Abstract:
Systems and methods for securing orthogonal frequency division multiplexing (OFDM) transmission of data are discussed herein. Transceivers can sense the signal strength of frequencies on an OFDM wireless channel. The transceivers can determine which of the frequencies are fading frequencies failing below a threshold. A transmitting transceiver can send artificial data on the failing frequencies and genuine data on the remaining frequencies. Artificial data can be designed to mitigate peak-average-power ratio (PAPR) to have an additional benefit without using extra resource. Finally, receiving transceiver can transceiver receive the genuine data while discarding any date received on the failing frequencies.

Description:
BACKGROUND 
     The broadcast nature of wireless mediums can be susceptible to threats regarding privacy and security of wireless communication. For example, an eavesdropper may be able to capture wireless signals. Depending on the type of wireless transmission, these signals can be available from a variety of distances, for example several yards, a mile, or across the globe. Data protection techniques can provide one form of security to communication. Encryption with a secret key is one method to conceal data. However, encryption techniques may not be sufficient for some scenarios, such as securely sharing a secret key with the intended nodes. Therefore, alternative means are needed to increase security. For example, physical layer security can provide enhanced security in the waveform domain. Security measures dealing with the waveform domain can prevent the extraction of the actual data properly. An eavesdropper can be prevented from extracting the actual data even if they can receive entire transmissions. 
     A multiple-input multiple-output (MIMO) system can have enhanced physical security by the addition of artificial noise to the actual signal. The noise signal can be designed to fall in the null space of the channel matrix between the antennas of legitimate devices using the MIMO system. Artificial noise can be eliminated while the signal is passing through the channel. Another device located in a different place has a different channel matrix than a legitimate transmitter. The artificial noise can distort the actual signal. The distortion of the actual signal can increase security. In a single-input single-output system (SISO) for single carrier-frequency domain equalization (SC-FDE) waveforms, artificial noise can be added to fading frequencies. For example, the artificial noise can be added to the fading frequencies instead of adding artificial noise to the null space. Fading frequency information can only be determined by legitimate transmitters. For example, an eavesdropper cannot separate the actual signal and artificial noise. As such, an eavesdropper can experience a significant error. However, unlike SC-FDE, where data symbols are carried in the time domain, OFDM systems carry data symbols in the frequency domain. Therefore, introducing a noise signal to an OFDM signal might not prevent eavesdroppers from detecting which frequencies are utilized and ignoring transmissions on those frequencies since noise has a different characteristic compared to the data. 
     In wireless communication systems, orthogonal frequency division multiplexing (OFDM) can provide numerous advantages, for example OFDM can transmit data with high bandwidth efficiency, OFDM can be implemented with fast Fourier transformation (FFT), and OFDM can be equalized simply by exploiting the advantage of cyclic prefix (CP). On the other hand, OFDM signals can suffer from high peak to average power ratio (PAPR) caused by the parallel transmission because of the non-linear characteristics of power amplifiers. An OFDM signal can be nonlinearly scaled, and in-band interference can occur at the transmitter. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
       For a more complete understanding of the embodiments and the advantages thereof, reference is now made to the following description, in conjunction with the accompanying figures briefly described as follows: 
         FIG. 1  is a drawing of a networked environment according to various embodiments of the present disclosure; 
         FIGS. 2A and 2B  illustrate frequency regions of data as seen by various receivers and transmitters according to various example embodiments; 
         FIG. 3  illustrates bit-error rates for various receivers and transmitters according to various example embodiments; 
         FIG. 4  illustrates PAPR reduction performance according to various example embodiments; 
         FIGS. 5A and 5B  are flow charts illustrating one example of functionality implemented on a transmitting transceiver and a receiving transceiver according to various embodiments; and 
         FIG. 6  illustrates an example schematic block diagram of a computing environment which can embody one or more of the system according to various embodiments. 
     
