Abstract:
A reception simplified, reduced cost spread spectrum transceiver device and method. The transmitting portion of the transceiver antipodally modulates a pseudorandom code and non-antipodally modulates binary bit data onto a carrier wave. The receiving portion of the transceiver performs a mathematical squaring function on the received signal combination of pseudorandom code and data. Squaring the pseudorandom code results in a cancellation of the code and the data remains in the absence of the pseudorandom code. The device and method of the invention eliminates the need for storing or generating a local reference for the spreading code or the need for any knowledge of key signal parameters at the receiver greatly simplifying receiver circuitry and allowing receiver operation at low energy density and low operational cost. Additionally, the device and method of the invention eliminates the synchronization constraint between the transmitted data and spreading code of conventional systems.

Description:
RIGHTS OF THE GOVERNMENT 
     The invention described herein may be manufactured and used by or for the Government of the United States for all governmental purposes without the payment of any royalty. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to the field of spread spectrum communications and more particularly to direct sequence spread spectrum transceivers. 
     Spread spectrum refers to a type of signal modulation in which the bandwidth required for transmission greatly exceeds the information bandwidth. Motivation for spread spectrum signals is based on the following: (1) the ability to reject intentional and unintentional jamming by interfering signals so that information can be communicated; (2) low probability of interception or detection since the power in the transmitted wave is “spread” over a large bandwidth or frequency extent; (3) message privacy since the signals cannot be readily demodulated without knowledge of the code; (4) tolerance of multipath reception; and (5) the ability to send many independent signals over the same frequency band. 
     One generic type of spread spectrum signal and the most often used is a direct sequence spread spectrum (DSSS) signal. In direct sequence modulation, each bit of an information-bearing signal, the information desired to be transmitted which is inherently random or at least unknown to the receiver and contains the information to be communicated bit by bit, is modulated by a higher frequency, pseudorandom code signal. Direct-sequence spread spectrum is commonly achieved by directly modulating a carrier with a high rate digital pseudorandom spreading code. For example, digital data at a much lower rate is modulo-2 added to a pseudorandom code. Modulo-2 addition uses symbols consisting of 0 or 1 and is equivalent to multiplication with symbols consisting of ±1&#39;s. Both the data and the pseudorandom code are multiplied by a carrier. This modulation method is often referred to as phase shift keying because the multiplication of data, code and carrier result in shifts of the carrier by 180 degrees. A bit value of one produces no phase shift and a bit value of minus one produces a 180 degree phase shift. 
     A generic DSSS signal can be represented as 
     
       
           s   i ( t )=cos(ω 0   t+d   i ( t )+ m ( t ))  (Eq. 1) 
       
     
     where ω 0  is the carrier frequency, d i (t) is the phase modulation due to a set of i data symbols, and m(t) is the phase modulation due to the spreading sequence. For conventional binary phase shift keying DSSS signaling this may be simplified to 
     
       
           s   i ( t )= d   i ( t ) m ( t )cos(ω 0   t ).  (Eq. 2) 
       
     
     Now d i (t) represents the data waveform consisting of the binary symbols ±1, and m(t) represents the pseudorandom code also consisting of the binary symbols ±1. For binary phase shift keying signaling, the combination of code modulo-2 added with data will produce 1 of 2 symbols: 
     
       
           s   1 ( t )=cos(ω 0   t )  (Eq. 3) 
       
     
     
       
           s   2 ( t )=cos(ω 0 +π)  (Eq. 4) 
       
     
     These waveforms are called ‘antipodal’ signals since s 1 (t)=−s 2 (t). This is shown vectorially in FIG.  1 . FIG. 1 shows an I or inphase-axis at  102  and a Q or quadature phase-axis at  101 . The signals s 1 (t) and s 2 (t), respectively are shown along the positive inphase-axis at  103  and the negative inphase-axis at  100 . The signals are antipodal by virtue of the fact that they are of equal magnitude and are 180 degrees apart in phase. 
     If the signal s i (t) is squared, which is done when a simplified receiver is used, as is described below, and the trigonometric identity cos 2 α=½+½ cos 2α is applied, the following results: 
     
