Abstract:
This invention discloses a method for communications faster than the Nyquist rate (FTN) and faster than the Shannon rate which method is Quadrature Layered Modulation (QLM). QLM properties include scaling the data symbol pulses to maintain the same error rate performance for all rates. QLM alternatively considers the increase in the data symbol rate to be a layering of additional communications over the same link. The Shannon bound is a limit on the capacity of a communication link when transmitting data symbols at the Nyquist rate. QLM observes one can communicate at FTN to transmit more information than the Shannon rate since the Nyquist rate captures the information in a frequency band and does not constraint the information. These properties describe QLM and a separate math proof-of-concept is disclosed. Implementation and performance data demonstrate QLM can support communications data rates which are at least double the Shannon rate.

Description:
This patent application is a continuation-in-part (CIP) of patent application Ser. No. 13/068,032. 
     BACKGROUND OF THE INVENTION 
     I. Field of the Invention 
     The present invention relates to cellular communications and also relates to the Nyquist rate for data symbol transmission, the Shannon bound on communications capacity, and symbol modulation and demodulation for high-data-rate satellite, airborne, wired, wireless, and optical communications and includes all of the communications symbol modulations and the future modulations for single links and multiple access links which include electrical and optical, wired, mobile, point-to-point, point-to-multipoint, multipoint-to-multipoint, cellular, multiple-input multiple-output MIMO, terrestrial networks, and satellite communication networks. In particular it relates to WiFi, WiFi 802.11ac, WiMax, long-term evolution LTE, 3G, 4G for cellular communications and satellite communications. WiFi, WiMax use orthogonal frequency division multiplexing OFDM on both links and LTE uses single carrier OFDM (SC-OFDM) on the uplink from user to base station and OFDM on the downlink form base station to user. WiMax occupies a larger frequency band than WiFi and both use OFDM waveforms. SC-OFDM LTE is a single carrier orthogonal waveform version of OFDM which uses orthogonal frequency subbands of varying widths. 
     II. Description of the Related Art 
     Bounds on current communications capacity are the communications Nyquist rate, the Shannon rate, and the Shannon capacity theorem. The Nyquist complex sample rate is 1/T=B where B is the signal bandwidth, the Shannon rate W≧B in Hz is the frequency band W=(1+α)B where α is the excess bandwidth required to capture the spillover of the signal spectrum beyond B with a representative value being α=0.25, and the Shannon capacity theorem specifies the maximum data rate C in Bps (bits/second) which can be supported by the communications link signal-to-noise power ratio SNR=S/N over W. 
     The Nyquist rate 1/T is the complex digital sampling rate 1/T=B that is sufficient to include all of the information within a frequency band B over a communications link. Faster than Nyquist rate communications (FTN) transmits data symbols at rates 1/T s ≧1/T wherein 1/T s  is the data symbol transmission rate in the frequency band B which means T s  is the spacing between the data symbols. FTN applications assume the communications links with a data symbol rate equal to the Nyquist rate 1/T s =1/T=B operate as orthogonal signaling with no intersymbol interference (ISI) between the demodulated data symbols. It is common knowledge that the communications data symbol rate 1/T s  for orthogonal signaling can be increased to as high as 25% above the Nyquist complex sample rate 1/T s =1/T=B with very little loss in E b /N o  and with no perceptible loss in some cases using simple data symbol modulations. Above 25% it has been observed that there is a rapid loss in signal strength. Only a few studies have addressed data symbol rates above 25%. An example in U.S. Pat. No. 8,364,704 transmits digital bit streams at FTN rates and depends on the transmit signal alphabets to be in distinct locations on receive to enable alphabet detection. There is no comparison or proof that the data rate performance is comparable to the performance using conventional orthogonal signaling. 
     The Shannon bound [1]-[5] on the maximum data rate C is complemented by the Shannon coding theorem, and are defined in equations (1). 
     Shannon bound and coding theorem 
     1 Shannon capacity theorem
 
 C=W  log 2 (1+SNR)  (1)
 
     2 Shannon coding theorem for the information bit rate R b  
         For R b &lt;C there exists codes which support reliable communications   For R b &gt;C there are no codes which support reliable communications
 
wherein C in Bps is the channel capacity for an additive white Gaussian noise AWGN channel in W, “log 2 ” is the logarithm to the base 2, and C is the maximum rate at which information can be reliably transmitted over a noisy channel where SNR=S/N is the signal-to-noise ratio in W.
       

