Patent Publication Number: US-11652679-B2

Title: Systems and methods for reliable chirp transmissions and multiplexing

Description:
TECHNICAL FIELD 
     Example embodiments generally relate to communications technologies and, more particularly, relate to apparatuses, systems, and methods for chirp transmissions and multiplexing. 
     BACKGROUND 
     Wireless communications devices in the form of cellular phones and the like have become commonplace. The desire to have wireless connectivity and communications capabilities is now demanded in almost any setting. In the context of the Internet of Things (IoT), on some level, almost all electronic devices are candidates for implementing wireless connectivity to support communications. Even moving vehicles, including land-based and aerial vehicles, are planned to be increasingly connected to support wireless communications. 
     While the ability to wirelessly connect devices in these contexts is currently available, there continues to be aspects where further innovation and improvements can be made. For instance, the demand for increased throughput to communicate and establish higher data transmission rates seems to have no end. Additionally, there continues to be a need for increased reliably and security in the context of wireless connectivity and communications. 
     BRIEF SUMMARY OF SOME EXAMPLES 
     According to some example embodiments, an apparatus is provided that leverages the use of chirp signals in communications. In this regard, the apparatus may comprise an antenna, a radio, and processing circuitry. The radio may be configured to transmit and receive wireless communications via the antenna. The processing circuitry may be configured to control the radio to establish a wireless communications link with a receiving communications device. In this regard, signaling transmitted by the antenna via the radio, as controlled by the processing circuitry, may comprise a plurality of sequenced chirp signals provided within an orthogonal frequency division multiplexing (OFDM) framework. 
     According to some example embodiments, the processing circuitry may be configured to control the radio to transmit the plurality of sequenced chirp signals. In this regard, a chirp signal of the plurality of sequenced chirp signals may comprise an up-chirp signal within OFDM symbols. The up-chirp signal comprises a signal with an increasing frequency with respect to time. 
     According to some example embodiments, the processing circuitry may be configured to control the radio to transmit the plurality of sequenced chirp signals. In this regard, a chirp signal of the plurality of chirp signals comprises a down-chirp signal within OFDM symbols. The down-chirp signal may comprise a signal with a decreasing frequency with respect to time. 
     According to some example embodiments, the processing circuitry may be configured to control the radio to transmit the plurality of sequenced chirp signals. In this regard, a chirp signal of the plurality of sequenced chirp signals may sweep across a frequency range from a first frequency to a second frequency within OFDM symbols. The first frequency may be lower than the second frequency or the first frequency may be higher than the second frequency. 
     According to some example embodiments, the processing circuitry may be configured to control the radio to transmit the plurality of sequenced chirp signals. In this regard, a chirp signal of the plurality of chirp signals may comprise an up-chirp signal or a down-chirp signal. The up-chirp signal may sweep across an range of frequencies with respect to time by increasing a frequency of the up-chirp signal, and the down-chirp signal may sweep across an range of frequencies with respect to time by decreasing a frequency of the down-chirp signal. 
     According to some example embodiments, the processing circuitry is configured to control the radio to transmit the plurality of sequenced chirp signals including a cyclic prefix via orthogonal chirp division multiplexing. According to some example embodiments, the processing circuitry may be configured to control the radio to maintain orthogonality by transmitting complementary sequences of chirp signals. 
     According to some example embodiments, the processing circuitry may be configured to control the radio to transmit the plurality of sequenced chirp signals. In this regard, the sequenced chirp signals may have a uniform, non-linear trajectory in time and frequency. 
     According to some example embodiments, the radio may comprise a power amplifier configured to provide amplification for transmission of the plurality of sequenced chirp signals and control instantaneous power fluctuations due to transmission of chirp signals associated with different frequencies within a band. 
     According to some example embodiments, the processing circuitry may be configured to control the radio to transmit the plurality of sequenced chirp signals via implementation of a Discrete Fourier Transform (DFT)-spreading scheme with OFDM and a frequency-domain spectral shaping (FDSS) filter that employs Bessel functions and Fresnel integrals used to develop the plurality of sequenced chirp signals as band-limited sinusoidal and/or linear chirp transmissions. 
     According to some example embodiments, the processing circuitry may be configured to control the radio to transmit the plurality of sequenced chirp signals. The plurality of sequenced chirp signals may be generated via chirp division multiplexing to encode trajectories of the chirp signals in frequency and time. 
     Additionally, according to some example embodiments, an example method is provided. The example method may comprise controlling, via processing circuitry, a radio to establish a wireless communications link via an antenna, and controlling the radio and the antenna to transmit communications as a plurality of sequenced chirp signals within an orthogonal frequency division multiplexing (OFDM) framework. 
     According to some example embodiments, the example method may further comprise controlling the radio to transmit the plurality of sequenced chirp signals, wherein a chirp signal of the plurality of sequenced chirp signals comprises an up-chirp signal within OFDM symbols or a down-chirp signal within OFDM symbols, wherein the up-chirp signal comprises a signal with an increasing frequency with respect to time and the down-chirp signal comprises a signal with a decreasing frequency with respect to time. 
     According to some example embodiments, the example method may further comprise controlling the radio to transmit the plurality of sequenced chirp signals within OFDM symbols. In this regard, a chirp signal of the plurality of chirp signals comprises an up-chirp signal or a down-chirp signal. The up-chirp signal may sweep across a range of frequencies with respect to time by increasing a frequency of the up-chirp signal, and the down-chirp signal may sweep across a range of frequencies with respect to time by decreasing a frequency of the down-chirp signal. 
     According to some example embodiments, the example method may further comprise controlling the radio to transmit the plurality of sequenced chirp signals including a cyclic prefix via orthogonal chirp division multiplexing. According to some example embodiments, the example method may further comprise controlling the radio to maintain orthogonality by transmitting complementary sequences of chirp signals. 
     According to some example embodiments, the example method may further comprise controlling the radio to transmit the plurality of sequenced chirp signals. In this regard, the sequenced chirp signals may have a uniform, non-linear trajectory in time and frequency. 
     According to some example embodiments, the example method may further comprise providing amplification via a power amplifier for transmission of the plurality of sequenced chirp signals and controlling instantaneous power fluctuations due to transmission of chirp signals associated with different frequencies within a band. 
     According to some example embodiments, the example method may further comprise controlling the radio to transmit the plurality of sequenced chirp signals via implementation of a Discrete Fourier Transform (DFT)-spreading scheme with OFDM and frequency-domain spectral shaping (FDSS) that employs Bessel functions and Fresnel intergrals used to develop the plurality of sequenced chirp signals as band-limited sinusoidal and/or chirp transmissions. 
     According to some example embodiments, the example method may further comprise controlling the radio to transmit the plurality of sequenced chirp signals, the plurality of sequenced chirp signals being generated via chirp division multiplexing to encode trajectories of the chirp signals in frequency and time. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S) 
       Having thus described some example embodiments in general terms, reference will now be made to the accompanying drawings, which are not necessarily drawn to scale, and wherein: 
         FIG.  1    illustrates an example communications system according to some example embodiments; 
         FIG.  2    illustrates an example communications device of a communications system according to some example embodiments; 
         FIG.  3    illustrates block diagram of an example transmitter according to some example embodiments; 
         FIG.  4    illustrates block diagram of an example receiver according to some example embodiments; 
         FIG.  5    is a graph of instantaneous power with respect to time for example chirp signals according to some example embodiments; 
         FIG.  6    is a spectrogram graph of frequency with respect to time for example chirp signals according to some example embodiments; 
         FIG.  7    is a graph of instantaneous power with respect to time for example chirp signals according to some example embodiments; 
         FIG.  8    is a spectrogram graph of frequency with respect to time for example chirp signals according to some example embodiments; 
         FIG.  9    is a graph of instantaneous power with respect to time for example chirp signals according to some example embodiments; 
         FIG.  10    is a spectrogram graph of frequency with respect to time for example chirp signals according to some example embodiments; 
         FIG.  11    is a graph of instantaneous power with respect to time for example chirp signals according to some example embodiments; 
         FIG.  12    is a spectrogram graph of frequency with respect to time for example chirp signals according to some example embodiments; 
         FIG.  13    is a graph of instantaneous power with respect to time for example chirp signals according to some example embodiments; 
         FIG.  14    is a spectrogram graph of frequency with respect to time for example chirp signals according to some example embodiments; 
         FIG.  15    is a graph of instantaneous power with respect to time for example chirp signals according to some example embodiments; 
         FIG.  16    is a spectrogram graph of frequency with respect to time for example chirp signals according to some example embodiments; 
         FIG.  17    is a graph of instantaneous power with respect to time for example chirp signals according to some example embodiments; 
         FIG.  18    is a spectrogram graph of frequency with respect to time for example chirp signals according to some example embodiments; 
         FIG.  19    illustrates an example transmitter according to some example embodiments; 
         FIGS.  20   a  and  20   b    are spectrogram graphs of frequency with respect to time for example chirp signals according to some example embodiments; 
         FIG.  21   a    is a spectrogram for plain DFT-s-OFDM pulses according to some example embodiments; 
         FIG.  21   b    is a spectrogram of linear chirps according to some example embodiments; 
         FIG.  21   c    is a spectrogram of sinusoidal chirps according to some example embodiments; 
         FIG.  21   d    is a graph of the amplitude of shaping coefficients for linear and sinusoidal chirp signals according to some example embodiments; 
         FIG.  21   e    is a graph of the power spectral density for linear and sinusoidal chirps according to some example embodiments; 
         FIG.  21   f    is a graph of peak-to-average power ratio distribution for linear and sinusoidal chirp signals according to some example embodiments; 
         FIG.  21   g    is a graph of coded bit error ratio for various chirp signals according to some example embodiments; 
         FIG.  21   h    is a graph of uncoded bit error ratio for various chirp signals according to some example embodiments; 
         FIG.  22    illustrates an example transmitter according to some example embodiments; 
         FIG.  23    illustrates a precoding matrix according to some example embodiments; 
         FIG.  24    illustrates an example transmitter according to some example embodiments; 
         FIG.  25    is a graph of tiles in frequency relative to tiles in time for a sinusoidal trajectory matrix according to some example embodiments; 
         FIG.  26    is a spectrogram graph of frequency relative to time for chirp signals according to some example embodiments; 
         FIG.  27    is a graph of tiles in frequency relative to tiles in time for another sinusoidal trajectory matrix according to some example embodiments; 
         FIG.  28    is a spectrogram graph of frequency relative to time for example chirp signals according to some example embodiments; 
         FIGS.  29 - 31    are spectrogram graphs of frequency relative to time for example chirp signals according to some example embodiments; 
         FIG.  32    is a graph of cross-correlations of example linear chirp signals according to some example embodiments; 
         FIG.  33    is a graph of instantaneous power with respect to time for example chirp signals according to some example embodiments; 
         FIG.  34    is a spectrogram graph of frequency with respect to time for example chirp signals according to some example embodiments; and 
         FIG.  35    is a flowchart of an example method for controlling a radio and antenna to conduct wireless communications involving chirp signals according to some example embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     Some example embodiments now will be described more fully hereinafter with reference to the accompanying drawings, in which some, but not all example embodiments are shown. Indeed, the examples described and pictured herein should not be construed as being limiting as to the scope, applicability, or configuration of the present disclosure. Rather, these example embodiments are provided so that this disclosure will satisfy applicable legal requirements. Like reference numerals refer to like elements throughout. 
     As used herein the term “or” is used as the logical or where any one or more of the operands being true results in the statement being true. As used herein, the phrase “based on” as used in, for example, “A is based on B” indicates that B is a factor that determines A, but B is not necessarily the only factor that determines A. 
     According to some example embodiments described herein, apparatuses, systems, and methods are provided that are configured to conduct wireless communications via reliable chirp transmissions and multiplexing, for example, within an orthogonal frequency division multiplexing (OFDM) framework. A chirp signal or transmission may be a signal where the frequency increases (up-chirp) or decreases (down-chirp) with respect to time and is, for example, provided within an OFDM symbol. In this regard, a chirp signal may be one that sweeps across a set frequency range. According to some example embodiments, such chirp transmissions and multiplexing may be implemented in a variety of contexts to conduct wireless communications. For example, such techniques may be employed in the context of interconnected devices that support the Internet of Things (IOT). Alternatively, such techniques may be employed when conducting wireless communications to and between vehicles (e.g., moving vehicles), such as land-based or aerial vehicles, to as part of an established communications link. According to some example embodiments, secure and reliable communications systems may be established that, for example, employ threshold levels of reliability using power-limited link budget devices. 
     Multi-chirp and/or multi-band communications may realize increased reliability and the chances of a signal not being received can be reduced. According to some example embodiments, the increased reliability of multi-chirp and/or multi-band-based communication systems may also operate to decrease the instantaneous power fluctuation at the transmitter through configured properties of the sequence signals that are being transmitted and DFT (discrete Fourier transform)-based waveforms that are defined and utilized. High instantaneous power fluctuation may reduce the power efficiency of a communication system, which leads to increased power demands and thus, in some instances, shortened battery life and smaller communication range for many power-limited devices, e.g., IoT devices, drones in aeronautical networks, or the like. As such, by exploiting time-frequency resources efficiently through orthogonal chirp division multiplexing with low-complexity discrete Fourier transform (DFT)-based operations, a robust communications system may be implemented that also supports increased communication link distance and battery life. 
     According to some example embodiments, chirp signals may be implemented in a manner where the signals sweep over a large spectrum while still being constant-envelope signals, thereby providing a significant robustness against non-linear distortions. As such, chirp signals may be applied in the context of, for example, 3GPP Fifth Generation (5G) New Radio (NR) and IEEE 802.11 Wi-Fi, to realize a number of advantages as described herein relating to applications involving short-range wireless sensing, simultaneous radar &amp; communications, and Internet-of-Things (IoT). However, according to some example embodiments, the physical layer of some communication systems may be based on orthogonal frequency division multiplexing (OFDM). According to some example embodiments, techniques for configuring communications devices to synthesize chirp signals within an OFDM framework is provided. 
     Chirp signals may, according to some example embodiments, involve the encoding bits as negative or positive slopes in the time-frequency (TF) plane. Chirp modulation can be implemented in the context of long-range air/ground communications in the high-frequency (HF) band. Further, according to some example embodiments, an orthogonal amplitude-variant linear chirp set may be defined where each chirp signal has a different chirp rate. Additionally, orthogonal chirps may be synthesized by introducing a term to the exponent of discrete Fourier transform (DFT) kernels. In this regard, chirp signals may be translated into the frequency domain and the signal bandwidth may increase with the number of chirps. To limit the bandwidth, additional up-sampling and filtering at baseband may be performed, which may fold the chirps in the frequency domain. A Fresnel transform and a fractional Fourier transform (FrFT) may be adopted to generate orthogonal chirp sets. Further, binary chirp spread spectrum (BCSS) signaling may be implemented. In this regard, non-linear chirps may be implemented to improve the bit-error ratio (BER) performance. Furthermore, the BER performance of quartic and linear chirps may be utilized in, for example, an empirical aeronautical channel model. Further, an iterative receiver may be used to improve the BER performance under frequency-selective fading channels and space-time coding schemes for orthogonal chirps. According to some example embodiments, a chirp spread spectrum (CSS) modulation, referred to as Long Range (LoRa), may be used in accordance with some example embodiments. 
