Abstract:
A method and apparatus for modulating a pulse signal with a bit stream is disclosed. The pulse signal is modulated by selectively inverting and delaying signal pulses within the pulse signal responsive to bits within the bit stream.

Description:
RELATED APPLICATION 
   This application claims the benefit of the filing date of provisional application No. 60/450,313 entitled “BIORTHOGONAL MODULATION FOR SUPPRESSING ULTRA-WIDEBAND SPECTRAL LINES” filed Feb. 27, 2003, the contents of which are incorporated herein by reference. 

   FIELD OF THE INVENTION 
   The present invention relates to communication systems and, more particularly, to methods and apparatus for modulating a pulse signal with a bit stream. 
   BACKGROUND 
   Ultra Wideband (UWB) technology, which uses base-band pulses of very short duration to spread the energy of transmitted signals very thinly from near zero to several GHz, is presently in use in military applications. Commercial applications will soon become possible due to a recent Federal Communications Commission (FCC) decision that permits the marketing and operation of consumer products incorporating UWB technology. 
   The key motivation for the FCC&#39;s decision to allow commercial applications is that no new communication spectrum is required for UWB transmissions because, when they are properly configured, UWB signals can coexist with other application signals in the same spectrum with negligible mutual interference. In order to ensure negligible mutual interference, however, the FCC has specified emission limits for the UWB applications. For example, a basic FCC requirement is that UWB systems do not generate signals that interfere with other narrowband communication systems. 
   The emission profile of a UWB signal can be determined by examining its power spectral density (PSD). The PSD for ideal synchronous data pulse streams based upon stochastic theory is well known and is described in an article by M. Z. Win, entitled “Spectral Density of Random Time-Hopping Spread-Spectrum UWB Signals with Uniform Timing Jitter”,  Proc. MICOM &#39;99, vol. 2, pp. 1196–1200, 1999. This article also provides a characterization of the PSD of the Time-Hopping Spread Spectrum signaling scheme in the presence of random timing jitter using a stochastic approach. 
   The power spectra of UWB signals consist of continuous and discrete components. Generally speaking, discrete components contribute more to the PSD than continuous components, which behave as white noise. Thus, discrete components cause more interference to narrowband wireless systems than continuous components. Accordingly, a basic objective in the design of UWB systems is to reduce the discrete component of the UWB power spectra. Another objective for UWB systems is to increase the power efficiency. 
   UWB communication system currently use one of two modulation techniques. These techniques include a pulse position modulation (PPM) technique and a bi-phase shift keying (BPSK) technique. The PPM technique has good power efficiency but a relatively high PSD. The BPSK technique, on the other hand, has a relatively low PSD but low power efficiency. 
   There is an ever present desire for efficient communication systems that transmit signals with low PSD. Accordingly, there is a need for improved modulation methods, apparatus, and systems that are not subject to the above limitations. The present invention fulfils this need among others. 
   SUMMARY OF THE INVENTION 
   The present invention is embodied in a method and an apparatus that modulates a pulse signal with a bit stream. The pulse signal is modulated by selectively inverting and delaying signal pulses within the pulse signal responsive to bits within the bit stream. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The invention is best understood from the following detailed description when read in connection with the accompanying drawings, with like elements having the same reference numerals. Included in the drawings are the following figures: 
       FIG. 1  is a block diagram of a transmitter for modulating a pulse signal with a bit stream in accordance with the present invention. 
       FIGS. 1A ,  1 B and  1 C are block diagrams of alternative exemplary pulse generators in accordance with the present invention for use in the transmitter of  FIG. 1 . 
       FIG. 2  is a flow chart of exemplary steps for modulating a pulse signal with a bit stream in accordance with the present invention. 
       FIG. 3  is a graph of amplitude versus frequency that shows the power-spectral density (PSD) of a single monocycle pulse. 
       FIG. 4  is a graph of amplitude versus frequency that shows the PSD of a pulse-position modulated ultra-wideband (UWB) signal in accordance with prior art; 
       FIGS. 5A and 5B  are graphs of amplitude versus time which illustrate a monocycle pulse and an inverted monocycle pulse, respectively, in accordance with prior art. 