    
    
     The drawings illustrate only example embodiments and are therefore not to be considered limiting of the scope described herein, as other equally effective embodiments are within the scope and spirit of this disclosure. The elements and features shown in the drawings are not necessarily drawn to scale, emphasis instead being placed upon clearly illustrating the principles of the embodiments. Additionally, certain dimensions or positionings may be exaggerated to help visually convey certain principles. In the drawings, similar reference numerals between figures designate like or corresponding, but not necessarily the same, elements. 
     DETAILED DESCRIPTION 
     In the following paragraphs, the embodiments are described in further detail by way of example with reference to the attached drawings. In the description, well known components, methods, and/or processing techniques are omitted or briefly described so as not to obscure the embodiments. 
     Although embodiments have been described herein in detail, the descriptions are by way of example. The features of the embodiments described herein are representative and, in alternative embodiments, certain features and elements may be added or omitted. Additionally, modifications to aspects of the embodiments described herein may be made by those skilled in the art without departing from the spirit and scope of the present invention defined in the following claims, the scope of which are to be accorded the broadest interpretation so as to encompass modifications and equivalent structures. 
     Discussed herein is a technique for securing OFDM signals. In addition, the technique can also be enhanced to mitigate PAPR. The reduction of PAPR can be accomplished without sacrificing additional spectral bandwidth. In contrast to introducing artificial noise, artificial data can be inserted into modulated symbols since an eavesdropping device may be able to determine the frequency location utilized as artificial noise. By inserting artificial data into modulated symbols, an eavesdropping device can be prevented from detecting artificial data frequencies. 
     In addition, the artificial data can be chosen to mitigate PAPR. By choosing PAPR mitigating artificial data, the PAPR can be reduced without consuming resources beyond those utilized for security. The artificial data can be selected based on the data to be transmitted. The data signal can include genuine data that corresponds to the frequencies that meet the channel gain threshold. In one example, the genuine data is information to be transmitted from the transmitting device to the receiving device. 
     The system can include an OFDM wireless channel that is between a pair of wireless transceivers. The wireless channel can include a plurality of frequencies bands. For example, in OFDM multiple orthogonal frequencies are used. A wireless transceiver can communicate over the OFDM wireless channel to another wireless transceiver. Each of the wireless transceivers can be configured to sense the signal strength or channel gain of each frequency on the OFDM wireless channel. The transceivers can also be configured to determine which of the frequencies meet a threshold, and this determination can be based on the signal strength of each of the frequencies. 
     While the threshold can be predefined, in some embodiments, the threshold can be set by a user of the system, and in other embodiments, the threshold can be negotiated between two devices. The threshold can be based on the amount of data to be transmitted and/or the sensitivity of the data. For example, if a transceiver is to transmit a short and very important data packet, the threshold can be higher. 
     The transmitting transceiver can be configured to generate artificial data that reduces the PAPR of the resulting transmission. The transceiver can also initiate transmission of the resulting data signal. The resulting data signal can include the artificial data corresponding to frequencies that are determined not to meet the threshold. The resulting data signal can include genuine data being mapped to frequencies that are determined to meet the threshold. 
     With reference to  FIG. 1 , shown is a networked environment  100  according to various embodiments. The networked environment includes a transmitting device  103 , a receiving device  106 , one or more eavesdropping device  109 , and an OFDM wireless channel  112 . The transmitting device  103 , the receiving device  106 , and the eavesdropping device  109  can include one or more transceiver  115 ,  118 , and  121 , respectively. Each of the transceivers  115 ,  118 , and  121  can transmit and receive a wireless OFDM signal on the OFDM wireless channel  112 . 
     The wireless OFDM channel  112  can have unique characteristics between any two of the transceivers  115 ,  118 , and  121 . The unique characteristics can be based on the location of each of the transceivers  115 ,  118 , and  121 . As an example, the wireless OFDM channel  112  between the transceiver  115  and the transceiver  118  has different characteristics in comparison to the wireless OFDM channel  112  between transceiver  115  and transceiver  121  or between transceiver  118  and transceiver  121 . 
     In OFDM, a block of N complex valued QAM symbols, for example X=[X 0 , X 1 , . . . , X N-1 ], can be mapped onto N subcarriers with the IFFT operation. The resulting signal can give the elements in a time domain representation, x=[x 0 , x 1 , . . . , x N-1 ]. In one embodiment, a sample of this signal can be written as: 
     
       
         
           
             
               
                 
                   
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                           0 
                         
                         
                           N 
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                           1 
                         
                       
                       ⁢ 
                       
                         
                           X 
                           k 
                         
                         ⁢ 
                         
                           
                             ⅇ 
                             
                               j2π 
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                                 / 
                                 N 
                               
