Devices, systems, and software including signal power measurements, and methods and software for measuring signal power

Abstract

The systems, devices, software, and methods of the present invention enable frequency-based signal power analysis in software suitable for signals with either stationary or non-stationary spectra. The methods, which can be used across various systems, including transmitters, receivers, repeaters, controllers, monitors, etc., and in software simulators, enable the calculation and analysis of various signal powers, such as frequency spectrum analysis, across operating systems, and are consistently applied to system design and operational simulations.

Description

[Technical Field]

[0001] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT This invention was made with government support under Award No. 1738453 awarded by the National Science Foundation. The government has certain rights in this invention. CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims the benefit of and priority to U.S. Provisional Patent Application No. 62 / 848,280, filed May 15, 2019, entitled "Devices, Systems, And Software Including Signal Power Measuring And Methods And Software For Measuring Signal Power," and U.S. Patent Application No. 16 / 819,126, filed March 15, 2020, each of which is incorporated by reference in its entirety. The present invention relates generally to measuring signal power, and more particularly to systems, devices, software, and methods for performing signal power measurements across a frequency spectrum for use in occupied bandwidth (OBW) and other frequency spectrum analyses. [Background technology]

[0003] The data transmission capacity of a communication system generally depends on the number of communication channels the system can support and the data transmission capacity of each communication channel. The number of channels available for use in a given frequency range depends on the frequency range or bandwidth occupied by each channel transmitted by the system. This is known as the channel's occupied bandwidth (OBW). System channel layout is based on the maximum expected OBW of each channel, with additional guard bands to determine channel spacing, which is usually defined by the frequency range separating the center frequencies of each channel.

[0004] A standard convention is to equate a signal's OBW to the frequency range containing 99% of the signal's power. This or a similar definition of OBW requires measuring the signal power across frequencies to obtain the power spectral density (PSD). This is most commonly achieved using the Fourier transform (FT) method, which is well known in the art. However, the FT method represents the signal spectrum in terms of a sine wave with a constant amplitude, i.e., as a stationary spectrum. This means that if the spectrum changes over time, i.e., is not stationary, the FT method inherently cannot correctly measure the OBW.

[0005] Research into time-frequency analysis has arisen due to this problem, including techniques such as the short-time Fourier transform (STFT), which generates the well-known spectrogram. The STFT uses a sliding time window, trading off time and frequency resolution. Joint time-frequency distribution is an alternative technique in which functions simultaneously distributed in the time and frequency domains are derived. However, these techniques are all based on Fourier transform (FT), and despite their high adaptability, they are fundamentally unsuitable for studying time-varying spectra. In effect, applying FT to time-varying (non-stationary) spectra requires finding periods where the spectrum can be considered approximately stationary and stitching them together (so that the FT assumptions are satisfied). Unfortunately, a priori knowledge of approximately stationary intervals is generally unknown. This may be one reason why spectra are measured.

[0006] A real-time spectrum analyzer ("RTSA") captures a signal at short intervals, which is then stored and analyzed using a Fast Fourier Transform (FFT) algorithm. Some RTSAs can sequence the FFT results to show spectral changes over time.

[0007] As the spectrum becomes increasingly non-stationary, as for example in the case of signals generated using continuously non-stationary spiral modulation, the stationary assumption becomes increasingly infeasible. For additional information regarding spiral modulation, see, for example, U.S. Patent No. 6,239,999 entitled "Telecommunication Signaling Using Non-Linear Functions," U.S. Patent No. 6,239,999 entitled "Methods and Systems for Communicating," U.S. Patent No. 6,239,999 entitled "Spiral Polynomial Division Multiplexing" (SPDM), the contents of which are incorporated herein by reference in their entirety except for the claims and any disclosure to the contrary.

[0008] Another approach to OBW measurement known in the art is the Hardware Swept-Tuned Spectrum Analyzer (HSTSA). Essentially, an HSTSA analyzes the physically transmitted signal and uses filters to sequentially isolate specific frequency ranges ("bins") of the spectrum. The power in each frequency bin is measured, and the distribution of power across the frequency bins can be used to measure OBW.