       FIG. 6  is a graph of amplitude versus frequency that shows the PSD of a bi-phase modulated UWB signal in accordance with prior art. 
       FIGS. 7A ,  7 B,  7 C and  7 D are graphs of amplitude versus time which illustrate a monocycle pulse, an inverted monocycle pulse, a delayed monocycle pulse, and an inverted and delayed monocycle pulse, respectively, in accordance with the present invention. 
       FIG. 8  is a graph of amplitude versus frequency that shows the PSD of a pulse signal in accordance with the present invention. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
     FIG. 1  depicts select components of an exemplary transmitter  100  for modulating a pulse stream with a bit stream for transmission in accordance with the present invention. A controller  102  includes a phase controller  112  and a delay controller  114 . In the illustrated embodiment, the phase controller  112  and the delay controller  114  are each coupled to the pulse generator  104  and to a shift register  110 . The phase controller  112  is coupled to a first position  110   a  of the shift register (e.g., to receive a first bit of the bit stream) and the delay controller  114  is coupled to a second position  110   b  of the shift register (e.g., to receive a second bit of the bit stream). It will be recognized by those of skill in the art that the designation of first and second positions  110   a  and  110   b  is for ease of description and not to suggest a particular bit processing order. For example, the phase controller  112  and delay controller  114  may be coupled to the second position  110   b  and the first position  110   a , respectively. Also, the first and second register positions referred to with numerals “ 110   a ” and “ 110   b ” may be interchanged. 
   In an exemplary embodiment, the shift register  110  is a shift register that shifts two bits of the bit stream during each shift such that the controller  102  receives two new bits of the bit stream for processing at each shift. In an alternative exemplary embodiment, the shift register shifts one bit at a time and the controller  102  is configured to pass the bits to the appropriate phase/delay controller  112 / 114 . Various other embodiments will be apparent to those of skill in the art from the description herein. The phase/delay controllers  112 / 114  may each be a latch that produces the value of the received bit at an output port. 
   A pulse generator  104  generates a pulse signal that includes a plurality of signal pulses. The pulse generator  104  is coupled to the phase controller  112  and the delay controller  114  to receive a phase signal and a delay signal, respectively, therefrom. In an exemplary embodiment, the pulse generator is a monocycle pulse signal generator that generates monocycle signal pulses such as a UWB pulse generator that generates UWB signal pulses. 
   The pulse generator  104  alters the signal pulses within the pulse signal responsive to the phase signal and the delay signal received, respectively, from the phase controller  112  and the delay controller  114 . In an exemplary embodiment, for a monocycle pulse signal, the pulse generator  104  selectively inverts the signal pulses responsive to the phase signal and selectively delays the signal pulse by a predefined amount responsive to the delay signal. Thus, the pulse generator  104  may alter the pulse signal by producing signal pulses with no delay or inversion, a delay, an inversion, or a delay and an inversion. In an exemplary embodiment, the signal pulse is delayed by an amount sufficient to substantially decorrelate a delayed signal pulse from the signal pulse prior to delay. For example, the signal pulses may be delayed such that a delayed signal pulse is orthogonal to the signal pulse prior to delay. The altering of the signal pulses is described in further detail below. 
     FIG. 1A  depicts an exemplary pulse generator  150  for use as the pulse generator  104  ( FIG. 1 ). The alternative exemplary pulse generator  150  includes a plurality of pulse generators  152  that each generate a signal pulse. A selector  154  is coupled to the plurality of pulse generators  152 , the phase controller  112  ( FIG. 1 ), and the delay controller  114  ( FIG. 1 ). The selector  154  is configured to select a signal pulse generated by a particular one of the plurality of pulse generators  152  responsive to phase and delay signals from the phase controller  112  and the delay controller  114 , respectively. The illustrated plurality of pulse generators  152  include a first pulse generator  156 , a second pulse generator  158 , a third pulse generator  160 , and a fourth pulse generator  162 . The first pulse generator  156  may be configured to generate a first signal pulse without either delay or inversion. The second pulse generator  158  may be configured to generate a second signal pulse that is inverted but not delayed with respect to the first signal pulse. The third signal pulse generator  160  may be configured to generate a third signal pulse that is delayed by not inverted with respect to the first signal pulse. The fourth pulse generator  158  may be configured to generate a fourth signal pulse that is both delayed and inverted with respect to the first signal pulse. By selecting one of the first through fourth signal pulses responsive to the phase and delay signals, the selector  154  is able to produce signal pulses that are selectively inverted and delayed. 