     As described herein, chirp signals, according to some example embodiments, may be synthesized and employed within an OFDM framework. In this regard, for example, methods and associated apparatuses to generate chirp signals based on discrete Fourier transform-spread orthogonal frequency division multiplexing (DFT-s-OFDM) adopted in 3GPP 5G NR and 3GPP Long-Term Evolution (LTE) are provided. As such, a DFT spreading scheme may be employed that is uses OFDM and a frequency-domain spectral shaping (FDSS) filter, that, as further described herein, employs Bessel functions and Fresnel integrals to develop a plurality of sequenced chirp signals as band-limited sinusoidal or linear chirp transmissions. In this regard, such methods and associated apparatuses may rely on the design of an FDSS function applied after DFT spreading to convert single-carrier pulses (e.g., Dirichlet sinc functions) to a set of chirp signals translated uniformly in time. Such methods and associated apparatuses may operate to limit the bandwidth of the chirp signals, for example, without additional up-sampling and filtering at baseband. As the chirp signals are generated within the given a number of subcarrier bins by point-to-point multiplications, existing DFT-s-OFDM transceivers may be configured to modulate and demodulate chirp signals while supporting multiple users. 
     A number of example methods, and associated apparatuses and systems configured to implement the example methods, for implementing multiple chirp transmissions and multiplexing to realize secure and reliable communication systems are provided herein. Such example methods, apparatuses, and systems may be applied within the context of any type of communication system, such as those that require threshold levels of reliability or systems that require low-power communications (e.g., utilizing power-limited link budget devices). In this regard, example embodiments may be implemented in the context of a structural configuration of a baseband/radio frequency chipset of a radio communication device. Further, example embodiments may be configured to operate in accordance with various prescribed communication standards, such as, for example, 4G, LTE, 5G, IEEE 802.11 WLAN, 3GPP NR, Bluetooth, and the like. According to some example embodiments, the methods, apparatuses, and systems described herein may be implemented in accordance with Wi-Fi sensing under the IEEE 802.11 Wi-Fi standard. 
     According to some example embodiments, example techniques for reducing instantaneous power fluctuation are provided in accordance with, for example, a multiple chirp transmission. Such a chirp transmission may have an arbitrary uniform non-linear trajectory in time and frequency. Additionally, orthogonality may be maintained between the chirp signals by using complementary sequences. In this regard, according to some example embodiments, low peak-to-average power ratio (PAPR) coded orthogonal chirp division multiplexing may therefore be implemented. 
     Additionally or alternatively, according to some example embodiments, the trajectories of different chirps (chirp signals) may be encoded jointly. Such encoding may be performed to keep instantaneous power fluctuations of the signal low. Accordingly, low PAPR trajectory-coded chirp division multiplexing may therefore be implemented. 
     Additionally or alternatively, according to some example embodiments, the instantaneous power fluctuations may be controlled when multiple signals over largely separated bands are transmitted through the same power amplifier (e.g., multi-band operation). Such a power amplifier may be a power amplifier of a radio of a transmitting device. Accordingly, in this regard, a multi-band complementary sequence-based encoder for orthogonal frequency division multiplexing may be implemented. 
     Additionally or alternatively, according to some example embodiments, the spectral efficiency of a chirp-based communication system may be increased by utilizing a Discrete Fourier Transform (DFT)-based scheme. In this regard, Bessel functions may be used to develop band-limited sinusoidal chirp transmissions. As such, band-limited circular-time-shift-based chirp transmissions may be implemented. 
     Additionally or alternatively, according to some example embodiments, an orthogonal frequency division multiplexing-based chirp transmissions may be implemented, where the trajectories in time and frequency are encoded based on the information in time and frequency. According to some example embodiments, orthogonal resources may be generated based on the implementation of multiple DFT precoding operations by using a constant-amplitude sequence in the frequency domain. As such, multi-cluster-DFT-based precoding for chirp division multiplexing may be implemented. 
     In view of the chirp transmissions and multiplexing techniques provided above and as further described herein, example embodiments may be implemented in the context of a communications system  10  as shown in  FIG.  1   . The communications system  10  may include a complex system of intermediate devices that support communications between communications device  100  and communications device  200  to form a communications link  150 , or the communications device  100  and communications device  200  may have a direct communications link formed as link  150 , as shown in  FIG.  1   . In either case, the communications devices  100  and  200  may be configured to support wireless communications. 
     In this regard, the system  10  may include any number of communications devices, including communications devices  100  and  200 . Although not shown, the communications devices may be physically coupled to a stationary unit (e.g., a base station or the like) or a mobile unit (e.g., a mobile terminal such as a cellular phone, a vehicle such as an aerial vehicle, a smart device with IoT capabilities, or the like). 
     The communications device  100  may comprise, among other components, processing circuitry  101 , radio  110 , and an antenna  115 . As further described below, the processing circuitry  101  may be configured to control the radio  110  to transmit and receive wireless communications via the antenna  115 . In the regard, a communications link  150  which may include a wireless component may be established between the antenna  115  and the antenna  215  of the communications device  200 . Similarly, the communications device  200  may comprise, among other components, processing circuitry  201 , radio  210 , and the antenna  215 . The processing circuitry  201  may be configured the same or similar to the processing  101 , and thus maybe configured to control the radio  210  to transmit and receive wireless communications via the antenna  215 . As further described below, the configuration of the communications device  200  may be the same or similar to the configuration of the communications device  100  to support communications involving chirp transmissions and multiplexing as described herein. 
     In this regard,  FIG.  2    shows a more detailed version of the communications device  100 , and, in particular, the processing circuitry  101 . Again, shown in  FIG.  2   , the communications device  100  may comprise the processing circuitry  101 , the radio  110 , and the antenna  115 . However, the link  150  is shown as being a communications link to communications device  200 , or as a communications link to the network  120 , which may be any type of wired or wireless communications network. 
     The processing circuitry  101  may be configured to receive inputs and provide outputs in association with the various functionalities of the communications device  100 . In this regard, the processing circuitry  101  may comprise, for example, a memory  102 , a processor  103 , a user interface  104 , and a communications interface  105 . The processing circuitry  101  may be operably coupled to other components of the communications device  100  or other components of a device that comprises the communications device  100 . 
     Further, according to some example embodiments, processing circuitry  101  may be in operative communication with or embody, the memory  102 , the processor  103 , the user interface  104 , and the communications interface  105 . Through configuration and operation of the memory  102 , the processor  103 , the user interface  104 , and the communications interface  105 , the processing circuitry  101  may be configurable to perform various operations as described herein. In this regard, the processing circuitry  101  may be configured to perform computational processing, memory management, user interface control and monitoring, and manage remote communications, signal development and generation, according to an example embodiment. In some embodiments, the processing circuitry  101  may be embodied as a chip or chip set. In other words, the processing circuitry  101  may comprise one or more physical packages (e.g., chips) including materials, components or wires on a structural assembly (e.g., a baseboard). The processing circuitry  101  may be configured to receive inputs (e.g., via peripheral components), perform actions based on the inputs, and generate outputs (e.g., for provision to peripheral components). In an example embodiment, the processing circuitry  101  may include one or more instances of a processor  103 , associated circuitry, and memory  102 . As such, the processing circuitry  101  may be embodied as a circuit chip (e.g., an integrated circuit chip, such as a field programmable gate array (FPGA)) configured (e.g., with hardware, software or a combination of hardware and software) to perform operations described herein. 
     In an example embodiment, the memory  102  may include one or more non-transitory memory devices such as, for example, volatile or non-volatile memory that may be either fixed or removable. The memory  102  may be configured to store information, data, applications, instructions or the like for enabling, for example, the functionalities described with respect to chirp transmissions and multiplexing. The memory  102  may operate to buffer instructions and data during operation of the processing circuitry  101  to support higher-level functionalities, and may also be configured to store instructions for execution by the processing circuitry  101 . The memory  102  may also store signaling schemes and techniques as described herein. According to some example embodiments, such data may be generated based on other data and stored or the data may be retrieved via the communications interface  105  and stored. 
     As mentioned above, the processing circuitry  101  may be embodied in a number of different ways. For example, the processing circuitry  101  may be embodied as various processing means such as one or more processors  103  that may be in the form of a microprocessor or other processing element, a coprocessor, a controller or various other computing or processing devices including integrated circuits such as, for example, an ASIC (application specific integrated circuit), an FPGA, or the like. In an example embodiment, the processing circuitry  101  may be configured to execute instructions stored in the memory  102  or otherwise accessible to the processing circuitry  101 . As such, whether configured by hardware or by a combination of hardware and software, the processing circuitry  101  may represent an entity (e.g., physically embodied in circuitry—in the form of processing circuitry  101 ) capable of performing operations according to example embodiments while configured accordingly. Thus, for example, when the processing circuitry  101  is embodied as an ASIC, FPGA, or the like, the processing circuitry  101  may be specifically configured hardware for conducting the operations described herein. Alternatively, as another example, when the processing circuitry  101  is embodied as an executor of software instructions, the instructions may specifically configure the processing circuitry  101  to perform the operations described herein. 
     The communication interface  105  may include one or more interface mechanisms for enabling communication by controlling the radio  110  to generate the communications link  150 . In some cases, the communication interface  105  may be any means such as a device or circuitry embodied in either hardware, or a combination of hardware and software that is configured to receive or transmit data from/to devices in communication with the processing circuitry  101 . The communications interface  105  may support wireless communications via the radio  110  using various communications protocols (802.11 WIFI, Bluetooth, cellular, WLAN, 3GPP NR, 4G, LTE, 5G, and the like or the like). 
     The user interface  104  may be controlled by the processing circuitry  101  to interact with peripheral devices that can receive inputs from a user or provide outputs to a user. In this regard, via the user interface  104 , the processing circuitry  101  may be configured to provide control and output signals to a peripheral device such as, for example, a keyboard, a display (e.g., a touch screen display), mouse, microphone, speaker, or the like. The user interface  104  may also produce outputs, for example, as visual outputs on a display, audio outputs via a speaker, or the like. 
     The radio  110  may be any type of physical radio comprising radio components. For example, the radio  110  may include components such as a power amplifier  112 , mixer, local oscillator, modulator/demodulator, and the like. The components of the radio  110  may be configured to operate in a plurality of spectral bands to support the transmission and receipt of chirp signals. Further, the radio  110  may be configured to receive signals from the processing circuitry  101  for transmission to the antenna  115 . In some example embodiments, the radio  110  may be a software-defined radio or a hybrid software/hardware-defined radio. 
     The antenna  115  may be any type of wireless communications antenna. The antenna  115  may be a configured to be controlled to transmit and receive at more than one frequency or band. In this regard, according to some example embodiments, the antenna  115  may be an array of antennas that may be configured by the radio  115  to support various types of wireless communications as described herein. 
     Having described aspects of the components of communications system  10 , the following describes the implementation of communications involving chirp transmissions and multiplexing via specific configurations of the processing circuitry  101  to control the radio  110  and the antenna  115 . In this regard, an example way of increasing the reliability and security of communication systems may be to establish the communication link  150  over multiple bands, e.g., L-Band (1-2 GHz) and C-Band (4-8 GHz) for aeronautical networks. However, simultaneous use of multiple bands can cause large instantaneous power fluctuations at the input of power amplifier  112  of the radio  110  due to the constructive or destructive additions of the signals on different bands. As a result of such large power fluctuations, the power efficiency of the overall communication system may be reduced, which may lead to a shortened battery life and smaller communication range for many power-limited devices, e.g., IoT devices, aerial drones, and the like. Attempts to fluctuate the instantaneous signal power by using, for example, traditional single-carrier waveforms or constant-amplitude schemes like minimum-shift keying (MSK) signals for each band have been shown to be ineffective in many cases because the superposition of these signals may not ensure low instantaneous power fluctuations. 
     According to some example embodiments, when multiple chirp signals are transmitted simultaneously, similar instantaneous power fluctuations can result when supporting multi-user communications or single-user designs for high data rate. While in isolation each individual chirp signal has no instantaneous power fluctuations, the superposition of the chirp signals can cause arbitrary power fluctuations. Such power fluctuations may require a reduction in the power output to the power amplifier  112  of the radio  110  to avoid saturation at the power amplifier  112 . If the power is not reduced (e.g., no power back-off), the power amplifier  112  may distort the signal and cause interference to any adjacent channels due to the spectral regrowth. Hence, the need to implement such power back-offs can offset the benefits of the chirp transmission, e.g., immunity against interference and time-varying multipath channel distortions. In other words, well-known power-efficient modulation schemes, which may be optimal for single band/channel, may lose their advantages in the context of multi-band/multi-channel transmission. 
     However, as described herein, some example embodiments provide new multi-channel power-efficient modulation schemes, for use by communications devices such as communications device  100 , that overcome these and other limitations of the prior art. According to various example embodiments, increased battery life and increased communication link distance can be realized while maintaining low-complexity operations at transmitter and receiver. 
     In this regard, example embodiments for implementing orthogonal chirp multiplexing in the context of, for example, low-complexity transmitters/receivers is provided. To do so, according to some example embodiments, a complementary sequence-based chirp spread spectrum approach may be implemented using coded orthogonal chirp division multiplexing, trajectory encoding with multi-DFT clusters and shift encoders. The implementation of such chirp transmissions and multiplexing as described herein may be applied in, for example power-limited communication devices such as internet-of-things (IoT) devices and permit the devices to transmit more information bits or transmit signals at further distances. Additionally, as described herein the implementation of chirp transmissions and multiplexing may also be applied in other communications contexts such as with aerial communications, including aerial drone communications, where again limitations on power usage may be considered but longer transmission distances may be required. 
     As such, the processing circuitry  101  of the communications device  100  may be configured to control the radio  110  and the antenna  115  to communicate via the implementation of complementary sequences for chirp transmissions as further described below. In this regard, a polynomial representation of a sequence a=(a 0 , a 1 , . . . , a N-1 ) is given by:
 