[0009] Because HSTSAs do not use FT, they essentially avoid the problems of FT-based OBW measurements described above. However, because HSTSAs sweep a frequency range, the assumption of a stationary spectrum remains throughout the sweep. Additionally, HSTSAs require physical signal transmission and are often expensive equipment that is not readily available outside of a laboratory.

[0010] Continuing demand for more sophisticated services from communication service customers of all types means that there is a continuing need for data transmission / communication systems, devices, software, and methods that enable those services. [Prior art documents] [Patent documents]

[0011] [Patent Document 1] U.S. Patent No. 8,472,534 [Patent Document 2] U.S. Patent No. 8,861,327, [Patent Document 3] U.S. Patent No. 10,069,664 [Non-patent literature]

[0012] [Non-Patent Document 1] Prothero, J., Islam, KZ, Rodrigues, H., Mendes, L., Gutierrez, J., and Montalban, J., Instantaneous Spectral Analysis. Journal of Communication and Information Systems, (2019), Vol. 34(1), pp. 12-26, https: / / doi.org / 10.14209 / jeis2019.2 Summary of the Invention

[0013] The systems, devices, software, and methods of the present invention enable frequency-based signal power analysis in software, thereby providing software frequency spectrum analyzers ("SSAs") and other power signal analyses. By enabling signal power and frequency spectral analysis to be performed in software, the present invention solves long-standing problems associated with making consistent spectral measurements and allows spectral analysis to be performed on both hardware and software-generated signals. In hardware, the present invention can be used to enable system design, feedback, and control based on the actual OBW performance of an operational system. In software, SSA can be used in software simulators to evaluate simulated system performance, such as analyzing the OBW of simulated signals. Because the present invention can be applied to signals generated by hardware and software, analysis can be performed consistently across hardware and software platforms.

[0014] The present invention can be used with signals having stationary or non-stationary spectra. The present invention is particularly useful for signals whose measured spectra are highly time-varying (non-stationary) due to the limitations of FT-based techniques. However, these methods are generally applicable and can be used in combination with or instead of FT-based techniques for stationary spectral signals.

[0015] The systems, devices, and methods of the present invention have an electrical signal input, which may be generated in hardware by receiving an information-carrying signal, such as an optical or radio signal, or in software by simulating prior reception. Analysis of the input signal may be performed according to various analysis inputs provided by a user. For example, before analyzing the input signal, a user may specify a frequency spectral range having a minimum frequency (f_min) and a maximum frequency (f_max) and define at least one frequency bin within the frequency spectral range, with each frequency bin having an associated frequency bin width f_width and source frequency f_source.

[0016] A target intermediate frequency (f_target) may be specified for each frequency bin. The target frequency may be the same for each frequency bin or may be changed by one skilled in the art. In some applications, it may be desirable to have the same target frequency for all frequency bins, so that the same filter can be used for each bin.

[0017] The input signal sample time interval dt defines the sampling frequency Fs=1 / dt. Also relevant is the total number of samples in the signal, sig_len. For each frequency bin, a software intermediate frequency bandpass filter ifbpf in terms of Fs, f_target, and f_width can be defined along with the input signal software bandpass filter INbf to remove frequencies outside the passband.

[0018] In operation, an input signal SIGin with power distributed across a range of signal frequencies is received. An input signal software bandpass filter INbf is applied to the input signal SIGin to produce a filtered input signal, which is normalized to produce a normalized signal power SIGin_norm that can be stored for use in various calculations.

[0019] For each frequency bin, a mixing frequency f_mix suitable for converting the frequency source f_source to the target IF frequency f_target may be calculated as f_mix=f_source−f_target.