     MIMO communications enable higher capacities to be supported with multiple independent links over the same bandwidth. This multiple-input multiple-output MIMO requires the physical existence of un-correlated multiple communications paths between a transmitter and a receiver. MIMO uses these multiple paths for independent transmissions when the transmission matrix specifying these paths has a rank and determinant sufficiently large to support the paths being used. In MIMO U.S. Pat. No. 7,680,211 a method is disclosed for constructing architectures for multiple input transmit and multiple output receive MIMO systems with generalized orthogonal space-time codes (C 0 ) which are generalization of space-time codes C and generalizations (H 0 ) of the transmission matrix (H) that enable the MIMO equation Y=Hf(C,X)+No to be written Y=H 0 C 0 X+N o  which factors out the input signal symbol vector X and allows a direct maximum-likelihood ML calculation of the estimate {circumflex over (X)} of X, and wherein Y is the received (Rx) symbol vector, N o  is the Rx noise vector, and f(C,X) is a non-separable encoding C of X. 
     OFDM waveform implement the inverse FFT (IFFT=FFT −1 ) to generate OFDM (or equivalently OFDMA which is orthogonal frequency division multiple access to emphasize the multiple access applications). OFDM uses pulse waveforms in time and relies on the OFDM tone modulation to provide orthogonality. SC-OFDM is a pulse-shaped OFDM that uses shaped waveforms in time to roll-off the spectrum of the waveform between adjacent channels to provide orthogonality, allows the user to occupy subbands of differing widths, and uses a different tone spacing, data packet length, and sub-frame length compared to OFDM for WiFi, WiMax. In addition to these applications the symbol modulations 4PSK, 8PSK, 16QAM, 64QAM, 256QAM are used for satellite, terrestrial, optical, and nearly all communication links and with maximum data symbol rates achieved using 256QAM. 
     SUMMARY OF THE INVENTION 
     This invention introduces quadrature layered modulation (QLM) which is a FTN layered communications that scales the signal strength to compensate for the inter-symbol (ISI) degradation of the signal strength and to ensure that QLM has the same error rate performance as the original communications link. The parameter n p  is the QLM increase in data symbol rate above the current baseband communications data symbol rate for the link being considered, and equivalently is considered to be the number n p  of communications layers of QLM. Unless otherwise stated the communications links being addressed for QLM transmit a data symbol rate at the Nyquist rate 1/T s =1/T=B and which means the n p  is referenced to the Nyquist rate. Layering of communications over the same link is a preferred way to analyze QLM since it enables one to consider each layer as a separate communications link with the overall capacity being the sum of the component capacities of the n p  layers and enables one to prove the scaled Shannon bound applies to each layer. 
     QLM recognizes that the Shannon bound is a bound on the information capacity supported by the performance metrics E b /N o  and SNR=S/N and proves the scaled Shannon bound on data capacity applies to each of the n p  layered communications channels. The E b  is the data pulse energy per information bit b, N o  is the noise power density, S is the signal power, N=N o /T s  is the noise power, and SNR is the signal-to-noise power ratio and equivalently is the signal energy to noise energy ratio of the data pulse. Inverting the Shannon bound and using the E b /N o  and SNR=S/N definitions enables one to use the Shannon bound to specify the performance metrics C/W, E b /N o , SNR as functions of the information bits b supported by the data pulses. This is equivalent to specifying that each QLM layer of communications obeys the Shannon bound modified to include the required scaling by n p  to maintain the same error rate performance. Communications data rate C/W in units of Bps/Hz is the normalized communications data rate C or communications efficiency which is equal to the communications data rate C in Bps per unit of the frequency band W in Hz. 
     Scaling requires the E b /N o  and SNR for the baseband communications link to be scaled to the values n p E b /N o  and n p   2 SNR for QLM. For n p =2 this is the exact scaling value and for larger values n p &gt;2 this is a lower bound on the scaling parameter and requires the addition of a small “loss” in dB to be added to this scaling bound in dB to yield the required QLM scaling loss in dB. 
     QLM requires transmitting each layer with a discriminating parameter which enables separation and decoding of each layer. Considering QLM to be an increase in the data symbol rate, the discriminating parameter is the increase in the data symbol rate above the orthogonal Nyquist rate. This increase is detected as a correlated interference by the data symbol detection in the receiver, and enables a demodulation algorithm to unscramble these received and detected correlated data symbol estimates to recover the data symbol information. Discriminating parameters include separate time, frequency, and beam offsets and combinations thereof. A common requirement is that the discriminating parameters provide a means to support a unique inverse so the demodulator can unscramble the detected data symbol estimates. 
     The QLM properties and performance prove that QLM can support data rates at least double the rates supported by the Shannon bound. A complementary math proof-of-concept is disclosed in  FIG. 19  which proves QLM supports a data rate which is 2× the Shannon rate. 
     A new derivation of the bound on the QLM communications capacity is presented which uses a geometric derivation of the Shannon bound modified to apply to QLM. Results are the same as previously derived in QLM patent U.S. Pat. No. 7,391,819 with the geometric derivation providing a new insight into the QLM bound. 
     Two sets of QLM demodulation algorithms disclosed are the maximum likelihood (ML) data packet and the trellis algorithms. Trellis algorithms are the ML and the co-state maximum a-posteriori (MAP) algorithms. ML algorithms have their roots in the current set of ML convolutional decoding and turbo-decoding algorithms. The MAP algorithms were introduced to provide the correct mathematical technology for these applications and with a decoding and trellis demodulation performance which is at least as-good-as the ML algorithms. These algorithms provide demodulation architectures and implementations of QLM for OFDM, SC-OFDM LTE, MIMO to support at least doubling of the data rates and with applicability to all communications. Performance estimates are the normalized data rate C/.W vs. E b /N o  and SNR=S/N for the QLM and Shannon bounds, and for the reference 4PSK, 8PSK, 16QAM, 64QAM, 256QAM, 4096QAM data symbol modulations. PSK is phase shift keying data symbol modulation and QAM is quadrature amplitude data symbol modulation. Measured bit-error-rate (BER) simulation data is disclosed for trellis demodulation of 4PSK with n p =1-6 layers and for ML data packet demodulation of 4PSK and 256QAM for n p =1-8 layers. 
     Representative QLM OFDM and QLM SC-OFDM architectures are disclosed for implementation of QLM transmitters and receivers using frequency offset as the differentiating parameter for QLM OFDM and using time offset as the differentiating parameter for QLM SC-OFDM, and implementing the QLM demodulation with ML trellis or MAP trellis algorithms. Frequency layering for QLM OFDM transmits the QLM layers as frequency offset FFT −1  waveforms which are layered over the reference inverse FFT waveform FFT −1  which implements OFDM. For SC-OFDM for LTE the time layering transmits the QLM layers as SC-OFDM waveforms which are offset in time compared to the original SC-OFDM waveform. 
     A representative MIMO cellular communications link architecture using QLM with MAP trellis demodulation is disclosed in this invention for the transmit and receive signal processing algorithms and supporting block diagrams are developed to illustrate the architecture and implementation. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS AND THE PERFORMANCE DATA 
       The above-mentioned and other features, objects, design algorithms, and performance advantages of the present invention will become more apparent from the detailed description set forth below when taken in conjunction with the drawings wherein like reference characters and numerals denote like elements, and in which: 
         FIG. 1  describes how to increase the data rate using a pulse waveform. 
         FIG. 2  describes how QLM increases the data rate for a pulse waveform at a constant frequency bandwidth. 
         FIG. 3  presents the QLM Lemma 1 which defines the scaling of the E b /N o  and SNR. 
         FIG. 4  presents the QLM Lemma 2 bounds on QLM C/W and E b /N o  for communications and presents the QLM communication link performance metrics C/W, E b /N o , SNR 
         FIG. 5  defines QLM and lists the QLM properties. 
         FIG. 6  calculates the time pulse correlation, candidate waveform ψ time response, and the ψ correlation in time. 
         FIG. 7  calculates the frequency pulse correlation, DFT waveform frequency response, and the DFT correlation in frequency. 
         FIG. 8  describes the QLM ML n s =2, 3, 4 data symbol packets in the baseband layer and illustrates the overlaying of the n p −1 QLM layers on these baseband layers. 
         FIG. 9  presents the ML data packet measured BER performance vs. the QLM scaled (E b /N 0 )/n p  for a pulse waveform with 4PSK modulation for QLM with n p =1,2,4,6,8 layers for the n s =2,3,4 data symbol packets. 
         FIG. 10  presents the ML data packet measured BER performance vs. the QLM scaled (E b /N 0 )/n p  for a pulse waveform with 256QAM modulation for QLM with n p =1,8 layers for the n s =3,4 data symbol packets, and for a Grey coded 256QAM performance bound. 
         FIG. 11  describes how the ML and MAP trellis algorithms select the best path j×i from trellis state S k−1  to trellis state S k . 
         FIG. 12  presents the Matlab code for calculating the best path from trellis state S k−1  to trellis state S k  and using this selection to update S k  for each of the possible mainlobe states. 
         FIG. 13A  is a flow diagram of a MAP trellis symbol demodulation algorithm. 
         FIG. 13B  is the continuation of the flow diagram of the MAP trellis symbol demodulation algorithm. 
         FIG. 13C  is the continuation of the flow diagram of the MAP trellis symbol demodulation algorithm. 
         FIG. 14  is a flow diagram of the MAP trellis symbol demodulation algorithm with iteration to include the sidelobes. 
         FIG. 15  presents the MAP trellis measured BER performance vs. the QLM scaled (E b /N o )/n p  for a pulse waveform with 4PSK modulation for QLM layers n p =1,2,3,4 and for n p =4,6 using a ½-word MAP trellis demodulation. 
         FIG. 16  calculates the communications data rate C/W versus E b /N o  for the new bound on QLM, Shannon bound, for PSK and QAM, using the demodulation parameters in  FIG. 18 . 
         FIG. 17  calculates C/W versus SNR for the new bound on QLM, Shannon bound, for PSK and QAM, using the demodulation parameters in  FIG. 18 . 
         FIG. 18  lists the MAP trellis and ML demodulation parameters used to calculate the performance expressed as C/W vs. the E b /N o  in  FIG. 16  and S/N in  FIG. 17   
         FIG. 19  presents the performance data base for C/W vs. E b /N o  which demonstrates that the QLM data rate is 2× the Shannon rate for the highest cellular data modulation 256QAM. 
         FIG. 20  is a representative transmitter implementation block diagram for QLM OFDM. 
         FIG. 21  is a representative receiver implementation block diagram for QLM OFDM. 
         FIG. 22  is a representative transmitter implementation block diagram for QLM LTE. 
         FIG. 23  is a representative receiver implementation block diagram for QLM LTE. 
         FIG. 24  is a schematic cellular network with the communications link between a base station and one of the users. 
         FIG. 25  is a representative architecture of this invention disclosure for MIMO systems for application to the communications link in  FIG. 24 . 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     OFDM, SC-OFDM, LTE applications of quadrature layered modulation QLM in this invention disclosure are illustrated by WiFi and WiMax which uses OFDM on both uplinks and downlinks between the user and base station for cellular communications as well as for communications with satellites, and by LTE which uses OFDM on the downlink and SC-OFDM on the uplink. QLM OFDM adds layers of the OFDM orthogonal data symbol tones at frequency offsets and QLM SC-OFDM adds layers of data symbol tones in separate channels. 
       FIG. 1  introduces QLM by considering a pulse waveform in the time domain. In 1 the pulse waveform is transmitted at the data symbol rate equal to 1/T s =B where T s  is the pulse length, B is the bandwidth, the signal power level 2 is P=A 2  where “A” is the signal amplitude, and the pulse modulation is phase shift keying PSK with “b” information bits per data symbol. To increase the data symbol rate to n p /T s  and the information rate to n p b/T s , the pulse waveform is shortened 3 to T s /n p  which increases the bandwidth to n p B wherein B=1/T s  and requires the transmitted (Tx) power to be increased 4 to P=n p A 2  in order to keep the same pulse energy per bit 5 is E b =A 2 T s /b. The corresponding energy-per-bit to noise power ratio 6 is E b /N o =A 2 /2σ 2 b where 7 N o =2σ 2 T s  is the noise power density and 2σ 2  is the “mean square” level of communication noise. 
       FIG. 2  introduces the QLM scaling law in  FIG. 3  QLM Lemma 1 which implements the  FIG. 1  increase in the data symbol rate using QLM communications without changing the bandwidth of the pulse waveform by extending the pulses 8 in  FIG. 2  over the original pulse length T s  and layering these extended data symbol waveforms on top of each other  11  while occupying the same bandwidth B=1/T s . The pulse waveforms in each layer  13  have E b /N o  values equal to n p  times the original E b /N o =A 2 /2σ 2 b due to the stretching of each pulse over T s  without changing the power level of the pulse. The layers are time synchronized for transmission at ΔT s =T s /n p , 2ΔT s , . . . , (n p −1)ΔT s  offsets  14  respectively for layers  2 , 3 , . . . , (n p −1) relative to the 1 st  layer at zero offset. This means the signal-to-noise power S/N over B=1/T s  is equal to n p ^2 times the original S/N due to the addition of the n p  pulse power levels  12  over each T s  interval and the scaling of E b /N o  by n p . This scaling of E b /N o  in each of the layered communications channels is summarized in equation (2) along with the corresponding scaling of the SNR=S/N over T s . We find for the scaling law in  FIG. 3  Lemma 1
 
 E   b   /N   o   =n   p   [E   b   /N   o ] for each layer or channel= n   p   [A   2 /2σ 2   ]/b 
 
             SNR   =       ∑     n   ⁢           ⁢   p       ⁢       bE   b     /     N   o               = n   p   2   [S/N]=n   p   2   [A   2 /2σ 2 ]  (2)
 
wherein [“o”] is the value of “o” for the baseband communications channel when there is no layering.
 