 p   a ( z )= a   N-1   z   N-1   +a   N-2   z   N-2   + . . . a   0   (1)
 
     Based on the polynomial representation, the following interpretations can be made. In this regard, 
               z   ∈     {         e       j   ⁢           ⁢   2   ⁢   π   ⁢           ⁢   t     T       ⁢     |     ⁢   0     ≤   t   &lt;   T     }       ,         
p a (z) may be equivalent to an OFDM signal in time where T is the OFDM symbol duration and the frequency domain coefficients are the elements of a where a 0  is mapped to the DC tone. Further,
 
               z   ∈     {         e       j   ⁢           ⁢   2   ⁢   π   ⁢           ⁢   t     T       ⁢     |     ⁢   0     ≤   t   &lt;   T     }       ,         
the instantaneous power of an OFDM symbol, can be calculated as |p a (z)| 2 =p a (z)p a ·(z −1 ) as p a ·(z −1 )=(p a (z))*. Thus, the peak-to-average-power ratio (PAPR) of p a (z) can be obtained by using |p a (z)| 2  within a period of
 
             z   =     e       j   ⁢           ⁢   2   ⁢   π   ⁢           ⁢   t     T             
where t=[0, T). To represent a sequence, let ƒ be a function that maps from    2   m ={(x 1 , x 2 , . . . , x m )|x i ∈{0,1}} to   as
 
ƒ( x   1   ,x   2   , . . . ,x   m ):   2   m →   (2)
 
     A sequence ƒ of length 2 m  may be associated with the function ƒ(x 1 , x 2 , . . . , x m ) by listing its values as (x 1 , x 2 , . . . , x m ) ranges over its 2 m  values in lexicographic order. In other words, the (x+1)th element of the sequence ƒ may be equal to ƒ(x 1 , x 2 , . . . , x m ) where x=Σ j=1   m x j 2 m-j  (i.e., the most significant bit is x 1 ). The sequence x and ƒ(x) may denote (x 1 , x 2 , . . . , x m ) and ƒ(x 1 , x 2 , . . . , x m ), respectively. Note that  =   2 , ƒ(x) may be a Boolean function, and ƒ =   H , ƒ(x) may be called a generalized Boolean function. 
     With respect an algebraic normal form (ANF), a generalized Boolean function can be uniquely expressed as a linear combination over    H  of the monomials as 
                     f   ⁡     (       x   1     ,     x   2     ,   …   ⁢           ,     x   m       )       =       f   ⁡     (   x   )       =         ∑     k   =   0         2   m     -   1       ⁢           ⁢       c   k     ⁢         ∏     j   =   1     m     ⁢           ⁢     x   j     k   j           ︸             ⁢     ith   ⁢           ⁢   monomial               =         c   0     ⁢   1     +         c   1     ⁡     (     x   1     )       2     +         c   2     ⁡     (     x   2     )       2     +     …   ⁢           ⁢         c     m   +   1       ⁡     (       x   1     ⁢     x   2       )       2       +   …                 (   3   )               
where the coefficient of each monomial belongs to    H  i.e., c k ∈   H  and k=Σ j=1   m k j 2 m-j  and x j ∈   2 . Note that monomials, e.g., 1, x 1 , x 2 , x 1 x 2 , . . . , and x 1 x 2  . . . x m  may be linearly independent. Linear independence can be proven by using the definition of linear independence, i.e., Σ i a i x i =0 if and only if a i =0 for all x.
 
     For example, let m=3 and H=4. Then
 
ƒ( x   1   ,x   2   ,x   3 )= c   0   x   1   0   x   2   0   x   3   0   +c   1   x   1   0   x   2   0   x   3   1   +c   2   x   1   0   x   2   1   x   3   0   +c   3   x   1   0   x   2   1   x   3   1   +c   4   x   1   1   x   2   0   x   3   0   +c   5   x   1   1   x   2   0   x   3   1   +c   6   x   1   1   x   2   1   x   3   0   +c   7   x   1   1   x   2   1   x   3   1 .  (4)
 
Assume that c 0 =3 and c 5 =2 and other c n =0 for n=1, 2, 3, 4, 6, 7. Then,
 
ƒ( x   1   ,x   2   ,x   3 )=3+2 x   1   x   3 .  (5)
 
     As described above, a sequence ƒ of length 2 m  may be associated with the function ƒ(x 1 , x 2 , . . . , x m ) by listing its values as (x 1 , x 2 , . . . , x m ) ranges over its 2 m  values in lexicographic order. In other words, the (x+1)th element of the sequence ƒ is equal to ƒ(x 1 , x 2 , . . . , x m ) where x=Σ j=1   m x j 2 m-j  (i.e., the most significant bit is x 1 )
 
ƒ( x   1 =0, x   2 =0, x   3 =0)=3+2 x   1   x   3 =3 mod 4=3
 
ƒ( x   1 =0, x   2 =0, x   3 =1)=3+2 x   1   x   3 =3 mod 4=3
 
ƒ( x   1 =0, x   2 =1, x   3 =0)=3+2 x   1   x   3 =3 mod 4=3
 
ƒ( x   1 =0, x   2 =1, x   3 =1)=3+2 x   1   x   3 =3 mod 4=3
 
ƒ( x   1 =1, x   2 =0, x   3 =0)=3+2 x   1   x   3 =3 mod 4=3
 
ƒ( x   1 =1, x   2 =0, x   3 =1)=3+2 x   1   x   3 =5 mod 4=1
 
ƒ( x   1 =1, x   2 =1, x   3 =0)=3+2 x   1   x   3 =3 mod 4=3
 
ƒ( x   1 =1, x   2 =1, x   3 =1)=3+2 x   1   x   3 =5 mod 4=1
 
     Therefore, ƒ(x 1 , x 2 , x 3 )=3+2x 1 x 3  leads to a sequence of ƒ=(3,3,3,3,3,1,3,1). All the possible monomials construct a basis for the generalized Boolean functions. Since there are 2 m  monomials for a given m, there are H 2     m    different generalized Boolean functions, each of which is a mapping  →   H . If ƒ(x 1 , x 2 , . . . , x m ) is over  , the coefficient of each monomial belongs to  , i.e., c k ∈  and the monomials construct a vector space over   and the dimensionality of the space is 2 m . Therefore, different sets of {c k |k=0, . . . , 2 m-1 } lead to different sequences. 
     Now with respect to aperiodic auto correlation (APAC) of the sequence, ρ a (k) may be the aperiodic autocorrelation of a complex sequence a of length N and ρ a (k) is expressed as 
     