[0020] Then the mixing stream cos_mix of length sig_len equal to the input signal length can be calculated by evaluating the cosine function in the range 0 to rads_per_signal in increments of rads_per_sample, which can be written as: cos_mix=eos(0:rads_per_sample:rads_per_signal), where: rads_per_sample=f_mix*dt*2*pi, and rads_per_signal=rads_per_sample(sig_len-l).

[0021] The formula for cos_mix above is also the notation used in MATLAB® to calculate cos_mix.

[0022] The intermediate frequency (IF) signal sig_IF may be generated using the formula sig_IF=SIGinjiorm*cos_mix.

[0023] A corresponding intermediate frequency bandpass filter ifbpf is applied to each intermediate frequency (IF) signal sig_IF to generate a filtered IF signal sigIF_filtered.

[0024] The bin power bin_power of the filtered IF signal sig_IF_filtered is calculated and stored by summing the squares of the amplitudes of sig_IF_filtered and dividing this sum by the time interval.

[0025] The frequency bin with the maximum bin_power, max_power, can be identified and used for various purposes. For example, the occupied bandwidth of the input signal can be calculated in various ways, such as by summing the frequency widths of the frequency bins based on the bin power, bin_power, at each frequency bin adjacent to the frequency bin with the maximum power.

[0026] In various embodiments, the OBW may be calculated as the double-sided occupied bandwidth of the input signal by starting with the frequency width of the maximum power bin and then successively summing the frequency widths of the maximum power frequency bins adjacent to either side of the maximum power bin or a previously summed frequency bin until the sum of the power in the frequency power bins equals 99% or some desired percentage of the input signal power. The single-sided occupied bandwidth may be calculated by dividing the double-sided occupied bandwidth by two to find the average, or by summing the power of frequency bins that span frequencies lower and higher than the maximum power frequency bin, respectively.

[0027] The systems, devices, software, and methods of the present invention can be applied at many, if not all, different design and operational stages. During the design stage, the present invention can be implemented in simulation software, as well as in device and system level designs and prototypes. During operation, the present invention can be implemented in transmitters, repeaters, receivers, standalone signal monitoring devices, and other devices where signal power measurements across the frequency spectrum may be useful, such as signal power monitor and control devices.

[0028] As may be disclosed, taught, and / or suggested to those skilled in the art herein, the present invention addresses the continuing need for hardware and / or software systems, devices, and methods for measuring signal power, such as the OBW of a signal in a software simulation or hardware transmission, which can be particularly important in the case of highly non-stationary spectra.

[0029] Advantages of embodiments of the present invention will be apparent from the following detailed description of exemplary embodiments thereof, which description should be considered in conjunction with the accompanying drawings, which are illustrative and not for purposes of limitation, but are included for purposes of exemplary illustration of various aspects of the invention. [Brief explanation of the drawings]