     FIG. 1B  depicts an alternative exemplary pulse generator  170  for use as the pulse generator  104  ( FIG. 1 ). The alternative exemplary pulse generator  170  includes a pair of pulse generators  172  that each generate a signal pulse. A selector  174  is coupled to the pair of pulse generators  172  and the phase controller  112  ( FIG. 1 ). A delay circuit  176  is coupled to the selector  174  and the delay controller  114  ( FIG. 1 ). The selector  174  is configured to select a signal pulse generated by a particular one of the pair of pulse generators  172  responsive to phase signals from the phase controller  112  and the delay circuit  176  is configured to selectively introduce delay responsive to delay signals from the delay controller  114 . The illustrated pair of pulse generators  172  include a first pulse generator  178  and a second pulse generator  180 . The first pulse generator  178  may be configured to generate a first signal pulse without inversion. The second pulse generator  180  may be configured to generate a second signal pulse that is inverted with respect to the first signal pulse. By selecting one of the first and second signal pulses responsive to the phase signal and selectively introducing delay to the selected signal pulse responsive to the delay signal, the selector  174  and delay circuit  176  are able to produce signal pulses that are selectively inverted and delayed. It will be understood by those skilled in the art that the delay circuit  176  may be positioned to introduce delay to the signal pulses generated by the pair of pulse generators  172  prior to selection of a particular signal by the selector  174 . Various alternative arrangements will be understood by those of skill in the art from the above description. 
     FIG. 1C  depicts an alternative exemplary pulse generator  190  for use as the pulse generator  104  ( FIG. 1 ). The alternative exemplary pulse generator  190  includes a single pulse generator  192  that generates a signal pulse. An inverter  194  is coupled to the pulse generator  192  and the phase controller  112  ( FIG. 1 ). A delay circuit  196  is coupled to the inverter  194  and the delay controller  114  ( FIG. 1 ). The inverter  194  is configured to selectively invert the signal pulse responsive to the phase signal from the phase controller  112  and the delay circuit is configured to selectively delay the signal pulse responsive to the delay signal from the delay controller  114  to produce signal pulses that are selectively inverted and delayed. It will be understood by those skilled in the art that the delay circuit  196  may be positioned to introduce delay to the signal pulse generated by the pulse generator  192  prior to selective inversion by the inverter  194 . Various alternative arrangements will be understood by those of skill in the art from the above description. 
   Referring back to  FIG. 1 , an optional time-hopping controller  106  introduces time-hopping to the pulse signal to position each signal pulse at a different time-hop index inside a frame. In the illustrated embodiment, the time-hopping controller  106  is coupled to the pulse generator  104 . In an exemplary embodiment, the time-hopping controller  106  introduces time-hopping to the pulse signal selectively inverted and delayed  102  in a conventional manner. 
   The antenna  108  transmits the pulse signal. In the illustrated embodiment, the antenna  108  is coupled to the time-hopping controller  106 . In this embodiment, the antenna  108  transmits a pulse signal as altered according to the pulse controller  102  and time-hopped by the time-hopping controller  106 . In embodiments where the pulse signal is not time-hopped, the antenna  108  is coupled to the pulse generator  104  for transmitting the pulse signal without time-hopping. The pulse signal as selectively inverted and delayed and, optionally, as time-hopped is transmitter via the antenna  108 . 
   A receiver (not shown) receives the pulse signal from the transmitter  100 . In an exemplary embodiment, the receiver uses a predefined pulse template to correlate incoming signals and, then, performs an integration over the pulse template. The template shifts forward and backward to find peaks of the integration. The position of the peak is used to identify the position of pulses in the PPM and the polarity of the peaks is used to determine phase (e.g., positive maximum value indicates normal phase and negative maximum value indicates inverted phase). If the optional time-hopping controller  106  is used in the transmitter  100 , it is desirable for the receiver to employ a complementary time-hopping controller (not shown) having the same time-hopping sequence used by the time-hopping controller  106  to locate each of the transmitted pulses in each frame so that the pulse signal can be recovered. 
