Patent Publication Number: US-9838170-B2

Title: Root non-orthogonal frequency division multiplexing (RNOFDM)

Description:
FIELD OF THE INVENTION 
     The aspects described herein relate to an improvement over Orthogonal Frequency Division Multiplexing (OFDM), namely Root Non-Orthogonal Frequency Division Multiplexing (RNOFDM), for use in wireless communication systems such as, for example, Wi-Fi (IEEE 802.11 standards) and LTE (3GPP standards). 
     BACKGROUND 
     Orthogonal Frequency Division Multiplexing (OFDM) is a known technology for wireless communication that has the benefit of low equalization complexity. Currently, Wi-Fi utilizes OFDM for both its uplink and downlink transmissions, and LTE utilizes OFDM for its downlink transmissions. 
     One way to increase the transmission rate of an OFDM signal without increasing its equalization complexity is to relax the orthogonal nature of the signal; however, this increased rate may come at the expense of a more complex detection algorithm at the receiver. One way to achieve this increased transmission rate is by increasing the number of subcarriers in one OFDM symbol, while preserving its duration and bandwidth. Such a modulation technique is referred to herein as Non-Orthogonal Frequency Division Multiplexing (NOFDM). In an example of a NOFDM system, a plurality of non-orthogonal carriers is utilized for transmission. As used herein, a plurality of non-orthogonal carriers is a set of carriers in which at least some of the carriers are non-orthogonal to each other. In such a system, the plurality of non-orthogonal carriers may also include carriers which are orthogonal to each other. 
     Aspects of NOFDM are known to those skilled in the art; however, NOFDM has not been adopted in any standards due to its inability to deliver significant improvement in capacity over the capacity delivered by OFDM. There is a need for a NOFDM technique that provides an improvement in capacity over the capacity delivered by OFDM, while maintaining a low equalization complexity. 
     SUMMARY 
     Systems, devices and methods for Root Non-Orthogonal Frequency Division Multiplexing (RNOFDM) are provided herein. 
     In an aspect, a method is provided for method for transmission by a transmitting node in a communication network using a plurality of non-orthogonal carriers. The method includes obtaining, by a processor in the transmitting node, an information element data set comprised of a first number of elements, applying, by the processor, a transform matrix to the information element data set to obtain an output samples data set comprised of a second number of elements, the transform matrix being based on a non-linear function applied to a non-orthogonal frequency division matrix comprised of a plurality of columns wherein each column is associated with one of the plurality of non-orthogonal carriers, and transmitting the output samples data set from a transmitter in the transmitting node. 
     Other features and advantages of the present invention should be apparent from the following description which illustrates, by way of example, aspects of the invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The details of the present invention, both as to its structure and operation, may be gleaned in part by study of the accompanying drawings, in which like reference numerals refer to like parts, and in which: 
         FIG. 1  is a block diagram of a communication network in which systems and methods disclosed herein may be implemented in accordance with aspects of the invention; 
         FIG. 2  is a block diagram of an access node in accordance with aspects of the invention; 
         FIG. 3  is a block diagram of a terminal node in accordance with aspects of the invention; 
         FIG. 4  is a block diagram of a transform module in accordance with aspects of the invention; 
         FIG. 5  is a block diagram of a transmitter chain in accordance with aspects of the invention; 
         FIG. 6  is a flowchart illustrating an example method for generating a data set transform for use in implementing root non-orthogonal frequency division multiplexing (RNOFDM) according to an aspect of the invention; 
         FIG. 7  is a flowchart illustrating an example method for applying a data set transform in the implementation of root non-orthogonal frequency division multiplexing (RNOFDM) according to an aspect of the invention; and 
         FIG. 8  is a graph depicting performance of RNOFDM in comparison to OFDM, according to an aspect of the invention. 
     