       FIG. 3  QLM Lemma 1 scaling law derived in (2) for the QLM performance metrics E b /N o  and SNR is re-derived from a signal detection viewpoint to complement the derivation of (2) using  FIG. 2 . QLM received (Rx) demodulation signal processing synchronizes and removes the Rx waveform by performing a convolution of the Rx waveform encoded data symbol with the complex conjugate of this waveform, to detect the correlated data symbols. This convolution is a correlation of the waveform with itself as illustrated in  FIG. 6  and  FIG. 7  since the waveforms are real and symmetric. Consider a QLM communications link receiver detecting the Tx signal s(t) 
                     ·     s   ⁡     (   t   )         =       ∑   k     ⁢       x   k     ⁢       Ψ   k     ⁡     (     t   -       kT   s     /     n   p         )                   (   3   )               
where x k ψ k  are the data symbol pulses, the x k  are the encoded data symbols, and the pulse waveforms ψ k  are orthogonal at the Nyquist rate 1/T s =1/T=B, are real and symmetric, are normalized with the square norm equal to unity, and have correlation functions behaving like the triangular correlation function in FIG.  6 , 7 . The Rx signal z(t)=s(t)+v(t) is detected to recover the estimates y k  of the Tx data symbols x k  by implementing the correlation of the Rx signal with the pulse waveform at the Rx data symbol rate. We find
 
                           y   k     =       ⁢     〈       ψ   k     ,     z   ⁡     (   t   )         〉                 =       ⁢     〈       ψ   k     ,       s   ⁡     (   t   )       +     v   ⁡     (   t   )           〉                 =       ⁢       ∑   n     ⁢       ⁢     〈         ψ   k     ⁡     (     t   -       kT   s     /     n   p         )       ,         x   n     ⁢       ψ   n     ⁡     (     t   -       nT   s     /     n   p         )         +     v   ⁡     (   t   )           〉                   =       ⁢       x   k     +       ∑   i     ⁢           ⁢       c   ⁡     (        i        )       ⁢     (       x     k   +   i       +     x     k   -   i         )         +     N   k                     (   4   )               
wherein &lt;ψ k ,z(t)&gt; is the inner product of the waveform ψ k  with the Rx signal z(t) after synchronization and translation to baseband, the inner product of ψ k  with z(t) is the correlation of the complex conjugate of ψ k  with z(t) evaluated at the reference time t=0 for y k , correlation coefficients c(|i|) of the waveforms are symmetric c(+i)=c(−i) and equal to the inner product of the waveforms c(i)=&lt;ψ k (t−kT s /n p ),ψ k+i (t−(k+i)T s /n p )&gt;, and the v(t), N k  are the additive white Gaussian noise (AWGN). At the peak the normalized correlation is unity c(0)=1. The first correlation is c(1)=1−1/n p , the second is c(2)=1−2/n p , . . . , and c(n p )=0. The Euclidian distance between the peak correlation value at c(0)=1 and the first correlated signal is equal to 1−c(1)=1/n p  and between the second is 1−c(2)=2/n p , and so forth. The energy of the first correlated signal is equal to c(1) 2 (|x k+1 | 2 +|x k−1 | 2 ) and behaves as unwanted interference with equal statistical components along and orthogonal to the axis of the detected signal energy E=|x k | 2 =A 2 ∀k. This means that the squared Euclidian distance between the detected signal x k  energy level and the first correlated signal energy level along this detection axis is equal to A 2 /n p   2  for a mainlobe triangular correlation. This requires the signal energy to be increased to n p   2 A 2  to make this shortened distance n p   2 (A 2 /n p   2 )=A 2  equal to the same value A 2  measured in the absence of any correlation interference for n p =1 and therefore the same error rate performance for n p =2 causing the first correlated interference. This is a lower bound for n p &gt;2 and is used for the scaling law in (2).
 
       FIG. 5  layered communications property of QLM is demonstrated. We need the QLM FTN architecture to be viewed as a layered communications link to help understand QLM. This layering for integer values of n p  is constructed as an overlay of n p  independent communications signals s 1 (t), s 2 (t), . . . , s np (t) in equation (3) and yields 
                           s   ⁡     (   t   )       =       ∑   k     ⁢       x   k     ⁢       Ψ   k     ⁡     (     t   -       kT   s     /     n   p         )                       =       ∑     n   =   1       n   ⁢           ⁢   p       ⁢       ∑   k     ⁢       x   k     ⁢       Ψ   k     ⁡     (     t   -       (     n   -   1     )     ⁢       T   s     /     n   p         -     kT   s       )                         =         ∑     n   =   1       n   ⁢           ⁢   p       ⁢         s   n     ⁡     (   t   )       ⁢           ⁢   where   ⁢           ⁢       s   n     ⁡     (   t   )           =       ∑   k     ⁢       x   k     ⁢       Ψ   k     ⁡     (     t   -       (     n   -   1     )     ⁢       T   s     /     n   p         -     kT   s       )                           (   5   )               
with each signal layer transmitting at the orthogonal data symbol Nyquist rate 1/T s =1/T=B. For non-integer values of n p  one of the layers has to be transmitted at a different rate.  FIG. 2  describes this layering of the communications channels for QLM. QLM is a layered topology for transmitting higher data rates than possible with each layer of communications and is implemented by transmitting each layer with a differentiating parameter which enables separation and decoding of each layer. Each layer or channel has a unique differentiating parameter such as time offset as in (5) or a frequency offset. Each layer or channel obeys the scaled Shannon&#39;s laws when using QLM scaling in equations (2).
 
     Shannon bound link metrics for the baseband layer n p =1 are derived by first observing that the Shannon bound in (1) is a bound on the communications performance metrics C/W, Eb/No for n p =1 and 1/T s =1/T=B=W, wherein C/W in Bps/Hz is the capacity per unit of the frequency band W. The Shannon bound specifies the following metrics for 1/T s =1/T=B=W
 
 C/W=b 
 
 SNR =(2^ b− 1)=(1/ T   s )( A   2 /2σ 2 )
 
 E   b   /N   o   =SNR/b =(2^ b− 1)/ b =(1/ T   s )( A   2 /2σ 2 )/ b   (6)
 
which set of metrics is equivalent to the Shannon bound. To prove this we start with the definition 1/T s =1/T=B for n p =1 which proves C/W=(b/T s )/(1/T s )=b. Inverting the expression for C in (1) and using C/W=b yields SNR=(2^b−1). This enables the definition E b /N o =SNR/b to yield E b /N 0 =(2^b−1)/b completing the proof. The set of metrics in (6) apply to the baseband communications link for n p =1 which is the first layer of QLM,.
 
       FIG. 4  QLM bounds are derived starting with the Shannon bound in (1) expressed as a bound on C/W=log 2 (1+SNR). For QLM the SNR is scaled by (2) to read SNR/n p ^2 and the scaled Shannon bound applies to a single layer “p” to read (C/W) p =log 2(1+SNR/n p ^2). The C/W is the sum of the (C/W) p  for each of the n p  layers and the maximum “max” with respect to n p  of the sum C/W over the n p  layers yields the QLM bound in (7). 
     
