 
       FIG. 1  shows the block diagram of one of the various embodiments of the invention. Referring to the FHFMSS (Frequency Hop Frequency Modulation Spread Spectrum) system transmitter  100  block diagram in  FIG. 1 , the input data  140  d(m) is inputted to the baseband modulator  145 . The data  140  d(m) may be binary valued taking values 0 and 1, wherein m denotes the discrete time. In various embodiments of the invention, the transmitter may be located at the base station of a mobile network and the data d(m) may be destined for a mobile user in the cell served by the base station. In various other embodiments of the invention, the transmitter may be part of the mobile user equipment with a plurality of data streams destined to the same base station, wherein the plurality of the data streams may be combined in to a single data stream d(m) by a parallel to serial converter, not shown in  FIG. 1 . 
     Referring to  FIG. 1 , the baseband modulator  145  segments the user input data  140  d(m) into groups of m binary valued data bits and maps each of the groups of the m binary data bits into one of the M=2 m , in general complex valued, information baseband symbols  150  s(m) with m selected equal to an integer greater than or equal to 1. The one to one mapping of the groups of m binary valued data bits into the corresponding baseband symbol may be based on any of the baseband modulation techniques, selected, for example, from the set of the MQAM (Multiple Quadrature Amplitude Modulation), the MPSK (Multiple Phase Shift Keying), and the MASK (Multiple Amplitude Shift Keying) modulation techniques. In various embodiments of the invention, the baseband modulator may also be comprised of an error correction code encoder and an interleaver. 
     Referring to  FIG. 1 , the information baseband symbol  150  s(m) is inputted to the FHFMSS modulator  101 . Referring to  FIG. 1 , the code generation subsystem  110  generates a multiplicity M code vector waveforms  115  c 1 (t), c 2 (t), . . . , c M (t) wherein each of the code vector waveforms c n (t) is a vector valued function of time t possibly taking values 0 and 1 with K B  denoting the dimension of the vector c n (t). In various embodiments of the invention, the various code vector waveforms may be generated by feedback shift registers of length N B  with K B  of the N B  stages&#39; outputs of the feedback shift register comprising the K B  elements of the vector waveforms c n (t),  1 ≦n≦M. In various alternative embodiments of the invention, the elements of the vector waveforms c n (t) may be periodic functions of time. 
     Referring to  FIG. 1 , the multiplicity M code vector waveforms  115  c 1 (t), c 2 (t), . . . , c M (t) are inputted to the multi frequency synthesizer  120 . The multi frequency synthesizer  120  generates a multiplicity M periodic waveforms ψ 1 (t), ψ 2 (t), . . . , ψ M (t) on the basis of the respective one of the code vector waveforms  115  c 1 (t), c 2 (t), . . . , c M (t). In various embodiments of the invention the waveform ψ n (t) may be a sinusoidal waveform with its possibly time varying frequency selected form 1 out of 2 K     B    possible values based on the 2 K     B    possible values taken by the corresponding code waveform c n (t) for n equal to 1 through M. For example with K B =8, the number of distinct frequencies of the waveforms ψ n (t) may be equal to 64. In various other embodiments of the invention the waveforms ψ n (t) may be different from the sinusoidal waveforms. 
     Referring to  FIG. 1 , the multiplicity M waveforms  125  ψ 1 (t), ψ 2 (t), . . . , ψ M (t) are inputted to the complex baseband FM (Frequency Modulation) modulator  130 . Referring to  FIG. 1 , the complex baseband FM modulator  130  is provided with the modulation coefficients  132  β 1f , β 2f , . . . , β Mf . The product of the modulation coefficient β nf  and the peak amplitude of the waveform ψ n (t) is equal to the peak frequency deviation in radians/sec due to the waveform ψ n (t) for n in the range of 1 through M. In various alternative embodiments of the invention, the FM modulator may be an IF (Intermediate Frequency) FM Modulator modulating an IF local oscillator signal  162  cos(2πf IF t) provided by the IF local oscillator  164 . 
     The complex baseband FM modulator  130  frequency modules the multiplicity M waveforms  125  ψ 1 (t), ψ 2 (t), . . . , ψ M (t) providing the complex valued FHFM (Frequency Hop Frequency Modulation) waveform  135  χ(t) given by (1). 
     
       
         
           
             
               
                 
                   
                     
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     Referring to  FIG. 1 , the information baseband symbols  150  s(m) with a symbol rate of R s =1/T s  sps (symbols per second) are inputted to the spread spectrum modulator  155  wherein T s  is the symbol period. The spread spectrum modulator  155  is provided with the complex valued FHFM waveform  135  χ(t) that modulates the information baseband symbol sequence  150  s(m) providing the baseband FHFMSS (Frequency Hop Frequency Modulation Spread Spectrum) signal  160  v b (t) at the output given by
 
v b (t)=s(m)χ(t); (m−1)T s ≦t&lt;mT s ; n=0, 1, . . . , N−1  (2)
 
     In various embodiments of the invention the waveforms ψ n (t), n=1, 2, . . . , M may be sinusoidal waveforms with their respective frequencies ω 1 , ω 2 , . . . , ω M  rad/sec and given by
 
ψ n (t)=cos (2πf n t); f n =(ω n /2π); n=1, 2, . . . , M  (3)
 
With the waveforms ψ n (t), n=1,2, . . . , M selected as the sinusoidal waveforms, the baseband FHFM waveform  135  χ(t) may be written as
 
     
       
         
           
             
               
                 
                   
                     
                       
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     In various embodiment of the invention, the frequencies f n  may all be selected equal to some integer multiples of the frequency f 0 =1/T 0  wherein T 0  is equal to the symbol period T s . In various alternative embodiments of the invention, the period T 0  may be equal to T s /N m  for some integer N m &gt;1. The modulation coefficients β nf  may be selected such that the equivalent phase modulation index β n  termed mod index for the n th  sinusoidal signal is equal to a constant β for all indices n between 1 through M. In the various embodiments of the invention the selection of the frequencies f n  and the frequency modulation coefficients β nf  in the aforesaid manner results in the FHFM waveform  135  χ(t) given by 
     
       
         
           
             
               
                 
                   
                     
                       
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     In (5) the x coefficients κ n  are some integers which may be all distinct and are equal to the normalized frequencies (f n /f 0 ). 
     Referring to  FIG. 1 , the baseband FHFMSS signal  160  v b (t) is inputted to the baseband to IF converter  165  that modulates the signal  160  v b (t) by the IF (Intermediate Frequency) local oscillator in phase and quadrature signals  162  cos(2πf IF t) and sin(2πf IF t) provided by the IF local oscillator (IFLO)  164  generating the IF FHFMSS band pass signal  170  v IF (t) given by
 
v IF (t)=Re{v b (t)exp[j2πf IF t]}=Re(v b (t))cos(2πf IF t)−Im(v b (t))sin(2πf IF t)  (6)
 
     In (6) f If  denotes the IF frequency and Re( ) and Im( ) denote the real part and imaginary part operators respectively. 
     Referring to  FIG. 1 , the IF FHFMSS band pass signal  170  v IF (t) is inputted to the RF stages block  175 . The RF stages block is inputted by the RF local oscillator signal  172  provided by the RF local oscillator (RFLO)  174  and may be comprised of an up converter, an RF band pass filter and power amplifier providing the RF FHFMSS signal  180  v RF (t) to the antenna  185 . The RF signal  180  is given by
 
v RF (t)=√{square root over (P G )}Re{v b (t)exp[j2πf c t]}  (7)
 
     In (7) f c  denotes the carrier frequency of the RF signal, and P G  is the power gain of the RF stages unit  175 . 
       FIG. 1A  shows a simplified diagram of a mobile communication network  1 . Referring to  FIG. 1A , the mobile communication network is comprised of a BS (base station)  10  and a number N u  of mobile subscriber units (MS)  20 . The mobile communication network of  Figure 1A  may operate in a FHFMSS multiple access mode wherein each of the MSs  20  may be equipped with a FHFMSS transmitter  100  and a FHFMSS receiver  700  shown in  FIG. 7 . The base station may be comprised of a multiplicity N u  of the FHFMSS transmitters and receivers that may share a common RF stage and a transmit and receive antenna and possibly a common baseband to IF converter unit wherein a multiplicity N u  of the baseband FHFMSS signals are added for providing a composite baseband FHFMSS signal that is inputted to the baseband to IF converter unit. 
     In the multiple access embodiments of the invention, a multiplicity N u  of FHFMSS transmitters located at the MS units  20   a  through  20 N u  may generate N u  distinct FHFM waveforms by selection of N u  different sets of κ coefficients κ un  with the FHFM (Frequency Hop Frequency Modulation) waveform generated by the u th  FHFMSS transmitter given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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                     N 
                     u 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     In various embodiments of the invention the number of waveforms M and the modulation coefficient β may be different for different FHFMSS transmitters. The correlation coefficient ρ u     1     u     2    between a pair of waveforms χ u     1    (t) and χ u     2    (t) may be evaluated as 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ρ 
                           
                             
                               u 
                               1 
                             
                             ⁢ 
                             
                               u 
                               2 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             
                               T 
                               0 
                             
                           
                           ⁢ 
                           
                             
                               ∫ 
                               0 
                               
                                 T 
                                 0 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   χ 
                                   
                                     u 
                                     1 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   χ 
                                   
                                     u 
                                     2 
                                   
                                   * 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     T 
                                     0 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ∫ 
                                     0 
                                     
                                       T 
                                       0 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       exp 
                                       ⁡ 
                                       
                                         [ 
                                         
                                           j 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           β 
                                           ⁢ 
                                           
                                             
                                               ∑ 
                                               
                                                 m 
                                                 = 
                                                 1 
                                               
                                               M 
                                             
                                             ⁢ 
                                             
                                               sin 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   2 
                                                   ⁢ 
                                                   
                                                     πκ 
                                                     
                                                       
                                                         u 
                                                         1 
                                                       
                                                       ⁢ 
                                                       m 
                                                     
                                                   
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   
                                                     f 
                                                     0 
                                                   
                                                   ⁢ 
                                                   t 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                     · 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   exp 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         - 
                                         j 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       β 
                                       ⁢ 
                                       
                                         
                                           ∑ 
                                           
                                             n 
                                             = 
                                             1 
                                           
                                           M 
                                         
                                         ⁢ 
                                         
                                           sin 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               2 
                                               ⁢ 
                                               
                                                 πκ 
                                                 
                                                   
                                                     u 
                                                     2 
                                                   
                                                   ⁢ 
                                                   n 
                                                 
                                               
                                               ⁢ 
                                               
                                                 f 
                                                 0 
                                               
                                               ⁢ 
                                               t 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   t 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     In various embodiments of the invention wherein M is equal to 1, each of the bracketed terms in (9) may be expanded into a complex Fourier series resulting in the correlation coefficient between the waveforms χ u (t) and χ v (t) given by 
     
       
         
           
             
               
                 
                   
                     ρ 
                     uv 
                   
                   = 
                   
                     
                       1 
                       
                         T 
                         0 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         
                           T 
                           0 
                         
                       
                       ⁢ 
                       
                         
                           { 
                           
                             
                               
                                 J 
                                 m 
                               
                               ⁡ 
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   
                                     - 
                                     ∞ 
                                   
                                 
                                 ∞ 
                               
                               ⁢ 
                               
                                 exp 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       κ 
                                       u 
                                     