[0030] [Figure 1] FIG. 1 illustrates an exemplary data transmission system. [Figure 2] FIG. 1 illustrates an exemplary data transmission system. [Figure 3] FIG. 1 illustrates an exemplary initial PSW alphabet. [Figure 4] FIG. 10 illustrates an exemplary initial PSW alphabet convolved with polynomials corresponding to Gaussian distributions with zero mean and sigma values ​​of 0.8, 1.0, 1.2, 1.6, and 2, respectively. [Figure 5] FIG. 10 illustrates an exemplary initial PSW alphabet convolved with polynomials corresponding to Gaussian distributions with zero mean and sigma values ​​of 0.8, 1.0, 1.2, 1.6, and 2, respectively. [Figure 6] FIG. 10 illustrates an exemplary initial PSW alphabet convolved with polynomials corresponding to Gaussian distributions with zero mean and sigma values ​​of 0.8, 1.0, 1.2, 1.6, and 2, respectively. [Figure 7] FIG. 10 illustrates an exemplary initial PSW alphabet convolved with polynomials corresponding to Gaussian distributions with zero mean and sigma values ​​of 0.8, 1.0, 1.2, 1.6, and 2, respectively. [Figure 8] FIG. 10 illustrates an exemplary initial PSW alphabet convolved with polynomials corresponding to Gaussian distributions with zero mean and sigma values ​​of 0.8, 1.0, 1.2, 1.6, and 2, respectively. [Figure 9]FIG. 10 illustrates the normalized difference between SSA and HSTSA OBW calculations as a function of bitstream length, number of frequency bins, and IF filter length using two methods of calculating OBW based on the power calculated in the frequency bins. [Figure 10] FIG. 10 illustrates the normalized difference between SSA and HSTSA OBW calculations as a function of bitstream length, number of frequency bins, and IF filter length using two methods of calculating OBW based on the power calculated in the frequency bins. [Figure 11] FIG. 10 illustrates the normalized difference between SSA and HSTSA OBW calculations as a function of bitstream length, number of frequency bins, and IF filter length using two methods of calculating OBW based on the power calculated in the frequency bins. [Figure 12] FIG. 10 illustrates the normalized difference between SSA and HSTSA OBW calculations as a function of bitstream length, number of frequency bins, and IF filter length using two methods of calculating OBW based on the power calculated in the frequency bins. [Figure 13] FIG. 10 illustrates the normalized difference between SSA and HSTSA OBW calculations as a function of bitstream length, number of frequency bins, and IF filter length using two methods of calculating OBW based on the power calculated in the frequency bins. [Figure 14] FIG. 10 illustrates the normalized difference between SSA and HSTSA OBW calculations as a function of bitstream length, number of frequency bins, and IF filter length using two methods of calculating OBW based on the power calculated in the frequency bins. [Figure 15] FIG. 10 illustrates the normalized difference between SSA and HSTSA OBW calculations as a function of bitstream length, number of frequency bins, and IF filter length using two methods of calculating OBW based on the power calculated in the frequency bins. [Figure 16]FIG. 1 illustrates eight polynomial symbol waveform (PSW) alphabets corresponding to a root-raised cosine (RRC) filtered 8-PSK symbol waveform alphabet. [Figure 17] OBW calculated using SSA and FT-based techniques are shown. [Figure 18] OBW calculated using SSA and FT-based techniques are shown. DETAILED DESCRIPTION OF THE INVENTION

[0031] In the drawings and detailed description, the same or similar reference numbers may identify the same or similar elements. It is understood that practices, features, etc. described with respect to an embodiment in a particular figure may be implemented with respect to other embodiments in other figures, unless explicitly stated or otherwise possible.

[0032] Aspects of the present invention are disclosed in the specification and related drawings that may relate to specific embodiments of the invention. Alternate embodiments may be devised without departing from the spirit or scope of the invention. Additionally, well-known elements of exemplary embodiments of the present invention will not be described in detail or will be omitted so as not to obscure the relevant details of the invention. Furthermore, explanations of some of the arguments used herein may be included to facilitate understanding of the description.

[0033] The word "exemplary" is used herein to mean "serving as an example, instance, or illustration," and not as limiting. Any embodiment described herein as "exemplary" is not necessarily to be construed as preferred or advantageous over other embodiments. Likewise, the term "embodiments of the invention" does not require that all embodiments of the invention include the discussed feature, advantage, or mode of operation.

[0034] Furthermore, many embodiments are described in terms of sequences of operations to be performed by, for example, elements of a computing device. It will be recognized that the various operations described herein can be performed by specific circuitry (e.g., an application-specific integrated circuit (ASIC)), a field-programmable gate array, program instructions executed by one or more processors, or combinations thereof. In addition, a sequence of operations described herein can be considered to be fully embodied in any form of computer-readable storage medium storing a corresponding set of computer instructions that, when executed, cause an associated processor to perform the functions described herein. Accordingly, various aspects of the present invention may be embodied in several different forms, all of which have been contemplated to be within the scope of the claimed subject matter. Additionally, for each of the embodiments described herein, the corresponding form of such an embodiment may be described herein as, for example, “logic configured” to perform the described operations. For example, it will be understood that transmitters, receivers, management systems, and other devices in the systems of the present invention can include one or more processors, memory, storage, inputs and components, communication interfaces, and other components, which may be interconnected via one or more buses and circuit boards, cards, etc., as desired by those skilled in the art.