       
         
           
             
               
                 
                   
                     
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       FIG. 4  QLM scaling law (2) and bound (7) are re-derived using a geometric approach. A college level geometric derivation of the Shannon bound is modified for QLM to derive the scaling law and bound on C/W for a communication links. The derivation starts by assuming we transmit a log 2 (M) bit equal-probable real signal M every T s  seconds over the Nyquist frequency band 1/T s =1/T=B=W for complex sampling. The Nyquist sampling theorem specifies there are n=2T s W independent real signal samples (T s W complex samples) in T s . Each Tx QLM signal energy in each layer is restricted to an n-dimensional hyperspace of radius [(S/n p )(T s /n p )]^n/2 about the origin where S/n p  is the signal power available in each layer and T s /n p  is the portion of the T s  that is available for demodulation of each layer allowed by the overlap of the signal which is equivalent to the scaling in (2). Each Rx QLM signal energy in each layer is restricted to a radius r=[(S/n p )(T s /n p )+N T s ]^n/2 where the noise energy is by definition N T s =N o . We assume a noise energy greater than NT will cause a signal detection error in estimating M. This means the number M of allowable signals that can be Rx without error in the n-dimensional signal space with volume r^n=[(S/n p )(T s /n p )+NT s ]^n/2 is bounded by
 
 M≦ [(( S/n   p )( T   s   /n   p )+ N T   s )/( N T   s )]^ n/ 2=[ SNR/n   p   2 +1]{circumflex over (])} n/ 2  (8)
 
which allows the channel capacity C p  in Bps in each layer p to be derived
 
 C   p =log 2 ( M )/ T   s   =W  log 2 [1+ SNR/n   p   2 ]  (9)
 
which derives the SNR scaling factor n p   2  equivalent to deriving the scaling factor n p  for E b /N o . The QLM channel capacity C is the sum of the capacity C p  in each layer which yields
 
 C=n   p   C   p   =n   p   W  log 2 [1+ SNR/n   p   2 ].  (10)
 
Re-arranging (10) and taking the maximum with respect to n p  yields the QLM bound in 1 in (7) for C/W.
 
       FIG. 4  Lemma 2 replaces the Shannon bound in (1) with the QLM capacity bounds on C/W 1 and E b /N o 2, and performance metrics 4 in (11) for 1/T s =1/T=B=W, and omits the new coding theorem since it has not been addressed in this application. The QLM replacement in (11) for the Shannon bound (1) is derived as follows. The maximum for C/W is derived in (7), the 
     New capacity bounds 15 in  FIG. 3  and coding theorem 
     
       
         
           
             
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     3 New coding theorem
         For R b &lt;C there exists codes which support reliable communications   For R b &gt;C there are no codes which support reliable communications       

     4. New performance metrics  26  in FIG.  3 
 
 C/W=n   p   b 
 
 SNR=n   p   2 (2^ b− 1)=( n   p   2   /T   s )( A   2 /2σ 2 )
 
 E   b   /N   o   =SNR /( C/W )= n   p (2^ b− 1)/ b =( n   p   /T   s )( A   2 /2σ 2 )  (11)
 
minimum E b /N o  in 2 is derived from (7) by solving for E b /No=SNR/(C/W)=n p   2 /(C/W)][2^((C/W)/n p )−1], in 4 the C/W=n p b/T s W=n p b is the data rate for the n p  layers per unit bandwidth of W, the SNR is the scaled value (2) in (6), and the E b /N o  is the scaled value (2) in (6). These bounds restate the communications bounds in U.S. Pat. No. 7,391,819 in a more useful format. Wavelet waveform from U.S. Pat. No. 7,376,688 reduces the excess bandwidth to α≈0 to meet these requirements for the bounds and performance metrics in addition to the pulse waveforms used to evaluate the QLM performance. The new coding theorem in 3 in equations (11) states that C is the upper bound on the information data rate R b  in bits/second for which error correcting codes exist to provide reliable communications with an arbitrarily low bit error rate BER wherein C is defined in 1 in equations (11) and upgrades the Shannon coding theorem 2 in equations (1) and new data symbol rate n p /T s  is n p  times the Nyquist rate 1/T=B for a single channel.
 
       FIG. 5  defines QLM to be a layered communications link equivalent to increasing the data symbol rate with the derived properties for the layering in (5), the Lemma 1 scaling in (2), Lemma 2 bounds in (11), and the Lemma 2 link metrics in (11). 
       FIG. 6  in time offset units of T s  calculates the ideal triangular correlation  10 , an example waveform  11  designated by ψ, and the waveform ψ correlation  12 . Parameters of interest for this example square-root raised-cosine waveform are the waveform length L=3, M=16, and excess bandwidth α=0.22, the mainlobe  13  which extends over a 2T s  interval, and the sidelobes  14  which fall outside of the mainlobe. Parameter L is the waveform length in units of M=16 samples and M is the number of digital samples between adjacent waveforms at a Nyquist symbol rate=1/T s  for which T s =MT where 1/T is the digital sample rate and α is a measure of the roll-off of the frequency response. The ideal triangular correlation is the correlation for the pulse waveform of length T s  in  FIG. 2  and  FIG. 7  demonstrates that for waveforms of interest for QLM the triangular correlation approximates the mainlobe correlations for QLM waveforms. 
       FIG. 7  in frequency offset units of 1/T s  presents a N=64 point discrete Fourier transform DFT for OFDM in frequency, the correlation function, and an ideal correlation function in frequency for a pulse waveform in frequency. The correlation function closely approximates the waveform and the ideal correlation with a triangular mainlobe, and has a mainlobe  22  extending over 2/T s =2 B Hz interval with low sidelobes  23  wherein the symbol rate 1/T s  is at the Nyquist rate and equal to the bandwidth B for each channel 
     QLM demodulation algorithms include ML and MAP trellis algorithms, ML data packet algorithms, sequential relaxation algorithms, equalization, super-resolution and other algorithms. Sequential relaxation algorithms process the stream of data symbol samples to find the best estimates of the data symbols using sequential relaxation methods. These are computationally intensive and the least developed of the other algorithms. We first consider the ML algorithms in the architecture of data packets. 
       FIG. 8  presents a representative set of ML QLM data packet architectures for time differentiating parameters which support implementable ML demodulation algorithms with acceptable demodulation losses and with applicability to a broad spectrum of communications systems. ML algorithms solve the QLM system equations for the correlated signals to find estimates of the data symbols in each of the QLM layers and are the least complex of the demodulation algorithms, are limited to a relatively few data symbols, and require the waveform correlation functions to approximate the ideal pulse correlations in FIG.  6 , 7 . The three data packet architectures in  FIG. 8  are a n s =2-symbol group  101 , a n s =3-symbol group  102 , and a n s =4-symbol group  103 , with each group respectively consisting of n s =2,3,4 contiguous symbols  104 , 106 , 108  for the first QLM layer n p =1. QLM layers for n p &gt;1 are overlayed on each group of symbols such that the additional QLM symbols stay within the composite bandwidth for each symbol group. These architectures assume the sidelobes of the data symbols will be included in the trellis algorithms by an iterative algorithm similar to the sidelobe algorithm disclosed in  FIG. 14 . The QLM ML packet demodulation algorithm finds an estimate {circumflex over (X)} of the Tx data symbol vector X in a QLM data packet in  FIG. 8  by solving the Rx vector equation (12) in the detected data symbol Rx vector Y. Each row “k” of the Rx vector equation (12) is the detection equation (4) which defines the detected signal y k  as a linear function of the correlated subset of Tx data symbols {d k |∀k} within the packet and the AWGN. The correlation coefficients {c(|i|)|i=0, . . . , N x −1} in (4) are the elements of the Rx system matrix H for the mainlobe correlation which is a symmetric N x  by N x  toeplitz correlation matrix over the ML data symbol packet in  FIG. 9  where N x  is the number of packet symbols N x =[(n s −1)n p +1] and n s =2,3,4 is the number of baseband symbols in a packet. Elements H(k,n) of H are the correlation values of the data symbol waveforms H(k,n)=&lt;ψ(t−kT/n p ), ψ(t−nT/n p )&gt;=c(|k−n|) with diagonal elements H(k,k)=c(0)=1 ∀k where indices “k,n” are over the data symbols in the data packet. The size of the packet is limited to n s =4 baseband data symbols to prevent the determinant of H from becoming too small and thereby causing the dispersion of the inverse H −1  of H to be too large for reliable estimation of the data symbols in AWGN. The ML waveforms used to measure the performance data are the square pulses in  FIG. 8  with triangular correlations in FIG.  6 , 7  and no sidelobes. 
     A ML data symbol packet (pulse group) in  FIG. 8  consists of n s =2, 3, 4 baseband square data symbols of length T s , 1/T s  for time, frequency square pulse waveforms. QLM overlays layers of data symbol pulses over each pulse group n s =2, 3, 4 and the matched filter detection in (2) recovers the elements of the Rx vector Y=[y 1  y 2 , . . . , y Nx ] T  in (12) of the Tx data symbol vector X=[x 1 , x 2 , . . . , x Nx ] T  where N is the AWGN noise vector and the elements {x k },{y k } are defined in (4). From (4) the Rx vector equation for the mainlobe correlation is
 
 Y=HX+N   (12)
 
     where
         Y=N x ×1 detected symbol vector   H=correlation matrix of data symbols   X=data symbol vector over layered Filters/channels   N=demodulation plus link noise vector   N x =number of QLM channels (filters)=(n s −1)n p +1
 
wherein the Rx data symbol vector Y has components {y i } over the set of channels or filters indexed on “i”, Tx data symbol vector X has components {x i } indexed over the same set of channels or filters, and Rx noise vector N has components {N i } indexed over the same set of channels or filters. The ML demodulation algorithm is derived in 4 in equations (13) for the system equations (12). We find
       

     ML demodulation algorithm 
     1 ML Cost
 
 J=[Y−HX]′Q   −1   [Y−HX]= (−) exponent of the ML probability density function
 
     2. ML solution {circumflex over (x)} minimizes J
 
 {circumflex over (x)}=[H′Q   −1   H]   −1   H′Q   −1   Y 
 
     3 The noise covariance Q is
 
 Q=E[NN′}= 2σ 2   H   (13)
 
since the noise is correlated by the same filter overlap as the signal. This reduces the ML solution 2 to:the reduced ML demodulation algorithm
 
 {circumflex over (x)}=H   −1   Y   (14)
 
wherein H′ is the conjugate transpose of H, the inverse H −1  exists for all waveforms of interest, and “2σ 2 ” is the root-mean-square rms data symbol detection noise.
 