                                     ⁢ 
                                     
                                       f 
                                       0 
                                     
                                     ⁢ 
                                     t 
                                   
                                   ] 
                                 
                               
                             
                           
                           } 
                         
                         ⁢ 
                         
                           { 
                           
                             
                               
                                 J 
                                 n 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   - 
                                   β 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   
                                     - 
                                     ∞ 
                                   
                                 
                                 ∞ 
                               
                               ⁢ 
                               
                                 exp 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     n 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       κ 
                                       v 
                                     
                                     ⁢ 
                                     
                                       f 
                                       0 
                                     
                                     ⁢ 
                                     t 
                                   
                                   ] 
                                 
                               
                             
                           
                           } 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     In (10) J n (β) is the Bessel function of order n and κ u  and κ v  are the normalized frequencies (f u /f 0 ) and (f v /f 0 ) of the waveforms ψ u (t) and ψ v (t) respectively for the FHFMSS transmitters with corresponding indices u and v. With the substitution of J n (=β)=J −n (β), the correlation coefficient ρ u,v  in (10) may be rewritten as 
     
       
         
           
             
               
                 
                   
                     ρ 
                     uv 
                   
                   = 
                   
                     
                       1 
                       
                         T 
                         0 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         
                           T 
                           0 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             
                               - 
                               ∞ 
                             
                           
                           ∞ 
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               
                                 - 
                                 ∞ 
                               
                             
                             ∞ 
                           
                           ⁢ 
                           
                             
                               
                                 J 
                                 m 
                               
                               ⁡ 
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 J 
                                 n 
                               
                               ⁡ 
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                             ⁢ 
                             
                               exp 
                               ⁡ 
                               
                                 [ 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     π 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           m 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             κ 
                                             u 
                                           
                                         
                                         - 
                                         
                                           n 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             κ 
                                             v 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     f 
                                     0 
                                   
                                   ⁢ 
                                   t 
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     After combining the terms of the same frequency, the expression for the correlation coefficient ρ u,v  in (11) may be simplified as 
     
       
         
           
             
               
                 
                   
                     ρ 
                     uv 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           
                             - 
                             ∞ 
                           
                         
                         ∞ 
                       
                       ⁢ 
                       
                         
                           
                             H 
                             m 
                           
                           ⁡ 
                           
                             ( 
                             β 
                             ) 
                           
                         
                         ⁢ 
                         
                           1 
                           
                             T 
                             0 
                           
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             
                               T 
                               0 
                             
                           
                           ⁢ 
                           
                             
                               exp 
                               ⁡ 
                               
                                 [ 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     mf 
                                     0 
                                   
                                   ⁢ 
                                   t 
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         H 
                         0 
                       
                       ⁡ 
                       
                         ( 
                         β 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     In (12) H m (β) is a summation of the products of the Bessel functions. For example H 0 (β) is given by 
                       H   0     ⁡     (   β   )       =       ∑     m   =     -   ∞       ∞     ⁢       ∫       n   =     -     ∞   :             n   ⁢           ⁢     κ   v       =     m   ⁢           ⁢     κ   u           ∞     ⁢         J   m     ⁡     (   β   )       ⁢       J   n     ⁡     (   β   )                     (   13   )               
The second equality in (12) follows form the fact that the integral in (12) for any non zero integer m is equal to 0.
 
     In various embodiments of the invention the κ coefficients κ u  and κ v  may be selected to be prime integers. Selection of the prime integer κ coefficients κ u  and κ v  results in most of the summation terms in (13) to be zero except those satisfying the condition that n and m are integer multiples of κ u  and κ v  respectively. For example, with κ u =7 and κ v =11, the integers m and n must be multiples of 11 and 7 respectively. For a given modulation index β, the magnitude of the function J m (β) as a function of the index m is bounded by a monotone decreasing function of m with J m (β) being relatively small in magnitude for the index m much higher than β. Under such condition, H 0 (β) in (13) may be approximated by
 
ρ uv =H 0 (β)≈J 0   2 (β)  (14)
 
     With a selection of β equal to a zero of the equation J 0 (x)=0, H 0 (β) in (13) is nearly equal to 0 with the result that the correlation coefficient between the two FHFM waveforms ρ u,v  in (9) is nearly equal to 0 showing that the two FHFM waveforms χ u (t) and χ v (t) are nearly orthogonal. 
     An evaluation of the correlation coefficient from (9) for the case of M equal to 1 shows that (14) provides a very good estimate of the correlation coefficients. For example, with N u =9, the set of N f =9 normalized frequencies κ n =(f n /f 0 ) given by S f =[5 7 11 13 17 19 23 29 31], and the modulation index β selected to be 2.4048, that is nearly equal to the first zero of the Bessel function J 0 ( ), the correlation coefficients are all close to 0. With β==2.4048, J 0 (β)=1.32×10 −5  and all of the 36 but one correlation coefficients ρ u,v  are equal to 1.78×10 −10  in good agreement with the approximation in (9). The remaining 36 th  coefficient is equal to 3.78×10 −10 . With β=2.4, J 0 (β)=0.0025 and the correlation coefficients are equal to 6.29×10 −6  compared to the value 6.25×10 −6  from the approximation (14). For the case of β equal to one of the zeros of the Bessel function J 0 ( ), and M=1, the waveforms χ u (t) in (8) are orthogonal waveforms over the interval (0, T 0 ). 
     For the case of M&gt;1, the N u  waveforms χ u (t) in (8) may be nearly orthogonal. In various multiple accessing embodiments of the invention, a subset of all possible M-tuples may be selected for minimizing the multiple access interference power resulting from non zero correlation coefficients. For example with N f =9 normalized frequencies given by the set S f  and M=2, the number of all distinct frequency pairs is equal to 
             N   =         !   9       !       2   !     ⁢   7         =   36.           
Due to non zero correlation coefficients ρ i,j  the total interference power for the user with the frequency pair i is given by
 
               P     I   ,   i       =       ∑       j   =   1       j   ≠   i       N     ⁢              ρ   ij          2     .             
With the signal power P s  equal to 1, a correlation detection of the FHFMSS signals results in the signal to total interference power ratio given by (Ps/P I,i ) and the normalized interference power (P I,i /Ps)=P I,i .
 
     By eliminating the frequency pairs for which the normalized interference power (P I,i ) is relatively high, the normalized interference power for the users with the remaining frequency pairs may be reduced. The process of elimination maybe performed in a recursive manner wherein a frequency pair corresponding to the highest normalized interference power (P I,i ) is eliminated from the set of N frequency pairs with the computation of the normalized interference power (P I,i ) performed for the remaining (N−1) frequency pairs. The procedure may be repeated a specified integer N E  times resulting in N u =(N−N E ) frequency pairs with the minimum normalized interference power. 
       FIG. 1B  shows the normalized interference power (P I,i ) for the case of M=2, N f =9 and N E =15 with N u =21, β=2.4048 and the set of normalized frequencies S f  given by [5 7 11 13 17 19 23 29 31]. The indices i 1  and i 2  in the set S f  of the normalized frequencies κ 1  and κ 2  in the N u  pairs of frequencies are given by Table 1. For example, the first pair in Table 1 has indices i 1 =1 and i 2 =2 with the normalized frequencies κ 1 =5 and κ 2 =7. From  FIG. 1 , the mean normalized interference power is equal to 0.1138. 
     
       
         
           
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 Indices in the set S f  of the normalized frequencies 
               
               
                 of N u  = 21 frequency pairs 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 i 1   
                 1 
                 1 
                 1 
                 1 
                 1 
                 1  
                 2  
                 2  
                 2 
                 3 
                 3 
                 3 
                 4 
                 4 
                 4 
                 5 
                 5 
                 6  
                 6 
                 7 
                 8 
               
               
                 i 2   
                 2 
                 3 
                 4 
                 5 
                 8 
                 9  
                 6  
                 7  
                 9 
                 5  
                 8 
                 9 
                 5 
                 7  
                 9 
                 8 
                 9 
                 7 
                 9 
                 9 
                 9 
               
               
                   
               
            
           
         
       
     