       
         [ s   i ( t )] 2   =d   i   2 ( t ) m   2 ( t )cos 2 (ω 0   t )  (Eq. 5) 
       
     
     then, for i=1 or 2 for binary signaling as in (3) and (4) above, 
       s   1   2 ( t )=cos 2 (ω 0   t +π)=½+½ cos(2ω 0   t ),  (Eq. 6) 
     and 
     
       
           s   2   2 ( t )=cos 2 (ω 0   t )=½+½ cos(2ω 0   t ).  (Eq. 7) 
       
     
     The result is a direct current component and a sinusoid at twice the carrier frequency, reduced by 3 dB. A power reduction of ½ results in a 3 dB loss. That is, 10 log (½)=−3.01 dB. The resulting waveform is devoid of both pseudorandom code and data components. This is a commonly known result of squaring binary phase shift keying antipodal signals. Traditional DSSS receivers, therefore, have not employed a mathematical squaring operation at the receiver because the antipodally modulated data would be eliminated in such operation. 
     FIGS. 2 and 3 show generally prior art arrangements of a DSSS transmitter and receiver, respectively. The transmitter of FIG. 2 shows a binary bit data sequence  200 , a pseudorandom spreading code  201  both combined by a mixer  202 . The signal is then modulated using a binary phase shift keying modulator  203  onto a carrier wave  204 . The signal is then transmitted at  205 . The prior art receiver of FIG. 3 receives the signal at  301 . Using a stored reference, the pseudorandom code at  303  is eliminated by the mixer  302  and the transmitted data signal is recovered and then demodulated using a binary phase shift keying demodulator  304 . The binary bit data signal is then outputted at  305  to be used in the specific application. 
     In traditional DSSS systems, for the receiver to function, a replica of the pseudorandom code must be generated in the receiver and must be brought into time and frequency alignment with the transmitted pseudorandom code. High implementation costs are associated with such receivers. A squaring loop is then used to recover the carrier while a parallel path retains the data for coherent demodulation. Their function is to track the carrier and thereby allow phase-coherent demodulation of the data, that is, multiplication by a signal exactly in phase with the received carrier. 
     One of the disadvantages of using DSSS is the cost of implementation due primarily to the difficulty in achieving synchronization with the high rate pseudorandom spreading sequence. Despreading of a DSSS signal without knowledge of the pseudonoise (pseudorandom) spreading sequence or code synchronization has been previously described by a technique known as “blind despreading”. This is achieved with knowledge of key signal parameters by either the “autocorrelation” or “cyclic autocorrelation” methods. 
     The present invention provides a method and device for blind despreading that requires only knowledge of carrier frequency and signal rate and is made possible by the use of novel non-antipodal phase shift keying phase modulation. This provides the advantages of both low energy density and low implementation costs. Low energy density is useful for applications such as space based systems that are required to maintain low power spectral densities on earth to reduce co-channel interference and to satisfy regulatory constraints. Implementation costs may be significantly lowered since this technique requires no embedded reference code or synchronization at the receiver. This suggests a variety of wireless applications where the advantages of spread spectrum modulation are desired and where numerous low cost receivers are required such as for computer local area networks (LAN), emergency alerting devices, and space-based broadcast systems of all types. 
     SUMMARY OF THE INVENTION 
     The present invention comprises a reception simplified, reduced cost direct sequence spread spectrum transceiver device and method. The transmitting portion of the transceiver antipodally modulates the pseudorandom code and non-antipodally modulates the data onto a carrier wave. The receiving portion of the transceiver performs a mathematical squaring function on the received signal combination of pseudorandom code and data. Squaring the pseudorandom code removes the code with the data remaining in the absence of the pseudorandom code. The device and method of the invention eliminates the need for storing or generating a local reference for the spreading code or the need for any knowledge of specific signal parameters at the receiver, greatly simplifying receiver circuitry. Additionally, the device and method of the invention eliminates the synchronization constraint between the transmitted data and spreading code of conventional systems allowing receiver operation at low energy density in receiver apparatus and low operational cost. 
     It is therefore an object of the present invention to provide a reduced cost spread spectrum transceiver device and method that provides simplified reception. 
     It is another object of the invention to provide a spread spectrum transceiver device operable in the absence of a stored reference at the receiver. 
     It is another object of the invention to provide a spread spectrum transceiver device operable in the absence of specific signal parameter knowledge at the receiver. 
     It is another object of the invention to provide a spread spectrum transceiver device and method employing non-antipodal phase shift keying modulation. 
     It is another object of the invention to provide a spread spectrum transceiver device and method for that allows for M-ary phase modulation data schemes. 
     It is another object of the invention to provide a spread spectrum transceiver device and method that eliminates the synchronization constraint between data and spreading code of conventional spread spectrum systems. 
     It is another object of the invention to provide a spread spectrum transceiver device and method that has a lower energy density in receiver apparatus than conventional systems and can be implemented at lower cost. 
     These and other objects of the invention are described in the description, claims and accompanying drawings and are achieved by a despreading simplified, receiver circuitry-minimized direct sequence spread spectrum transceiver device comprising: 
     a binary bit data interfacing circuit; 
     a binary bit pseuodrandom code generating circuit; 
     a non-antipodal modulation preprogrammable digital synthesis phase modulating device communicating with said binary bit data interfacing circuit and outputting a binary bit data sequence of preselected frequency; 
     a binary bit pseudorandom code and binary bit data sequence mixing device, said output of said mixing device modulated onto a spread spectrum carrier wave; 
     an operator determinative frequency translating circuit communicating with said mixing device; 
     a mathematical squaring function spread spectrum receiving circuit, said receiving circuit receiving output from said operator determinative frequency translating circuit and performing a mathematical squaring function thereon, said squaring function eliminating said binary bit psuedorandom code and preserving said binary bit data sequence; and 
     said mathematical squaring function capability receiving circuit including an operator determinative binary bit data demodulating circuit. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 shows a prior art mathematical vector relationship of antipodal signals. 
     FIG. 2 shows a prior art antipodal binary phase shift keying transmitter. 
     FIG. 3 shows a prior art antipodal binary phase shift keying receiver. 
     FIG. 4 shows a mathematical vector relationship of squared non-antipodal signals. 
     FIG. 5 shows a non-antipodal binary phase shift keying transmitter. 
     FIG. 6 shows a non-antipodal binary phase shift keying receiver. 
     FIG. 7 shows a graphical representation of non-antipodal data modulation angles. 
     FIG. 8 shows a graphical representation of the phase angles of FIG. 7 entering a demodulator. 
     FIG. 9 shows an example of a non-antipodal spread spectrum quadature phase shift keying receiver. 
     FIG. 10 shows a mathematical vector relationship of non-antipodal signals. 
     FIG. 11 shows a possible circuit arrangement for a digital synthesis device. 
    