    
    
     DETAILED DESCRIPTION 
     Systems, devices and methods for conducting interference resolution in a communication system are provided. 
     The systems and methods disclosed herein can be applied to various types of communication systems, including wireless communication systems and wired communication systems. For example, the systems and methods disclosed herein may be used with Cellular 4G (including Long Term Evolution (LTE), LTE Advanced, and WiMAX), 5G, cellular backhaul, Wi-Fi, Ultra Mobile Broadband (UMB), and other point-to-point or point-to-multipoint wireless technologies, including communication systems used in wireless personal area networks (WPAN), wireless local area networks (WLAN), wireless metropolitan area networks (WMAN) and wireless wide area networks (WWAN). The systems and methods disclosed herein may also be implemented in wired communication systems including, but not limited to, hybrid fiber-coax cable modem systems. For concise exposition, the descriptions provided herein use terminology and aspects of particular communication technologies and standards; however, the devices, systems and methods described herein are also broadly applicable to other communication technologies and standards, both wired and wireless. 
       FIG. 1  is a block diagram of a communication network in which systems and methods disclosed herein may be implemented in accordance with aspects of the invention. A macro base station (access node)  110  is connected to a core network  102  through a backhaul connection  170 . In an embodiment, the backhaul connection  170  is a bidirectional link or two unidirectional links. The direction from the core network  102  to the macro base station  110  is referred to as the downstream or downlink (DL) direction. The direction from the macro base station  110  to the core network  102  is referred to as the upstream or uplink (UL) direction. Subscriber stations (terminal nodes)  150   a  and  150   d  can connect to the core network  102  through the macro base station  110 . Wireless links  190  between subscriber stations  150  and the macro base station  110  are bidirectional point-to-multipoint links, in an embodiment. The direction of the wireless links  190  from the macro base station  110  to the subscriber stations  150  is referred to as the downlink or downstream direction. The direction of the wireless links  190  from the subscriber stations  150  to the macro base station  110  is referred to as the uplink or upstream direction. Subscriber stations are sometimes referred to as user equipment (UE), users, user devices, handsets, terminal nodes, or user terminals and are often mobile devices such as smart phones or tablets. The subscriber stations  150  access content over the wireless links  190  using base stations (access nodes), such as the macro base station  110 , as a bridge. 
     In the network configuration illustrated in  FIG. 1 , an office building  120 ( 1 ) causes a coverage shadow  104 . A pico station (access node)  130  can provide coverage to subscriber stations (terminal nodes)  150   b  and  150   e  in the coverage shadow  104 . The pico station  130  is connected to the core network  102  via a backhaul connection  170 . The subscriber stations  150   b  and  150   e  may be connected to the pico station  130  via links that are similar to or the same as the wireless links  190  between subscriber stations  150   a  and  150   d  and the macro base station  110 . 
     In office building  120   b , an enterprise femto base station (access node)  140  provides in-building coverage to subscriber stations (terminal nodes)  150   c  and  150   f . The enterprise femto base station  140  can connect to the core network  102  via an internet service provider network  101  by utilizing a broadband connection  160  provided by an enterprise gateway  103 . 
       FIG. 2  is a functional block diagram of a base station  275  in accordance with aspects of the invention. In various embodiments, the base station  275  may be a mobile WiMAX base station, an LTE evolved Node B (eNB or eNodeB), or other wireless base station or access node of various form factors. For example, the macro base station  110 , the pico station  130 , the enterprise femto base station  140  of  FIG. 1  may be provided, for example, by base station  275  of  FIG. 2 . Base station  275  includes a processor  281  that is coupled to a transmitter-receiver (transceiver)  279 , a backhaul interface  285 , and storage  283 . 
     Transmitter-receiver  279  is configured to transmit and receive communications wirelessly with other devices. Base station  275  generally includes one or more antennae for transmission and reception of radio signals. The communications of transmitter-receiver  279  may be with one or more terminal nodes. 
     Backhaul interface  285  provides communication between the base station  275  and a core network. This communication may include communications directly or indirectly (through intermediate devices) with other base stations, for example to implement the LTE X2 interface. The communication may be over a backhaul connection such as, for example, the backhaul connection  170  of  FIG. 1 . Communications received via the transmitter-receiver  279  may be transmitted, after processing, on the backhaul connection via backhaul interface  285 . Similarly, communication received from the backhaul connection via backhaul interface  285  may be transmitted by the transmitter-receiver  279 . Although the base station  275  of  FIG. 2  is shown with a single backhaul interface  285 , other embodiments of the base station  275  may include multiple backhaul interfaces. Similarly, the base station  275  may include multiple transmitter-receivers. In such a scenario, the multiple backhaul interfaces and transmitter-receivers may operate according to different protocols. Communications originating within the base station  275 , such as communications with other base stations, may be transmitted on one or more backhaul connections by backhaul interface  285 . Similarly, communications destined for base station  275  may be received from one or more backhaul connections via backhaul interface  285 . 
     Processor  281  can process communications being received and transmitted by the base station  275 . Storage  283  stores data for use by the processor  281 . Storage  283  may also be used to store computer readable instructions for execution by processor  281 . The computer readable instructions can be used by base station  275  for accomplishing the various functions of base station  275 . In an aspect, storage  283 , or parts of storage  283 , may be considered a non-transitory machine readable medium. For concise explanation, base station  275  or aspects of base station  275  are described as having certain functionality. It will be appreciated that in some embodiments, this functionality is accomplished by processor  281  in conjunction with storage  283 , transmitter-receiver  279 , and backhaul interface  285 . Furthermore, in addition to executing instructions, processor  281  may include specific purpose hardware to accomplish some functions. 