     FIG. 2  depicts a flow chart  200  of exemplary steps for modulating a pulse signal for transmission. The exemplary steps are described with reference to  FIG. 1 . At block  202 , the shift register  110  receives a bit stream. In an exemplary embodiment, the shift register  110  shifts the bit stream through the shift register  110  two bits at a time. 
   At block  204 , the phase controller  112  generates a phase signal responsive to the bit stream. In an exemplary embodiment, the phase controller  112  generates a phase signal responsive to a first bit of the bit stream and every other bit thereafter. In an alternative exemplary embodiment, the phase controller  112  generates the phase signal responsive to a second bit of the bit stream and every other bit thereafter. The phase signal may be a binary signal that is set to a relatively high (low) value when a bit is high (i.e., a logical one “1”) and is set to a relatively low (high) value when the bit is low (i.e., a logical zero “0”). 
   At block  206 , the delay controller  114  generates a delay signal responsive to the bit stream. In an exemplary embodiment, the delay controller  114  generates a delay signal responsive to a second bit of the bit stream (i.e., the next consecutive bit following the first bit) and every other bit thereafter. In an alternative exemplary embodiment, the delay controller  114  generates the delay signal responsive to the first bit of the bit stream and every other bit thereafter. The delay signal may be a binary signal that is set to a relatively high (low) value when a bit is high and is set to a relatively low (high) value when the bit is low. 
   At block  208 , the pulse generator  104  generates the pulse signal responsive to the phase signal and the delay signal received from the pulse controller  102 . The pulse generator  104  selectively inverts signal pulses responsive to the phase signal and selectively delays signal pulses responsive to the delay signal. In an exemplary embodiment, where a single pulse generator  192  (see  FIG. 1C ) is used in the pulse generator  104 , the pulse generator  104  does not alter the signal pulse if the phase signal and the delay signal are both relatively high (low), delays the signal pulse if the phase signal is relatively low (high) and the delay signal is relatively high (low), inverts the signal pulse if the phase signal is relatively high (low) and the delay signal is relatively low (high), and inverts and delays the signal pulse if both the phase signal and the delay signal are relatively high (low). 
   In an alternative exemplary embodiment, where a plurality of pulse generators  152  (see  FIG. 1A ) are used in the pulse generator  104 , the selector  154  selects a first signal pulse (which is not delayed and not inverted) produced by the first pulse generator  156  if the phase signal and the delay signal are both relatively high (low), selects a second signal pulse (which is delayed and not inverted) produced by the second pulse generator  158  if the phase signal is relatively low (high) and the delay signal is relatively high (low), selects a third signal pulse (which is inverted and not delayed) produced by the third pulse generator  160  if the phase signal is relatively high (low) and the delay signal is relatively low (high), and selects a fourth signal pulse (which is delayed and inverted) produced by the fourth pulse generator  162  if both the phase signal and the delay signal are relatively high (low). 
   In an alternative exemplary embodiment, where a pair of pulse generators  172  (see  FIG. 1B ) are used in the pulse generator  104 , the selector  174  selects a first signal pulse (which is not inverted) produced by the first pulse generator  178  if the phase signal is relatively high (low) and selects a second signal pulse (which is inverted with respect to the first signal pulse) produced by the second pulse generator  180  if the phase signal is relatively low (high). The delay circuit  176  then introduces delay to the selected signal if the delay signal is relatively high (low) and does not introduce delay if the delay signal is relatively low (high). 
   Exemplary pulses are shown in  FIGS. 7A ,  7 B,  7 C, and  7 D.  FIG. 7A  depicts a monocycle pulse  710 .  FIG. 7B  depicts an inverted monocycle pulse  712 .  FIGS. 7C and 7D  depict a delayed monocycle pulse  714  and a delayed and inverted monocycle pulse  716 , respectively. Because there are four possible pulses, each pulse may represent two bits of the bit stream. The pulses may be assigned as shown in Table 1. 
   