    
     DETAILED DESCRIPTION 
     The direct sequence spread spectrum transceiver device and method of the invention eliminates the need for a stored reference or the need for knowledge of key signal parameters at the receiver, greatly simplifying spread spectrum receiver circuitry and thereby allowing receiver operation at low energy density and low operational cost. The basic components of the transceiver include a source for a binary bit data sequence such as a computer, a pseudorandom spreading code generator, means for non-antipodally modulating the transmitted data sequence, a mixer for combining the modulated data sequence and pseudorandom code onto a carrier sequence or wave, and a filter for eliminating unwanted signal artifacts. Finally, the transceiver includes a receiver for receiving the transmitted signal, performing a mathematical squaring function on the received signal and means for demodulating the recovered data sequence. 
     A binary bit data sequence or signal comprised of 0&#39;s and 1&#39;s is outputted by a computer or other data source. In order to transmit as a direct sequence spread spectrum signal the signal must be modulated and spread. In conventional analog systems modulation of both the data and pseudorandom code is generally accomplished by binary phase shift keying or antipodal phase modulation. That is, the phase is modulated in increments of 180 degrees (a mathematical squaring would eliminate all the data and code). In conventional systems, binary phase shift keying is accomplished using a double balance mixer. A double balance mixer is an analog-based operating device and has the capability to modulate, or change the phase of a signal by 180 degrees. A receiver receiving such a modulated signal requires specific knowledge of the data signal being transmitted in order to despread the data signal. 
     In contrast to conventional systems, a significant aspect of the present invention is directed to accomplishing phase shift keying of a binary bit data signal using a digital synthesis device. One such device is manufactured by Analog Devices, part number AD9850. A direct digital synthesis device operates by programming a preselected length of binary bit words into the device which in turn outputs desired frequency. The AD9850 and other similar devices permit the selection of incremental phase values. The AD9850 allows increments of 180 degrees, 90 degrees, 45 degrees, 22.5 degrees and 11.25 degrees and any combination thereof. Other devices may permit finer resolution or a direct digital synthesis device could be designed to produce a specific resolution. The structure and characteristics of a digital synthesis device are well known to those skilled in the art. 
     FIG. 11 illustrates a possible circuit arrangement for a direct digital synthesis device. The direct digital synthesis device of FIG. 11 comprises a phase comparator at  1111 , a filter at  1112 , a voltage oscillator at  1113  and a component for varying phase at  1115 . Generally, the circuit of FIG. 11 is a closed loop system that compares the phase of two signals. The reference frequency signal at  1110  enters the phase comparator  1111 , the signal is filtered at  1112  and then actuates a voltage oscillator  1113 . A pre-programmed digital, binary bit word  1116  of preselected length varies the phase of the signal until an acceptable variation is achieved and the signal of desired frequency f o  at  1114  is output. 
     Use of a digital synthesis device in the transceiver of the present invention is significant due to its capability to modulate signals in incremental phase values, or in other words, modulate non-antipodally in increments other than 180 degrees. Phase modulation from a direct digital synthesis device is non-antipodal, meaning the phase modulated signals do not cancel each other out when squared. This allows the phase modulated signal to be despread at the receiver by a mathematical squaring operation which retains the data and eliminates the pseudorandom code. Recovering the transmitted data by a simple squaring operation eliminates the need for a stored reference and other key signal parameters at the receiver. 
     To appreciate non-antipodal phase shift keying modulation as used in the present invention, consider binary symbols and let d i (t), be a phase modulation signal due to a set of data symbols as used in Eq. 1, let d i (t)ε{π/4,−π/4}, or equivalently d(t) π/4, where d(t) is ±1. With m(t) being pseudorandom code as in Eq.2 the signal may now be written as 
       s   i ( t )= m ( t )cos(ω 0   t+d ( t )π/4).  (Eq. 8) 
     This results in quaduature phase components: 
     