       FIG. 1C  shows the plot of the magnitudes of the N u (N u −1)/2=210 correlation coefficients among the signals χ u (t) corresponding to the selected N u =21 frequency pairs.  FIG. 1D  plots the probability distribution of the magnitudes of the  210  correlation coefficients. As may be inferred from  FIG. 1D , most of the correlation coefficients are less than 0.05 in magnitude. 
     In an alternative embodiment of the invention the number of tones M may be selected equal to 3. As an example of the M=3 case,  FIG. 1E  shows the normalized interference power (P I,i ) for the case of M=3, N f =9, N=84, and N E =40 with N u =44, β=5.52 and the set of normalized frequencies S f  given by [5 7 11 13 17 19 23 29 31]. The N u =44 frequency triplets are selected from a totality of N=84 possible distinct frequency triplets by minimization of the normalized interference power as for the case of M=2. The mean value of the normalized interference power for the users assigned the N u  frequency triplets is equal to 0.1242. The interference power may be further reduced by increasing N E  beyond 40. Alternatively N u  may be increased at the cost of some increase in the interference power. In various embodiments of the invention, the multiple access interference may be reduced to nearly 0 for the case of M&gt;1 by selecting the code vector sequences c 2 (t), c 3 (t), . . . , c M (t) to be the same for all of the N u  multiple accessing users with the c 1 (t) sequence different for the various users. 
       FIG. 1F  plots the probability distribution of the magnitudes of the N u (N u −1)/2=946 correlation coefficients. As may be inferred from  FIG. 1F , most of the correlation coefficients are less than 0.1 in magnitude. 
       FIG. 1G  plots the power spectral density of the FHFM waveform χ u (t) corresponding to one of the selected N u  normalized frequency triplets with indices in the set S f  equal to [1 4 7] for the case of M=3 with κ 1 =5, κ 2 =13 and κ 3 =23. The power spectral density is comprised of impulse functions at frequencies that are integer multiples of the frequency f 0 .  FIG. 1G  shows the magnitudes of the impulse functions versus the normalized frequency (f/f 0 ). The sum of magnitudes of the impulse functions is equal to the average signal power of the waveform that is equal to 1. 
       FIG. 1H and 1J  respectively show the plots of the power spectral density in dBW for a selected pair of the FHFM waveforms corresponding to the normalized frequency triplets equal to [5 13 23] and [5 23 29] respectively for the case of M=3. As may be observed for  FIGS. 1H and 1J , the signal power spectral density is spread over the entire signal bandwidth that may be approximated by Carson&#39;s rule wherein the one sided bandwidth is approximately equal to (Δf+f max ) wherein Δf is the peak frequency deviation in Hz and f max  is the maximum modulation frequency. For the case of  FIG. 1H , κ m =23 with f max =23f 0 , the bandwidth by Carson&#39;s rule is equal to 249.3f 0  compared to 99.9% power bandwidth of 242 f 0  evaluated from the power spectral density of the FHFM waveform. For the case of  FIG. 1J , κ m =29 with f max =29f 0 , the bandwidth by Carson&#39;s rule is equal to 343.6f 0  compare to 99.9% power bandwidth of 309 f 0  evaluated from the power spectral density of the FHFM waveform. Bandwidth approximation based on Carson&#39;s rule is somewhat more conservative than the evaluated power bandwidth due to the dense nature of the power spectral density of the FHFM waveform. 
       FIG. 1K  shows the plot of the power spectral density in dBW versus normalized frequency for a FHFM waveforms corresponding to the indices in the set S f  of the normalized frequency pair given by [2 6] for the case of M=2 and N f =9, N=36, and N E =15 with N u =21, β=2.4048 and the set of normalized frequencies S f  given by [5 7 11 13 17 19 23 29 31]. The bandwidth estimate by Carson&#39;s rule is equal to 81.5 f 0  that is nearly equal to the corresponding 99.5% power bandwidth of 82f 0  evaluated from the power spectral density of the FHFM waveform. 
     The power spectral density (PSD) of the FHFM waveform of user u for the case of M=1, not shown, is a sparse function of the frequency in that the PSD has discrete spectral lines only at the integer multiples of the frequency κ u f 0  where κ u  is the normalized frequency of the ψ u -waveform of the user u. For example with κ u =19, the power spectral density has discrete spectral lines only at the frequencies 19f 0 , 38f 0 , . . . Increasing the value of M results in making the power spectral density of the FHFM waveform a more dense function of the frequency. 
       FIG. 2  shows the block diagram of an alternative embodiment of the FHFMSS transmitter of the invention. Referring to the FHFMSS transmitter  200  in  FIG. 2 , the input data  140  d(m) is inputted to the baseband modulator  145 . The data  140  d(m) may be binary valued taking values 0 and 1, wherein m denotes the discrete time. In various embodiments of the invention, the transmitter may be located at the base station of a mobile network and the data d(m) may be destined for a mobile user in the cell served by the base station. In various other embodiments of the invention, the transmitter may be part of the mobile user equipment with a plurality of data streams destined to the same base station, wherein the plurality of the data streams may be combined in to a single data stream d(m) by a parallel to serial converter, not shown in  FIG. 2 . 
     The baseband modulator  145  segments the user input data  140  d(m) into groups of m binary valued data bits and maps each of the groups of the m binary data bits into one of the M=2 m , in general complex valued, information baseband symbols  150  s(n), n denotes discrete time index, with m selected equal to an integer greater than or equal to 1. The one to one mapping of the groups of m binary valued data bits into the corresponding baseband symbol may be based on any of the baseband modulation techniques, selected, for example, from the set of the MQAM (Multiple Quadrature Amplitude Modulation), the MPSK (Multiple Phase Shift Keying), and the MASK (Multiple Amplitude Shift Keying) modulation techniques. In various embodiments of the invention, the baseband modulator may also be comprised of an error correction code encoder and an interleaver. 
     Referring to  FIG. 2 , the code generation subsystem  210  is comprised of code clock unit  205  for providing a clock signal  208  at the clock rate of f 0 . In various alternative embodiments of the invention, the clock rate of the clock signal  208  may be an integer sub multiple of f o . Referring to  FIG. 2 , in some embodiments of the invention, the clock rate signal inputted to the multiplicity M code generators may be different. Referring to  FIG. 2 , the code clock signal  208  is provided to a multiplicity M of PN code generators  210  a, . . .  210 M for generating a multiplicity M binary valued code vector sequences  215   a  through  215 M c 1 (n), c 2 (n), c M (n). Throughout this application, the notations 1, 2, . . . , N, or a, b, N, or a through N are all equivalent and denote the enumeration of integers 1 to N for any integer N. Each of the code vectors  215  a through M is of dimension K B  wherein each component of the vectors c i (n),  1 ≦i≦M may take values 0 and 1 and n denotes discrete time index. The dimension K B  is selected such that 2 K     B   ≧N f  wherein N f  is the number of periodic ψ-waveforms or the number of frequencies when the ψ-waveforms are selected to be the sinusoidal waveforms. 
     In various embodiments of the invention, the code generators  215  may be implemented with feedback shift registers of length N B  with K B  of the N B  stages&#39; outputs of the feedback shift register comprising the K B  elements of the code vectors c i (n) termed sub state code vectors. Different code sequences may be generated by using different feedback connections in the feedback shift registers. In various embodiments of the invention the multiplicity M binary valued code vector sequences  215   a  through  215 M c 1 (n), c 2 (n), c M (n) may be generated as appropriate delayed versions of a single sequence binary valued code vector sequences c 1 (n). 
     In various multiple access systems embodiments of the invention, the multiple access interference may be reduced to approximately 0 for M&gt;1 by selecting the code vector sequences c 2 (n), c 3 (n), . . . , c M (n) to be the same for the multiplicity N u  multiple access users with different code sequences c 1 (n) for different users wherein the different code sequences c 1 (n) are mapped into the same set S 1  of the frequency selection indices. The sets of frequency selection indices corresponding to the multiplicity M code sequences may be all disjoint sets. Such an assignment of the code vector and frequency selection indices to the multiplicity N u  multiple access users results in the correlation coefficient among any pair of the FHFM waveforms χ u (t) and χ v (t) with a corresponding pair of M tuples of the ψ-waveforms for a pair of multiple access users u and v to be equal to the correlation coefficient of a corresponding pair of χ(t) waveforms each with only a single ψ-waveform that are distinct resulting in an approximately 0 correlation coefficient. Approximately 0 correlation coefficient between the FHFM waveforms may thus be achieved while the FHFM waveforms possess a dense power spectral density function. 
     Referring to  FIG. 2 , the multiplicity M code vector sequences  215   a  through  215 M c 1 (n), c 2 (n), c M (n) are inputted to the multiplexer  220  that time multiplexes the M inputs in to a single multiplexed code vector sequence  222  c(l) wherein 1 is the discrete time index. A multiplicity of different symbols to designate the discrete time index may refer to different time scales in various discrete time sequences. Referring to  FIG. 2 , the multiplexed code vector sequence  222  c(l) is inputted to a code converter unit  225 . The code converter unit  225  may map the 1 out of 2 K     B    possible set of values taken by the elements of the multiplexed code vector  222  c(l) into a K F  bit frequency selection indices with a one to one correspondence providing the output frequency selection indices sequence  227  c a (l). 
     In various embodiments of the invention, the frequency selection indices may be selected from the set of normalized frequencies κ n =f n /f 0  in the set S f  and the number of bits K F  may be selected such that 2 K     F   ≧κ max  wherein κ max  is the maximum normalized frequency in the set S f  for the case of sinusoidal waveforms ψ n (t). In various embodiments of the invention, the multiplicity M code vector sequences may be mapped into disjoint sets of frequency selection indices by the code converter. The frequency selection indices may be selected from the set of prime integers. The mapping of the code vectors into a frequency selection index may be performed by a table look up. 
     Referring to  FIG. 2 , the frequency selection indices sequence  227  c a (l) is inputted to the demultiplexer  230 . The demultiplexer  230  time demultiplexes the input frequency selection indices sequence into the multiplicity M frequency selection indices subsequences c a     i   (n), i=1, 2, . . . , M. 
     Referring to  FIG. 2 , the multiplicity M frequency selection indices subsequences c a     1   (n), c a     2   (n), . . . , c a     M    (n)  235   a, b , . . . , M are inputted to the time multiplexed direct digital multi frequency synthesizer (TMDDFS)  240 . The multi frequency synthesizer  240  generates a discrete time waveform ψ I (η) that is a time multiplexed version of the multiplicity M waveforms ψ 1   I (η), ψ 2   I (η), . . . , ψ M   I (η) with η denoting the time index and ψ i   I (η) is the discrete time version with K s  number of bits per sample of the waveform ψ i   I (t) given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ψ 
                           i 
                           I 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       = 
                       
                         
                           α 
                           i 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             t 
                           
                           ⁢ 
                           
                             
                               
                                 ψ 
                                 i 
                               
                               ⁡ 
                               
                                 ( 
                                 τ 
                                 ) 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅆ 
                               τ 
                             
                           
                         
                       
                     
                     ; 
                     
                       i 
                       = 
                       1 
                     
                   
                   , 
                   2 
                   , 
                   
                     … 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     M 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     In (15) α i  is a scale factor selected such that the peak magnitude of the waveform ψ i   I (t) is equal to 1. 
     In various embodiments of the invention the waveform ψ i (t) and ψ i   I (t) may be sinusoidal waveform with their possibly time varying frequencies selected from 1 out of 2 K     B    possible values based on the 2 K     B    possible values taken by the elements of the corresponding code vector c i (l) for i equal to 1 through M. For example with K B =8, the number of distinct frequencies of the waveforms ψ i (t) may be equal to 64. In various other embodiments of the invention waveforms ψ i (t) may be different from the sinusoidal waveforms. In various alternative embodiments of the invention, the multiplicity M waveforms ψ 1   I (η), ψ 2   I (η), . . . , ψ M   I (η) may be generated by one of the various frequency synthesizers such as the indirect frequency synthesizer based on phase lock loops, direct synthesizers, etc. 
     Referring to  FIG. 2 , the discrete time waveform  244  ψ I (η) is inputted to the demultiplexer  245 . The demultiplexer  245  time demultiplexes the waveform  244  ψ I (η) into multiplicity M discrete time waveforms  246   a, b , . . . , M ψ 1   I (n), ψ 2   I (n), . . . , ψ M   I (n). Referring to  FIG. 2 , the multiplicity M discrete time waveforms  246   a, b , M ψ 1   I (n), ψ 2   I (n), . . . , ψ M   I (n) are inputted to the digital baseband FM (Frequency modulation) modulator  250 . The baseband FM modulator  250  is provided with the mod indices  248   a, b , . . . , M β 1 , β 2 , . . . , β M  corresponding to the waveforms ψ 1 (t), ψ 2 (t), . . . , ψ M (t). 
     Referring to  FIG. 2 , the multiplicity M discrete time waveforms  246   a, b , . . . , M ψ 1   I (n), ψ 2   I (n), . . . , ψ M   I (n) are inputted to the scalar multipliers  247   a,b , . . . , M that multiply the waveforms  246   a, b , . . . , M by the respective mod indices  248   a, b , M β 1 , β 2 , . . . , β M  providing the resulting scaled waveforms  249  a through M to the adder  255 . The adder provides the resulting sum waveform  256  φ(n) to the rate converter unit  260 . In various alternative embodiments wherein the mod indices β 1 , β 2 , . . . , β M  are all equal to the same mod index β, the multiplicity M discrete time waveforms  246   a, b , M ψ 1   I (n), ψ 2   I (n), . . . , ψ M   I (n) may be added first by the adder  255  followed by scalar multiplication by β by a single scalar multiplier  248  for providing the waveform  256  φ(n) to the rate converter unit  260 . 
     The sampling rate of various waveforms is determined by the bandwidths of the waveforms to avoid aliasing errors. According to Nyquist theorem the sampling rate must be at least two times the bandwidth of the waveform termed the Nyquist rate. For example, for the case of sinusoidal waveform ψ m (t) with frequency f m =κ m f 0 , the sampling rate must be greater than or equal to 2κ m  per code chip period T 0 . The sampling rate for the multiplicity M discrete time waveforms  246   a, b , M ψ 1   I (n), ψ 2   I (n), . . . , ψ M   I (n) may be selected to be 4κ max  per code chip period T 0  wherein κ max  is the maximum normalized frequency. The bandwidth B s  of the FHFM waveform χ(t) given by (5) may be estimated by Carson&#39;s rule as B s =(Δf+f max ) where Δf is the peak frequency deviation equal to β(κ 1 +. . . +κ M )f 0 , and f max =κ m f 0  is the maximum modulation frequency with κ m  equal to the maximum of the normalized frequencies κ 1 , . . . , κ M . The sampling rate of the FHFM waveform may be determined on the basis of an upper bound on the bandwidth B s =(1+Mβ)κ max f 0  and must be sampled at a rate greater than or equal to 2B s  samples/sec. Referring to  FIG. 2 , the rate converter unit  260  increases the sampling rate of the waveform  256  φ(n) by a factor of α(1+Mβ) wherein the scalar a depends upon the rates selected for the waveforms ψ m (t) and χ(t) and is equal to 1 if both the waveforms ψ m (t) and χ(t) are sampled at their respective Nyquist rates. 
     In various embodiments of the invention, the sampling rate of the waveforms for the multiplicity M discrete time  246   a, b , M ψ 1   I (n), ψ 2   I (n), . . . , ψ M   I (n) may be selected to be greater than or equal to the Nyquist rate of the waveform χ(t) eliminating the need for the rate converter  260 . However, this will require the operation of the TMDDFS unit  240 , the scalar multipliers  247  and the adder  255  at the increased sampling rate. Both the waveforms ψ m (t) and ψ m   I (n), m=1, 2, . . . , M are here to fore also referred to as the ψ-waveforms. 
     Referring to  FIG. 2 , the rate converted waveform  262  φ(k) is inputted to the exponential function unit  265  that provides the discrete time FHFM waveform  268  χ(k) with sampling period t s  given by (16). 
     