       
         
           
             
               
                 
                   
                     
                       ρ 
                       a 
                     
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   ⁢ 
                   
                     = 
                     Δ 
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   ρ 
                                   a 
                                   + 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     ρ 
                                     a 
                                     + 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       - 
                                       k 
                                     
                                     ) 
                                   
                                 
                                 * 
                               
                               , 
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               k 
                               ≥ 
                               0 
                             
                           
                         
                         
                           
                             
                               k 
                               &lt; 
                               0 
                             
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       where 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ρ 
                       a 
                       + 
                     
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   ⁢ 
                   
                     = 
                     Δ 
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     a 
                                     i 
                                     * 
                                   
                                   ⁢ 
                                   
                                     a 
                                     
                                       i 
                                       + 
                                       k 
                                     
                                   
                                 
                               
                               , 
                             
                           
                           
                             
                               0 
                               ≤ 
                               k 
                               ≤ 
                               
                                 N 
                                 - 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               0 
                               , 
                             
                           
                           
                             otherwise 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     To generate the complementary sequences, a Golay complementary pair and a Golay sequence may be considered. In this regard, the pair of (a,b) is called a Golay complementary pair (GCP) if
 
ρ a ( k )+ρ b ( k )=0,  k≠ 0.  (8)
 
The sequence a=(a 0 , a 1 , . . . , a N-1 ) may be defined as a complementary sequence (CS) if there exists another sequence b=(b 0 , b 1 , . . . , b N-1 ) which complements a as ρ a (k)+ρ b (k)=0, k≠0. Additionally, the PAPR of a CS can be less than 3 dB.
 
     With respect to a complementary sequence encoder, the following theorem for constructing complementary sequences is provided. Theorem: Let wC denote any permutation {1, 2, . . . , m} and (a,b) be a Golay complementary pair (GCP) of length N and calculate 
                       f   o     ⁡     (     x   ,   z     )       =     (           p   a     ⁡     (   z   )       ⁢     (     1   -     x     π   ⁡     (   1   )           )       +         p   b     ⁡     (   z   )       ⁢     x     π   ⁡     (   1   )             )             (   9   )                   f   r     ⁡     (   x   )       =       e   0     +       e   m     ⁢     x     π   ⁡     (   m   )           +       ∑     l   =   1       m   -   1       ⁢           ⁢       e   l     ⁡     (       x     π   ⁡     (   l   )         +     x     π   ⁡     (     l   +   1     )           )                   (   10   )                   f   i     ⁡     (   x   )       =       k   0     +       ∑     l   =   1     m     ⁢           ⁢       k   l     ⁢     x     π   ⁡     (   l   )             +       H   2     ⁢       ∑     l   =   1       m   -   1       ⁢           ⁢       x     π   ⁡     (   l   )         ⁢     x     π   ⁡     (     l   +   1     )                         (   11   )                   f   s     ⁡     (   x   )       =       ∑     n   =   1     m     ⁢           ⁢       d   n     ⁢     x     π   ⁡     (   n   )                     (   12   )               
where x=(x 1 , x 2 , . . . x m ) and x=Σ j   2     m   x j 2 m-j  for x j ∈   2 , e n ∈ , k n ∈[0, H), d n ∈  for n=0, 1, . . . , m. Then, the sequence c where its polynomial representation is given by
 
                       p   c     ⁡     (   z   )       =       ∑     x   =   0         2   m     -   1       ⁢           ⁢         f   o     ⁡     (     x   ,   z     )       ×     e         2   ⁢   π     H     ⁢     (         f   r     ⁡     (   x   )       +       jf   i     ⁡     (   x   )         )         ×     z         f   s     ⁡     (   x   )       +   xN                   (   13   )               
is a complementary sequence (CS).
 
     The polynomial p c (z) forms an OFDM symbol for 
             z   =     e       2   ⁢   π   ⁢   t     T             
and limits the peak-to-average-power ratio to be less than or equal to 2 (i.e., approximately 3 dB) as the sequence c is a CS. Additionally, the parameters, i.e., e n , k n , d n  can be selected, for example, based on information bits and demonstrations for random bit mappings.
 
     Now, with respect to a first example embodiment that involves the implementation of low-PAPR coded orthogonal chirp division multiplexing, M information bits may be mapped to e n , k n , d n ∈  for n=0, 1, 2, . . . , m. The calculated parameters may be processed by the amplitude, phase, and shift encoders as 
                       f   r     ⁡     (   x   )       =       e   0     +       e   m     ⁢     x     π   ⁡     (   m   )           +       ∑     l   =   1       m   -   1       ⁢       e   l     ⁡     (       x     π   ⁡     (   l   )         +     x     π   ⁡     (     l   +   1     )           )                   (   14   )                   f   i     ⁡     (   x   )       =       k   0     +       ∑                         l   =   1         m     ⁢       k   l     ⁢     x     π   ⁡     (   l   )                       (   15   )                   f   s     ⁡     (   x   )       =       d   0     +         ∑   m       n   =   1       ⁢       d   n     ⁢     x     π   ⁡     (   n   )                       (   16   )               
where x=(x 1 , x 2 , . . . x m ) and x=Σ j   2     m   x j 2 m-j  for x j ∈   2 , w denotes any permutation {1, 2, . . . , m}, and (a,b) be a GCP of length N. A multiple chirp waveform can be expressed as
 
                       p   c     ⁡     (   t   )       =       ∑     x   =   0         2   m     -   1       ⁢           ⁢         f   o     (     x   ,     e       j   ⁢           ⁢   2   ⁢   π   ⁢           ⁢   t     T         )     ×     e     j   ⁢           ⁢   π   ⁢           ⁢       f   sign     ⁡     (   x   )           ×     e       α   ⁢           ⁢       f   r     ⁡     (   x   )         +     j   ⁢           ⁢   β   ⁢           ⁢       f   i     ⁡     (   x   )             ×     e       ϕ   ⁡     (   t   )         ︸       j   ⁢           ⁢   2   ⁢   π   ⁢           ⁢     t   T     ⁢     (         f   s     ⁡     (   x   )       +   xN     )       +     j   ⁢           ⁢   2   ⁢   π   ⁢           ⁢   ψ   ⁢           ⁢     (   t   )                           (   17   )               
where ƒ o (x,z)=(p a (z)(1−x π(1) )+p b (z)x π(1) ) and ƒ sign (x)=Σ l=1   m-1 x π(l) x π(l+1) , α and β are non-zero values to scale the output of encoders, and ψ(t) is a function that determines the trajectory of the chirps in time and frequency. Note that p a (z) and p b (z) are the polynomial representation of the sequence a and the sequence b form a GCP. Since the complex exponential term in p c (t) is a function of ψ(t), for a given x, each
 
                 f   0     ⁡     (     x   ,     e       j   ⁢   2   ⁢   π   ⁢   t     T         )       ×     e         j   ⁢   π   ⁢   f     sign     ⁡     (   x   )         ×     e       α   ⁢       f   r     ⁡     (   x   )         +         j   ⁢   β   ⁢   f     i     ⁡     (   x   )                 
changes its location in time and frequency depending on ψ(t). The corresponding trajectory in time-frequency for
 
                   f   0     ⁡     (     x   ,     e       j   ⁢     2   ⁢   π     ⁢   t     T         )       ×     e         j   ⁢   π   ⁢   f     sign     ⁡     (   x   )         ×     e       α   ⁢       f   r     ⁡     (   x   )         +         j   ⁢   β   ⁢   f     i     ⁡     (   x   )             ,     i   .   e   .     ,       F   x     ⁡     (   t   )       ,         
can be obtained by calculating the instantaneous frequency as
 
     
       
         
           
             
               
                 
                   
                     
                       F 
                       x 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         1 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           d 
                           ⁢ 
                           
                             ϕ 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         
                           d 
                           ⁢ 
                           t 
                         
                       
                     
                     = 
                     
                       
                         
                           1 
                           T 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 f 
                                 s 
                               
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             + 
                             
                               x 
                               ⁢ 
                               N 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           d 
                           ⁢ 
                           
                             ψ 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         
                           d 
                           ⁢ 
                           t 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     For example, if 
                 ψ   ⁡     (   t   )       =       M   n     ⁢       (     t   T     )     n         ,         
with
 
             B   ⁢     =   ^     ⁢     M   /   T           
and T real constants representing bandwidth and symbol duration and n is a non-negative integer, respectively, the corresponding trajectory can be calculated as
 
     
       
         
           
             
               
                 
                   
                     
                       
                         d 
                         ⁢ 
                         
                           ψ 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       
                         d 
                         ⁢ 
                         t 
                       
                     
                     = 
                     
                       
                         
                           M 
                           ⁢ 
                           
                             
                               t 
                               
                                 n 
                                 - 
                                 1 
                               
                             
                             
                               T 
                               n 
                             
                           
                         
                         ⇒ 
                         
                           
                             F 
                             x 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       = 
                       
                         
                           1 
                           T 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 f 
                                 s 
                               
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             + 
                             
                               x 
                               ⁢ 
                               N 
                             
                             + 
                             
                               
                                 M 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     T 
                                   
                                   ) 
                                 
                               
                               
                                 n 
                                 - 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   . 
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     Therefor, when t=T, the maximum deviation from the initial frequency, i.e. 
                     f   s     ⁡     (   x   )       +     x   ⁢   N       T     ,         
will be
 
               B   =     M   T       ⁢     Hz   .           
Since, in some example embodiments, every time instant of the generated signal originates from a CS, the PAPR of the generated signal may, according to some example embodiments, also be less than or equal to 2 (i.e., approximately 3 dB).
 
     Additionally several non-linear trajectories may be defined by changing ψ(t) for uncoded multi-user chirp transmission while the orthogonality between the chirp signals may be lost. However, it is worth noting that the basis functions in the expression for p c (t), i.e., 
                 B   k     ⁡     (   t   )       =     e       2   ⁢   π   ⁢     t   T     ⁢   k     +     j   ⁢   2   ⁢     πψ   ⁡     (   t   )                   
for k∈  may form a complete set of orthogonal functions for any ψ(t) that is not a function of k. This can be proven by using inner-product of B k (t) and B n (t) as
 
                           〈         B   k     ⁡     (   t   )       |       B   n     ⁡     (   t   )         〉     ⁢     =   ^     ⁢         ∫   0   T     ⁢         B   k     ⁡     (   t   )       ⁢       B   n   *     ⁡     (   t   )       ⁢   dt       =       ∫   0   T     ⁢       e       2   ⁢   π   ⁢     t   T     ⁢   k     +       j   ⁢   2πψ     (   t   )         ⁢     e         -   2     ⁢   π   ⁢     t   T     ⁢   n     -       j   ⁢   2πψ     (   t   )         ⁢   dt                     =         ∫   0   T     ⁢       e         j   ⁢     2   ⁢   π     ⁢   t     T     ⁢     (     k   -   n     )         ⁢   dt       =     δ   nk                     (   20   )               
where δ nk  is Kronecker delta function. Hence, ψ(t) may be flexibly chosen depending on the application without losing the orthogonality of the basis functions in this embodiment. For example, it may a sinusoidal, linear, nth-order polynomial.
 