     
       
             
             
           
         
             
               TABLE 1 
             
             
                 
             
             
               Value 
                 
             
             
               (first bit, second bit) 
               Pulse 
             
             
                 
             
           
           
             
               0, 0 
               monocycle pulse 
             
             
               0, 1 
               inverted monocycle pulse 
             
             
               1, 0 
               delayed monocycle pulse 
             
             
               1, 1 
               delayed and inverted monocycle pulse 
             
             
                 
             
           
        
       
     
   
   Referring back to  FIG. 2 , at block  210 , the optional time-hopping controller  106  time-hops the pulse signal. In an exemplary embodiment, the time-hopping controller  106  places each modulated signal pulse at a different time-hop index inside a frame. Each modulated signal pulse may be repeated inside the frame or across multiple frames in order to provide redundancy for a multiple-access environment. In an alternative exemplary embodiment, the pulse signal is not time-hopped and the step in block  210  can be eliminated. 
   At block  212 , the transmitter  100  transmits the pulse signal via the antenna  108 . In an exemplary embodiment, the pulse signal, as selectively inverted, delayed, and time-hopped, is transmitted. In an alternative exemplary embodiment, the pulse signal as selectively inverted and delayed is transmitted without time-hopping. 
   Additional details regarding modulation techniques for use with UWB signals will now be described. UWB signals can be modeled as shown in equation (1). 
                   S   ⁡     (   t   )       =       ∑     l   =     -   ∞       ∞     ⁢       1   N     ⁢       ∑     i   =   0       N   -   1       ⁢       A   l     ·       X   pulse     ⁡     (     t   -     i   ·     T   PPM       -     l   ·     T   symbol         )                       (   1   )               
In equation (1), A i  and T PPM  represent the data, and T symbol  represents the symbol index that is being transmitted. X pulse  represents the waveform including pulse shape and transmission power. There are different techniques for transmitting data over an UWB channel. These methods are now described.
 
   The PSD for the monocycle pulse typically used in UWB communications is shown as plot  300  in  FIG. 3 . This plot and the other PSD plots depicted in  FIGS. 4–8  were generated using a simulation in which each pulse is represented by 64 samples and each frame is 64 times longer than a single pulse, thus, a frame includes 2048 samples. Each simulation was run for 3,000 repetitions. The number of points in the fast Fourier transform (FFT) was 262,144. In each repetition, the data being sent was randomized 128 times. The randomization included both bi-phase and pulse position randomness. All of the PSD plots are generated using the Bartlett periodogram method described in a text by J. G. Proakis et al. entitled  Digital Signal Processing , Prentice Hall, third edition, 1996. 
   Pulse position modulation (PPM) is now described. PPM is one of the most popular modulation methods used in UWB communication systems. The major advantage of PPM is its power efficiency, i.e., as the number of levels (M) increase, there is no corresponding increase in power. The number of levels indicate the number of modulation positions. For example, for M-PPM, where M=2, two (2) modulation positions are needed; where M=4, four (4) modulation positions are needed. As shown in the following equations, however, PPM modulation has relatively high power spectral lines in its PSD. Therefore, if data is transmitted using PPM modulation, the average power per pulse may need to be reduced for the power spectral density of the pulse to be within emission limits specified by the FCC for UWB communications (referred to herein as the FCC mask), which is undesirable. 
   Equation (2) below, is taken from a textbook by S. Wilson entitled  Digital Modulation and Coding Prentice Hall , 1995. This equation is used to calculate the PSD for the PPM modulation. 
   