       FIG. 3  is a functional block diagram of a terminal node  355  in accordance with aspects of the invention. In various embodiments, terminal node  355  may be a mobile WiMAX subscriber station, an LTE user equipment, or other wireless terminal node of various form factors. The subscriber stations  150  of  FIG. 1  may be provided, for example, by terminal node  355  of  FIG. 3 . Terminal node  355  includes a processor  361  that is coupled to a transmitter-receiver (transceiver)  359 , a user interface  365 , and storage  363 . 
     Transmitter-receiver  359  is configured to transmit and receive communications with other devices. For example, transmitter-receiver  359  may communicate with base station  275  of  FIG. 2  via its transmitter-receiver  279 . Terminal node  355  generally includes one or more antennae (not shown) for transmission and reception of radio signals. Although terminal node  355  of  FIG. 3  is shown with a single transmitter-receiver  359 , other embodiments of terminal node  355  may include multiple transmitter-receivers. In such a scenario, the multiple transmitter-receivers may operate according to different protocols. 
     Terminal node  355 , in many aspects, provides data to and receives data from a person (user). Accordingly, terminal node  355  includes user interface  365 . User interface  365  includes functionality for communicating with a person. User interface  365 , in an aspect, includes a speaker and a microphone for voice communications with the user, a screen for providing visual information to the user, and a keypad for accepting alphanumeric commands and data from the user. In some aspects, a touch screen may be used in place of or in combination with the keypad to allow graphical inputs in addition to alphanumeric inputs. In an alternate aspect, user interface  365  includes a computer interface, for example, a universal serial bus (USB) interface, to interface terminal node  355  to a computer. For example, terminal node  355  may be in the form of a dongle that can be connected to a notebook computer via user interface  365 . The combination of computer and dongle may also be considered a terminal node. User interface  365  may have other configurations and include functions such as vibrators, cameras, and lights. 
     Processor  361  can process communications being received and transmitted by terminal node  355 . Processor  361  can also process inputs from and outputs to user interface  365 . Storage  363  stores data for use by processor  361 . Storage  363  may also be used to store computer readable instructions for execution by processor  361 . The computer readable instructions may be used by terminal node  355  for accomplishing the various functions of terminal node  355 . In an embodiment, storage  363 , or parts of storage  363 , may be considered a non-transitory machine readable medium. For concise explanation, terminal node  355  or aspects of terminal node  355  are described as having certain functionality. It should be appreciated that in some aspects, this functionality is accomplished by processor  361  in conjunction with storage  363 , transmitter-receiver  359 , and user interface  365 . Furthermore, in addition to executing instructions, processor  361  may include specific purpose hardware to accomplish some functions. 
       FIG. 4  is a block diagram of transform module  400  that is used in implementing root non-orthogonal frequency division multiplexing (RNOFDM) according to aspects of the invention. Transform module  400  receives, or accesses, information element data set  410  that may be comprised of, for example, data elements that are to be transferred by a transmitter chain in a communication system. Information element data set  410  may be arranged in any type of known data arrangement or structure such as, for example, an array, a matrix, a record, an object, a data set, etc. Transform module  400  applies a data set transform to generate output samples data set  420  which may be transmitted in a communication system by a transmitter chain, thereby implementing root non-orthogonal frequency division multiplexing (RNOFDM). The data set transform that is applied by transform module may be a matrix, a table, or other known data structure. Output samples data set  420  may be arranged in any type of known data arrangement or structure such as, for example, an array, a matrix, a record, an object, a data set, etc. 
       FIG. 5  is a block diagram of a transmitter chain  501  in accordance with aspects of the invention. Transmitter chain  501  is used herein to process and transmit an input stream of data according to aspects of the invention, implementing root non-orthogonal frequency division multiplexing (RNOFDM). Transmitter  501  comprises encoder  510 , scrambler  511 , a modulation mapper  512 , an FFT or DFT module  514  (optional), a resource mapper  517 , a transform  518 , a cyclic prefix (CP) adder module  519  and a filter  520 . Filter  520  may be one or more of various types of known filters, such as a raised-cosine time-domain window based filter, and filter  520  may be optionally applied. Scrambler  510  and encoder  511  may be optional. As seen in  FIG. 5 , the output of transmitter chain  501  is transmitted via antenna  550 . In the case of a wired communication system, the output of transmitter chain  501  is transmitted via a communication wire or cable. One skilled in the art would know how to apply the single-in-single-out (SISO) transmitter chain  501  illustrated in  FIG. 5  to a multiple-input-multiple-output (MIMO) application/environment (e.g., multiple “RNOFDM” SISO chains). 
       FIG. 6  is a flowchart illustrating an exemplary method for generating a data set transform (such as the transform  400  of  FIG. 4 ) for use in implementing root non-orthogonal frequency division multiplexing (RNOFDM) according to aspects of the invention. It should be noted that, in an aspect, the generation of a data set transform may be performed before operation of the communication system, a priori, such as for example before implementation of the transmitter chain. In an aspect, multiple data set transforms may be generated beforehand and then an appropriate one of the data set transforms can be selected during operation of the communications system. In step  601 , a Non-Orthogonal Frequency Division Multiplexing (NOFDM) matrix, h NOFDM , is defined, wherein each column of the matrix is associated with one of a multiple of non-orthogonal subcarriers. 
     In this regard, in order to increase the transmission rate of OFDM, the orthogonality condition that is imposed on the OFDM subcarriers may be relaxed. Several techniques are known in the art to generate a non-orthogonal modulation based on OFDM. In one example, the number of subcarriers that are multiplexed (i.e., transformed) into one symbol is chosen, beyond the number, N, of orthogonal subcarriers. The increase in the number of subcarriers is denoted by a multiple M, which is selected for convenience to be a power of 2. In other words, one OFDM symbol, {right arrow over (y)}, of duration T and consisting of N multiplexed orthogonal subcarriers, is replaced by one NOFDM symbol, {right arrow over (y)}′, also of duration T and consisting of NM multiplexed subcarriers. In this case, the NOFDM symbol, {right arrow over (y)}′ may be represented as:
 