     In view of this approach, a transmitter  300  may be defined as shown in  FIG.  3   , which may be implemented by the processing circuitry  101  of the communications device  100 . In this regard, the transmitter  300  may be configured to implement low-PAPR coded orthogonal chirp division multiplexing as provided herein. The transmitter  300  may include a bit mapper  310  configured to map bits received by the bit mapper  310  to the amplitude encoder  312 , the phase encoder  314 , and the shift encoder  316 . A sign control  318  may also operate as an input. The outputs of these various components, as shown in the  FIG.  3   , may be mixed and ordered to generate a polynomial representation of the complementary sequence. In this example, s n  is determined by the ordering block that calculates ƒ o (x,z)=(p a (z)(1−x π(1) )+p b (z)x π(1) ). Hence, the polynomial representation of the sequence s n  is p s     n   (z)=(p a (z)(1−x π(1) )+p b (z)x π(1) ), where z=ej2πt/T. 
     Accordingly,  FIG.  4    illustrates a receiver  400  that may be implemented by the processing circuitry  101  of the communications device  100 . In this regard, the receiver  400  may support receipt of communications in the context of low-PAPR coded orthogonal chirp division multiplexing. In this regard, the receiver  400  may be configured to exploit the orthogonality of the basis functions. The receiver  400  may be configured to calculate K inner-product operations with the received signal and each basis function B n (t) for n=0, . . . , K−1, where K is the total number of possible chirps. Then, the receiver  400  may be configured to estimate D 0 , D 1 , . . . , D 2     m-1    and decode e n , k n , d n ∈  for n=0, 1, . . . , m and calculate the received bits. 
     For the transmitter  300 , the baseband signal may be generated through the samples of the waveform p c (t) and a fast Fourier transform (FFT)-based implementation may be used. 
     The following provides some example implementations of sequences for chirp transmissions and multiplexing as described above and with respect to  FIGS.  3  and  4   . In one example embodiment, the sample rate of the system  10  may be ƒ s =80 MHz and where there are N s =1024 samples in the signal. Therefore, the symbol duration 
             T   =         N   s       f   s       =     1   ⁢     2   .   8     ⁢   μ   ⁢     s   .               
Let m=3, d 0 =−300, d 1 =50, d 2 =200, d 3 =300, β=2π/4, π=0 (i.e., no amplitude encoder), k 0 =1, k 1 =0, k 2 =1, k 3 =1, π=(3,2,1), and (a,b)=(1,1). Therefore, the shift encoder leads to (D 0 , D 1 , . . . , D 7 )=(−300, −250, −100, −50, 0, 50, 200, 250). The frequency deviation may be 10 MHz, with the following functions ψ(t)
 
     
       
         
           
             
               ψ 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   B 
                   n 
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       t 
                       T 
                     
                     ) 
                   
                   n 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 where 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 n 
               
               = 
               
                 
                   2 
                   → 
                   
                     Instantaneous 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     frequency 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         F 
                         x 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
                 = 
                 
                   
                     1 
                     T 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         D 
                         x 
                       
                       + 
                       xN 
                       + 
                       
                         
                           M 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               T 
                             
                             ) 
                           
                         
                         1 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               ψ 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   B 
                   n 
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       t 
                       T 
                     
                     ) 
                   
                   n 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 where 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 n 
               
               = 
               
                 
                   4 
                   → 
                   
                     Instantaneous 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     frequency 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         F 
                         x 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
                 = 
                 
                   
                     1 
                     T 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         D 
                         x 
                       
                       + 
                       xN 
                       + 
                       
                         
                           M 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               T 
                             
                             ) 
                           
                         
                         3 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               ψ 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   B 
                   n 
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       t 
                       T 
                     
                     ) 
                   
                   n 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 where 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 n 
               
               = 
               
                 
                   11 
                   → 
                   
                     Instantaneous 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     frequency 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         F 
                         x 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
                 = 
                 
                   
                     1 
                     T 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         D 
                         x 
                       
                       + 
                       xN 
                       + 
                       
                         
                           M 
                           ⁡ 
                           
                             ( 
                             
                               t 
                               T 
                             
                             ) 
                           
                         
                         10 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Since the frequency deviation is 
                 B   =     M   T       ⁢   Hz     ,         
M is obtained as M=10 MHz×12.8 μs=128. In  FIGS.  5 - 10   , the instantaneous signal power and trajectories for polynomial ψ(t) for different orders are illustrated. In all cases, the orthogonality of the basis functions is retained and the PAPR of the signal is less than or equal to 3 dB. In this regard, the graph  500  of  FIG.  5    is the instantaneous power with respect to time where n=2 and the graph  600  of  FIG.  6    is the frequency with respect to time where n=2. The graph  700  of  FIG.  7    is the instantaneous power with respect to time where n=4 and the graph  800  of  FIG.  8    is the frequency with respect to time where n=4. Further, the graph  900  of  FIG.  9    is the instantaneous power with respect to time where n=11 and the graph  1000  of  FIG.  10    is the frequency with respect to time where n=11.
 
     Further in this regard, it may be considered that 
                 ψ   ⁡     (   t   )       =       M     4   ⁢   π       ⁢     sin   ⁡     (     2   ⁢   π   ⁢     t   T       )           .         
As such, the instantaneous frequency for the xth component can be calculated as
 
     
       
         
           
             
               
                 
                   
                     
                       F 
                       x 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         1 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           d 
                           ⁢ 
                           
                             ϕ 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         
                           d 
                           ⁢ 
                           t 
                         
                       
                     
                     = 
                     
                       
                         1 
                         T 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             ( 
                             
                               
                                 
                                   f 
                                   s 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               + 
                               
                                 x 
                                 ⁢ 
                                 N 
                               
                             
                             ) 
                           
                           + 
                           
                             
                               M 
                               2 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     t 
                                     T 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Therefore, the minimum and the maximum instantaneous signal frequency can be calculated as 
               1   T     ⁢     (       (         f   s     (   x   )     +     x   ⁢   N       )     -     M   2       )     ⁢         and   ⁢           1   T     ⁢       (       (         f   s     (   x   )     +     x   ⁢   N       )     +     M   2       )     .           
Further, the occupied bandwidth may be
 
               M   T     ⁢     Hz   .           
As such, for 5 MHz bandwidth, M=64; for 10 MHz bandwidth, M=128; for 10 MHz bandwidth, M=128; and for 15 MHz bandwidth, M=192.
 
     In  FIGS.  11 - 16   , the instantaneous signal power and trajectories for a sinusoidal ψ(t) for different frequency bandwidths of each chirp signal are illustrated. According to some example embodiments, in all cases, the orthogonality of the basis functions is kept and the PAPR of the signal is less than or equal to 3 dB. 
     In this regard, the graph  1100  of  FIG.  11    is the instantaneous power with respect to time where the bandwidth is 5 MHz and the graph  1200  of  FIG.  12    is the frequency with respect to time where the bandwidth is 5 MHz. The graph  1300  of  FIG.  13    is the instantaneous power with respect to time where the bandwidth is 10 MHz and the graph  1400  of  FIG.  14    is the frequency with respect to time where the bandwidth is 10 MHz. Further, the graph  1500  of  FIG.  15    is the instantaneous power with respect to time where the bandwidth is 15 MHz and the graph  1600  of  FIG.  16    is the frequency with respect to time where the bandwidth is 15 MHz. 
     Additionally or alternatively, with respect to a second example embodiment, a low-PAPR trajectory-coded chirp division multiplexing technique may be implemented by the configured processing circuitry. Low-PAPR trajectory-coded chirp division multiplexing may be implemented by the processing circuitry  101  of the communications device  100 . In this regard, according to some example embodiments, M information bits may be mapped to e n , k n , d n (t)∈  for n=0, 1, . . . , m. The calculated parameters may be processed by the amplitude, phase, and shift encoders as 
     
       
         
           
             
               
                 
                   
                     
                       f 
                       r 
                     
                     ( 
                     ϰ 
                     ) 
                   
                   = 
                   
                     
                       e 
                       0 
                     
                     + 
                     
                       
                         e 
                         m 
                       
                       ⁢ 
                       
                         ϰ 
                         
                           π 
                           ⁡ 
                           ( 
                           m 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           1 
                         
                         
                           m 
                           - 
                           1 
                         
                       
                         
                       
                         
                           e 
                           l 
                         
                         ( 
                         
                           
                             ϰ 
                             
                               π 
                               ⁡ 
                               ( 
                               l 
                               ) 
                             
                           
                           + 
                           
                             ϰ 
                             
                               π 
                               ⁡ 
                               ( 
                               
                                 l 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       f 
                       i 
                     
                     ( 
                     ϰ 
                     ) 
                   
                   = 
                   
                     
                       k 
                       0 
                     
                     + 
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           1 
                         
                         m 
                       
                         
                       
                         
                           k 
                           l 
                         
                         ⁢ 
                         
                           ϰ 
                           
                             π 
                             ⁡ 
                             ( 
                             l 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       f 
                       s 
                     
                     ( 
                     
                       ϰ 
                       , 
                       t 
                     
                     ) 
                   
                   = 
                   
                     
                       
                         d 
                         0 
                       
                       ( 
                       t 
                       ) 
                     
                     + 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         m 
                       
                         
                       
                         
                           
                             d 
                             n 
                           
                           ( 
                           t 
                           ) 
                         
                         ⁢ 
                         
                           ϰ 
                           
                             π 
                             ⁡ 
                             ( 
                             n 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     where x=(x 1 , x 2 , . . . x m ) and x=Σ j   2m x j 2 m-j  for x j ∈   2 , π denotes any permutation {1, 2, . . . , m}, and (a,b) be a GCP of length N. A multiple chirp waveform which can be expressed as 
                       p   c     (   t   )     =       ∑     ϰ   =   0         2   m     -   1                 f   o     (   ϰ   )     ×     e       jπf   sign     (   ϰ   )       ×     e       α   ⁢       f   r     (   ϰ   )       +     j   ⁢   β   ⁢       f   i     (   ϰ   )           ×     e     j   ⁢         2   ⁢   π   ⁢     t   T     ⁢     (         f   s     (     ϰ   ,   t     )     +   ϰN     )       )       ︸     ϕ   ⁡   (   t   )                         (   25   )               
where ƒ o (x,z)=(p a (z)(1−x π(1) )+p b (z)x π(1) ) and ƒ sign (x)=Σ l=1   m-1 x π(l) x π(l+1) , α and β are non-zero values to scale the output of encoders and ψ(t,ƒ s (x)) is a function that determines the trajectory of the chirps in time and frequency. Note that p a (z) and p b (z) are the polynomial representation of the sequence a and the sequence b, and the sequence a and the sequence b form a GCP. The corresponding trajectory for the xth component can be calculated from the angle of the complex exponential as
 
     
       
         
           
             
               
                 
                   
                     
                       F 
                       ϰ 
                     
                     ( 
                     t 
                     ) 
                   
                   = 
                   
                     
                       
                         1 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           d 
                           ⁢ 
                           
                             ϕ 
                             ⁡ 
                             ( 
                             t 
                             ) 
                           
                         
                         dt 
                       
                     
                     = 
                     
                       
                         1 
                         T 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               ϰ 
                               ⁢ 
                               N 
                             
                             + 
                             
                               
                                 
                                   df 
                                   s 
                                 
                                 ( 
                                 
                                   ϰ 
                                   , 
                                   t 
                                 
                                 ) 
                               
                               dt 
                             
                           
                           ) 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     In this case, the orthogonality between basis function may not be guaranteed. However, the PAPR of the signal may still be less than or equal to 3 dB because of the structure of the encoders. As a difference as compared to the approach provided above in association with  FIGS.  3  and  4   , the values of d n (t)∈  may be functions of time and these values may jointly determine the final trajectory. Further, the transmit bits may encode the trajectories. According to some example embodiments, the information bits may encode the slope of the parameters d n (t)∈  for n=0, 1, . . . , m. In some example embodiments, the information bits may encode the order of the polynomials. As such, the approach described here may be used for increasing the security of the communication. 
     The following provides some examples of such a trajectory-encoded approach. In this regard, for example, the sample rate of the system is ƒ s =80 MHz and there may be N s =1024 samples in the signal. Therefore, the symbol duration described as 
             T   =         N   s       f   s       =     12.8           µs   .               
Let
 
               m   =   3     ,         d   0     (   t   )     =       -   300     +     30   ⁢       (     t   T     )     2           ,         d   1     (   t   )     =     50   -     10   ⁢       (     t   T     )     2           ,   
         d   2     (   t   )     =     200   +     20   ⁢       (     t   T     )     2           ,         d   3     (   t   )     =     300   -     100   ⁢     t   T           ,         
β=2π/4, α=0 (i.e., no amplitude encoder), k 0 =1, k 1 =0, k 2 =1, k 3 =1, π=(3,2,1), and (a,b)=(1,1). Therefore, the shift encoder leads to time-varying (D 0 (t), D 1 (t), . . . , D 7 (t)) and as shown in  FIGS.  17  and  18   , the PAPR of the signal is less than or equal to 3 dB while the chirp signals follow different directions. In this regard, the graph  1700  of  FIG.  17    is the instantaneous power with respect to time for a signal that has been generated using trajectory-coded chirp division multiplexing and the graph  1800  of  FIG.  18    is the frequency with respect to time for a signal that has been generated using trajectory-coded chirp division multiplexing.
 