     
       
         
           
             
               
                 
                   
                     
                       
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                     H     2   -   ppm       ⁡     (   f   )       =       ∑     k   =   0     1     ⁢     Exp   ⁡     [       -   j     ⁢           ⁢     ω   ⁡     (         T   p     4     +         T   p     2     ⁢   k       )         ]                 (   3   )               
In equation (3), T p  is the pulse time.
 
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   It is noted that equation (4) includes two components: a discrete component and a continuous component. The discrete component, which represents the spectral lines encountered in PPM modulation, is shown in equation (5). 
                   S     yy   -   disc       =       1     4   ⁢     T   f   2         ⁢       ∑     n   =     -   ∞       ∞     ⁢         S   2     ⁡     (     n     T   f       )       ⁢       cos   2     ⁡     (     π   ⁢           ⁢       nT   p       2   ⁢     T   f           )       ⁢     δ   ⁡     (     f   -     n     T   f         )                     (   5   )               
The continuous component, which represents the pulse shape and the pulse position modulation filter, is shown in equation (6).
 
                   S     yy   -   cont       =       1     T   f       ⁡     [         S   2     ⁡     (   f   )       -       1   2     ⁢       S   2     ⁡     (   f   )       ⁢       cos   2     ⁡     (     ω   ⁢           ⁢       T   p     4       )           ]               (   6   )               
The PSD derived from the PPM signal is shown in  FIG. 4 .  FIG. 4  clearly shows both the power spectrum  400  of the pulse shape and the power spectrum of the discrete components  410 . The spectral lines  410  are relatively high. Accordingly, the average power per pulse is desirably reduced to fit the signal within the FCC mask for UWB transmission.
 
   Bi-phase shift keying (BPSK) modulation in now described. With BPSK modulation the spectral lines associated with the power spectral density of the signal itself are reduced, which is an advantage over PPM modulation; however, power efficiency with BPSK modulation decreases for M≧4. BPSK modulation uses a monocycle pulse and its inverse to transmit data. The monocycle pulse represents one logic state, for example, logic one (“1”) and the inverse pulse represents the other state, for example, logic zero (“0”).  FIGS. 5A and 5B  show a waveform  510  of the monocycle pulse and the waveform  520  of its inverse, respectively. 
   The PSD for a BPSK modulated signal is now derived. The data stream transmitted using the BPSK signal is assumed to be perfectly random and, thus, the BPSK modulation may be represented as shown in equation (7).
 
 S   i ( f )=Σ i=0   1 (2 i −1)Φ( f )  (7)
 
In equation (7), Φ(f) represents the pulse shape. Introducing this model into the above power spectrum equation, produces equation (8).
 
                         S   yy     =       ⁢         1     4   ⁢     T   f   2         ⁢       ∑     n   =     -   ∞       ∞     ⁢                ∑     i   =   0     1     ⁢       (       2   ⁢   i     -   1     )     ⁢     Φ   ⁡     (   f   )                2     ⁢     δ   ⁡     (     f   -     n     T   f         )             +                     ⁢       1     T   f       ⁡     [         1   2     ⁢       ∑     i   =   0     1     ⁢              (       2   ⁢   i     -   1     )     ⁢     Φ   ⁡     (   f   )              2         -              ∑     i   =   0     1     ⁢       (       2   ⁢   i     -   1     )     ⁢     Φ   ⁡     (   f   )                2       ]                   =       ⁢       1     T   f       ⁡     [       Φ   2     ⁡     (   f   )       ]                     (   8   )               
The power spectrum  600  generated from the simulation using equation (8) is shown in  FIG. 6 .
 