 {right arrow over (y)}′=h   NOFDM   {right arrow over (x)}′   (1)
 
Where
 
     {right arrow over (x)}′≡NM×1 vector, is the input to the NOFDM multiplexer; 
     {right arrow over (y)}′≡N×1 vector, is the output from the NOFDM multiplexer; and 
     h NOFDM =N×NM matrix, is the NOFDM Matrix, defined as 
                       h     NOFDM   ⁢               ⁢   ⁢       1     NM       ⁡     [         1       1       1       …       1           1         ω     1   M             ω     2   M           …         ω       NM   -   1     M               1         ω     2   M             ω     4   M           …         ω       2   ⁢     (     NM   -   1     )       M               ⋮       ⋮       ⋮                   ⋮           1         ω       N   -   1     M             ω       2   ⁢     (     N   -   1     )       M           …         ω         (     N   -   1     )     ⁢     (     NM   -   1     )       M             ]         ⁢     
     ⁢   where   ⁢           ⁢   ω   ⁢   ⁢           ⁢       e     j2π   N       .             (   2   )               
In other words, the l th  column, h NOFDM,l , of h NOFDM , is a subcarrier which carries the information corresponding to the l th  element of and {right arrow over (x)}, may be defined as:
 
     
       
         
           
             
               
                 
                   