       
         
           
             
               
                 
                   
                     χ 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     exp 
                     ⁡ 
                     
                       [ 
                       
                         j 
                         ⁢ 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             M 
                           
                           ⁢ 
                           
                             
                               β 
                               n 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   k 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     f 
                                     n 
                                   
                                   ⁢ 
                                   
                                     t 
                                     s 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Referring to  FIG. 2 , the information baseband symbols  150  s(n) is inputted to the spread spectrum modulator  275  for modulation by the FHFM waveform  268  χ(k). Referring to  FIG. 2 , the information baseband symbols  150  s(n) is inputted to the rate converter unit  272  that increases the sampling rate of the information baseband symbols  150  s(n) from R s  to the sampling rate of the FHFM waveform χ(k) that is greater than or equal to 2 B S  with B S  equal to the bandwidth of the FHFM waveform. The rate converted symbol sequence  273  is inputted to the modulator  274 . Referring to  FIG. 2 , the FHFM waveform  268  χ(k) is inputted to the modulator  274  for modulating the rate converted symbol sequence s(k)  273  providing the discrete time baseband FHFM spread spectrum signal  276  v b (k) given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         v 
                         b 
                       
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       
                         
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                           ⁡ 
                           
                             ( 
                             k 
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                         · 
                         
                           χ 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       = 
                       
                         
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                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         · 
                         
                           exp 
                           ⁡ 
                           
                             [ 
                             
                               j 
                               ⁢ 
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   M 
                                 
                                 ⁢ 
                                 
                                   
                                     β 
                                     n 
                                   
                                   ⁢ 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         2 
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                                         ⁢ 
                                         
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                                           n 
                                         
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                             ] 
                           
                         
                       
                     
                   
                   ; 
                   
                     j 
                     = 
                     
                       
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     The power spectral density (PSD) of the baseband FHFM spread spectrum signal  276  v b (k) is given by the convolution of the PSD of the FHFM waveform with the PSD of the information baseband signal  150  s(m). The one sided bandwidth B SS  of the spread spectrum signal  276  v b (k) is equal to the sum of the one sided bandwidth B S  of the FHFM waveform and the one sided bandwidth B I  of the information baseband symbol  150  s(m) wherein B S  may be orders of magnitude higher than B I . For any selected baseband modulation technique the bandwidth B I  may be evaluated in terms of the symbol rate R s . For example, for MQAM modulation B I  is equal to R s . 
     Referring to  FIG. 2 , the discrete time baseband FHFM spread spectrum signal  276  v b (k) is inputted to the digital to analog converter  280  for providing the baseband FHFMSS signal  282  v b (t) at the output. 
     Referring to  FIG. 2 , the baseband FHFMSS signal  282  v b (t) is inputted to the baseband to IF converter  165  that modulates the signal  282  v b (t) by the IF (Intermediate Frequency) local oscillator in phase and quadrature signals  162  cos(2πf IF t) and sin(2πf IF t) provided by the IF local oscillator (IFLO)  164  generating the IF FHFMSS band pass signal  286  v IF (t) given by
 
v IF (t)=Re{v b (t)exp[j2πf IF t]}=Re(v b (t))cos(2πf IF t)−Im(v b (t))sin(2πf IF t)  (18)
 
     In (18) f If  denotes the IF frequency and Re( )and Im( )denote the real part and imaginary part operators respectively. The baseband to IF converter unit  165  may also be comprised of a band limiting IF filter that may limit the bandwidth of the IF signal to be smaller than two times the 99.9% bandwidth B SS  of the baseband FHFM spread spectrum signal v b (k). 
     Referring to  FIG. 2 , the IF FHFMSS band pass signal  286  v IF (t) is inputted to the RF stages block  175 . The RF stages block is inputted by the RF local oscillator signal  172  provided by the RF local oscillator (RFLO)  174  and may be comprised of an up converter, an RF band pass filter and power amplifier providing the RF FHFMSS signal  285  v RF (t) to the antenna  185 . The RF signal  285  is given by
 
v RF (t)=√{square root over (P G )}Re{v b (t)exp[j2πf c t]}  (19)
 
     In (19) f c  denotes the carrier frequency of the RF signal, and P G  is the power gain of the RF stages unit  175 . 
       FIG. 3  is the block diagram of an embodiment of the time division direct digital frequency synthesizer (TMDDFS)  240 . Referring to  FIG. 3 , the multiplicity M frequency selection indices subsequences  235   a,  M c a     m    (n), m=1, 2, . . . , M are inputted to the respective ones of the frequency register blocks  310   a , . . . , M of the TMDDFS  240 . The frequency register block  310   m,  m=1, 2, . . . , M stores the frequency selection index  235   m  c a     m    (n) held constant during any code chip period T c  that may be equal to T 0 . In various embodiments of the invention, the frequency selection index  235   m  c a     m    (n) may be equal to the normalized frequency κ m  during the n th  code chip period. Referring to  FIG. 3 , the normalized frequency  315   m  κ m  provided by the frequency register  310   m  is inputted to phase accumulator  325   m.  The phase accumulator  325   m  modulo N s  accumulates the normalized frequency κ m  at a sampling rate equal to the sampling rate of the ψ-waveforms that may be selected to be N s  times the frequency f 0 , wherein N s  is an integer, providing the normalized accumulated phase  322   m  ξ m (n) at the output with n denting the time index at the sampling rate N s f 0 . Referring to  FIG. 2 , in various embodiments of the invention, the code clocks inputted to the PN code generators  210  a through M may have clock periods T c  that are different integer multiples of T 0 . In some embodiments of the invention, the period of the multiplicity M code vector sequences may be equal to 1 wherein the frequencies of the ψ-waveforms generated by the synthesizer do not very with time. 
     Referring to  FIG. 3 , the normalized frequency  315   m  κ m  is inputted to the adder  316   m . The output of the adder  316   m  is inputted to the modulo N s  unit  320   m.  The modulo N s  unit  320   m  performs the integer division of the input by the integer N s  and provides the remainder at the output  321   m  to the phase register  323   m.  Referring to  FIG. 3 , the output of the phase register  323   m  is delayed by one sample time by the delay unit  324   m  with the delayed output provided to the adder  316   m.  With N s  selected to be the integer multiple of κ m , m=1, 2, . . . , M, during any period T 0  comprised of N s  samples of the ψ-waveform, the phase accumulator  325   m  output is given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ξ 
                           m 
                         
                         ⁡ 
                         
                           ( 
                           n 
                           ) 
                         
                       
                       = 
                       
                         
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                             N 
                             s 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             n 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
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                               m 
                             
                           
                           ) 
                         
                       
                     
                     ; 
                     
                       n 
                       = 
                       0 
                     
                   
                   , 
                   1 
                   , 
                   … 
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     For example, for the case of κ m =1 and κ m =2, ζ m (n) sequence is given by
 
ξ m (n)=0, 1, . . . , N s −1, 0 1, . . . ; κ m =1  (22a)
 
ξ m (n)=0, 2, . . . , N s −2, 0 1, . . . N s −2, 0, 2, . . . ; κ m =2  (22b)
 
     For κ m  equal to any sub integer multiple m of N s , there are m periods each of length (N s /m) during the N s  samples at the output of the phase accumulator. 
     Referring to  FIG. 3 , the outputs  322   m  ξ m (n), m=1, 2, . . . , M of the multiplicity M phase accumulators  325   1  through M are inputted to the multiplexer  330 . The multiplexer  330  time multiplexes the M outputs  322   m  ξ m (n) providing the multiplexed output  332  Θ(η) at a sampling rate of MN s f 0  sps (samples per second). Referring to  FIG. 3 , the multiplexed output  332  Θ(η) is inputted to the memory address register (MAR)  335 . The memory address register stores the multiplexer output  332  Θ(η) and may add a constant offset term, for example 1, providing the address  338  ζ(η) of the data stored in the ROM  340 . 
     Referring to  FIG. 3 , the ROM memory unit  340  stores the data at N s  locations with the content of the location at the address  338  ζ(η) given by sin(2πk/N s ) with k equal to the address  338  ζ(η) minus any offset introduced by the MAR, that is, the N s  memory locations store the N s  samples of one period of a sine wave of period T 0 . Referring to  FIG. 3 , the output  342  Ψ I (η) of the ROM memory  340  is stored in the data buffer  345  that provides the multiplexed ψ waveform  244  Ψ I (η) at the output. The number of bits K S  for each data word at any one of the memory addresses may be determined on the basis of the signal to quantization noise power ratio that is approximately given by 6K S  dB. For example, a selection of K S =8 results in 48 dB signal to quantization noise ratio. The number of samples N s  may be selected to be an integer v≧4 times the maximum normalized frequency η max . For example, with η max =100, a memory size of  400  words may be sufficient. 
       FIG. 4  is the block diagram of an embodiment of the code generation subsystem  400  wherein a multiplicity M of PN (pseudo random) codes are generated by appropriate delayed versions of a single PN code. Referring to  FIG. 4 , the feedback shift register is comprised of N B  shift register stages  410 - 1  through  410 -N B . The outputs of the N B  shift register stages  410 - 1  through  410 -N B  are multiplied by the respective N B  binary valued feedback coefficients  420 - 1  through  420 -N B  g 1 , g 2 , . . . , g NB  by multipliers  415 - 1  through  415 -N B . Referring to  FIG. 4 , the outputs of the multipliers  415 - 1  through  415 -N B  are added by the modulo  2  adders  420 - 1  through  420 -N B  providing the binary sum  416  to the input of the multiplier  415 - 0 . The operations of all the units in the code generation subsystem  400  is in modulo  2  arithmetic or in binary field. Referring to  FIG. 4 , the binary sum  416  is multiplied by the binary coefficient  420 - 0  g 0  for providing the output to the input  430   a - 0  of the shift register  410 . The binary valued coefficients  420 - 0  through  420 -N B  g 0 , g 1 , g 2 , . . . , g N     B    may be selected such that the polynomial g(D)=g 0 +g 1 D+g 2 D 2 +. . . +g N     B    D N     B    in the delay operator D is a primitive polynomial of degree N B  for generation of a maximal length sequence with period 2 N     B   −1 at the output x i    420 -N B  . 
     In prior art frequency hopped spread spectrum systems, the outputs of all of the N B  stages of the feedback shift register may be used to select the hop frequency, with a one to one mapping of the shift register states into the synthesizer frequencies, wherein the number of distinct frequencies, equal to the number of states of the feedback register, that need to be generated by the frequency synthesizer is equal to the period 2 N     B   −1. The state of the shift register is the length N B  vector comprised of the binary outputs of the N B  shift register stages. The number of shift register states and the period of the sequence of frequencies generated by the frequency synthesizer based on the state of the shift register when g(D) is a primitive polynomial are both equal to 2 N     B   −1. 
     In the FHFMSS transmitter of the invention, the complexity of the frequency synthesizer may be reduced by reducing the required number of distinct frequencies to be generated by the frequency synthesizer without reducing the period of the sequence. In order to reduce the number of distinct frequencies to be generated by the frequency synthesizer while keeping the period of the sequence of frequencies unchanged, only a subset K B  of the N B  stages of the feedback shift register are used to select the synthesizer frequency in  FIG. 4 . Referring to  FIG. 4 , the K B , K B ≦N B , outputs x 1 , x 2 , . . . , x K     B    of the stages  410 -(N B −K B +1) through  410 -N B  of the shift register constitute the elements of the first code vector c 1 (k), k is the discrete time index, provided at the output of the code generation subsystem  400 . That the sequence of frequencies generated by one to one map of the 2 K     B    values of the sub state vector [x 1 , x 2 , . . . , x K     B   ] into the synthesizer frequencies has a period 2 N     B   −1 may be seen as follows. If the period were less than 2 N     B   −1 then necessarily the component x 1  of the sub state vector will have a period less than 2 N     B   −1. However, x 1  is the PN code at the output of the feedback shift register and has period 2 N     B   −1, thus the period of the sequence of frequencies generated by one to one map of the  2   K     B    values of the sub state vector must be 2 N     B   −1. 
     Referring to  FIG. 4 , a multiplicity M−1 code vector sequences may be generated as some selected delayed versions of the code vector sequence c 1 (n). Delaying of the component x i  by L chip periods may be implemented by delaying the input  430   a - 0  to the shift register  410  by L−N B  code chips if L&gt;N B  and (2 N     B   +L−N B ) chips if L&lt;N B . Delaying of the input sequence  430   a - 0  denoted b(k) by any integer L may be implemented by multiplying the corresponding polynomial b(D) by the polynomial h(D) wherein
 