       
           s   1 ( t )=cos(ω 0   t+π/ 4) for  m ( t )= d ( t )=1  (Eq. 9) 
       
     
     
       
           s   2 ( t )=cos(ω 0   t− 3π/4) for  m ( t )=−1 and  d ( t )=(Eq. 10) 
       
     
     
       
           s   3 ( t )=cos(ω 0   tπ/ 4) for  m ( t )=1 and  d ( t )=−1 (Eq. 11) 
       
     
     
       
           s   4 ( t )=cos(ω 0   t+ 3π/4) for  m ( t )= d ( t )=−1.  (Eq. 12) 
       
     
     FIG. 10 shows the vector relationship of signals represented mathematically by Equations 9-12 phase shifted non-antipodally. The Q or quadature phase axis is shown at  1000  and the I or inphase axis is shown at  1001 . The vector representation of the signal of Eq. 9 is shown at  1002  in FIG. 10, the signal of Eq. 10 is shown at  1004 , the signal of Eq. 11 is shown at  1003 , and the signal represented by Eq. 12 is shown at  1005 . Notice that the signal s 1 (t) at  1002  and the signal s 3 (t) at  1003  are antipodal since s 1 (t)=−s 3 (t). Similarly, signal s 2 (t) at  1004  and s 4 (t) at  1005  are antipodal since s 2 (t)=−s 4 (t). This phase relationship is referred to as quadrature waveform. Squaring the quadrature waveform signals produces, 
     
       
           s   1   2 ( t )= s   2   2 ( t )=cos 2 (ω 0   t +π/4)=½+½ cos(2ω 0   t+π/ 2)  (13) 
       