[0035] FIG. 1 illustrates an exemplary system 10 including an exemplary transmitter 12 and receiver 14 pair that may be used in a transmission or communication system, as further illustrated in FIG. 2. Bits, typically representing data / information, transmitted as signals through system 10 may be encoded in a channel encoder 16 section of transmitter 12, as well as having other signal processing performed to prepare the signal for transmission. The encoded bits may then be used to modulate a carrier having a center frequency provided by a carrier source 20 or frequency source using an external modulator 22, as shown in FIG. 1, or to directly modulate carrier / frequency source 20 to generate the transmission signal. The signal may be transmitted using one carrier or multiple carriers simultaneously, such as when implemented with instantaneous spectrum analysis ("ISA"), see above-incorporated U.S. Pat. No. 10,069,664.

[0036] Encoder 16 and decoder 18 are shown as single blocks in Figure 1. However, encoder 16 and decoder 18 may include one or more stages / components used to process information passing through system 10. The encoding and decoding functions may be performed internal and / or external to transmitter 12 and receiver 14, as desired by those skilled in the art.

[0037] In receiver 14, detector 24 may detect the transmitted signal and provide it to a signal processor, which may include a decoder 18 to perform any decoding necessary to output bits. The bits output from system 10 may or may not be in the form of data and clock signals. In various embodiments, system 10 may be monitored and / or controlled locally and / or remotely by a management system 25, as known in the art. System 10 may be deployed as part of a local private point-to-point network, as well as as part of a global terrestrial and satellite infrastructure, and managed accordingly.

[0038] 2 shows an exemplary system 10 including multiple transmitters 12 and receivers 14 that may be deployed in a variety of transmission and communication systems employing a variety of wired and wireless transmission media 26 and that may include the PSW technology of the present invention. Signal monitoring may be performed at transmitters, receivers, amplifiers / repeaters, and other devices within the system 10 and / or using standalone signal monitoring devices 27 deployed at various locations within the system 10. In the various devices, the present invention may be implemented as software, including firmware, embedded logic, etc., stored on computer-readable media such as memory, drives, and other storage devices and executed by processes running in proximity to (or near) the devices as known in the art.

[0039] 1 and 2 and other systems may be deployed in a variety of electrical and optical wireline transmission and communication networks, as well as satellite and terrestrial wireless networks. In various systems, the transmission signals may be multiplexed in a multiplexer 28 before transmission and may require demultiplexing after transmission before detection in a demultiplexer 30, as is commonly performed in wireline and wireless systems carrying multiple channels.

[0040] The systems, devices, and methods of the present invention have an electrical signal input, which may be generated in hardware by receiving a data / information-carrying signal, such as an optical or radio signal, or in software by simulating prior reception. Analysis of the input signal may be performed according to various analysis inputs provided by a user. For example, before analyzing the input signal at a center frequency f_c, a user may specify a frequency spectrum range (f_range) having a minimum frequency (f_min) and a maximum frequency (f_max). The analysis is performed in software, and the input signal may be stored so that a user can change the frequency range of interest (f_range) and re-run the analysis to evaluate the sensitivity of the analysis to various user inputs. Those skilled in the art will understand that a range of user inputs may provide suitable results. However, increasing the number of calculations performed to generate a result may affect the latency of the result.

[0041] Within a frequency spectrum range (f_range), a user can define at least one frequency bin having a frequency width f_width. This can be done by specifying the number of frequency bins and segmenting the frequency range of interest (f_range) accordingly. The width of each frequency bin is called f_width. While it is typically desirable to segment the frequency range evenly for efficiency and ease of use, there may be scenarios where it is desirable to segment the frequency range unevenly, such as a frequency range where signal power is expected to change rapidly (narrow bin) and not change significantly (wide bin).