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
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     After passing through the wireless channel, such as OFDM wireless channel  112 , each subcarrier can be multiplied by the channel gain of the corresponding frequency, and received signal in frequency domain can be given as
 
 Y   k   =H   k   X   k ,  (2)
 
where Y k  is the received symbol and H k  is the channel gain at the frequency of kth subcarrier. The channel gain on the OFDM wireless channel  112  at a frequency can be dependent on the location of the communicating devices. For example, the channel gain at a frequency can be based on the location of the transmitting device  103 , the receiving device  106 , and the medium between the transmitting device  103  and the receiving device  106 . As the location of the transmitting device  103  or receiving device  106  changes, the OFDM wireless channel  112  between them can also change.
 
     In addition, the OFDM wireless channel  112  between two communicating transceivers is reciprocal. According to one example, the characteristics of the OFDM wireless channel  112  between the transceiver  115  and the transceivers  118  have the same characteristics when viewed from either transceiver  115  or  118 . Both the transceiver  115  and the transceiver  118  can experience the same OFDM wireless channel  112  throughout the coherence time of the medium. Further, transceiver  121  cannot experience the same OFDM wireless channel  112  throughout the coherence time of the medium between the transceivers  115  and  118 . 
     In an SC-FDE system, artificial noise can be inserted on fading frequencies. Fading frequencies can be one or more frequency and/or one or more bands of frequencies. The fading frequencies can be identified as having low channel gains. The fading frequencies can also be identified as causing large power losses as compared to other frequencies. Information regarding which frequencies can be faded on any given channel will only be available to the receiver and transmitter sharing the channel. An eavesdropping device cannot separate the actual signal and artificial noise. Therefore, the data carrying signal from the eavesdropping device can be distorted with the artificial noise. Meanwhile, the transmitting device and receiving device can agree to ignore artificial signal noise because of the channel reciprocity. 
     However, unlike in the SC-FDE system, where data symbols are carried in the time domain, OFDM systems carry data symbols in the frequency domain. Therefore, introducing a noise signal to an OFDM signal may not prevent eavesdroppers from detecting which frequencies are utilized and ignoring transmissions on those frequencies. However, inserting artificial data as opposed to artificial noise can solve this deficiency of artificial noise. For example, if a QPSK modulation is utilized, the transmitting device  103  can select symbols from four complex numbers. If a particular region of a signal is filled with arbitrary samples while the remaining part can be filled with QPSK symbols, the eavesdropping device  109  may be able to understand artificial noise carrying frequencies through observation. However, the transmitting device  103  can choose irrelevant QPSK symbols rather than artificial noise, which is referred to herein as artificial data insertion. Further, the transmitting device  103  can select the artificial data to reduce PAPR. In fact, a large PAPR can correspond to a large variance in the signal, which can be one of the major drawbacks in OFDM systems. 
     As PAPR increases for an OFDM signal over OFDM wireless channel  112 , a power amplifier can output outside of its linear range, which can cause signal distortion. This distortion can degrade communication performance. This can be prevented by reducing the PAPR of a signal. Alternatively, this can be mitigated by utilizing a more expensive power amplifier having a larger dynamic range in transceivers  115  and  118 . When artificial data is added to the fading frequencies on a channel on OFDM wireless channel  112  for security, the transmitting device  103  can choose the artificial data to reduce PAPR to mitigate PAPR problems without additional cost. For example, the transmitting device  103  can reduce the the PAPR without utilizing additional bandwidth on the OFDM wireless channel  112  beyond the fading frequencies utilized for security. 
     By doing this, the transmitting device  103  can reduce PAPR and secure the communication to the receiving device  106  over OFDM wireless channel  112  without consuming any extra resources. As a non-limiting example, the artificial data inserted OFDM signal in time domain can be expressed in equation 3 as follows: 
                       x   ^     m     =       1   N     ⁢       ∑     k   =   0       N   -   1       ⁢       (         a   k     ⁢     X   k       +       b   k     ⁢       X   ~     k         )     ⁢     ⅇ     j2π   ⁢           ⁢   k   ⁢           ⁢     m   /   N                       (   3   )               
where {tilde over (X)} k  is the artificial data samples in frequency, a k  is 1 when k is the index of an information data, otherwise it is 0 and b k  is 1 when k is the index of an artificial data, otherwise it is 0. Then we can form our objective function as,
 
                       X   ~     k     =         arg   ⁢           ⁢   min         X   ~     k       ⁢            x   ^          ∞               (   4   )                 subject   ⁢           ⁢   to   ⁢           ⁢       X   ~     k       ∈   Γ           (   5   )               
where ∥{circumflex over (x)}∥ ∞  represents the infinite norm operation and r is the set of modulation symbols, e.g., {e jπ/4 , e j3π/4 , e −j3π/4 , e −jπ/4 } for QPSK type of modulation.
 