     The ML algorithm for QLM in equations (12),(13),(14) equally applies when using partial symbol ΔT s  integration wherein the integration intervals are over the separation between the consecutive layers of communications so that ΔT s =T s /n p . The correlation matrix for a full symbol integration is a symmetric Toeplitz matrix whereas using a ΔT s =T s /n p  integration interval reduces the matrix to a triangular form thereby offering a decrease in computational complexity and a more robust solution since the determinant of the correlation matrix will not become too small with relatively large data packets. However, the requirement to demodulate with a matched waveform makes it extremely difficult to use partial symbol ΔT s  integration. 
       FIG. 9  measures the ML bit error rate BER performance for uncoded 4PSK pulse waveforms for no QLM wherein n p =1 and for n p =1,2,4,8 layers of of QLM modulation using n s =2,3,4 data symbol groups and implementing the ML symbol demodulation algorithm in (14). Performance validates QLM technology and scaling. QLM performance is plotted as bit error rate BER versus the QLM normalized value (E b /N o )/n p  of the E b /N o . Measured performance values are from a direct error count Monte Carlo simulation of the trellis algorithm and are plotted as discrete measurement points. 
       FIG. 10  measures the ML BER performance for uncoded 256QAM pulse waveforms for no QLM wherein n p =1 and for n p =8 layers of QLM modulation using n s =3,4 data symbol groups and implementing the ML symbol demodulation algorithm in (14). The 256QAM bound assumes a Grey code assignment of the data bits to the 256 data points in the complex plane and therefore provides improved performance compared to the 256 point regular data mapping used in the simulations. 
       FIG. 11  introduces the trellis path selection in the ML and MAP trellis demodulation algorithms which unscramble the detected correlated signals in (4) to recover the information in the Tx data symbols. After down-conversion and synchronization the Rx QLM signal z(t) is processed as described in (4) to strip off the waveform ψ k  by convolving the z(t) with the complex conjugate of ψ k , to recover the correlated data symbol y k  estimate of the Tx data symbol x k  . Trellis processing of the y k  recovers the m k =1,2, . . . , 2^b s  modulation states (alphabet elements) for each x k  wherein b s  is the number of uncoded bits in each data symbol x k , the information bits b=R b s  wherein R is the code rate, and the modulation states m k  are the encoded complex symbols representing x k . 
     This convolution is a correlation of the waveform with itself as illustrated in  FIG. 6 , since the waveforms are real and symmetric. QLM trellis demodulation algorithms process the mainlobes of the waveform correlation algorithms in the first pass and pick up the impacts of the sidelobes in subsequent passes. This iterative architecture provides a means to demodulate the stream of over-lapping and correlated QLM data symbol waveforms with reasonable computational requirements by processing the mainlobes and including the sidelobes in subsequent iterations. Correlation data in  FIG. 6 ,  7  suggest this approach since the sidelobes are relatively low compared to the mainlobes for the bandwith efficient waveforms being addressed. 
     MAP trellis algorithms have the correct architecture for reducing the number of trellis paths to enable higher order modulations to be used for larger numbers of overlapped QLM channels, compared to the maximum likelihood ML trellis algorithms which are derived from the ML trellis algorithms used for convolutional and Turbo decoding. Another critical property disclosed in U.S. Pat. No. 7,337,383 is that the MAP trellis algorithms are at least-as-good-as the ML trellis algorithms 
       FIG. 12  trellis paths jxi for the ML and MAP trellis algorithms are paths from the set of xi nodes defining the previous state S k−1  to the set of jx nodes defining the current state S k . Path jxi reads from left to right, “j” is the new data symbol state, “i” is the last data symbol state, “x” is the string of path data symbol states between “j” and “i”, and “k” is step k corresponding to the detected symbol y k  in (4). Nodes designated by circles in  FIG. 12  are the sets of symbols identified by their indices, state S k−1  is the set of xi nodes for state k−1 and state S k  is the set of jx nodes for state k. The data symbols in the transition paths jxi between nodes are the mainlobe correlation values in FIG.  6 , 7  encoded with the data modulations m x . For a QLM signal which has n p  simultaneous communications channels, there are (2n c −1) correlation values over the mainlobe which means each state S k−1 , S k  has 2^ b s (2n p −2) nodes identified as xi,jx respectively and there are n s ^(2n p −1)=2^b s (2n c −1) jxi paths from state S k−1  to state S k  data symbols wherein 2^b s  is the number of possible modulaton symbols m x  or un-encoded states of each data symbol, Sidelobe correlations for n c &gt;n p  are included in iterative MAP trellis demodulation algorithms and wherein n c  is the number of one-sided correlation values including the sidelobes. As illustrated in  FIG. 12  the ML algorithm selects the best path  25  by finding the xi which provides the best transitioning metric jxi from xi to jx from among the possible choices  26  for xi. The MAP algorithm selects the best path  27  by finding the jx which provides the best transitioning jxi from xi to jx from among the possible choices  28  for jx. The MAP is a co-state formulation of the trellis algorithm since it replaces the search over jx with a search over xi thereby enabling the number of paths specified by xi to be reduced along with the number of states in the trellis diagram and the computational complexity, and with minimal impact on performance. 
     MAP and ML trellis algorithms implement the decisioning rules in (15) for updating the path and state metric α k (S k ) for each node in the trellis diagram
 
α k ( S   k )=min{α k−1 ( S   k−1 )+ R   k ( S   k−1   −&gt;S   k )= R   k ( jxi )}=new metric for state  k   (15)
 
     wherein 
                             α   k     ⁡     (     S   k     )       =       ⁢     min   ⁢     {           α     k   -   1       ⁡     (     S     k   -   1       )       +       R   k     ⁡     (       S     k   -   1       -&gt;     S   k       )         =       R   k     (     j   ⁢           ⁢   x   ⁢           ⁢   i     ⁢           )       }                   =       ⁢     new   ⁢           ⁢   metric   ⁢           ⁢     fo   ⁢   r     ⁢           ⁢   state   ⁢           ⁢   k                   (   15   )               
wherein (o)* is the complex conjugate of (o), the choice of a “min” or a “max” depends on the definition of R(jxi), the y k  refers to the correlated data symbol measurement at state or step k, the estimate ŷ k  of y k  is the correlated sum of the estimates {{circumflex over (x)} k } of the data symbols {x k }, and the R k (S k−1 −&gt;S k )=R k (jxi) is the state k transition decisioning metric for the transition written symbolically as S k−1 −&gt;S k  and realized in Matlab notation as “jxi”.
 