                     
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     Both output vectors, {right arrow over (y)} and {right arrow over (y)}′, respectively, may be oversampled by a multiple equal to M, or equivalently, to force the total number of samples for both {right arrow over (y)} and {right arrow over (y)}′ to equal NM using a sampling frequency, f′ s , equal to Mf s  for both. The oversampling does not change the duration of the output vectors, {right arrow over (y)} and {right arrow over (y)}′ since the oversampling is still equal to 
               NM     Mf   s       =       N     f   s       =     T   .             
The choice of oversampling by a multiple equal to M forces h NOFDM  to be square. In this scenario, oversampling the NOFDM symbol, {right arrow over (y)}′, by a multiple M corresponds to having a multiplexing matrix defined as:
 
                     h     NOFDM   ⁢               ⁢   ⁢       1       N     ⁢   M       ⁡     [         1       1       1       …       1           1         ω     1   M             ω     2   M           …         ω       NM   -   1     M               1         ω     2   M             ω     4   M           …         ω       2   ⁢     (     NM   -   1     )       M               ⋮       ⋮       ⋮                   ⋮           1         ω       NM   -   1     M             ω       2   ⁢     (     NM   -   1     )       M           …         ω         (     NM   -   1     )     ⁢     (     NM   -   1     )       M             ]               (   4   )               
where
 
     h NOFDM ≡NM×NM square matrix, is the oversampled NOFDM Matrix; 
               ω   ⁢   ⁢     e       -   j     ⁢       2   ⁢   π     NM           ;         
and
 
     {right arrow over (y)}′≡NM×1 vector, is the output vector from the NOFDM multiplexer. 
     In this case, the l th  column of h NOFDM ,h NOFDM,l , is a subcarrier which carries the information corresponding to the l th  element of {right arrow over (x)}′, and may be which is defined as: 
     
       
         
           
             
               
                 
                   
                     
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     In step  602 , the NOFDM matrix, h NOFDM , is multiplied by its complex conjugate transpose matrix (i.e. Hermitian) to obtain a matrix product h NOFDM h NOFDM *. In an aspect, in order to improve the spectral efficiency of NOFDM, in particular when the spectral threshold level is ≦−40 dBr, the variance between the eigenvalues of h NOFDM h NOFDM * may be reduced. One way to reduce the variance is by replacing h NOFDM  with 
                   h   NOFDM     ⁢     h   NOFDM   *       n     ,         
or by replacing Equation (1) with:
 
                       y   →     ′     =           h   NOFDM     ⁢     h   NOFDM   *       n     ⁢           ⁢       x   →     ′               (   6   )               
where n is selected to be an integer. Such a modulation may be referred to as Root-NOFDM (RNOFDM). In an aspect, the product h NOFDM h NOFDM * corresponds to a square NM×NM matrix, and the duration of the RNOFDM symbol {right arrow over (y)} e  is equal to that of the OFDM symbol {right arrow over (y)}, T.
 
     In step  603 , a non-linear function (e.g., a root function such as an n th  root function 
               •   n     )         
is applied to the matrix product to generate a data set transform (e.g., to generate
 
                     h   NOFDM     ⁢     h   NOFDM   *       n     )     .         
The data set transform (e.g., transform module  400  of  FIG. 4 , and transform  518  of  FIG. 5 ) may then be used to transform information elements (such as information element data set  410  of  FIG. 4 , and output of resource mapper  517  of  FIG. 5 ) into an output samples data set (such as information element data set  420  of  FIG. 4 , and output of transform  518  of  FIG. 5 ) for transmission according to RNOFDM, as described below with respect to  FIG. 7  according to aspects of the invention.
 