     In this way, transmitting device  103  can reduce the PAPR and secure the OFDM wireless channel  112  to prevent attacks by the eavesdropping device  109 . For example, the transmitting device  103  can prevent the eavesdropping device  109  from determining the genuine data because the eavesdropping device  109  cannot determine which frequencies are carrying the artificial data. 
     According to one embodiment, the transmitting device  103  and a receiving device  106  are identical devices. The receiving device  106  transmits data over the OFDM wireless channel  112  to the transmitting device using the security and PAPR reduction techniques discussed herein. For example, the receiving device  106  can transmit artificial data chosen to reduce PAPR in frequencies that fail to meet a threshold and transmit genuine data in the remaining frequencies. The transmitting device  103  receives the wireless transmission over the OFDM wireless channel  112  from the receiving device  106  using the techniques discussed herein. For example, the transmitting device  103  determines which frequencies fail to meet the threshold and discard data on those frequencies. 
     With reference to  FIG. 2A , shown is an illustration of the fading frequencies of the OFDM wireless channel  112  as seen by the transceivers  115  and  118  according to various embodiments. The OFDM wireless channel  112  as seen by transceivers  115  and  118  includes genuine data frequencies  203  and artificial data frequencies  206 . The transmitting device  103  and the receiving device  106  can sense the frequencies having signal characteristics, such as signal strength, that fails to meet the threshold. 
     The transmitting device  103  and the receiving device  106  sense the strength of the signal using transceivers  115  and  118 . The transmitting device  103  and the receiving device  106  can identify those frequencies as artificial data frequencies. In communications, the transmitting device  103  and the receiving device  106  can use the frequencies meeting the threshold signal strength to carry genuine data and use the frequencies failing to meet the threshold signal strength to carry artificial data. 
     With reference to  FIG. 2B , shown is an illustration of the fading frequencies of the OFDM wireless channel  112  as seen by the transceiver  121  according to various embodiments. The eavesdropping device  109  can receive the OFDM wireless data transmitted by either transmitting device  103  or receiving device  106 . However, the eavesdropping device  109  experiences different fading frequencies as compared to the transmitting device  103  and the receiving device  106  when receiving the OFDM wireless data. The OFDM wireless channel  112  as seen by transceivers  121  and  118  cannot differentiate between genuine data frequencies  212  and artificial data frequencies  215 . 
       FIGS. 3 and 4  illustrate measured results according to an example embodiment. In this example embodiment, the multiple parameters can be determined. For example, QPSK can be utilized for modulation, a number of subcarriers can be determined as 32, a 16-tap Rayleigh channel can be used for having the required selectivity in frequency, the average channel power can be normalized to 1, and one of three different threshold values, 0.1, 0.2, and 0.3, can be chosen in order to select artificial data insertion frequencies. According to one example, the thresholds of 0.1, 0.2, and 0.3 correspond to utilizing of 9%, 18%, or 26% respectively for spectral resource for artificial data based on the performed simulations. 
       FIG. 3  illustrates a bit error rate for a variety of users and thresholds according to the example embodiment. Bit-error-rate (BER) results for a legitimate user using receiving device  106 , Bob, and an eavesdropper using eavesdropping device  109 , Eve, are shown in  FIG. 3 . The receiving device  106  has a lower error probability rate than standard communications because the fading frequencies are used for artificial data insertion. This is because the majority of bit errors that occur in OFDM can be within data transmitted over the fading frequencies. In contrast in  FIG. 3 , the eavesdropping device  109  has a large BER even for high signal-to-noise ratio (SNR) values. For example, the BER does not noticeably reduce as E b /N 0  (dB) increases for the signal sensed by the eavesdropping device  109  at any threshold level. In addition, the transmitting device  103  can use a larger threshold can degrade BER performance of the eavesdropping device  109  while enhancing BER performance of the receiving device  106  at the expense of more spectral resource usage. 