b(D)=b 0 +b 1 D+b 2 D 2 +. . . ; h(D)=mod(D L )g(D)  (23)
 
     In (23) {b 0 , b 1 , . . . } denotes the sequence at the input  430   a - 0  and modulo g(D) operation refers to the remainder polynomial obtained after the Euclidean division of D L  by g(D). 
     Referring to  FIG. 4 , the sequence b(k) is delayed sequentially by a multiplicity M−1 integers L 1 , L 2 , . . . L M−1  providing the delayed versions of b(k) with cumulative delays L 1 , (L 1 +L 2 ), . . . , (L 1 +L 2 +. . . +L M−1 ) with the corresponding polynomials h 1 (D) . . . , h M−1 (D). Referring to  FIG. 4 , the sequence  430   a - 0  b(k) and its delayed versions  430   a - 1  through  430   a -N B  are multiplied by the respective binary coefficients of the polynomial h 1 (D)  435   a - 0  through  435   a -N B  h 0   1 , h 1   1 , . . . , h N     B     1  by the multipliers  438   a - 0  through  438   a -N B  providing the result of multiplication  440   a - 0  through  440   a -N B  to the adder  442   a.  The output  430   b - 0  of the adder  442   a  is the L 1  chips delayed version of b(k). The binary coefficients  435   a - 0  through  435   a -N B  h 0   1 , h 1   1 , . . . , h N     B     1  are the coefficients of the polynomial h 1 (D) that is the remainder polynomial obtained after the Euclidean division of D L     1    by g(D). Referring to  FIG. 4 , the first delayed sequence  430   b - 0  is inputted to the shift register  410   b - 1  through  410   b -N B . The outputs of the stages 1 through K B  x 1   2 , x 2   2 , . . . , x K     B     2  of the shift register  410   b  constitute the elements of the second code vector c 2 (k). In the like manner, the first delayed sequence  430   b - 0  may be delayed by a delay of L 2  chips providing the third code vector c 3 (k) and so on providing the code vectors c 1 (k), . . . , c M (k) at the output of the code generation subsystem  400  for inputting to the multiplexer unit  220 . 
       FIG. 4A  shows the block diagram of an illustration of the code generation subsystem  400  for the case of M=2, N B =6, K B =4 and the delay L 1 =6 with the primitive polynomial g(D)=1+D 6 . The polynomial h 1 (D) is given by Euclidean division of D 6  by g(D) and is equal to (1+D). Referring to  FIG. 4A , the first code generator is comprised of the shift register  450  with the shift register feedback comprised of the modulo  2  adder  465 . Referring to  FIG. 4A , the modulo  2  adder  465  is inputted with the outputs of the stages  450 - 1  x 6  and  450 - 6  x 1 . The outputs of the last four stages  450 - 3  through  450 - 6  of the shift register comprise of the elements of the first code vector c 1 (k). 
     Referring to  FIG. 4A , the sequence  460 - 0  b(k) at the input of the shift register  450  is delayed by L 1 =6 chips by multiplying the polynomial b(D) corresponding to the sequence  460 - 0  b(k) by the polynomial h 1 (D)=1+D. Referring to  FIG. 4A , the polynomial division by h 1 (D) is implemented by adding the sequence b(k)  460 - 0  with the output  460 - 1  of the first stage of the shift register  450  via the adder  468 . The output of the adder  468  is inputted to the shift register  470 . Referring to  FIG. 4A , the outputs of the last four stages  470 - 3  through  470 - 6  of the shift register  470  comprise of the elements of the second code vector c 2 (k). 
     Referring to  FIG. 2 , in various embodiments of the invention the code converter may map the code vector  222  c(l) into the normalized frequency κ(l) with a BCD (binary coded decimal) code. Referring to  FIG. 4A , with the shift register  450  initial state selected to be 0 . . . 0 1, the sequence of the normalized frequencies corresponding to the first code is given by [1 0 0 8 12 14 15 15 15 7 11 5 10 5 10 13 6 3 9 12 6 11 13 14 7 11 13 6 11 5 2 9 4 2 9 12 14 7 3 1 8 4 10 13 14 15 7 3 9 4 10 5 2 1 8 12 6 3 1 0 8 4 2] 
     The period of the normalized frequencies sequence is 63 that is equal to the period of the maximal length sequence of the PN code generated by a six stage feedback shift register. The distribution V of the normalized frequencies x is give by Table 2. 
     
       
         
           
               
             
               
                 TABLE 2 
               
               
                   
               
               
                 Distribution of the normalized frequency κ 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 κ 
                 0 
                 1  
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
               
                 ν 
                 3  
                 4  
                 4  
                 4  
                 4  
                 4 
                 4 
                 4  
                 4 
                 4 
                  4 
                  4 
                  4 
                  4 
                  4 
                  4 
               
               
                   
               
            
           
         
       
     
     Table 2 shows that the distribution of the normalized frequencies κ is nearly uniform over the interval [0, 15] that is a property of random sequences.  FIG. 5  plots the correlation coefficient versus the delay of the normalized frequency sequence corresponding to the code vector c 1 (k) of  FIG. 4A . Referring to  FIG. 5 , the correlation coefficient is periodic in delay with a period of 64 and has a relatively small value of −0.0439 for the delay in the range of 4 to 60. The BCD frequency indices may also be mapped into a set of prime integers wherein the generated frequency sequences may inherit similar properties of the random sequences. 
     In various multiple access systems embodiments of the invention, the generation of different code vectors by feedback shift registers in the multiplicity N u  multiple accessing FHFMSS transmitters may result in a collision among various FHFMSS signals in that with some nonzero probability, more than one FHFMSS signals may use the same M-tuple of periodic ψ-waveforms. Such a collision may result in an erasure of the information baseband symbol transmitted during the period of collision. In prior art FHSS systems, error correction codes with a capability of correcting for an expected number of such erasures are required for a successful transmission. In various embodiments of the invention, such collisions and erasures may be avoided by the generation of the i th  code vector sequence for the multiple accessing user j c ij (n) by a cyclic permutation of the corresponding i th  code vectors c il (n), or that of the sub state vectors comprised of the K B  stages of the feedback shift register generating the code vectors c il (n), for the multiple accessing user  1 . Different multiple accessing users are assigned the corresponding different cyclic permutation of the code vectors c il n). The cyclic permutation may be implemented by the modulo 2 K B addition of an integer between 1 and 2 K     B   −1 to the BCD (Binary Coded Decimal) equivalent of the sub state vector. For example, with K B =6, 2 K     B   =64, the BCD equivalent of the sub state vector of user  1  at any one time may take one of the values 0, 1, 2, . . . , 63. Modulo  64  addition of integer  1  to the BCD equivalent of the sub state vector for the generation of the code for user  2 , the sub state vector for user  2  has the corresponding values 1, 2, 3, . . . , 64, 0 respectively and corresponds to a cyclic permutation of the sub state vectors of user  1 . In a like manner, modulo  64  addition of integers  2  through  63  results in different cyclic permutations of the sub state vectors of user  1 . Generation of the code vector sequences in this manner ensures that no two users will have the same sub state vector or the same code vector at any time thereby avoiding any possibility of collision among the FHFMSS signals of the various multiple accessing users. 
     In various multiple access systems embodiments of the invention, one or more of the multiplicity M code vector sequences may be common for the multiplicity N u  multiple accessing users. In some of the embodiments, the common sequences may have a period equal to 1. The normalized frequency indices corresponding to the common sequences may be relatively prime to the normalized frequencies corresponding to the other sequences. For example with M=3, the first normalized frequencies may take values from the set of prime numbers {5, 7, 11, 13, . . . } whereas the second and third sequences may have a period equal to 1 with the normalized frequencies equal to 8 and 9 respectively resulting in correlation coefficient ρ uv  among any pair of multiple accessing user FHFM waveforms equal to the correlation coefficient obtained for the case of M=1 periodic ψ-waveform in the generation of the FHFM waveforms of the two users that is nearly equal to 0. However, unlike the M==1 case, the FHFM waveforms for the case of M=3 periodic ψ-waveforms possess dense power spectral density functions. For example,  FIG. 5A  shows the plot of the PSD of the FHFM waveform for the case of the normalized frequency triplet taken from the above example and equal to [19, 4, 9] with the corresponding mod indices equal to β 1 =2.4048, β 2 =β 3 =1. As may be inferred from the  FIG. 5A , the power spectral density of the FHFM waveform is a relatively dense function of frequency with a 99.9% power bandwidth equal to  77 f 0  comparable to the bandwidth equal to  77 . 7 f 0  estimated from Carson&#39;s rule. In comparison the sparse power spectral density function shown in  FIG. 5B  of the FHFM waveform for the case of a single normalized frequency of 19 only has nonzero spectral lines spaced at multiples of  19 f 0  with the 99.9% power bandwidth equal to  76 f 0 . 
       FIG. 6  shows an embodiment of the FHFMSS receiver  600  of the invention. Referring to  FIG. 6 , the transmitted FHFMSS signal at the center frequency f c  is received by the receive antenna  610  providing the received RF (Radio frequency) signal  615  v RF (t) to the RF front stage unit  620 . The RF front stage unit  620  is provided with the RF local oscillator signal  628  cos(2πf LO   RF t) generated by the RF local oscillator  625  wherein the RF local oscillator frequency is given by f LO   RF =(f c −f IF ). The RF front stage unit  620  may be comprised of a band pass filter to reject out of band noise and interference, a down converter to down convert the center frequency of the input RF signal  307  to some appropriate intermediate frequency (IF) f IF , and the RF and IF amplifiers. 
     Referring to  FIG. 6 , the RF front stage unit  620  provides the IF signal  630  v IF (t) to the IF to complex baseband converter  635 . The IF to complex baseband converter  635  is provided with the IF local oscillator signal in phase and quadrature signals  642  cos(2πf IF t) and sin(2πf IF t) generated by the NCO (Numerically Controlled Oscillator)  640 . The IF to complex baseband converter  635  may be comprised of demodulating the IF signal  630  v IF (t) by the in phase and quadrature signals  642  cos(2πf IF t) and sin(2πf IF t) and filtering the resulting signals by the low pass filters. The outputs of the low pass filters comprise of the real and imaginary components of the complex baseband signal  645  v b (t) at the output of the IF to complex baseband converter  635 . 
     