       FIG. 7  is a flowchart illustrating an exemplary method for applying a data set transform (such as transform module  400  of  FIG. 4 ) in the implementation of RNOFDM according to an aspect of the invention. In step  701 , an information element data set, {right arrow over (x)}′, is obtained or accessed, wherein the information element data set is comprised of a first number of elements. 
     A data set transform is applied to the information element data set in step  702  to obtain an output samples data set, {right arrow over (y)}″, comprised of a second number of elements M. The data set transform may, for example, be generated as described above with respect to  FIG. 6 . 
     In step  702 , a filter (such as filter  520  of  FIG. 5 ) may be optionally implemented. For example, in an aspect, a filter may be applied in which a raised-cosine time-domain window is used. A raised-cosine window is defined by its roll-off factor: the larger the roll-off factor of the window, the faster the rate of decay of its Fourier Transform. In an aspect, other types of windows may be used instead of the raised-cosine window. Examples of other windows include the Hamming window, the Hanning window, the Blackman-Harris window, and others. In addition, other techniques exist that may be used to speed the rate of decay of the power spectral density (PSD) of RNOFDM. Examples of such other techniques include the Filter Bank Multi-Carrier (FBMC) technique, which avoids using a Cyclic Prefix, and the Universal Filtered Multi-Carrier (UFMC) which is effective for short packets. FBMC performs a subcarrier filtering of the OFDM signal, while UFMC performs a sub-band filtering of the OFDM signal. Both techniques may be used for filtering the RNOFDM signal. 
     In an aspect, the PSD of NOFDM may be forced to decay below a spectral threshold using a cyclic Step-wise Gaussian Band Pass Filter (BPF) with a baseband-equivalent Transfer Function, H(f), which may be expressed mathematically as: 
                     H   ⁡     (   f   )       ⁢           ⁢   ⁢           ⁢     {           e       -       (     [       f   n     +     0.5   M       )     ]     2         2   ⁢     σ   f   2                   for   ⁢           ⁢     f   n       ≤     -     0.5   M                 1           for   ⁢           -     0.5   M       &lt;     f   n     &lt;     0.5   M                 e       -       (     [       f   n     +     0.5   M       )     ]     2         2   ⁢     σ   f   2                   for   ⁢           ⁢     f   n       ≥     +     0.5   M                         (   7   )               
where σ f  is the normalized standard deviation of the Gaussian filter, normalized by f s ′ and f n  is the normalized frequency, also normalized by f s ′. It should be noted that other types of cyclic BPF exist and may be used instead of the Gaussian BPF in Equation (7) in other aspects. In aspects of the invention, the RNOFDM symbols described herein may be band-limited using the cyclic baseband-equivalent Band Pass Filter described in Equation (7).
 
     In step  703 , the output samples data set, {right arrow over (y)}′, is transmitted over a communications channel. In this manner, RNOFDM as described herein is utilized to achieve improved performance as compared to OFDM or known NOFDM techniques. 
     In this regard,  FIG. 8  is a graph depicting the performance of RNOFDM in comparison to OFDM, according to aspects of the invention. In  FIG. 8 , the spectral efficiency, η OFDM , for the non-filtered OFDM signal using a raised-cosine window with a 12.5% roll-off factor (blue curves) is compared with that of the filtered RNOFDM signal using the BPF described in Equation (7) with σf=0.022097 and using a raised-cosine time window with a 12.5% roll-off factor after taking into consideration their relative bandwidths, 
                   W   OFDM       W   NOFDM       ⁢     ❘       -   40     ⁢   dBr         ,         
defined at a spectral threshold level of −40 dBr, for NM=64, 128, 256, 512, M=8 and n=5. The relative improvements in spectral efficiency that NOFDM offers over OFDM are shown in  FIG. 7 , namely 11% at 10 dB SNR, 25% at 20 dB SNR and 36% at 30 dB SNR, for NM=64 subcarriers.
 
     In one aspect, the performance results described above are obtained under the following example assumptions and constraints. 
     Example Assumptions:
         A1: The signal, {right arrow over (y)}′, is transmitted using one antenna over a channel that is contaminated by an Additive White Gaussian Noise (AWGN), and is received by one antenna;   A2: The OFDM signals that are selected as bases to develop the RNOFDM signal have N=8, 16, 32 and 64 subcarriers;   A3: RNOFDM uses a multiple M=8 where NM is the number of subcarriers in the initial NOFDM signal and its number of samples;   A4: The duration, T, of the RNOFDM symbol is identical to the duration, T, of the OFDM symbol;   A5: The total signal energy used in one OFDM signal is identical to the total signal energy used in one NOFDM signal, which is also equal to the total signal energy used in one RNOFDM signal; and   A6: Results in  FIG. 8  are obtained for the spectral efficiency versus the received normalized signal-to-noise ratio (SNR) for the OFDM signal and the RNOFDM signal. The normalized signal-to-noise ratio (SNR) is obtained for both signals, the OFDM signal and the RNOFDM signal, by dividing their total SNR by N.       

     Example Constraint:
         C1: The spectral threshold level is set at −40 dBr.       