       FIG. 4  illustrates PAPR values for legitimate user Bob using receiving device  106  and Eve on eavesdropping device  109  at various threshold values according to the example embodiment. The additional data can be selected to optimize PAPR. Shown are the PAPR curves for the aforementioned threshold values. The transmitting device  103  can use a flipping algorithm to find a sub-optimum artificial data set that reduces PAPR. The complexity of this technique can be expressed as proportional to M×K where M is the number of modulation symbols and K is the number of subcarriers used for artificial data in one symbol. As shown in  FIG. 4 , as the threshold increases, the PAPR reduces. In addition, utilizing a flipping algorithm may not add significant complexity because a flipping algorithm for such optimization problems can be implemented with low complexity. In other embodiments, more complex algorithms can be selected by the transmitting device  103  to obtain better PAPR reduction performance. 
     Referring next to  FIG. 5A , shown is a flowchart that provide examples of the operation of a receiving device  106  ( FIG. 1 ) according to various embodiments. It is understood that the flowchart of  FIG. 5A  provides merely an example of the many different types of functional arrangements that can be employed to implement the operation of the portion of the receiving device  106  as described herein. As an alternative, the flowchart of  FIG. 5A  can be viewed as depicting examples of steps of a method implemented in a receiving device  106  according to one or more embodiments. 
     The process in  FIG. 5A  can be used to receive data with security in the physical layer and a reduced PAPR on the transmitted signal according to one or more embodiments. In step  503 , the process involves sensing the signal strength of each of one or more frequencies on a wireless channel. For example, a transceiver  118  ( FIG. 1 ) can be configured to communicate utilizing closely spaced orthogonal sub-carrier signals on parallel data frequencies. The transceiver  118  can be configured to sense the signal strength on each of the frequencies. 
     In step  506 , the process involves identifying frequencies that meet a threshold signal strength on the wireless channel. In some embodiments, the process can involve identifying fading frequencies that fail to meet the threshold signal strength. For example, the receiving device  106  can compare the signal strengths measured in  506  to a threshold frequency. The receiving device  106  can identify the frequencies that meet the threshold as frequencies to use to receive genuine data. The receiving device  106  can identify the frequencies that fail to meet the threshold as frequencies to use for artificial data. 
     In step  509 , the process involves receiving an OFDM transmission from the wireless channel. For example, the transceiver  118  can listen on the OFDM wireless channel  112  for a transmission and record the transmission upon receipt. In step  512 , the process involves processing data from frequencies that meet the threshold. The receiving device  106  can identify data from frequencies that fail to meet the threshold as artificial data and processing data from the frequencies that meet the threshold. The receiving device  106  can discard data received on the frequencies failing to meet the threshold. 
     Referring next to  FIG. 5B , shown is a flowchart that provide an example of the operation of a transmitting device  103  ( FIG. 1 ) according to various embodiments. It is understood that the flowchart of  FIG. 5B  provides merely an example of the many different types of functional arrangements that can be employed to implement the operation of the portion of the transmitting device  103  as described herein. As an alternative, the flowchart of  FIG. 5B  can be viewed as depicting examples of steps of a method implemented in a transmitting device  103  according to one or more embodiments. 
     The process in  FIG. 5B  can be used to transmit data with security in the physical layer and a reduced PAPR on the transmitted signal according to one or more embodiments. In step  515 , the process can involve sensing the signal strength of each of one or more frequencies on a wireless channel. For example, a transceiver  115  ( FIG. 1 ) of the transmitting device  103  can be configured to communicate utilizing closely spaced orthogonal sub-carrier signals on parallel data frequencies. The transceiver  115  can be configured to sense the signal strength on each of the frequencies. 
     In step  518 , the process involves identifying frequencies that meet a threshold signal strength on the wireless channel. The transmitting device  103  can identify fading frequencies that fail to meet the threshold signal strength on the OFDM wireless channel  112 . For example, the transmitting device  103  can compare the signal strengths measured in  515  to a threshold frequency. The transmitting device  103  can identify the frequencies that meet the threshold as frequencies to use to transmit genuine data. The transmitting device  103  can identify the frequencies that fail to meet the threshold as frequencies to use for artificial data. 