     Additionally or alternatively, in a third example embodiment, a low-PAPR multiband CS-encoded OFDM technique may be implemented. In this regard, the processing circuitry  101  and the communications device  100  may be configured to implement a low-PAPR multiband CS-encoded OFDM approach. In this regard, according to some example embodiments, the shift encoder of CS encoder may be tuned such that the part of the encoded sequence is transmitted through different bands. For example, let w denote any permutation {1, 2, . . . , m} and (a,b) be a GCP of length N and calculate 
                       f   o     (     ϰ   ,   z     )     =     (           p   a     (   z   )     ⁢     (     1   -     ϰ     π   ⁢     (   1   )           )       +         p   b     (   z   )     ⁢     ϰ     π   ⁢     (   1   )             )             (   27   )                               f   r     (   ϰ   )     =       e   0     +       e   m     ⁢     ϰ     π   ⁢     (   m   )           +       ∑     l   =   1       m   -   1               e   l     (       ϰ     π   ⁢     (   l   )         +     ϰ     π   ⁢     (     l   +   1     )           )                 (   28   )                               f   i     (   ϰ   )     =       k   0     +       ∑     l   =   1     m             k   l     ⁢     ϰ     π   ⁢     (   l   )             +       H   2     ⁢       ∑     l   =   1       m   -   1               ϰ     π   ⁢     (   l   )         ⁢     ϰ     π   ⁢     (     l   +   1     )                         (   29   )                               f   s     (   ϰ   )     =       d   0     +       ∑     n   =   1     m             d   n     ⁢     ϰ     π   ⁢     (   n   )                       (   30   )               
where x=(x 1 , x 2 , . . . x m ) and x=Σ j   2     m   x j 2 m-j  for x j ∈   2 , e n ∈ , k n ∈[0, H), d n ∈  for n=0, 1, . . . , m. Then, the sequence c where its polynomial representation is given by
 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       c 
                     
                     ( 
                     t 
                     ) 
                   
                   = 
                   
                     
                       ∑ 
                       
                         x 
                         = 
                         0 
                       
                       
                         
                           2 
                           m 
                         
                         - 
                         1 
                       
                     
                     
                       
                         
                           f 
                           o 
                         
                         ( 
                         
                           x 
                           , 
                           
                             e 
                             
                               
                                 
                                   j 
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 ⁢ 
                                 t 
                               
                               T 
                             
                           
                         
                         ) 
                       
                       × 
                       
                         e 
                         
                           
                             
                               2 
                               ⁢ 
                               π 
                             
                             H 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   f 
                                   r 
                                 
                                 ( 
                                 x 
                                 ) 
                               
                               + 
                               
                                 
                                   jf 
                                   i 
                                 
                                 ( 
                                 x 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                       × 
                       
                         e 
                         
                           j 
                           ⁢ 
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                             t 
                             T 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   f 
                                   s 
                                 
                                 ( 
                                 x 
                                 ) 
                               
                               + 
                               
                                 x 
                                 ⁢ 
                                 N 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
     Since ƒ s (x) admits any real number, the frequency separation can be arbitrarily large. In some example embodiments, by adjusting the separation to be large enough, a radio may transmit 
                 f   o     (     x   ,     e       j   ⁢   2   ⁢   π   ⁢   t     T         )     ×     e         2   ⁢   π     H     ⁢     (         f   r     (   x   )     +       jf   i     (   ϰ   )       )               
on different bands, such as L and C bands (mentioned above), through the same power amplifier  112  by choosing large d n ∈  for n=0, 1, . . . , m. The separation between the bands may be on the level of gigahertz.
 
     Additionally or alternatively, according to a fourth example embodiment, a band-limited circular-time-shift-based chirp transmission technique may be used. In this regard, the processing circuitry  101  and the communications device  100  may be configured to implement band-limited circular-time-shift-based chirp transmission. According to some example embodiments, the information may be encoded over a set of a circularly time-shifted versions of the chirp signal, and the chirp signals may be transmitted to increase time-frequency efficiency. The amount of the shift in time may be selected such that the circularly time-shifted versions of a band-limited chirp signal are orthogonal to each other. 
     According to some example embodiments, the circularly time-shifted chirp signals may be sinusoidal signals. For example, the transmitted signal p(t) may be a combination of the circularly time-shifted sinusoidal chirp signals with the bandwidth of 
             M   T         
Hz as
 
                     p   ⁡   (   t   )     =         ∑   u         d   u     ⁢       B     τ   u       (   t   )         =       ∑   u         d   u     ⁢     e       j   ⁢     M   2     ⁢     sin   (     2   ⁢   π   ⁢       (     t   -     τ   u       )     T       )         ︸       ϕ   u     (   t   )                         (   32   )               
where d u ∈  is the uth modulation symbol or pilot symbols or any information encoding number (e.g., ON-OFF keying, or quadrature amplitude modulation symbols), and τ u ∈[0, T) is the amount of the circular-time shift. The instantaneous frequency of the kth basis function B τ     u   (t) is
 
     
       
         
           
             
               
                 
                   
                     
                       F 
                       u 
                     
                     ( 
                     t 
                     ) 
                   
                   = 
                   
                     
                       
                         1 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           d 
                           ⁢ 
                           
                             
                               ϕ 
                               u 
                             
                             ( 
                             t 
                             ) 
                           
                         
                         
                           d 
                           ⁢ 
                           t 
                         
                       
                     
                     = 
                     
                       
                         1 
                         T 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             M 
                             2 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             ( 
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       τ 
                                       u 
                                     
                                   
                                   ) 
                                 
                                 T 
                               
                             
                             ) 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     Hence, the effective bandwidth of each basis function is 
             M   T         
Hz. The amount of the circular-time shifts, i.e. {τ u |u=0, 1, . . . , U−1}, may be uniformly chosen between 0 and T, where U is the number chirps considered in the scheme. For example,
 
     
       
         
           
             
               τ 
               u 
             
             = 
             
               
                 u 
                 U 
               
               × 
               
                 T 
                 . 
               
             
           
         
       
     
     According to some example embodiments, this approach may be implemented by 1) calculating the U-point DFT of a data vector, i.e., [d 0 , d 1 , . . . , d U-1 ], 2) multiplying each element of the output of DFT with a Bessel function of the first kind of the order of n, denoted by J n (⋅), and 3) calculating N-point IDFT of the multiplied sequence as shown in by the transmitter  1900  of  FIG.  19   . A cyclic prefix may also be appended to the beginning of the symbol (e.g., the OFDM symbol) to handle the synchronization issues. This can be proven by using 
                       e     i   ⁢     M   2     ⁢       sin   (     2   ⁢   π   ⁢       t   -     τ   u       T       )     =         ⁢       ∑     k   =     -   ∞       ∞               J   k     (     M   2     )     ⁢     e     j   ⁢   2   ⁢   π   ⁢       t   -     τ   u       T     ⁢   k             ≅       ∑     k   =     L   d         L   u             J   k     (     M   2     )     ⁢     e     j   ⁢   2   ⁢   π   ⁢       -     τ   u       T     ⁢   k       ⁢     e     j   ⁢   2   ⁢   π   ⁢     t   T     ⁢   k                   (   34   )               
as
 
               J   n     (     M   2     )         
is a decaying function for n goes to zero, where L u −L d +1=U. For
 
                 τ   u     =       u   U     ×   T       ,         
the nth sample of the waveform p(t) may be expressed as
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           p 
                           [ 
                           n 
                           ] 
                         
                         = 
                           
                         
                           p 
                           ⁡ 
                           ( 
                           
                             
                               n 
                               N 
                             
                             ⁢ 
                             T 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         
                           
                             
                               ∑ 
                               
                                 
                                   u 
                                   = 
                                   0 
                                 
                                 , 
                                 1 
                                 , 
                                 … 
                                   
                                 , 
                                 
                                   U 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   
                                     L 
                                     d 
                                   
                                 
                                 
                                   L 
                                   u 
                                 
                               
                               
                                 
                                   
                                     J 
                                     k 
                                   
                                   ( 
                                   
                                     M 
                                     2 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   d 
                                   u 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     j 
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                       
                                         - 
                                         
                                           τ 
                                           u 
                                         
                                       
                                       T 
                                     
                                     ⁢ 
                                     k 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     j 
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                       t 
                                       T 
                                     
                                     ⁢ 
                                     k 
                                   
                                 
                               
                             
                           
                           = 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         
                           
                             ∑ 
                             
                               
                                 u 
                                 = 
                                 0 
                               
                               , 
                               1 
                               , 
                               … 
                                 
                               , 
                               
                                 U 
                                 - 
                                 1 
                               
                             
                           
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 
                                   L 
                                   d 
                                 
                               
                               
                                 L 
                                 u 
                               
                             
                             
                               
                                 
                                   J 
                                   k 
                                 
                                 ( 
                                 
                                   M 
                                   2 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 d 
                                 k 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     u 
                                     U 
                                   
                                   ⁢ 
                                   k 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     n 
                                     N 
                                   
                                   ⁢ 
                                   k 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 
                                   L 
                                   d 
                                 
                               
                               
                                 L 
                                 u 
                               
                             
                             
                               
                                 
                                   
                                     
                                       J 
                                       n 
                                     
                                     ( 
                                     
                                       M 
                                       2 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         ∑ 
                                         
                                           
                                             u 
                                             = 
                                             0 
                                           
                                           , 
                                           1 
                                           , 
                                           … 
                                             
                                           , 
                                           
                                             U 
                                             - 
                                             1 
                                           
                                         
                                       
                                       
                                         
                                           d 
                                           k 
                                         
                                         ⁢ 
                                         
                                           e 
                                           
                                             
                                               - 
                                               j 
                                             
                                             ⁢ 
                                             2 
                                             ⁢ 
                                             π 
                                             ⁢ 
                                             
                                               u 
                                               U 
                                             
                                             ⁢ 
                                             k 
                                           
                                         
                                       
                                     
                                     
                                       ︸ 
                                       
                                         U 
                                         - 
                                         point 
                                         ⁢ 
                                             
                                         DFT 
                                       
                                     
                                   
                                 
                                 
                                   ︸ 
                                   Multiplication 
                                 
                               
                               ⁢ 
                               
                                 
                                   e 
                                   
                                     j 
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                       n 
                                       N 
                                     
                                     ⁢ 
                                     k 
                                   
                                 
                                 . 
                               