     
     
       
           s   3   2 ( t )= s   4   2 ( t )=cos 2 (ω 0   t−π/ 4)=½+½ cos(2ω 0   t −π/2).  (14) 
       
     
     The result now is a direct current component, a sinusoid at twice the carrier frequency reduced by half, and data keying the carrier phase ±π/2. The pseudorandom code has been removed since it gave rise to keying between s 1  and s 3  and between s 2  and s 4 , which are found to overlap when squared. In contrast, in a conventional system the signals would be antipodally modulated, that is, they would be shifted by 180 degrees and application of a squaring operation would result in masking the code due to overlap. However, in the device and method of the present invention transmitted data remains because it gives rise to keying between s 1  and s 3  and between s 2  and s 4  which are non-antipodal. 
     Vectorially, this is shown in FIG.  4 . The inphase axis component, I, is shown along the axis at  401  and the quadature phase component, Q, is shown along the axis at  402 . The signals s 1   2  and s 2   2  are represented by the bold line at  404  on the positive Q axis and signals s 3   2  and s 4   2  are represented by the bold line  403  on the negative Q axis. This illustrates that after squaring the signal, which removes the code, the data becomes antipodal. 
     The direct digital synthesis device may be used for modulating both antipodal signals and nonantipodal signals. Accordingly, antipodal modulation of the pseudorandom code may also be accomplished by a direct digital synthesis device. 
     FIGS. 5 and 6 show a possible hardware arrangement for a non-antipodal phase shift keying DSSS system. FIG. 5 shows a non-antipodal binary phase shift keying transmitter. A binary bit input data sequence is shown at  500 , a non-antipodal binary phase shift keying modulator is shown at  501  and a carrier is shown at  502 . The non-antipodal modulator  501  modulates the input data by phase increments of π/4 onto a carrier wave cos (ω 0 t). The modulated data is then mixed at  503  with a antipodally modulated pseudorandom code g(t) represented at  504 . The signal is then transmitted as represented at  505 . 
     FIG. 6 shows a non-antipodal phase shift keying receiver. The signal is received at the antenna  600  and the circuit capable of performing the squaring function is represented at  601 . The squaring operation performed by the receiver removes the pseudorandom code, but leaves the binary bit data sequence intact. The binary bit data sequence is then demodulated at  602 . The data is then output at  603 . 
     An example quadrature non-antipodal phase shift keying receiver arrangement according to the invention is shown in FIG.  9 . The non-antipodally modulated binary bit data signal is received at  900  and a mathematical squaring operation is performed at  901 . The mathematical squaring function operation eliminates the pseudorandom code and the modulated binary bit digital data signal continues transmission through the receiver. The I-Q down converter  902  reduces the received carrier frequency to a lower frequency that can be handled easier by the demodulator while keeping the inphase and quadature components of the signal in two separate channels. The I-Q down demodulator  903  demodulates the data and the desired estimated data  904  is transmitted for use in the specific application. 
     The method and device of the present invention allows for a wide range of modulation capability. In theory, signal modulation could occur at phase increments of any degree and is only limited by the capabilities of the digital synthesis device performing the modulation. The criteria for determining the modulation, that is the signal phase shift or angle, depends upon the application. Simply stated, the data modulation angles, d i (t), must be chosen so that when they are doubled modulo 2π, none overlap. The following provides a rule for selecting modulation angles. Consider a pair of modulation angles d i (t) ε{φ 1 ,φ 2 } to be used in a DSSS signaling scheme. With m(t)=0 or π for antipodal modulation of the pseudorandom code, the signal of Eq. 1 now becomes 
     
       
           s   i ( t )=(cosω 0   t+d   i ( t )+ m ( t )).  (15) 
       