   As shown in  FIG. 6 , the PSD of a BPSK modulated random data signal is essentially the same as the PSD of the monocycle signal without any spectral lines. The difference in power between BPSK and PPM modulation as may be seen from  FIGS. 4 and 6  is about 10 dB, which illustrates the advantage of bi-phase modulation. Moreover, because the PSD is essentially the PSD of the monocycle pulse, different pulse shapes can be tried in the simulation to identify shapes useful for UWB transmissions. 
   The modulation scheme of the present invention is now described. This scheme merges PPM and BPSK modulation to produce a modulation scheme referred to herein as biorthogonal modulation. The merging of these two techniques provides improved power efficiency and reduces/eliminates PSD spectral lines from the pulse shape. In an exemplary embodiment, biorthogonal modulation uses the monocycle pulses shown in  FIGS. 7A ,  7 B,  7 C and  7 D. 
   Because the data stream is assumed to be perfectly random, the biorthogonal signal may be modeled as shown in equation (9). 
                     S   i     ⁡     (   f   )       =       ∑     i   =   0     1     ⁢       ∑     k   =   0     1     ⁢       (       2   ⁢   i     -   1     )     ⁢     Φ   ⁡     (   f   )       ⁢     Exp   ⁡     [       -   j     ⁢           ⁢     ω   ⁡     (         T   p     4     +         T   p     2     ⁢   k       )         ]                     (   9   )               
In this equation, Φ(f) represents the pulse shape.
 
   Introducing this model into the above power spectrum equation (2) produces the power spectrum equation (10). 
   
     
       
         
           
             
               
                 
                   
                     
                       
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   From equation (10), it is noted that biorthogonal modulation provides the PSD of the monocycle pulse essentially without any spectral lines (which is similar to the bi-phase modulation described above). This results from the random changing of the polarity of the pulse such that even when used in conjunction with PPM modulation there is no correlation between any of the present and future pulses. Therefore, because there is no correlation between pulses, the resulting PSD  800  is shown in  FIG. 8 . Thus, the PSD for biorthogonal modulation is essentially the same as for a BPSK signal with time-hopping in which the pulse is not repeated (i.e., one in which the polarity of the pulses are randomly changed over the hops). 
   This biorthogonal modulation method achieves several important results. By way of non-limiting example, this method provides the following four advantages. First, a user may send two bits per symbol instead of one bit as in a PPM or a BPSK modulation scheme. Second, this method may achieve the power efficiency of a PPM signal because the number of levels, M ppm , may be increased (e.g., by increasing the number of possible delays for sending the bi-phase pulse) without increasing the average power. Third, this method substantially eliminates spectral lines due to no/low correlation between signal pulses. Fourth, this method enables multi-user access. From the simulations, it is seen that biorthogonal modulation achieves the same result as bi-phase modulation with an added feature that a user is sending two bits per symbol instead of one. Another advantage of biorthogonal modulation is that the power spectral density of the pulse is achieved. Therefore, one can meet the FCC mask requirement merely by using a pulse that has a PSD which meets the mask requirements. 
   The PSD for biorthogonal modulation is essentially the PSD of the monocycle pulse for biorthogonal modulated signals with time-hopping sequences when the pulse is not repeated between time-hops. Essentially from one time-hopping index to the next, a different modulated pulse is sent. This means, that no matter what pulse is used, it is expected that the PSD will be the PSD of the pulse shape by itself. The inventors have determined that the PSD of the monocycle pulse (i.e., the 2nd derivative of a Gaussian pulse) may not meet the PSD mask requirements imposed by the FCC. The inventors have also determined that higher derivatives of the Gaussian pulse (e.g. 5th order and above) meet these requirements. Because biorthogonal modulation maintains the PSD of the pulse shape, any pulse shape that has a PSD within the FCC mask may be used to meet the FCC&#39;s PSD mask requirements for the biorthogonal modulated signal without the need of any other pulse manipulation. 
   Although the invention has been described in terms of a transmitter  100  including a pulse controller  102 , pulse generator  104 , and time-hopping controller  106 , it is contemplated that the invention may be implemented in software on a computer (not shown), such as a general purpose computer, special purpose computer, digital signal processor, microprocessor, microcontroller, or essentially any device capable of processing signals. In this embodiment, one or more of the functions of the various components may be implemented in software that controls the computer. This software may be embodied in a computer readable carrier, for example, a magnetic or optical disk, a memory-card or an audio frequency, radio-frequency, or optical carrier wave. 
   In addition, although the invention is illustrated and described herein with reference to specific embodiments, the invention is not intended to be limited to the details shown. Rather, various modifications may be made in the details within the scope and range of equivalents of the claims and without departing from the invention.