       FIG. 12  lists the Matlab code in a pseudo-code format for the calculation of the best trellis paths in and for the state updates for the ML and MAP trellis algorithms. For the best trellis path, the code translates the trellis algorithms in (15) written as a state metric update equation, into a Matlab code format for calculation of the state S k  metric α k (S k ) for the best trellis transition path. ML trellis algorithm finds the last symbol “i” with the best path “jx” to the new symbol “j” whereas the MAP algorithm finds the new symbol “j” with the best path “xi” from the last symbol “i” to support a reduction in the number of paths “xi” when necessary to reduce the computational complexity. For the state update for k≧D the state metric S k  is upgraded for this new path jxi by the update operations S k (:,jx)=[y k (jxi); S k−1 (1:D−1, xi)], S k (:,xi)=[y k (jxi); S k−1 (1:D−1, jx)] respectively for ML, MAP using Matlab notation which replaces the column jx, xi vector with the column xi, jx vector after the elements of xi, jz have been moved down by one symbol and the new estimated symbol {circumflex over (x)} k  added to the top of the column vector which is the row 1 element. 
       FIG. 13A  is a summary implementation flow diagram of a representative MAP trellis data symbol demodulation algorithm with error correction decoding and with the option of including the sidelobes in subsequent iterations. Basic elements of the algorithm are the trellis parameters  170 , MAP trellis algorithm  171  structured by the parameter set in  170 , and followed by trellis error correction decoding  172  which also could be combined with the trellis demodulation, and with the option  175  to include the sidelobes in an iteration(s) or to end the iteration(s) and handover  174  the data estimates to the receiver. The parameter set  170  is applicable to the mainlobes of the data symbol waveforms with two examples illustrated in FIG.  6 , 7 . The correlation function for the data symbol waveforms defined in FIG.  6 , 7  have low sidelobes compared to the mainlobes which suggests it is sufficient to consider mainlobe correlation in the MAP trellis algorithm with the sidelobes included in subsequent iterations. A one-sided correlation vector C is used since the correlation is symmetrical about the origin, whose elements are the mainlobe correlation coefficients C=[c(0), c(1), . . . , c(n p −1)] wherein c(0)=1 with waveform normalization. Waveform modulations emphasized are 4PSK, 8PSK, 16QAM, 64QAM, 256QAM with b s =2, 3, 4, 6, 8 uncoded bits per data symbol and n s =2^b s =4, 8, 16, 64, 256 states per data symbol. 
     FIG.  13 B, 13 C is a detailed implementation flow diagram of the MAP trellis data symbol demodulation algorithm in  FIG. 13A  for mainlobe demodulation.  FIG. 13B  uses the trellis parameters  170  in the initialization  150  prior to the start k=0 with an empty shift register (SR) which normally contains the 2n p −1 correlated data symbols for the mainlobe for each node being addressed. At k=0 the path metric α 0 (xi) is set equal to a negative or zero initialization constant for all of the nodes xi=0,1,2, . . . , (n s ^(2n p −2)−1} of the trellis diagram where α 0 (xi) is the logarithm of the state S 0  path metric at k=0 for node xi, the receive Rx symbols are indexed over k with k=0 indicating the initial value prior to the Rx symbol k=1, nodes of the trellis diagram are the states of the shift register SR, and state S k  refers to the trellis diagram paths and metrics at symbol k in the trellis algorithm. 
     In  FIG. 13B  loop  151  processes the Rx symbols k=1,2, . . . ,n where the index k also refers to the corresponding algorithm steps and the states of the trellis algorithm. In  152  the Rx signals are pulse detected to remove the carrier frequency and waveform to recover the normalized correlated data symbol y k . For each Rx symbol y k  the state transition decisioning metrics R k (jxi) are calculated by the transition metric equations {R k (jxi)=−|y k −ŷ k (jxi)|^2} for a ML metric, {R k (jxi)=|ŷ k |^2−2Real(y k ŷ k (jxi)))*} for a MAP metric wherein (o)* is the complex conjugate of (o), for {R k (jxi)=Metric(y k ,ŷ k )} for another metric, for all possible transition paths {jxi} from the previous state S k−1 (xi) at node xi to the new state S k (jx) at node jx in the trellis diagram snd where ŷ k (jxi) is the hypothesized normalized detected correlated symbol k for the path jxi. For a mainlobe correlation function the ŷ k (jxi) is defined by the equation ŷ k (jxi)=c(n p −1)[sr(1)+sr(2n p −1)]+ . . . +c(1)[sr(n p −1)+sr(n p +1)]+c(0)[sr(n p )] which calculates ŷ k (jxi) as the correlated weighted sum of the elements of the shift register SR=[sr(1),sr(2), . . . , sr(2n p −1)]′ with {circumflex over (x)} k =sr(n p ), {circumflex over (x)} k−1 =sr(n p −1), {circumflex over (x)} k+1 =sr(n p +1), . . . where c(0)=1, the normalized data symbol estimates {{circumflex over (x)} k } correspond to the transition index jxi, and the state k estimated symbol {circumflex over (x)} k  is the SR center element sr(n p ) with correlation coefficient c(0)=1. Symbols move from left to right starting with “j” with each new Rx symbol or step in the trellis recursion algorithm, and ending with “i”. With this convention “j” is indexed over the states of sr(1), “x” is indexed over the current states of sr(2), . . . , sr(2n p −2), and “i” is indexed over the states of sr(2n p −1). Index over the paths of the trellis diagram is defined by the equation jxi=sr(1)+n s sr(2)+n s ^2sr(3) + . . . +n s ^(2n p −2) sr(2n p −1)−1=0,1,2, . . . , n s ^(2n p −1)−1 when the contents of the SR elements are the indices corresponding to the assumed data symbol state values. 
     In  FIG. 13B  loop  153  calculates the best trellis transition paths from state S k−1  to the new state S k  for xi=0,1,2, . . . , n s ^(2n p −2)−1. In  154  the path metric α k (S k ) is defined by the recursive logarithm equation α k (S k )=α k−1 (S k−1 )+R k (S k−1 −&gt;S k ) which can be rewritten as α k (xi)=α k−1 (jx)+R k (jxi) since α k (S k )=p(S k |y k )=α k (xi), α k−1 (S k−1 )=p(S k−1 |y k−1 )=α k−1 (jx), R k (S k−1 −&gt;S k )=R k (jxi) from equation (5). The best path metric α k (xi) for each new node jx is chosen by the decisioning equation α k (xi)=min{α k−1 (jx)+R k (jxi)} with respect to the admissible “j”. For each jx, the corresponding xi yielding the highest value of the path metric α k (xi) is used to define the best path and new symbol {circumflex over (x)} k−D . 
       FIG. 14  is a flow diagram of an iterative version of the MAP data symbol trellis algorithm in  FIG. 13  wherein the iteration is used to incorporate the effects of the sidelobes of the correlation function C into the trellis state transition metric R k (jxi) in  152  in  FIG. 13B  with a relatively small increase in computational complexity. In  FIG. 14  the algorithm is initialized with the parameter set in  176  which is the parameter set in  170  in  FIG. 13A  with the partitioning of the correlation vector C into the mainlobe vector C 0  plus the sidelobe vector C 1  and adding the specification of the stopping rule for the iterations. For a correlation vector C=[c(0),c(1), . . . , c(n p −1),c(n p ), . . . ,c(n c )] consisting of n c  correlation coefficients, the mainlobe vector is C 0 =[c(0),c(1), . . . , c(n p −1),0,0, . . . , 0] and the sidelobe vector is C 1 =[0,0, . . . ,0,c(n p ), . . . , c(n c )] to partition C=C 0 +C 1 . The algorithm starts  177  by implementing the MAP algorithm in  FIG. 12 . 
     In  FIG. 14  the output data symbol estimates  164  in  FIG. 13C  are the outputs from the MAP algorithm in  177  and are used to calculate the a-priori estimated sidelobe contribution ŷ k|1  in  179  to ŷ k (jxi) which is the hypothesized normalized detected correlated symbol k for the path jxi in the calculation of the metric R k (jxi) in  152  in  FIG. 13B . In this implementation  180  of the trellis algorithm  171  in  FIG. 14A , the ŷ k (jxi)=ŷ k|0 (jxi)+ŷ k|1  is the sum of the hypothesized mainlobe contribution ŷ k|0 (jxi) using the non-zero C 0  coefficients as described in  FIG. 13A  and the sidelobe contribution ŷ k|0  using the data symbol estimates from  177  and the non-zero C 1  coefficients and wherein the subscripts “k|0” reads “index k given C 0 ” and “k|1” reads “index k given C 1 ”. From  152  in  FIG. 13B  we find the ŷ k|0 (jxi) is defined by the equation ŷ k|0 (jxi)=c(n p −1)[sr(1)+sr(2n p −1)]+ . . . +c(1)[sr(n p −1)+sr(n p +1)]+c(0)[sr(n p )] which calculates ŷ k|0 (jxi)−ŷ k (jxi) in  FIG. 13B  as the correlated weighted sum of the elements of the shift register SR=[sr(1),sr(2), . . . , sr(2n p −1)]′ with {circumflex over (x)} k =sr(n p ), {umlaut over ({circumflex over (x)} k−1 =sr(n p −1), {circumflex over (x)} k+1 =sr(n p +1), . . . where c(0)=1, the normalized data symbol estimates {{circumflex over (x)} k } correspond to the transition index jxi, and the state k estimated symbol {circumflex over (x)} k  is the SR center element sr(n p ) with correlation coefficient c(0)=1. Symbols move from left to right starting with “j” with each new Rx symbol or step in the trellis recursion algorithm, and ending with “i”. With this convention “j” is indexed over the states of sr(1), “x” is indexed over the current states of sr(2), . . . , sr(2n p −2), and “i” is indexed over the states of sr(2n p −1). Index over the paths of the trellis diagram is defined by the equation jxi=sr(1)+n s sr(2)+n s ^2sr(3)+ . . . +n s ^(2n p −2) sr(2n p −1)−1=0,1,2, . . . , n s ^(2n p −1)−1 when the contents of the SR elements are the indices corresponding to the assumed data symbol state values. The sidelobe contribution is equal to ŷ k|1 =c(n p )({circumflex over (x)} k−n     p   +{circumflex over (x)} k+n     p   )+c(n p +1)({circumflex over (x)} k−1−n     p   +{circumflex over (x)} k+1+n     p   )+c(n p +2)({circumflex over (x)} k−2−n     p   +{circumflex over (x)} k+2+n     p   )+ . . . until the end of the sidelobe correlation coefficients or the end of the data symbol bit estimates and wherein {circumflex over (x)} k−n     p    is the data symbol estimate in  180  for symbol k−n p . A stopping rule in  182 ,  184  is used to decide if another iteration is required. When another iteration is required the data symbol bit estimates are used to update the calculation of the a-priori contribution ŷ k|1  of the sidelobes to the ŷ k  (jxi) in the modified trellis algorithm  184 . After the iteration is stopped the outputs are error corrected decoded  183  and handed over to the receiver. A possible alternative path is to use the decoded output estimates  185  to re-start  184  the MAP trellis demodulation algorithm with sidelobes. 