     
     The signal set is shown vectorially in FIG.  7 . In FIG. 7 the inphase axis I is shown at  700  and the quadature phase axis Q is shown at  701 . The modulation angle φ 1  is shown at  704  and the modulation angle φ 2  is shown at  703  with the modulated signals φ 1 +π and φ 2 +π shown, respectively, at  706  and  705 . 
     Following the squaring operation performed on the non-antipodally modulated signals, the modulation angles are doubled and the spreading code modulation is removed. The phases and signals entering the demodulator are shown in FIG.  8 . The inphase axis, I, is shown at 800 and the quadature phase axis Q is shown at  801 . The decision boundary, that is, the imaginary line that divides the space where each of the two transmitted symbols are equally spaced is shown at  802 , phase angle  2 φ 1  is shown at  805  and phase angle  2 φ 2  is shown at  804 , with the associated signals shown at  803  and  806 , respectively. 
     If only nearest neighbor errors are considered, then the probability of error in selecting phase modulation angles is minimized by maximizing the angular separation of the signals entering the demodulator. In a binary signaling system, the maximum separation is π, or 180°. This indicates that the separation between the data modulation angles (the phases prior to entering the squarer) should be π/2. The example above used ±π/4, however any pair of angles separated by 90° would yield the same performance. The choice of different pairs of angles only impacts the orientation of the decision boundary used by the demodulator. For higher order M-ary systems, the rule remains the same: optimal separation for the data modulation angles is π/M. For instance, the binary system based on ±π/4 could be converted to an optimal 4-ary system by adding 0 and π/2 to the data set, i.e. d i (t)ε{0, π/2, π/4, −π/4}. The signal entering the demodulator based on these angles would appear to be a standard quadature phase shift keying signal. Note that some form of quadrature demodulation must be used to recover the data. 
     This type of receiver, one possible arrangement being shown in FIG. 9, would be required for any system where the decision boundary does not lie along the quadrature axis in the phase plane. For binary systems where the decision boundary does lie along the quadrature axis, d i (t)ε{0, π/2} for example, a simple integrate and compare-to-zero demodulator may be used. 
     The theoretical performance of the system is most easily understood as follows. Consider an optimal binary modulation scheme in the presence of additive white Gaussian noise (AWGN) 
     
       
           r ( t )=A cos(ω 0   t+d   i ( t )+ m ( t ))+ n ( t ),  (16) 
       
     
     where A is a scale factor for power, ω 0  is the carrier frequency, d i (t)ε{0, π/2}, and m(t) is ±π. When the received signal r(t) is squared we get the result                        r   2          (   t   )       =                    A   2     /   2                +         A   2     /   2                   cos                   (       2                   ω   0        t     +     2          d   i          (   t   )         +     2        m        (   t   )           )       +       n   2          (   t   )       +                              2        An        (   t   )          cos                     (         ω   0        t     +       d   i          (   t   )       +     m        (   t   )         )     .                     (   17   )                                
     This signal is then down converted by cos(2ω 0 t) resulting in                        r   d   2          (   t   )       =                      A   2     /   4                   cos                   (     2          d   i          (   t   )         )       +         A   2     /   2                   cos                   (     2                   ω   0       )       +                                    A   2     /   4                   cos                   (       4                   ω   0       +     2          d   i          (   t   )           )       +         n   2          (   t   )                     cos                   (     2                   ω   0       )       +                                An        (   t   )       [       cos                   (       3        ω   0        t     +       d   i          (   t   )       +     m        (   t   )         )       +                                  cos                   (         -     ω   0          t     +       d   i          (   t   )       +     m        (   t   )         )       ]     .                 (   18   )                                
     Note that the 2m(t) argument has been removed since it is equal to ±2π and thus has no effect on the signal. Next, this signal is lowpass filtered to remove content at frequencies above R d , the data rate. There are now three terms which contribute to the noise: the squared noise term n 2 (t) modulated at 2ω 0 , and the scaled noise terms An(t) at the carrier frequency and its third harmonic. The tone at 2ω 0  and the narrowband signal at 4ω 0  are assumed to be removed by the filter. 
     The interference due to the noise can be analyzed by examining the power spectral density of the signal entering the demodulator. First consider the autocorrelation of the signal entering the demodulator, 
     