     In step  521 , the process can involve generating a signal including artificial data on frequencies that fail to meet the threshold and genuine data on frequencies that meet the threshold requirement. The process can also involve generating the artificial data as data that minimized the PAPR of the generated signal. For example, the transmitting device  103  can determine artificial data that will lower a PAPR value of a resulting signal when substituted for genuine data. The transmitting device  103  can also place artificial data into bandwidth frequencies that, based on various real world factors, require greater power to or that have reduced quality or intensity at a corresponding receiving device  106 . As an example, the various real world factors can include the medium between the transmitting device  103  and the receiving device  106  and the distance between the transmitting device  103  and the receiving device  106 . 
     In step  524 , the process can involve transmitting the resulting signal over the OFDM wireless channel. For example, the transmitting device  103  can transmit the signal over the OFDM wireless channel  112 . The transmitting device  103  can utilize the transceiver  115  to transmit the signal to a receiving device  106 . 
       FIG. 6  illustrates an example schematic block diagram of a computing architecture  600  that may be employed as either of the transceivers according to various embodiments described herein. The computing architecture  600  may be embodied, in part, using one or more elements of a specific purpose processing circuit or computing device. The computing architecture  600  includes a processor  610 , a random access memory (RAM)  620 , a Read Only Memory (ROM)  630 , a transceiver  640 , a memory device  650 , and an Input Output (I/O) interface  660 . The elements of computing architecture  600  are communicatively coupled via one or more local interfaces  602 . The elements of the computing architecture  600  are not intended to be limiting in nature, as the architecture may omit elements or include additional or alternative elements. 
     In various embodiments, the processor  610  may include or be embodied as a general purpose arithmetic processor, a state machine, or an ASIC, for example. The processor  610  may include one or more circuits, one or more microprocessors, ASICs, dedicated hardware, or any combination thereof. In certain aspects and embodiments, the processor  610  is configured to execute one or more software modules which may be stored, for example, on the memory device  650 . 
     The RAM and ROM  620  and  630  may include or be embodied as any random access and read only memory devices that store computer-readable instructions to be executed by the processor  610 . The memory device  650  stores computer-readable instructions thereon that, when executed by the processor  610 , direct the processor  610  to execute various aspects of the embodiments described herein. 
     As a non-limiting example group, the memory device  650  includes one or more non-transitory memory devices, such as an optical disc, a magnetic disc, a semiconductor memory (i.e., a semiconductor, floating gate, or similar flash based memory), a magnetic tape memory, a removable memory, combinations thereof, or any other known non-transitory memory device or means for storing computer-readable instructions. The I/O interface  660  includes device input and output interfaces, such as keyboard, pointing device, display, communication, and/or other interfaces. The one or more local interfaces  602  electrically and communicatively couples the processor  610 , the RAM  620 , the ROM  630 , the memory device  650 , and the I/O interface  660  so that data and instructions may be communicated among them. 
     In certain aspects, the processor  610  is configured to retrieve computer-readable instructions and data stored on the memory device  650 , the RAM  620 , the ROM  630 , and/or other storage means and copy the computer-readable instructions to the RAM  620  or the ROM  630  for execution, for example. The processor  610  is further configured to execute the computer-readable instructions to implement various aspects and features of the embodiments described herein. In embodiments where the processor  610  includes a state machine or ASIC, the processor  610  may include internal memory and registers for maintenance of data being processed. 
     The transceiver  640  can be any one of transceiver  115 ,  118 , or  121  ( FIG. 1 ). The transceiver  640  can transmit and receive data wirelessly on various frequencies. The transceiver  640  can also be configured to sense the signal strength on each frequency of a wireless channel. The transceiver  640  can communicate utilizing wireless LAN radio interfaces; digital radio system, such as DAB/EUREKA 147, DAB+, Digital Radio Mondiale, HD Radio, T-DMB, and ISDB-TSB; terrestrial digital TV systems, such as DVB-T and ISDB-T, terrestrial mobile TV systems, such as DVB-H, T-DMB, ISDB-T, and MediaFLO forward link; cellular system, such as 3G, 4G, LTE, BWA, MBWA, and E-UTRA; and other wireless frequencies.