       FIG. 15  measures the trellis decoding performance for uncoded 4PSK pulse waveforms for no QLM wherein n p =1 and for n p =2,3,4 layers of QLM modulation using the trellis data symbol decoding algorithm in  FIG. 13  and for n p =4,6 layers by implementing a ½-word trellis data symbol decoding algorithm. Performance validates QLM technology and scaling. QLM performance is plotted as BER versus the QLM normalized value (E b /N o )/n p  of the E b /N o . Measured performance values are from a direct error count Monte Carlo simulation of the trellis algorithm and are plotted as discrete measurement points. It is expected that higher order modulations 8PSK, 16QAM, 64QAM, 256QAM will have the same relative trellis coding performance for n p =1,2,3,4,5,6 layers of communications. 
     FIG.  16 , 17  calculate the QLM data rate C/W vs. E b /N o , SNR respectively using the MAP trellis demodulation and ML packet demodulation algorithms. Performance calculated are the Shannon bound, new bound, 4PSK, 8PSK, 16QAM, 64QAM, 256QAM, 4096QAM, MAP trellis demodulation, and ML demodulation communications. Equation (1) is used to calculate the Shannon bound, equation (7) is used to calculate the new bound, modulation performance for 4PSK, 8PSK, 16QAM, 64QAM, 256QAM, 1024QAM assumes turbo coding performance provides a performance almost equal to the Shannon bound and with information rate b=1.5,2,3,4,6,8 bits=Bps/Hz respectively assuming coding rates R=¾, ⅔, ¾, ⅔, ¾, ⅔ and bits per symbol b s =1.5,2,3,4,6,8 bits respectively for 4PSK, 8PSK, 16QAM, 64QAM, 256QAM, 4096QAM. 
       FIG. 18  parameters  121  are used to calculate the MAP trellis and ML demodulation performance. MAP parameter values  122  are selected to provide best values for b while avoiding excessive numbers of layers n p  which increase the sidelobe induced demodulation loss and increase the required synchronization accuracy to support the n p . Plot point  5  lists two alternative candidates to achieve b=24 Bps/Hz which are: 64QAM requires n p =6 layers and E b /N o 17.3 dB, and 256QAM requires a lower n p =4 layers and a slightly higher E b /N o 18.2 dB. The 256QAM appears to be the preferred candidate since the n p  is somewhat lower. ML parameter values 1,2,3 for the n s =3 data symbol group assume n p =4 QLM layers and the resulting performance in FIG.  16 , 17  is considerable lower than the MAP performance in FIG.  16 , 17 . ML performance approaches the MAP performance in FIG.  16 , 17  for a n s =4 data symbol group with increased number of QLM layers to n p =6. 
       FIG. 19  is a proof-of-concept demonstration that QLM can provide a data rate which is substantially faster than the Shannon rate in (1), (11) with n p =1, and in  FIG. 17 ,  18 . We use as a reference the performance for 256QAM at the Shannon rate which is C/W=6 Bps/Hz in  FIG. 17  for E b /N o =10.21 dB which is the highest data rate mode for cellular communications and for almost all of the other communications links snd applications. We observe that starting with 16QAM at the Shannon rate which is C/W=3 Bps/Hz for E b /N o =3.68 dB we can increase the data symbol rate to n p =2, 3 times the 3 Bps/Hz whereupon the QLM performance is C/W=6, 9 Bps/Hz respectively for E b /N o =6.69, 8.95 dB which C/W is observed to be substantially above the Shannon bound or rate. Likewise it is observed that starting with 64QAM at the Shannon rate which is C/W=4 Bps/Hz for E b /N o =5.74 dB we can increase the data symbol rate to n p =2, 3 times the 4 Bps/Hz whereupon the QLM performance is C/W=8, 12 Bps/Hz respectively for E b /N o =8.75, 11.01 dB which C/W is observed to be substantially above the Shannon bound or rate. This completes the proof-of-concept demonstration which complements the proof-of-concept in  FIG. 17 ,  18 . 
       FIG. 20  is an OFDM transmitter block diagram modified to support OFDM QLM with frequency offsets as the differentiating parameter to increase the symbol transmission rate from 1/T s  to the QLM rate n p /T s  and with an increase in transmitter power to support this increased data rate.  FIG. 20  differs from the IEEE WiFi 802.16 standard in the deletion of the 4.0 μs delay between the FFT output NT and the Tx FFT output NT s =NT+4.0 μs, and in the use of the complete FFT band for transmission, in order to apply to all of the IEEE standard OFDM systems. Ideal OFDM modulates N input data symbols at the sample rate 1/T s  over the time interval NT s  with an N-point inverse fast fourier transform FFT −1  to generated N harmonic waveforms e^j2πkn/N with each modulated by the corresponding data symbol wherein the normalized frequencies k=0,1, . . . N−1 correspond to channels 0,1, . . . , N−1, “j”=√(−1), “π=pi”, and “n” is a time index,. Data symbol output rates are 1/NT s  per channel and the N channels have a total symbol rate equal to N/NT s =1/T s =B=(Nyquist sample rate). Signal processing starts with the stream of user input data words (d k }  46  with k indexed over the words. Frame processor  47  accepts these data words and performs turbo error correction encoding, error detection cyclic redundant encoding CRC, frame formatting, and passes the outputs to the symbol encoder  48  which encodes the frame data words into data symbols for handover to the OFDMA QLM signal processing. QLM transmits in parallel N Rx data symbols for each of the n p  FFT −1  signal processing steams. Each set of Rx N data symbols are offset in frequency by 0, Δk, 2Δk, . . . , (n p −1)Δk with Δk=1/n p  using the normalized frequency index k and are implemented in  49  by the frequency translation operator with FFT −1  time sample index n. Following this frequency translation and FFT −1  signal processing, the output streams of the OFDMA encoded symbols for the n p  frequency offsets are summed  51  and waveform encoded. The output stream of up-sampled complex baseband signal samples  52  {z(t i )} at the digital sample times t i  with digitization index i, is handed over to the digital-to-analog converter DAC, and the DAC output analog signal z(t) is single sideband SSB upconverted  52  to RF and transmitted as the analog signal v(t) wherein v(t) is the real part of the complex baseband signal z(t) at the RF frequency. Non-ideal OFDMA has a separation interval between contiguous FFT −1  data blocks to allow for timing offsets and the rise and fall times of the channelization filter prior to the FFT −1  processing. 
       FIG. 21  is a OFDM receiver block diagram modified to support OFDM QLM from the OFDM QLM transmitter in  FIG. 20 . Receive signal processing for QLM demodulation starts with the wavefronts  54  incident at the receiver antenna for the n u  users u=1, . . . , n u ≦N c  which are combined by addition in the antenna to form the receive Rx signal {circumflex over (v)}(t) at the antenna output  55  where {circumflex over (v)}(t) is an estimate of the Tx signal v(t)  52  in  FIG. 20  that is received with errors in time Δt, frequency Δf, and phase Δθ. This Rx signal {circumflex over (v)}(t) is amplified and downconverted to baseband by the analog front end  56 , synchronized (synch.) in time t and frequency f, waveform removed to detect the Rx QLM signal at the QLM symbol rate, inphase and quadrature detected (I/Q), and analog-to-digital ADC converted  57 . ADC output signal is demultiplexed into n p  parallel signals  58  which are offset in frequency by 0, −Δk, −2Δk, . . . , −(n p −1)Δk wherein Δk=1/n p  and processed by the FFT&#39;s. Outputs are trellis decoded  59  and further processed  60 , 61  to recover estimates of the Tx data d k  with k indexed over the data words. 