       
           R (τ)= E{r   d   2 ( t ) r   d   2 ( t −τ)},  (19) 
       
     
     where E{.} is the expected value operation. Expanding R(τ) yields 36 terms, but with the assumptions that the noise and transmitted signal are uncorrelated and that the expected value of cos (nω 0 t) is zero, then R(τ) reduces to                      R        (   τ   )       =                E        {           A   4     8        cos                   (     2                   ω   0        τ     )       +         A   2     4            n   2          (     t   -   τ     )                     cos                   (     2                   ω   0        τ     )       +                                        A   4       16                     cos                   (     2          d   i          (   t   )         )        cos                   (     2          d   i          (     t   -   τ     )         )       +                                    A   4     32        cos                   (       4        ω   0        τ     +     2          d   i          (     t   -   τ     )         -     2          d   i          (   t   )           )       +                                    A   4     4            n     2                       (   t   )                     cos                   (     2        ω   0        τ     )       +       1   2            n   2          (   t   )                         n   2          (     t   -   τ     )          cos                   (     2        ω   0        τ     )       +                                    A   2     2          n        (   t   )            n        (     t   -   τ     )          cos                   (       3        ω   0        τ     +     m        (   t   )       -     m        (     t   -   τ     )       +       d   i          (   t   )       -       d   i          (     t   -   τ     )         )       +                                    A   2     2          n        (   t   )            n        (     t   -   τ     )          cos                   (         ω   0        τ     -     m        (   t   )       +     m        (     t   -   τ     )       -       d   i          (   t   )       +       d   i          (     t   -   τ     )         )       }     .                 (   20   )                                
     Taking the expected value over t results in                      R        (   τ   )       =                      A   4       8                     cos                   (     2        ω   0        τ     )       +         A   2     4            R   nn          (   0   )          cos                   (     2        ω   0        τ     )       +         A   4     16            R   dd          (   τ   )         +                                    A   4       32                     cos                   (     4        ω   0        τ     )                       R   dd          (   τ   )         +         A   2     4            R   nn          (   0   )                       cos        (     2        ω   0        τ     )         +                                    1   2            R   nn   2          (   0   )                     cos                   (     2        ω   0        τ     )       +         R   nn   2          (   0   )            ∂     (   τ   )         +       A   2            R   nn          (   0   )            ∂     (   τ   )           ,                   (   21   )                                
     where R nn (τ) is the autocorrelation of the AWGN, and R dd (τ) is the autocorrelation of the squared baseband data signal. Taking the Fourier transform of the R(τ) gives the resulting PSD lower spectral density as                      S        (   ω   )       =                      A   4       8                         S   d          (   ω   )         +       R   nn   2          (   0   )       +       A   2            R   nn          (   0   )         +         A   4     32            S   d          (     ω   ±     4        ω   0         )         +                                ∂       (       ±   2          ω   0       )          [         A   4     16     +         A   2     4            R   nn          (   0   )         +       1   4            R   nn   2          (   0   )           ]         ,                   (   22   )                                
     where S d (ω) is the PSD of the squared baseband data signal. The spectral components at 2ω 0  and the image of the data at 4ω 0  are removed by lowpass filtering. The components of the filtered PSD can be factored into the input (prior to squaring) and output (ust before demodulation) signal-to-noise ratios giving                    SNR   out       SNR   in       =       A   2       8        (         R   nn          (   0   )       +     A   2       )           ,           (   23   )                                
     which indicates a 9 dB loss in SNR for signals whose power is significantly larger than the variance of the additive white gaussian noise. This 9 dB loss is the price incurred is for including the squaring operation which removes the synchronization and reference code generation requirements at the receiver. 
     It is understood that certain modifications to the invention as described may be made, as might occur to one with skill in the field of the invention, within the scope of the appended claims. All embodiments contemplated hereunder which accomplish the objects of the invention have therefore not been shown in complete detail. Other embodiments may be developed without departing from the spirit of the invention or from the scope of the appended claims.