                             
                           
                           
                             ︸ 
                             
                               N 
                               - 
                               point 
                               ⁢ 
                                   
                               IDFT 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     In a general case, the circularly time-shifted the chirp signals may be any function, e.g., a linear function in time. In this case, by calculating its Fourier series, i.e., c k , as 
                     e     i   ⁢     ψ   ⁡   (   t   )         =           ∑     k   =     -   ∞       ∞         c   k     ⁢     e     j   ⁢   2   ⁢   π   ⁢     t   T     ⁢   k           ⇒     c   k       =       1   T     ⁢       ∑     k   =     -   ∞       ∞         e     i   ⁢     ψ   ⁡   (   t   )         ⁢     e       -   j     ⁢   2   ⁢   π   ⁢     t   T     ⁢   k                       (   36   )               
and replacing
 
               J   k     (     M   2     )         
with c k  for k=L d , . . . , L u  as shown in the design of the transmitter  1900  of  FIG.  19   , the same IDFT-based diagram may be utilized.
 
     As such, an example implementation of circularly time-shifted chirp signals is now provided. In this regard, let T=12.8 μs and M=128. Hence, the bandwidth of 
                 B     τ   u       (   t   )     =     e     j   ⁢         M   2     ⁢   sin   ⁢     (     2   ⁢   π   ⁢       (     t   -     τ   u       )     T       )         ︸       ϕ   u     (   t   )                   
is
 
                 M   ×     1   T       =     1   ⁢   0       ⁢           MHz   .           
The basis signal B τ     u   (t) deviates between
 
               -     M     2   ⁢   T         =         -   5     ⁢         MHz   ⁢         and   ⁢           M     2   ⁢   T         =     5   ⁢         MHz             
around the center frequency. The IDFT size may be N=512 and DFT precoding size may be U=144.
 
     For a simulation, the following parameters were used: M=128, L u =72 and L d =−71. The spectrogram of the signal for d 0 =1 and d u≠0 =0 is given in graph  2000  of  FIG.  20   a   , which illustrates the expected sinusoidal chirp signal as frequency with respect to time. In the graph  2010  of  FIG.  20   b   , also rendered as frequency with respect to time, the spectrogram of the signal for d 0,51 =1 and d u≠0,51 =0 is shown. Since d 0,51 =1, there are two active chirp signals, where the cyclic shift for the second chirp is 
     
       
         
           
             
               
                 
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     Further, according to some example embodiments, the processing circuitry  101  may be configured to transmit the data symbols over the basis functions {B τ     0   (t), B τ     1   (t), . . . , B τ     M-1   (t)} constructed by translating a chirp signal circularly in time, where τ m  is the amount of circular shift. According to some example embodiments, an assumption can be made that the shifts in time are uniformly spaced between 0 and T s , i.e., τ m =m/M×T s , where T s  is the chirp duration. The complex baseband signal p(t) can then be expressed as: 
                 p   ⁡   (   t   )     =         ∑     m   =   0       M   -   1           d   m     ⁢       B     r   m       (   t   )         =       ∑     m   =   0       M   -   1           d   m     ⁢     e     j   ⁢       ϕ   m     (   t   )                 ,         
where d m ∈  is the mth modulation or a pilot symbol and ψ m (t) is the phase of the carrier for the mth basis function. Therefore, the instantaneous frequency of the mth chirp signal B τ     m   (t) around the carrier frequency ƒ c  can be obtained as
 
                 F   m     (   t   )     =       1     2   ⁢   π       ⁢   d   ⁢         ψ   m     (   t   )     /   dt             
Hz. The waveform p(t) may be a simple linear combination of the basis functions related to chirp signals. However, the waveform may be implicitly related to DFT-s-OFDM. This relation can be shown as follows. Let B τ     m   (t)=e jψ     m     (t)  be an arbitrary band-limited function with the period of T s . Hence, the relation can be expressed as:
 
               e     j   ⁢       ψ   m     (   t   )         =       ∑     k   =     -   ∞       ∞         c   k     ⁢     e     j   ⁢   2   ⁢   π   ⁢   k   ⁢       t   -     τ   m         T   s                     
where c k  is the kth Fourier coefficient given by:
 
               c   k     =       1     T   s       ⁢       ∫     T   s           e     j   ⁢       ψ   0     (   t   )         ⁢     e       -   j     ⁢   2   ⁢   π   ⁢   k   ⁢     t     T   0           ⁢     dt   .                 
By using the equations above, τ m =m/M×T s , p(t) can be expressed as:
 
                 p   ⁡   (   t   )     =       ∑     m   =   0       M   -   1           d   m     ⁢       ∑     k   =     -   ∞       ∞         c   k     ⁢     e     j   ⁢   2   ⁢   π   ⁢   k   ⁢       t   -     r   m         T   s                     ,     ≈       ∑     k   =     L   s         L   u           c   k     ⁢       ∑     m   =   0       M   -   1           d   m     ⁢     e       -   j     ⁢   2   ⁢   π   ⁢   k   ⁢     m   M         ⁢       e     j   ⁢   2   ⁢   π   ⁢   k   ⁢     t     T   s           .                     
where L d &lt;0 and L u &gt;0 are integer values. The approximation is due to the fact that B τ     m   (t) is a band-limited function, i.e., c k  is a decaying Hermitian symmetric function as k goes to positive or negative infinity. Finally, by sampling p(t) with the period T s /N, the discrete-time signal can be obtained as:
 
     
       
         
           
             
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     In other words, p(t) may be a special DFT-s-OFDM symbol that can be implemented by 1) determining the M-point DFT of a data vector, i.e., [d 0 , d 1 , . . . , d M-1 ], 2) multiplying each element of the output of DFT with the corresponding Fourier coefficient, i.e., FDSS or windowing in frequency, and 3) determining the N-point inverse DFT (IDFT) of the shaped sequence after padding it with N−(L u −L d +1) zero symbols (i.e., guard subcarriers in OFDM). Without loss of generality, it can be assumed that L u −L d +1=M to ensure that the FDSS occurs within the bandwidth spanned by M subcarriers. Therefore, the chirp bandwidth should be less than or equal to. 
     In 3GPP LTE and 3GPP 5G NR uplink, FDDS is described as an implementation-specific option to reduce peak-to-average-power ratio (PAPR) further for DFT-s-OFDM. Based on 
               p   ⁡   (       nT   s     N     )     ,         
the same DFT-s-OFDM transmitter can also generate arbitrary chirp signals by selecting the shaping coefficients properly without compromising the other features of the physical layer design in these communication systems. According to some example embodiments, such an approach may not cause any bandwidth expansion as the chirps can be circularly shifted versions of each other in the time domain while eliminating additional processing to avoid aliases.
 
     According to some example embodiments, the chirp signals may be sinusoidal chirps. In this regard, let the instantaneous frequency of B τ     0   (t) around the carrier frequency θ c  be a sinusoidal function given by: 
                 F   0     (   t   )     =       D     2   ⁢     T   s         ⁢     cos   ⁡   (     2   ⁢   π   ⁢     t     T   s         )             Therefore, 
                 ψ   0     (   t   )     =       D   2     ⁢       sin   ⁡   (     2   ⁢   π   ⁢     t     T   s         )     .             
Further, e jψ     0     (t)  can be decomposed as:
 
               e     j   ⁢     D   2     ⁢     sin   (     2   ⁢   π   ⁢     t     T   o         )         =       ∑     k   =     -   ∞       ∞           J   k     (     D   2     )     ⁢     e     j   ⁢   2   ⁢   π   ⁢   k   ⁢     t     T   s                     
J k (⋅) is the Bessel function of the first kind of order k. Hence, c k  may be equal to
 
               J   k     (     D   2     )         
for sinusoidal chirps. The maximum frequency deviation of each basis function and the effective bandwidth of the transmitted signal p(t) may be equal to Δƒ=D/2T s  Hz and D/T Hz, respectively. Therefore, D≤M can hold true to form the sinusoidal chirp in
 
     
       
         
           
             
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     According to some example embodiments, the chirp signals may be linear chirps. In this regard, assume that the instantaneous frequency of B τ     0    around the carrier frequency θ c  changes from 
             -     D     2   ⁢     T   s               
Hz to
 
             D     2   ⁢     T   s             
Hz, i.e.
 
                 F   0     (   t   )     =       D     2   ⁢     T   s         ⁢     (         2   ⁢   t       T   o       -   1     )             which results in 
                 ψ   0     (   t   )     =         π   ⁢   D       T   s       ⁢       (         t   2       T   s       -   t     )     .             
The shaping coefficients c k  can be obtained as
 
                 c   k     =           π   D       ⁢       e         -   j     ⁢         (     2   ⁢   π   ⁢   k     )     2       2   ⁢   D         -     j   ⁢   π   ⁢   k         (     C   ⁡   (     x   1     )     )       +     C   ⁡   (     x   2     )     +     j   ⁢     S   ⁡   (     x   1     )       +     jS   ⁡   (     x   2     )         )         
where C(⋅) and S(⋅) are the Fresnel integrals with cosine and sine functions, respectively, and
 
 x   1 =( D/ 2+2π k )/√{square root over (π D )}
 
and
 
 x   2 ( D/ 2−2π k )/√{square root over (π D )}
 
Similar to the sinusoidal chirp, the condition D≤M can be satisfied to form the linear chirp.
 