       
         
           
             
               
                 
                   
                     
                       v 
                       b 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           ∑ 
                           
                             u 
                             = 
                             1 
                           
                           
                             N 
                             u 
                           
                         
                         ⁢ 
                         
                           
                             
                               s 
                               u 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               χ 
                               u 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                       + 
                       
                         n 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             u 
                             = 
                             1 
                           
                           
                             N 
                             u 
                           
                         
                         ⁢ 
                         
                           
                             
                               s 
                               u 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           ⁢ 
                           
                             exp 
                             ⁡ 
                             
                               [ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 β 
                                 ⁢ 
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       1 
                                     
                                     M 
                                   
                                   ⁢ 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         2 
                                         ⁢ 
                                         
                                           πκ 
                                           un 
                                         
                                         ⁢ 
                                         
                                           f 
                                           0 
                                         
                                         ⁢ 
                                         t 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                       + 
                       
                         n 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     In ( 24 ) N u  is the number of multiple accessing users and n(t) is comprised of the receiver noise and any external interference, and s u (t) is the continuous time version of the information symbol sequence s u (k) of the multiple accessing user u. The signal s u (t) is equal to the information symbol s u (k) during the period kT s ≦t&lt;(k+1)T s , k=0, 1, . . . with T s  denoting the symbol period. Referring to  FIG. 6 , the complex baseband signal  645  v b (t) is inputted to the spread spectrum demodulator  650 . 
     Referring to  FIG. 6 , the code generation subsystem  110  generates a multiplicity M code vector waveforms  115  c 1 (t), c 2 (t), . . . , c M (t) wherein each of the code waveforms c n (t) is a vector valued function of time t possibly taking values 0 and 1 with K B  denoting the dimension of the vector c n (t). In various embodiments of the invention, the various code waveforms may be generated by feedback shift registers of length N B  with K B  of the N B  stages&#39; outputs of the feedback shift register comprising the K B  elements of the vector waveforms c n (t). In various other elements of the invention, the elements of the vector waveforms c n (t) may be periodic functions of time. 
     Referring to  FIG. 6 , the multiplicity M code vector waveforms  115  c 1 (t), c 2 (t), . . . , c M (t) are inputted to the multi frequency synthesizer  120 . The multi frequency synthesizer  120  generates a multiplicity M waveforms ψ 1 (t), ψ 2 (t), . . . , ψ M (t) on the basis of the respective one of the code waveforms  115  c 1 (t), c 2 (t), . . . , c M (t). In various embodiments of the invention the waveform ψ n (t) may be a sinusoidal waveform with its possibly time varying frequency selected form 1 out of 2 K     B    possible values based on the 2 K     B    possible values taken by the corresponding code waveform c n (t) for n equal to 1 through M. For example with K B =8, the number of distinct frequencies of the waveforms ψ n (t) may be equal to 64. In various other embodiments of the invention waveforms ψ n (t) may be different from the sinusoidal waveforms. In multiple accessing system embodiments of the invention, various users may have different code waveforms  115  c 1 (t), c 2 (t), . . . , c M (t) with corresponding different selection of the waveforms ψ 1 (t), ψ 2 (t), . . . , ψ M (t). 
     Referring to  FIG. 6 , the multiplicity M waveforms  125  ψ 1 (t), ψ 2 (t), . . . , ψ M (t) are inputted to the complex baseband FM (Frequency Modulation) modulator  130 . Referring to  FIG. 1 , the complex baseband FM modulator  130  is provided with the modulation coefficients  132  β 1f , β 2f , . . . , β Mf . The product of the modulation coefficient β nf  and the peak amplitude of the waveform ψ n (t) is equal to the peak frequency deviation in radians/sec due to the waveform ψ n (t) for n in the range of 1 through M. 
     The complex baseband FM modulator  130  frequency modules the multiplicity M waveforms  125  ψ 1 (t), ψ 2 (t), . . . , ψ M (t) providing the complex valued FHFM (Frequency Hop Frequency Modulation) waveform  135  ψ u (t) for the multiple accessing user u given by (25). 
     
       
         
           
             
               
                 
                   
                     
                       
                         χ 
                         u 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       exp 
                       ⁡ 
                       
                         [ 
                         
                           j 
                           ⁢ 
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               t 
                             
                             ⁢ 
                             
                               
                                 { 
                                 
                                   
                                     
                                       β 
                                       
                                         1 
                                         ⁢ 
                                         f 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         ψ 
                                         
                                           u 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         τ 
                                         ) 
                                       
                                     
                                   
                                   + 
                                   … 
                                   + 
                                   
                                     
                                       β 
                                       Mf 
                                     
                                     ⁢ 
                                     
                                       
                                         ψ 
                                         uM 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         τ 
                                         ) 
                                       
                                     
                                   
                                 
                                 } 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 τ 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   ; 
                   
                     j 
                     = 
                     
                       
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     In (25) ψ u1 (t), ψ u2 (t), . . . , ψ uM (t) denote the specific selection of the multiplicity M waveforms ψ 1 (t), ψ 2 (t), . . . , ψ M (t) for the multiple accessing user u. With the waveforms ψ un (t), n=1, 2, . . . , M selected as the sinusoidal waveforms with frequencies f un , the baseband FHFM waveform χ u (t) in (25) may be written as 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           χ 
                           u 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       = 
                       
                         exp 
                         ⁡ 
                         
                           [ 
                           
                             j 
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   1 
                                 
                                 M 
                               
                               ⁢ 
                               
                                 
                                   β 
                                   n 
                                 
                                 ⁢ 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       2 
                                       ⁢ 
                                       π 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         f 
                                         un 
                                       
                                       ⁢ 
                                       t 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                     ; 
                     
                       
                         β 
                         n 
                       
                       = 
                       
                         
                           β 
                           nf 
                         
                         / 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               f 
                               un 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     n 
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   M 
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     In various embodiments of the invention β n  for n between 1 and M may be equal to a constant β, the frequency f un  may be an integer κ un  multiple of a frequency f 0  for 1≦n≦M and the output  135  χ u (t) of the complex baseband FM modulator  130  provided to the complex conjugate operation unit  652  is given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         χ 
                         u 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       exp 
                       ⁡ 
                       
                         [ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           β 
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 1 
                               
                               M 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   
                                     πκ 
                                     un 
                                   
                                   ⁢ 
                                   
                                     f 
                                     0 
                                   
                                   ⁢ 
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   ; 
                   
                     
                       κ 
                       un 
                     
                     = 
                     
                       
                         f 
                         un 
                       
                       / 
                       
                         f 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     Referring to  FIG. 6 , the output  655  χ u *(t) of the complex conjugate operation unit  652  is inputted to the spread spectrum demodulator block  650 . The spread spectrum demodulator block  650  correlates the complex baseband signal  645  v b (t) with the FHFM waveform χ u (t) providing the despread signal  660  v ou (l) given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           v 
                           ou 
                         
                         ⁡ 
                         
                           ( 
                           l 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             1 
                             
                               T 
                               0 
                             
                           
                           ⁢ 
                           
                             
                               ∫ 
                               
                                 
                                   ( 
                                   
                                     l 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   T 
                                   0 
                                 
                               
                               
                                 lT 
                                 0 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   v 
                                   b 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   χ 
                                   u 
                                   * 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                         
                         = 
                         
                           
                             
                               s 
                               pu 
                             
                             ⁡ 
                             
                               ( 
                               l 
                               ) 
                             
                           
                           + 
                           
                             
                               n 
                               o 
                             
                             ⁡ 
                             
                               ( 
                               l 
                               ) 
                             
                           
                         
                       
                     
                     ; 
                     
                       l 
                       = 
                       0 
                     
                   
                   , 
                   1 
                   , 
                   … 
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     In (28) s pu  is the complex amplitude of the symbol waveform s u (t) during the interval (l−1)T 0 ≦t&lt;|T 0 , l=0, 1, . . . The subsystem for the generation of the FHFM waveform  135  and the spread spectrum demodulator  650  constitute the despreading subsystem of the receiver. 
     Referring to  FIG. 6 , the spread spectrum demodulator output  660  v ou (l) is inputted to the symbol detector  665 . The symbol detector is comprised of an integrator and a decision device, not shown. The integrator averages out the N m , samples of v ou (l) during any symbol interval T s =N m T 0  and inputs the resulting estimate ŝ u (m) to the decision device. For the case of N m =1, the spread spectrum demodulator output  660  v ou (l) is inputted to the decision device of the symbol detector. 
     Referring to  FIG. 6 , the decision device in the symbol detector detects the information baseband symbols s(m) on the basis of the signal constellation diagram of the baseband modulator  145  at the FHFMSS transmitter of  FIGS. 1-2  by mapping the estimate ŝ u (m) into one of the points of the signal constellation diagram of the baseband modulator  145  in the two dimensional signal space and provides the detected symbol  670  s u (m); m=0, 1, . . . , to the baseband demodulation subsystem  675 . 
     For example, the baseband modulator may be an M=64 QAM modulator with 64 points in the signal constellation diagram. The detection of the information baseband symbols may be based on, for example, the maximum likelihood criteria or the minimum distance criteria in the two dimensional signal space. 
     Referring to  FIG. 6 , the detected baseband symbol s u (m) is inputted to the baseband demodulator  675 . The baseband demodulator  675  maps the detected baseband symbols s u (m) into groups of m binary digits, wherein m=log 2  (M) assumed to be an integer, using the inverse of the map from group of m binary digits into 1 out of M possible information baseband symbols used in the baseband modulator  145  at the FHFMSS transmitter. The baseband demodulator block  675  may finally concatenate the groups of M binary digits into the serial stream  680  d(m) that constitutes the detected version of the user n input data  140  at the FHFMSS transmitter. In various embodiments of the invention, the baseband demodulator may also be comprised of an error correction decoder and a deinterleaver. 
     Various modifications and other embodiments of the invention applicable to various problems in Communication and other fields will be readily apparent to those skilled in the art in the field of invention. For example, the ψ-waveform may be selected to be different than the sinusoidal waveform in various embodiments of the invention. 
     The number of orthogonal FHFM waveforms in a given bandwidth B S  may be increased by a factor of two by augmenting the set of FHFM waveforms 
                 χ     s   ⁢           ⁢   κ       =     exp   ⁡     [     j   ⁢           ⁢   β   ⁢       ∑     n   =   1     M     ⁢     sin   ⁡     (     2   ⁢     πκ   n     ⁢     f   0     ⁢   t     )           ]         ,         
κ n  prime integers, with the FHFM waveforms obtained by the FM modulation of the periodic ψ-waveforms different than the sin functions of time. For example, the augmented set of the FHFM waveforms may include the waveforms
 
               χ     c   ⁢           ⁢   κ       =     exp   ⁡     [     j   ⁢           ⁢   β   ⁢       ∑     n   =   1     M     ⁢     cos   ⁡     (     2   ⁢     πκ   n     ⁢     f   0     ⁢   t     )           ]             
wherein κ n  are prime integers.
 