     A summary of the spectral efficiency ratios, 
                   η   OFDM       η   NOFDM       ⁢     ❘       -   40     ⁢   dBr         ,         
is shown in Table III below with NM=64. From Table III, it may be seen that the improvements that Equation (6) offers over Equation (1) may reach a peak value when n=4
 
     
       
         
           
               
               
               
               
             
               
                 TABLE III 
               
               
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             η 
                             OFDM 
                           
                           
                             η 
                             NOFDM 
                           
                         
                         ⁢ 
                         
                           | 
                           
                             
                               - 
                               40 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             dBr 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             η 
                             OFDM 
                           
                           
                             η 
                             NOFDM 
                           
                         
                         ⁢ 
                         
                           | 
                           
                             
                               - 
                               40 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             dBr 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             η 
                             OFDM 
                           
                           
                             η 
                             NOFDM 
                           
                         
                         ⁢ 
                         
                           | 
                           
                             
                               - 
                               40 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             dBr 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                 @SNR = 10 dB, 
                 @SNR = 20 dB, 
                 @SNR = 30 dB, 
               
               
                   
                 NM = 64 
                 NM = 64 
                 NM = 64 
               
               
                   
               
             
            
               
                 n = 1 
                  3% 
                  9% 
                 14% 
               
               
                 n = 2 
                  3% 
                 10% 
                 18% 
               
               
                 n = 3 
                  8% 
                 19% 
                 28% 
               
               
                 n = 4 
                 15% 
                 27% 
                 38% 
               
               
                 n = 5 
                 11% 
                 25% 
                 36% 
               
               
                 n = 6 
                 12% 
                 27% 
                 39% 
               
               
                   
               
            
           
         
       
     
     It should be noted that the above-described assumptions, constraints and performance data and comparisons are exemplary only and that aspects of the invention as described herein are not limited by such assumptions, constraints and performance data. 
     The foregoing systems and methods and associated devices and modules are susceptible to many variations. Additionally, for clarity and concision, many descriptions of the systems and methods have been simplified. For example, the figures generally illustrate one of each type of device (e.g., one access node, one terminal node), but a communication system may have many of each type of device. Similarly, descriptions may use terminology and structures of a specific wireless standard, such as WiFi or LTE. However, the disclosed systems, devices and methods are more broadly applicable to wireless and wired communication systems, including for example, to hybrid fiber-coax cable modem systems. 
     Those of skill will appreciate that the various illustrative logical blocks, modules, units, and algorithm steps described in connection with the embodiments disclosed herein can often be implemented as electronic hardware, computer software, or combinations of both. To clearly illustrate this interchangeability of hardware and software, various illustrative components, blocks, modules, and steps have been described above generally in terms of their functionality. Whether such functionality is implemented as hardware or software depends upon the particular constraints imposed on the overall system. Skilled persons can implement the described functionality in varying ways for each particular system, but such implementation decisions should not be interpreted as causing a departure from the scope of the invention. In addition, the grouping of functions within a unit, module, block, or step is for ease of description. Specific functions or steps can be moved from one unit, module, or block without departing from the invention. 
     The various illustrative logical blocks, units, steps and modules described in connection with the embodiments disclosed herein can be implemented or performed with a processor. As used herein a processor may be a general purpose processor, a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA) or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any portion or combination thereof that is capable of performing the functions described herein. A general purpose processor can be a microprocessor, but in the alternative, the general purpose processor can be any processor, controller, microcontroller, or state machine. A processor can also be implemented as a combination of computing devices, for example, a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration. 
     The steps of a method or algorithm and the processes of a block or module described in connection with the embodiments disclosed herein can be embodied directly in hardware, in a software module executed by a processor, or in a combination of the two. A software module can reside in RAM memory, flash memory, ROM memory, EPROM memory, EEPROM memory, registers, hard disk, a removable disk, a CD-ROM, or any other form of storage medium. An exemplary storage medium can be coupled to the processor such that the processor can read information from, and write information to, the storage medium. In the alternative, the storage medium can be integral to the processor. The processor and the storage medium can reside in an ASIC. Additionally, device, blocks, or modules that are described as coupled may be coupled via intermediary device, blocks, or modules. Similarly, a first device may be described as transmitting data to (or receiving from) a second device when there are intermediary devices that couple the first and second device and also when the first device is unaware of the ultimate destination of the data. 
     The above description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles described herein can be applied to other embodiments without departing from the spirit or scope of the invention. Thus, it is to be understood that the description and drawings presented herein represent a presently preferred embodiment of the invention and are therefore representative of the subject matter that is broadly contemplated by the present invention. It is further understood that the scope of the present invention fully encompasses other embodiments that may become obvious to those skilled in the art and that the scope of the present invention is accordingly limited by nothing other than the claims.