     A receiver, according to some example embodiments, for the proposed scheme may be based on a typical DFT-s-OFDM receiver. After the cyclic prefix (CP) is discarded, the DFT of the received signal may be determined. In the frequency domain, the impact of the channel may be removed with a single-tap minimum mean square error (MMSE)-frequency-domain equalization (FDE). The modulation symbols may be obtained after an M-IDFT operation on the equalized signal vector. For a practical receiver, the shaping coefficients may be considered as part of the channel frequency response and estimated through channel estimation procedure. From this aspect, the proposed scheme does not require any change from a practical DFT-s-OFDM receiver. On the other hand, if the shaping coefficients are available at the receiver a priori, the receiver may perform better as coefficients may not need to be estimated. 
     For simulations associated with the approach, according to some example embodiments, an OFDM framework may be considered where the symbol duration Ts=16.67 μs, i.e., the subcarrier spacing is 60 kHz. The CP duration can be set to 2.34 μs. The transmitter may be assumed to exploits M=336 subcarriers, i.e., the bandwidth is 20.16 MHz. For the linear and sinusoidal chirps, D may be equal to 318 to not distort chirp signals due to the truncation in frequency. The data symbols may be generated based on quadrature phase shift keying (QPSK) modulation and M=336 basis functions may be used simultaneously unless otherwise stated. The multi-path channel may be generated based on an Extended Vehicular A (EVA) power delay profile. For the channel coding, IEEE 802.1 lay low-density parity check (LDPC) code with the rate of 1/2 may be employed where the codeword length is 672. 
     In  FIGS.  21   a - 21   c   , spectrograms are provided for plain DFT-s-OFDM (i.e., no FDSS) at  2100 , linear chirps at  2110 , and sinusoidal chirps  2120 , when only d 0  and d 75  are set to 1. Since the plain DFT-s-OFDM is a form of single carrier waveform, the symbols d 0  and d 75  appear as two pulses in time as provided at  2100  in  FIG.  21   a   . In contrast, the same symbols result in two linear and sinusoidal chirps transmitted simultaneously as given at  2110  in  FIG.  21   b    and at  2120  in  FIG.  21   c   , respectively. As shown at  2110 , a sudden frequency change may occur for linear chirps. On the other hand, the abrupt instantaneous frequency changes may be avoided for the sinusoidal chirps as shown at  2120 , due to the sinusoidal chirp being a continuous periodic function while the linear chirp is a discontinuous periodic function in e jψ     m     (t) . In  FIG.  21   d    at  2130 , the amplitude of the shaping coefficients are shown for the linear and sinusoidal chirp signals. As shown at  2130 , the chirp signals do not distribute symbol energy to the subcarriers evenly. While the amplitude variation for linear chirp signals are relatively mild except the edge subcarrier, the amplitude variation can be large for sinusoidal chirps. A large amplitude variation can cause ripples in the spectrum and noise enhancement. As shown in  FIG.  21   e    at  2140 , the main lobe of the spectrum is not flat for linear and sinusoidal chirps. Particularly, a majority of the symbol energy may be carried over the edge subcarrier bins for sinusoidal chirps. Secondly, large ripples can affect the BER performance. For example, the amplitude of the shaping coefficients can be very small values for sinusoidal chirps. Therefore, the corresponding bins may be more prone to noise as compared to the other bins with large shaping coefficients. 
     In  FIG.  21   f    at  2150 , PAPR distributions are compared. The introduction of FDSS for generating chirp signals does not keep the low PAPR benefit of DFT-s-OFDM. This is because the symbol energy is distributed in both time and frequency for the chirp signals. When multiple chirp signals are transmitted in parallel, the chirp signals can constructively or destructively add up, which increases the PAPR. 
     In  FIG.  21   g    at  2160  and  FIG.  21   h    at  2170 , coded BER and uncoded BER curves, respectively, are provided for additive white Gaussian noise (AWGN) and multi-path fading channels. For uncoded bits, a large degradation for the sinusoidal chirps in both AWGN and multipath channels occurs, as compared to DFT-s-OFDM. However, the difference between linear chirps and DFT-s-OFDM is approximately 1 dB for uncoded bits. When the channel encoder is introduced, the degradation reduces to approximately 0.5 dB while it is approximately 4 dB for the sinusoidal chirps in AWGN and multi-path fading channels. Thus, linear chirps provide more reliable links as compared to sinusoidal chirps for this particular design. 
     In accordance with the above, the processing circuitry  101  may be configured to generate chirp signals with, for example, DFT-s-OFDM via well-designed shaping coefficients. A benefit of the approach is that a typical DFT-s-OFDM receiver with a single-tap FDE-MMSE (frequency domain equalization-minimum mean square error) can decode the modulated chirp signals, according to some example embodiments. The numerical results may show that the amplitude variations in the shaping coefficients adversely affect the BER performance due to the noise enhancement during equalization. While the signal-to-noise ratio (SNR) degradation is approximately 0.5 dB for linear chirps, the degradation reaches 4 dB for the sinusoidal chirps due to the large variations, as compared to DFT-s-OFDM without any FDSS. As such, via the approach, waveforms for radar and communications and IoT applications can be synthesized, for example, without introducing major modifications to the physical layer of wireless communication standards. 
     Additionally or alternatively, according to a fifth example embodiment, multi-cluster-DFT-based precoding for chirp division multiplexing may be implemented. In this regard, according to some example embodiments, the processing circuitry  101  of the communications device  100  may be configured to implement multi-cluster-DFT-based precoding for chirp division multiplexing. In this regard, according to some example embodiments, the information bits may be mapped to a trajectory in time-frequency plane divided into M tile ×N tile  grids, i.e., T∈   M     tile     ×N     tile    where the ith row and jth column of T is d i,j , and the signal x∈   N×1  given by
 
 x=F   H   M vec{ DT   H }  (37)
 
where F H  is the inverse N-point DFT matrix, M∈   N×M     tile     N     tile    is the subcarrier mapping matrix, D∈   M×M  is the M-point DFT matrix, and vec{⋅} is the vectorization operation. A block diagram of a transmitter  2200  for chirp signal transmission is provided in  FIG.  22   , along with an illustration of an example multi-cluster DFT precoding technique. The information bits are first mapped to the trajectory matrix T. For example, for a linear chirp, d l,l , may be a non-zero value and d i,j|i≠j =0 if M tile =N tile . The information bits may be mapped to orthogonal trajectories. For example, a circular shift of the matrix T may be used for orthogonal chirps. Trajectory matrix may encode multiple information bits through orthogonal/non-orthogonal chirps. The values of d i,j  may be a modulation symbol, or simply indicate an on/off keying. After trajectory matrix is generated, vec{DT H } may be calculated through M tile  simultaneous N tile -point DFT precoding operations. The calculated vector may be mapped a set of subcarriers based on the resource allocation. After the mapping, the inverse DFT of the sequence may be calculated. A cyclic prefix may be appended to the generated signal in association with a symbol. The calculated signal may be transmitted through a transmitter.
 
     According to some example embodiments, the trajectory matrix may be chosen such that only one of the inputs of each DFT in the precoding matrix  2300  of  FIG.  23    is non-zero. In other words, d i,j=l     i   =1 and d i,j≠l     i   =0 for a given ith row of T 0 . In this case, vec{DT 0   H } may yield to a constant-amplitude sequence in frequency domain. Hence, orthogonal chirp signals may be generated by circularly shifting the signal. According to some example embodiments, the circular time shift may be implemented by introducing phase rotations in frequency domain. For a given trajectory matrix T 0  under this condition, the transmitted signal may be expressed as 
                   x   =       F   H     ⁢   M   ⁢       ∑   k         b   k     ⁢     P   k     ⁢   vec   ⁢     {     DT   0   H     }                   (   38   )               
where
 
               P   k     =     diag   ⁢     {     e       -   j     ⁢   2   ⁢   π   ⁢       k       M   tile     ⁢     N   tile         [     0   ,   1   ,   …       ,         M   tile     ⁢     N   tile       -   1       ]         }             
and b k  may be a modulation symbol or a fixed symbol. As such, a block diagram of a transmitter  2400  based on this approach is shown in  FIG.  24   . The processing circuitry  101  may be configured to implement the transmitter  2400  which is capable of generating the chirp signals with multi-cluster DFT pre-coding and orthogonal chirps with circular shifts.
 
     As such, an example of an approach comprising multi-cluster-DFT-based precoding for chirp division multiplexing can be provided. In this regard, T=12.8 μs and M tile =N tile =24. IDFT size may be set to N=2048. In this example case, the maximum bandwidth is 
                     M   tile     ×     N   tile       T     =     4   ⁢   5       ⁢           MHz   .           
The sinusoidal-like trajectories for the trajectory matrix T and the corresponding spectrograms are given in  FIGS.  25 - 28   . In this regard, the graph  2500  of  FIG.  25    shows tiles in frequency with respect to tile in time as the sinusoidal T. The graph  2600  of  FIG.  26    shows a spectrogram output of frequency with respect to time associated with the sinusoidal T as provided in graph  2500 . Similarly, the graph  2700  of  FIG.  27    shows tiles in frequency with respect to tile in time as a different sinusoidal T. The graph  2800  of  FIG.  28    shows a spectrogram output of frequency with respect to time associated with the different sinusoidal T as provided in graph  2700 .
 
     Further,  FIGS.  29 - 31    show additional spectrograms for different values of k for x k =F H MP k vec{DT 0   H }. In this regard, the graph  2900  of  FIG.  29    shows a spectrogram where k=0, graph  3000  of  FIG.  30    shows a spectrogram where k=100, and graph  3100  of  FIG.  31    shows a spectrogram where k=300. In the graph  3200  of  FIG.  32   , the linear chirp signals are illustrated for d i,i =1 and d i,j|i≠j =0 for T 0  by using equation (38). The cross-correlation between different linear chirps (x k  and x l ) are given in graph  3200 , which demonstrates that the generated chirp signals construct an orthogonal set, where the x and y axes show different indices of x k . 
     Additionally, as shown in  FIGS.  33  and  34   , a complementary-sequence encoded signal consisting of the sum of 8 chirp signals with linear time-frequency trajectories is provided. In this regard, the instantaneous signal power never exceeds 2, as shown in the graph  3300  of  FIG.  33   , although the chirps follow different directions in time-frequency with different rates, as shown in the spectrogram of graph  3400  of  FIG.  34   . In this regard, graph  3300  shows instantaneous power vs. time for the sum of 8 chirp signals, and graph  3400  shows a spectrogram of the 8 chirp signals. 
     Having described various example embodiments that implement communications techniques involving chirp signals, an example method is described with respect to  FIG.  35   . In this regard, the example method may comprise controlling, via processing circuitry, a radio to establish a wireless communications link via an antenna at  3500 . Further, the example method may comprise controlling the radio and the antenna to transmit communications as a plurality of sequenced chirp signals within an orthogonal frequency division multiplexing (OFDM) framework at  3510 . 
     According to some example embodiments, the example method may also include controlling the radio to transmit the plurality of sequenced chirp signals, wherein a chirp signal of the plurality of sequenced chirp signals comprises an up-chirp signal within OFDM symbols or a down-chirp signal within OFDM symbols. In this regard, the up-chirp signal may comprise a signal with an increasing frequency with respect to time and the down-chirp signal may comprise a signal with a decreasing frequency with respect to time. Additionally or alternatively, according to some example embodiments, the example method may include controlling the radio to transmit the plurality of sequenced chirp signals within OFDM symbols, wherein a chirp signal of the plurality of chirp signals comprises an up-chirp signal or a down-chirp signal. In this regard, the up-chirp signal may sweep across a range of frequencies with respect to time by increasing a frequency of the up-chirp signal, and the down-chirp signal sweeps across a range of frequencies with respect to time by decreasing a frequency of the down-chirp signal. 
     Additionally or alternatively, according to some example embodiments, the example method may include controlling the radio to transmit the plurality of sequenced chirp signals including a cyclic prefix via orthogonal chirp division multiplexing. Additionally or alternatively, according to some example embodiments, the example method may include controlling the radio to maintain orthogonality by transmitting complementary sequences of chirp signals. Additionally or alternatively, according to some example embodiments, the example method may also include controlling the radio to transmit the plurality of sequenced chirp signals, the sequenced chirp signals having a uniform, non-linear trajectory in time and frequency. Additionally or alternatively, according to some example embodiments, the example method may also include providing amplification via a power amplifier for transmission of the plurality of sequenced chirp signals and controlling instantaneous power fluctuations due to transmission of chirp signals associated with different frequencies within a band. 
     Additionally or alternatively, according to some example embodiments, the example method may include controlling the radio to transmit the plurality of sequenced chirp signals via implementation of a Discrete Fourier Transform (DFT)-spreading scheme with OFDM and a frequency-domain spectral shaping (FDSS) filter that employs Bessel functions and Fresnel integrals used to develop the plurality of sequenced chirp signals as band-limited sinusoidal or linear chirp transmissions. Additionally or alternatively, according to some example embodiments, the example method may include controlling the radio to transmit the plurality of sequenced chirp signals, the plurality of sequenced chip signals being generated via orthogonal frequency division multiplexing to encode trajectories of the chirp signals in frequency and time. 
     Many modifications and other embodiments of the inventions set forth herein will come to mind to one skilled in the art to which these inventions pertain having the benefit of the teachings presented in the foregoing descriptions and the associated drawings. Therefore, it is to be understood that the inventions are not to be limited to the specific embodiments disclosed and that modifications and other embodiments are intended to be included within the scope of the appended claims. Moreover, although the foregoing descriptions and the associated drawings describe exemplary embodiments in the context of certain exemplary combinations of elements or functions, it should be appreciated that different combinations of elements or functions may be provided by alternative embodiments without departing from the scope of the appended claims. In this regard, for example, different combinations of elements or functions than those explicitly described above are also contemplated as may be set forth in some of the appended claims. In cases where advantages, benefits or solutions to problems are described herein, it should be appreciated that such advantages, benefits or solutions may be applicable to some example embodiments, but not necessarily all example embodiments. Thus, any advantages, benefits or solutions described herein should not be thought of as being critical, required or essential to all embodiments or to that which is claimed herein. Although specific terms are employed herein, they are used in a generic and descriptive sense only and not for purposes of limitation.