     For the case of M=1, the set of FHFM waveforms {χ sκ , χ cκ ; κ a prime integer} forms an orthogonal set in that the correlation coefficient of any pair of waveforms foromm the set is near zero except for a non zero coefficient ρ sc  between the waveforms χ sκ , χ cκ  with the same value of κ. The non zero correlation coefficient ρ sc  is a function of β but does not depend upon κ. For example, for β=2.4048, ρ sc =−0.3645. The extended set of the FHFM waveforms may be made orthogonal by replacing the FHFM waveforms χ sκ , χ cκ  by a pair of their weighted combinations given by (29).
 
χ 1,κ =P 11 χ s,κ +P 12 χ c,κ   (29a)
 
χ 2,κ =P 21 χ s,κ +P 22 χ c,κ   (29b)
 
     In (29) P ij ; i, j=1, 2 are the elements of a 2×2 orthogonalization matrix P that is inverse of the square root factor of a matrix R given by (30). 
     
       
         
           
             
               
                 
                   
                     R 
                     = 
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             
                               ρ 
                               sc 
                             
                           
                         
                         
                           
                             
                               ρ 
                               sc 
                               * 
                             
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                   
                     
                       QQ 
                       H 
                     
                     = 
                     R 
                   
                   ; 
                   
                     P 
                     = 
                     
                       Q 
                       
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     In (30) * denotes complex conjugate and the superscript H denotes complex conjugate transpose of a matrix. For the case of β=2.4048, the matrix P is given by 
     
       
         
           
             
               
                 
                   P 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             - 
                             0.8870 
                           
                         
                         
                           
                             - 
                             0.8870 
                           
                         
                       
                       
                         
                           
                             - 
                             0.6053 
                           
                         
                         
                           0.6053 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
     Referring to  FIG. 2 , in various embodiments of the invention, the TMDDFS unit  240  may generate discrete time version of both the ψ-waveforms sin(2πκ n f 0 t) and sin(2πκ n f 0 t) wherein the integers κ n  are determined by the frequency selection indices  235  c am (n) for 1≦m ≦M. Referring to  FIG. 3 , in various embodiments of the invention, the ROM word size for the ROM unit  340  may be selected to be 2K S  bits wherein the first K S  bits store the samples of the function sin(2πf 0 t) with the last K S  bits storing the samples of the cos(2πf 0 t) function. Referring to  FIG. 3 , the data buffer unit  345  may be comprised of a data splitter that may split the ROM word read from the memory into two outputs and may time multiplex the two outputs corresponding to the samples of the ψ-waveforms sin(2πκ n f 0 t) and sin(2πκ n f 0 t). 
     Referring to  FIG. 2 , in the FM modulator  250  unit, the mod. coefficients corresponding to the sin ψ I -waveforms may be set to a constant β with the mod. coefficients corresponding to the cos ψ I -waveforms set to 0 wherein the exponentiation unit  265  may include a gain equal to P 11  providing the FHFM waveform P 11 χ s (k) at the output. A second baseband FM modulator unit, not shown, inputted with the ψ I -waveforms  246  may set the mod. coefficients corresponding to the sin ψ I -waveforms to 0 with the mod. coefficients corresponding to the cos ψ I -waveforms set to β wherein the gain in its exponentiation unit is set equal to P 12  with the second baseband FM modulator unit providing the FHFM waveform P 12 χ c (k) at the output. An adder, not shown, adds the two FHFM waveforms P 11 χ s (k) and P 12 χ c (k) providing the sum waveform χ 1 (k) to the spread spectrum modulator  275 . 
     In various multiple access systems embodiments of the invention, a second transmitter may replace the gains P 11  and P 12  of the exponentiation units of the two FM modulator units by P 21  and P 22  respectively for providing the waveform χ 2 (k) to the spread spectrum modulator  275  resulting in the orthogonality of the FHFM waveforms generated by the two FHFMSS transmitters. 
     In various multiple access systems embodiments of the invention, wherein the FHFMSS transmitter is located, for example, at the base station of a cellular communication network, the transmitter may generate a multiplicity N u  FHFM waveforms similar to the generation of the FHFM waveform  268  in the transmitter of  FIG. 2  that modulate the respective ones of the multiplicity N u  information baseband signals  150  for the generation of a multiplicity N u  baseband FHFMSS signals, similar to  276  in  FIG. 2 , that may be added by an adder, not shown, to provide a composite baseband FHFMSS signal that is inputted to the cascade of the DAC  280 , baseband to IF converter  165 , and the RF stages unit  175  for the generation of the RF FHFMSS signal. In case when a number N p  of the FHFM waveforms are generated by the periodic ψ-waveforms of the same normalized frequency κ, those N p  waveforms are orthogonalized before modulating the corresponding information baseband signals. The orthogonalization may be performed by pre multiplying the vector comprised of the N p  FHFM waveforms by the orthogonalization matrix P given by (31) for the case of N p =2. The elements of the resulting product vector constitute the corresponding multiplicity N p  orthogonalized FHFM waveforms. 
     The number of orthogonal FHFM waveforms in a given bandwidth B S  may be further increased by a factor of two by augmenting the set of FHFM waveforms with the FHFM waveforms corresponding to a different values of the mod indices. For example, for the case of M=1, the augmented set of the FHFM waveforms may include the waveforms: 
     
       
         
           
             
               
                 
                   
                     
                       χ 
                       
                         
                           s 
                           1 
                         
                         ⁢ 
                         κ 
                       
                     
                     = 
                     
                       exp 
                       ⁡ 
                       
                         [ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             β 
                             1 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 1 
                               
                               M 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   
                                     πκ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     f 
                                     0 
                                   
                                   ⁢ 
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   ; 
                   
                     
                       χ 
                       
                         
                           s 
                           2 
                         
                         ⁢ 
                         κ 
                       
                     
                     = 
                     
                       exp 
                       ⁡ 
                       
                         [ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             β 
                             2 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 1 
                               
                               M 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   
                                     πκ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     f 
                                     0 
                                   
                                   ⁢ 
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     32 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       χ 
                       
                         
                           c 
                           1 
                         
                         ⁢ 
                         κ 
                       
                     
                     = 
                     
                       exp 
                       ⁡ 
                       
                         [ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             β 
                             1 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 1 
                               
                               M 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   
                                     πκ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     f 
                                     0 
                                   
                                   ⁢ 
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   ; 
                   
                     
                       χ 
                       
                         
                           c 
                           2 
                         
                         ⁢ 
                         κ 
                       
                     
                     = 
                     
                       exp 
                       ⁡ 
                       
                         [ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             β 
                             2 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 1 
                               
                               M 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   
                                     πκ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     f 
                                     0 
                                   
                                   ⁢ 
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     32 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     In (32) κ is a prime integer, and β 1  and β 2  are two different mod indices and preferably are the zeros of the Bessel function J 0 ( ). 
     The extended set of multiplicity N p =4 FHFM waveforms {χ s     1     κ , χ s     2     κ , χ c     1     κ , χ c     2     κ ; κ a prime integer} forms an orthogonal set in that the correlation coefficient of any pair of waveforms from the set is near zero except for a non zero coefficients between the waveforms with the same value of κ. The non zero correlation coefficients are a function of β 1  and β 2  but do not depend upon κ. For example, for β 1 =2.4048, and β 2 =5.52 the correlation coefficient matrix R of the FHFM waveforms {χ s     1     κ , χ s     2     κ , χ c     1     κ , χ c     2     κ } for any prime integer κ is given by 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       = 
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               a 
                             
                             
                               b 
                             
                             
                               c 
                             
                           
                           
                             
                               a 
                             
                             
                               1 
                             
                             
                               c 
                             
                             
                               d 
                             
                           
                           
                             
                               b 
                             
                             
                               c 
                             
                             
                               1 
                             
                             
                               a 
                             
                           
                           
                             
                               c 
                             
                             
                               d 
                             
                             
                               a 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                     ; 
                     
                       a 
                       = 
                       
                         - 
                         0.2966 
                       
                     
                   
                   , 
                   
                     b 
                     = 
                     
                       - 
                       0.3645 
                     
                   
                   , 
                   
                     c 
                     = 
                     0.1564 
                   
                   , 
                   
                     d 
                     = 
                     .2141 
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     As for the case of N p =2, the extended set of the FHFM waveforms may be made orthogonal by replacing the FHFM waveforms {χ s     1     κ , χ s     2     κ , χ c     1     κ , χ c     2     κ } by their weighted combinations with weighting coefficients given by the rows of the matrix P given by (34). 
     
       
         
           
             
               
                 
                   
                     P 
                     = 
                     
                       [ 
                       
                         
                           
                             
                               - 
                               0.4634 
                             
                           
                           
                             
                               - 
                               0.8457 
                             
                           
                           
                             0.4634 
                           
                           
                             0.8457 
                           
                         
                         
                           
                             0.8873 
                           
                           
                             0.2037 
                           
                           
                             0.8873 
                           
                           
                             0.2037 
                           
                         
                         
                           
                             
                               - 
                               0.1417 
                             
                           
                           
                             0.6173 
                           
                           
                             
                               - 
                               0.1417 
                             
                           
                           
                             0.6173 
                           
                         
                         
                           
                             0.4883 
                           
                           
                             
                               - 
                               0.2675 
                             
                           
                           
                             
                               - 
                               0.4883 
                             
                           
                           
                             0.2675 
                           
                         
                       
                       ] 
                     
                   
                   ; 
                   
                     
                       QQ 
                       H 
                     
                     = 
                     R 
                   
                   ; 
                   