[0042] For each frequency bin, a target intermediate frequency (f_target) and a corresponding software intermediate frequency bandpass filter ifbpf can be defined in terms of f_target, sampling frequency Fs, and f_width. Those skilled in the art can select various target intermediate frequencies depending on the particular application of the present invention. By using an intermediate target frequency, those skilled in the art can convert the signal power to a lower frequency. For example, for an RF signal in the 900 MHz range, the target intermediate frequency can be in the 100 KHz range. The intermediate frequency bandpass filter can be implemented using various techniques known to those skilled in the art. For example, using the MathWorks® MATLAB® filter design tool, which may include the designfilt function.

[0043] In signal power analysis, the sample time interval dt of the input signal can be selected to provide a sampling frequency Fs=1 / dt. The signal length sig_len can be determined from the signal length in samples. The sample time interval can be changed by one skilled in the art based on preference and the bit rate of the signal, as further described herein.

[0044] Depending on the particular application, it may be desirable to provide an input signal software bandpass filter INbf to remove frequencies outside the passband. Bandpass filters can be implemented using a variety of techniques known to those skilled in the art, as described above.

[0045] In operation, an input signal SIGin having power distributed across a signal frequency range is received. An input signal software bandpass filter INbf can be applied to the input signal SIGin to produce a filtered input signal, which is normalized to produce a normalized signal power SIGin_norm that can be stored for use in various calculations.

[0046] For each frequency bin, the mixing frequency f_mix suitable for converting the source frequency f_source associated with the frequency bin to the target IF frequency f_target by mixing, i.e., pairwise multiplication of the samples of the two streams, can be determined by applying the equation 2*cos(a)*cos(b)=cos(ab)+cos(a+b), defining a=f_source and b=f_mix, where: f_mix=f_source-f_target.

[0047] Then the mixing stream cos_mix of length sig_len equal to the input signal length can be calculated by evaluating the cosine function in the range 0 to rads_per_signal in increments of rads_per_sample, which is cos_mix=cos(0:rads_per_sample:rads_per_signal), where rads_per_sample=f_mix*dt*2*pi, and It can be written as rads_per_signal = rads_per_sample(sig_len-l).

[0048] The formula for cos_mix above is also the notation used in MATLAB® to calculate cos_mix.

[0049] The intermediate frequency (IF) signal sig_IF is It can be generated using the formula sig_IF=SIGin_norm*cos_mix.

[0050] A corresponding bandpass filter ifbpf is applied to each intermediate frequency (IF) signal sig_IF to generate a filtered IF signal sig_IF_filtered.

[0051] The bin power bin_power of the filtered IF signal sig_IF_filtered is calculated and stored by summing the squares of the amplitudes of sig_IF_filtered and dividing this sum by the time interval.

[0052] The frequency bin with the maximum bin_power, max_power, can be identified and used for various purposes. For example, the occupied bandwidth of the input signal can be calculated in various ways, such as by summing the frequency widths of the frequency bins based on the bin power, bin_power, of each frequency bin neighboring the frequency bin with the maximum power.

[0053] In various embodiments, the OBW may be calculated as the double-sided occupied bandwidth of the input signal by starting with the frequency width of the maximum power bin and then successively summing the frequency widths of the maximum power frequency bins adjacent to either side of the maximum power bin or a previously summed frequency bin until the sum of the power in the frequency power bins equals 99% or some desired percentage of the input signal power. The single-sided occupied bandwidth may be calculated by dividing the double-sided occupied bandwidth by two to find the average, or by summing the power of frequency bins that span frequencies lower and higher than the maximum power frequency bin, respectively.

[0054] The OBW can be calculated by successively summing the power from the lowest frequency bin to the highest frequency bin and calculating the 99% OBW as the difference between the frequencies at which 0.5% and 99.5% power are reached. This method can be used with the present invention. However, the above method, which starts from the peak power frequency bin and extends successively to adjacent frequency power bins, may more accurately reflect the desire to define the OBW centered around the peak power. If the spectrum is symmetrical around the peak power, the two methods may be equivalent; however, if the spectrum is not symmetrical around the peak power for some reason, they may diverge.