       FIG. 22  is an ideal LTE SC-OFDM transmitter block diagram modified to support SC-OFDM QLM with timing offsets as the differentiating parameter to increase the symbol transmission rate from 1/T s  to the QLM rate n p /T s  and with an increase in transmitter power to support this increased data rate.  FIG. 22  is ideal in the assumption of the use of orthogonal Wavelet multiplexing OWDM disclosed in U.S. Pat. No. 7,376,688 wherein the Wavelets provide ideal communications performance in that the symbol rates 1/T s  are equal to the bandwidth B=1/T s  and they are orthogonal in frequency offsets 1/T s  and time offsets T s , and the transmission is over a single channel and bandwidth in order to apply to all of the IEEE standard SC-OFDM LTE systems. Obviously, the transmitter block diagram applies to other waveforms equivalent to the Wavelet. Signal processing starts with the stream of user input data words (d k }  111  with k indexed over the words. Frame processor  112  accepts these data words and performs the turbo error correction encoding, error detection cyclic redundant encoding CRC, frame formatting, and passes the outputs to the symbol encoder  113  which encodes the frame data words into data symbols for handover to the OWDM QLM transmit signal processing. The n p  time delays 0, ΔT s , 2ΔT s , 3ΔT s , . . . , (n p −1)ΔT s  wherein ΔT s =T s /n p , are performed  114  and the output streams of the OWDM waveform encoded  115  symbols for the n p  time delays are summed  116  and passband waveform encoded and the up-sampled output stream of complex baseband signal samples  117  {z(t i )} at the digital sample times t i  with digitization index i, is handed over to the DAC and the DAC output analog signal z(t) is single sideband SSB upconverted  117  to RF and transmitted as the analog signal v(t) wherein v(t) is the real part of the complex baseband signal z(t) at the RF frequency. It is convenient to refer to the OFDM LTE communications as LTE(OFDM). 
       FIG. 23  is a LTE SC-OFDM QLM receiver block diagram modified to support SC-OFDM QLM from the transmitter in  FIG. 22 . Receive signal processing for QLM demodulation starts with the wavefronts  131  incident at the receiver antenna for the n u  users u=1, . . . , n u ≦N c  which are combined by addition in the antenna to form the receive Rx signal {circumflex over (v)}(t) at the antenna output  132  where {circumflex over (v)}(t) is an estimate of the Tx signal v(t)  117  in  FIG. 22  that is received with errors in time Δt, frequency Δf, and phase Δθ. This Rx signal {circumflex over (v)}(t) is amplified and downconverted to baseband by the analog front end  133 , synchronized (synch.) in time t and frequency f, waveform removed to detect the Rx QLM signal at the QLM symbol rate, inphase and quadrature detected (I/Q) and analog-to-digital ADC converted  134 . ADC output signal is demultiplexed into n p  parallel signals  135  which are offset in time by 0, ΔT s , 2ΔT s , . . . , (n p −1)ΔT s  and processed by the OWDMA decoders. Outputs are trellis decoded  136  with an algorithm comparable to the algorithm defined in  FIG. 9 . Outputs are further processed  137 , 138  to recover estimates of the Tx data d k  wherein k is indexed over the data words. 
       FIG. 24  illustrates a representative segment of current cellular communications networks with a schematic layout of part of a cellular network which depicts cells  1 , 2 , 3 , 4  that partition this portion of the area coverage of the network, depicts a user (network user)  5  located within a cell with forward and reverse communications links  6  with the cell-site base station (access point/hub)  7 , depicts the base station communication links  8  with the MSC (mobile switching center) or the WSC (wireless switching center)  9 , and depicts the MSC/WSC communication links with another base station (access point/hub)  17 , with another MSC/WSC  16 , and with external elements  10 , 11 , 12 , 13 , 14 , 15 . One or more base stations are assigned to each cell or multiple cells or sectors of cells depending on the application. One of the base stations  9  in the network serves as the MSC/WSC which is the network system controller and switching and routing center that controls all of user timing, synchronization, and traffic in the network and with all external interfaces including other MSC&#39;s. External interfaces could include satellite  10 , PSTN (public switched telephone network)  11 , LAN (local area network)  12 , PAN (personal area network)  13 , UWB (ultra-wideband network)  14 , and optical networks  15 . As illustrated in the figure, base station  7  is the nominal cell-site station for cells i−2, i−1, i, i+1 identified as 1,2,3,4, which means it is intended to service these cells with overlapping coverage from other base stations (access points/hubs). The cell topology and coverage depicted in the figure are intended to be illustrative and the actual cells could be overlapping and of differing shapes. Cells can be sub-divided into sectors. Not shown are possible subdivision of the cells into sectors and/or combining the cells into sectors. Each user in a cell or sector communicates with a base station which should be the one with the strongest signal and with available capacity. When mobile users cross over to other cells and/or are near the cell boundary a soft handover scheme is employed for CDMA in which a new cell-site base station is assigned to the user while the old cell-site base station continues to service the user for as long as required by the signal strength. 
       FIG. 25  depicts a MIMO scenario for the cellular networks in  FIG. 24  (2) using the multiple access OFDM QLM and SC-OFDM QLM communications encoding of the encoded data symbols transmitting signal information, followed by space-time coding H 0 ,C O ,L as disclosed in U.S. Pat. No. 7,680,211 for the communications architecture consisting of N Tx antennas  106  communicating through a scintillated, dispersive, fading, and multipath links  107  to M Rx antennas  108  for the forward communications links for the cellular communications  6  from the access point or hub  7  to the network user  5  in  FIG. 24 . Transmission starts with the input Tx data d  101  which is encoded, interleaved, formatted, and data symbol encoded  102  followed by the OFDM QLM and SC-OFDM QLM encoding  103  to generate the input signal (symbol) vector X  104  whose elements are the encoded data symbols from  103 . This encoding includes the Tx signal processing required to support Rx signal processing equalization such as frequency domain equalization (FDE) and Wavelet domain equalization (WDE) to correct for scintillation, dispersion, fading, and multipath. Tx symbol vector X is space-time encoded  105  using H 0 /H,C O ,L where (H 0 /H) is equal to H 0  with the elements of H removed since they are generated by the Tx-to-Rx communication links, and handed over to the Tx antenna subsystem which performs the digital-to-analog conversion, symbol waveform modulation, single-side-band upconversion, power amplification, and transmission by the antenna elements for each of the N Tx transmissions by the access point or hub. Rx signals received by the network user antennas are amplified, filtered, downconverted, detection filtered and analog-to-digital converted to recover the Rx symbol vector Y  109 . Space-time decoding  110  defined in scenarios 1,2,3,4 is implemented to recover the ML estimates {circumflex over (x)}  111  of X which are processed by the OFDM QLM and SC-OFDM QLM decoding and equalization  112  followed by deinterleaving and turbo decoding  113  to recover the ML estimates {circumflex over (d)}  114  of the Tx data d  101 . 
     A multi-scale MS code can be implemented with modest complexity in order to improve the bit-error-rate BER performance of OFDM/LTE(OWDM), WiFi, WiMax, SC-OFDM QLM by spreading each Tx data symbol over the data band and over the data packet. This technique has been disclosed in U.S. Pat. No. 7,907,512. Jensen&#39;s inequality from mathematical statistics proves that this uniform spreading of the transmit signals using MS provides the best communications BER performance. 
     This patent covers the plurality of everything related to QLM generation for WiFi, WiMax, LTE, OFDM, and SC-OFDM waveforms, QLM demodulation for WiFi, WiMax, LTE, OFDM, and SC-OFDM waveforms, to QLM generation and data recovery of QLM, to the corresponding bounds on QLM, to all QLM inclusive of theory, teaching, examples, practice, and to implementations for related technologies. The representative trellis and ML algorithms for QLM demodulation are examples to illustrate the methodology and validate the performance and are representative of all QLM demodulation algorithms including all maximum likelihood ML architectures, maximum a posteriori MAP trellis and ML trellis architectures, maximum a priori, finite field techniques, direct and iterative estimation techniques, trellis symbol and iterative trellis symbol and with/without simplifications, trellis bit and iterative trellis bit and with/without simplifications and with/without bit error correction coding, sequential relaxation, equalization, super-resoluion, and all other related algorithms whose principal function is to recover estimates of the Tx symbols and information for QLM layered modulation and QLM increase in the data symbol rates as well as data recovery related to QLM and the QLM bounds. 
     This patent covers the plurality of everything related to QLM which has been disclosed in this and previous patents to consist of all methods and means to increase the communications data rates while introducing scaling orother possible performance improvements or combinations thereof to compensate for the impact of ISI and other communications link imperfections in reducing the available signal energy for demodulation and decoding to recover the communications information. The example applications in this invention disclosure have used a restricted version of QLM in order to derive the data rate performance to compare with the Shannon rate.