                     P 
                     = 
                     
                       Q 
                       
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     In some multiple accessing systems embodiments of the invention, the vector {χ s     1     κ , χ s     2     κ , χ c     1     κ , χ c     2     κ } T  may be pre multiplied by the matrix P for the orthogonalization of the Np=4 FHFM waveforms as in the case of N p =2. The number of orthogonal waveforms in the set of FHFM waveforms is thereby increased by a factor of N p . 
     In the extended set of the FHFM waveforms, relatively low values of κ may be included for the higher value of β so that there is no significant increase in the bandwidth B s  over that for the case of single mod index β. Increase in the number of orthogonal FHFM waveforms is thus achieved without an increase in the signal bandwidth B SS  of the spread spectrum signal. The number of different mod indices may be increased to more than 2 at the cost of some additional complexity for a further increase in the set of orthogonal waveforms. 
     In various multiple accessing systems embodiments of the invention, wherein the FHFMSS transmitters generate orthogonalized FHFM waveforms, the FHFMSS receiver  600  in  FIG. 6  may detect any of the multiplicity N u  information baseband signals by despreading the baseband FHFMSS signal  645  by the spread spectrum demodulator  650  wherein the FHFM waveform  135  may be the corresponding one of the orthogonalized waveforms. The orthogonalized FHFM waveform at the receiver may be generated as at the transmitter and may be comprised of a multiplicity N p  complex baseband FM modulators  130  generating the FHFM waveforms {χ s     1     κ , χ s     2     κ , χ c     1     κ , χ c     2     κ } that are weighted combined with the weights given by the rows of the matrix P. 
     In various multiple accessing systems embodiments of the invention, wherein the FHFMSS receiver is located, for example, at the base station of a mobile communication network, the MS (mobile stations) transmitters may transmit the FHFMSS signals generated by spreading with the extended set of FHFM waveforms such as {χ s     1     κ , χ s     2     κ , χ c     1     κ , χ c     2     κ } without the orthogonalization. The multiple accessing interference due to non zero correlation coefficient among some of the FHFM waveforms may be eliminated at the FHFMSS receiver. The FHFMSS receiver at the base station may generate a multiplicity N p  FHFM waveforms {χ s     1     κ , χ s     2     κ , χ c     1     κ , χ c     2     κ } similar to the generation of the FHFM waveform  135  χ(t) shown in the receiver of  FIG. 6  and may comprise of a multiplicity N p  spread spectrum demodulators similar to  650  of  FIG. 6  that may be inputted with the respective ones of the multiplicity N p  FHFM waveforms. The multiplicity N p  spread spectrum demodulators are inputted with the baseband FHFMSS signal  645 . A vector comprised of the outputs of the multiplicity N p  spread spectrum demodulators may be pre multiplied by the matrix P in (34) to mitigate the multiple accessing interference due to the non zero correlation coefficients among the FHFM waveforms. The elements of the product vector comprise of the interference mitigated multiplicity N p  despread signals similar to the signal  660  in  FIG. 6 . The multiple versions of various subsystems are not shown in  FIG. 6  for a clear understanding of the invention. 
     The frequency hopped frequency modulation spread spectrum multiple accessing architectures of the invention can be readily modified and applied to various fields where such an architecture is applicable. Examples of such fields include Radars, sonar, digital audio systems and so on. 
     It is to be understood that the figures and descriptions of the present invention have been simplified to illustrate elements that are relevant for a clear understanding of the present invention, while eliminating other elements, for purposes of clarity. Those of ordinary skill in the art will recognize that these and other elements may be desirable. However, because such elements are well known in the art and because they do not facilitate a better understanding of the present invention, a discussion of such elements is not provided herein. 
     In general, it will be apparent that the embodiments described herein may be implemented in many different embodiments of software, firmware, and/or hardware, for example, based on Field Programmable Gate Array (FPGA) chips or implemented in Application-Specific Integrated Circuits (ASICS). The software and firmware code may be executed by a computer or computing device comprising a processor (e.g., a DSP or any other similar processing circuit) including, for example, the computing device described below. The processor may be in communication with memory or another computer-readable medium comprising the software code. The software code or specialized control hardware that may be used to implement embodiments is not limiting. For example, embodiments described herein may be implemented in computer software using any suitable computer software language type, using, for example, conventional or object-oriented techniques. Such software may be stored on any type of suitable computer-readable medium or media, such as, for example, a magnetic or optical storage medium. According to various embodiments, the software may be firmware stored at an EEPROM and/or other non-volatile memory associated with a DSP or other similar processing circuit. The operation and behavior of the embodiments may be described without specific reference to specific software code or specialized hardware components. The absence of such specific references is feasible, because it is clearly understood that artisans of ordinary skill would be able to design software and control hardware to implement the embodiments based on the present description with no more than reasonable effort and without undue experimentation. 
       FIG. 7  shows an example of a computing device  700  according to one embodiment. For the sake of clarity, the computing device  700  is illustrated and described here in the context of a single computing device. However, it is to be appreciated and understood that any number of suitably configured computing devices can be used to implement a described embodiment. For example, in at least some implementations, multiple communicatively linked computing devices may be used. One or more of these devices can be communicatively linked in any suitable way such as via one or more networks. One or more networks can include, without limitation: the Internet, one or more local area networks (LANs), one or more wide area networks (WANs) or any combination thereof. 
     In the example of  FIG. 7 , the computing device  700  comprises one or more processor circuits or processing units  702 , one or more memory and/or storage circuit component(s)  704  and one or more input/output (I/O) devices  706 . Additionally, the computing device  700  comprises a bus  708  that allows the various circuit components and devices to communicate with one another. The bus  708  represents one or more of any of several types of bus structures, including a memory bus or memory controller, a peripheral bus, an accelerated graphics port, and a processor or local bus using any of a variety of bus architectures. The bus  708  may comprise wired and/or wireless buses. 
     The processing unit  702  may be responsible for executing various software programs such as system programs, application programs, and/or program modules/blocks to provide computing and processing operations for the computing device  700 . The processing unit  702  may be responsible for performing various voice and data communications operations for the computing device  700  such as transmitting and receiving voice and data information over one or more wired or wireless communications channels. Although the processing unit  702  of the computing device  700  is shown in the context of a single processor architecture, it may be appreciated that the computing device  700  may use any suitable processor architecture and/or any suitable number of processors in accordance with the described embodiments. In one embodiment, the processing unit  702  may be implemented using a single integrated processor. The processing unit  702  may be implemented as a host central processing unit (CPU) using any suitable processor circuit or logic device (circuit), such as a general purpose processor. The processing unit  702  also may be implemented as a chip multiprocessor (CMP), dedicated processor, embedded processor, media processor, input/output (I/O) processor, co-processor, microprocessor, controller, microcontroller, application-specific integrated circuit (ASIC), field programmable gate array (FPGA), programmable logic device (PLD), or other processing device in accordance with the described embodiments. 
     As shown, the processing unit  702  may be coupled to the memory and/or storage component(s)  704  through the bus  708 . The bus  708  may comprise any suitable interface and/or bus architecture for allowing the processing unit  702  to access the memory and/or storage component(s)  704 . Although the memory and/or storage component(s)  704  may be shown as being separate from the processing unit  702  for purposes of illustration, it is worthy to note that in various embodiments some portion or the entire memory and/or storage component(s)  704  may be included on the same integrated circuit as the processing unit  702 . Alternatively, some portion or the entire memory and/or storage component(s)  704  may be disposed on an integrated circuit or other medium (e.g., hard disk drive) external to the integrated circuit of the processing unit  702 . In various embodiments, the computing device  700  may comprise an expansion slot to support a multimedia and/or memory card, for example. 
     The memory and/or storage component(s)  704  represent one or more computer-readable media. The memory and/or storage component(s)  704  may be implemented using any computer- readable media capable of storing data such as volatile or non-volatile memory, removable or non-removable memory, erasable or non-erasable memory, writeable or re-writeable memory, and so forth. The memory and/or storage component(s)  704  may comprise volatile media (e.g., random access memory (RAM)) and/or non-volatile media (e.g., read only memory (ROM), Flash memory, optical disks, magnetic disks and the like). The memory and/or storage component(s)  704  may comprise fixed media (e.g., RAM, ROM, a fixed hard drive, etc.) as well as removable media (e.g., a Flash memory drive, a removable hard drive, an optical disk). Examples of computer-readable storage media may include, without limitation, RAM, dynamic RAM (DRAM), Double-Data-Rate DRAM (DDRAM), synchronous DRAM (SDRAM), static RAM (SRAM), read-only memory (ROM), programmable ROM (PROM), erasable programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), flash memory (e.g., NOR or NAND flash memory), content addressable memory (CAM), polymer memory (e.g., ferroelectric polymer memory), phase-change memory, ovonic memory, ferroelectric memory, silicon-oxide-nitride-oxide-silicon (SONOS) memory, magnetic or optical cards, or any other type of media suitable for storing information. 
     The one or more I/O devices  706  allow a user to enter commands and information to the computing device  700 , and also allow information to be presented to the user and/or other components or devices. Examples of input devices include data ports, analog to digital converters (ADCs), digital to analog converters (DACs), a keyboard, a cursor control device (e.g., a mouse), a microphone, a scanner, a touch sensitive screen, and the like. Examples of output devices include data ports, ADCs, DACs, a display device (e.g., a monitor or projector, speakers, a printer, a network card). The computing device  700  may comprise an alphanumeric keypad coupled to the processing unit  702 . The keypad may comprise, for example, a QWERTY key layout and an integrated number dial pad. The computing device  700  may comprise a display coupled to the processing unit  702 . The display may comprise any suitable visual interface for displaying content to a user of the computing device  700 . In one embodiment, for example, the display may be implemented by a liquid crystal display (LCD) such as a touch- sensitive color (e.g., 76-bit color) thin-film transistor (TFT) LCD screen. The touch-sensitive LCD may be used with a stylus and/or a handwriting recognizer program. 
     The processing unit  702  may be arranged to provide processing or computing resources to the computing device  700 . For example, the processing unit  702  may be responsible for executing various software programs including system programs such as operating system (OS) and application programs. System programs generally may assist in the running of the computing device  700  and may be directly responsible for controlling, integrating, and managing the individual hardware components of the computer system. The OS may be implemented, for example, as a Microsoft® Windows OS, Symbian OS™, Embedix OS, Linux OS, Android system, Binary Run-time Environment for Wireless (BREW) OS, JavaOS, or other suitable OS in accordance with the described embodiments. The computing device  700  may comprise other system programs such as device drivers, programming tools, utility programs, software libraries, application programming interfaces (APIs), and so forth. 
     In various embodiments disclosed herein, a single component may be replaced by multiple components, and multiple components may be replaced by a single component to perform a given function or functions. Except where such substitution would not be operative, such substitution is within the intended scope of the embodiments. 
     While various embodiments have been described herein, it should be apparent that various modifications, alterations, and adaptations to those embodiments may occur to persons skilled in the art with attainment of at least some of the advantages. The disclosed embodiments are therefore intended to include all such modifications, alterations, and adaptations without departing from the scope of the embodiments as set forth herein. 
     Embodiments may be provided as a computer program product including a non-transitory machine-readable storage medium having stored thereon instructions (in compressed or uncompressed form) that may be used to program a computer (or other electronic device) to perform processes or methods described herein. The machine-readable storage medium may include, but is not limited to, hard drives, floppy diskettes, optical disks, CD-ROMs, DVDs, read-only memories (ROMs), random access memories (RAMs), EPROMs, EEPROMs, flash memory, magnetic or optical cards, solid-state memory devices, or other types of media/machine-readable medium suitable for storing electronic instructions. Further, embodiments may also be provided as a computer program product including a transitory machine-readable signal (in compressed or uncompressed form). Examples of machine-readable signals, whether modulated using a carrier or not include, but are not limited to, signals that a computer system or machine hosting or running a computer program can be configured to access, including signals downloaded through the Internet or other networks. For example, the distribution of software may be an Internet download. 
     Although embodiments have been described in language specific to structural features and/or methodological acts, it is to be understood that the disclosure is not necessarily limited to the specific features or acts described. Rather, the specific features and acts are disclosed as illustrative forms of implementing the embodiments. Conditional language, such as, among others, “can,” “could,” “might,” or “may,” unless specifically stated otherwise, or otherwise understood within the context as used, is generally intended to convey that certain embodiments could include, while other embodiments do not include, certain features, elements, and/or steps. Thus, such conditional language is not generally intended to imply that features, elements, and/or steps are in any way required for one or more embodiments or that one or more embodiments necessarily include logic for deciding, with or without user input or prompting, whether these features, elements, and/or steps are included or are to be performed in any particular embodiment.