[0055] The system 10 and devices of the present invention include software and / or hardware for analyzing signal power over a range of frequencies, which may be useful for OBW and other signal processing and control purposes. Signal analysis methods may be implemented at various points in the system 10, including at the transmitter, receiver, and locations along the link, as needed, for example, for feedback loops and link performance analysis.

[0056] The performance of the SSA of the present invention was compared to an HP 8590B hardware-based swept-tuned spectrum analyzer (HSTSA) for time-amplitude sequences generated by the same polynomial symbol waveform (PSW) alphabet using a time-varying signal spectrum. For details about the PSW alphabet, see the above-incorporated U.S. Patent No. 10,069,664 and U.S. patent application Ser. No. 16 / 735,655, filed March 6, 2020, entitled "Devices, Systems, and Methods Employing Polynomial Symbol Waveforms," ​​the contents of which are incorporated by reference in their entirety except for the claims and any disclosure to the contrary. The comparison examines the effects of varying the stream / signal length, the number of frequency bins, and the bandpass filter length.

[0057] For comparison, an initial set of eight "polynomial symbol waveforms" (PSWs) was constructed as shown in Figure 3, which may be collectively referred to as the PSW alphabet. To create PSW alphabets that would generate signals with different OBW properties, the initial PSW alphabet was convolved with polynomials corresponding to Gaussian distributions with mean zero and sigma values ​​of 0.8, 1.0, 1.2, 1.6, and 2, as shown in Figures 4-8, respectively. The initial PSW alphabet (with its very high OBW) was not tested; only the convolved PSW alphabets were tested.

[0058] In each of the convolutional PSW alphabets, each PSW is assigned a 3-bit sequence.

[0059] In the following discussion, "bitstream length" refers to the number of transmitted bits. OBW measurements are based on the analysis of the time-amplitude sequences generated by converting a random sequence of bits into a corresponding random sequence of PSWs from a particular PSW alphabet. For the purposes of this study, all PSW alphabets have eight symbols corresponding to bit sequences of length three, and each polynomial symbol waveform is represented by 25 sample points.

[0060] The following parameters apply to all analyses reported below: Symbol time dt=1 microsecond, f_c=922MHz, where f_c is the center frequency of the signal. f_min=f_c-10MHz, f_max=f_c+10MHz, 30 trials per condition. Unilateral OBW is reported.

[0061] OBW is measured relative to the 99% power spectral density (PSD) width. Two approaches to OBW measurement have been implemented, although those skilled in the art may choose other approaches.

[0062] Method I: Add frequency power bins. Find the range of frequency power bins that contains 99% of the signal power. The results are shown in Figures 9 to 12.

[0063] Method II: Find the frequency difference between the frequency bin of the maximum power point and the frequency bin of 1% signal power. The results are shown in Figures 13 to 15.

[0064] Method I generally yielded better agreement with the HSTSA and much better test-retest agreement (lower standard deviation), as further described herein. The second approach yields reasonable results and may prove more suitable for other applications.

[0065] The following data from the trial reported in more detail below indicates that the agreement between SSA and HSTSA is less than 10%. A summary of the comparison is given in Table 1 below. Table 1: Method I: Comparison of SSA and HSTSA OBW JPEG0007762418000001.jpg69164

[0066] The data show that the concordance between SSA and HSTSA is less than 10%. Effect of bitstream length on OBW measurements

[0067] Figure 9 shows several comparisons made to see how SSA performance is affected by input stream length (100, 1,000, 10,000) (in bits). In the trials reported below, the SSA IF filter length is 500. The number of frequency bins is 100.

[0068] The data used in Figure 9 above is given below. PSW=Gaussian Sigma 0.8,bit stream len=100,Norm. OBW Diff=0.054167(0.027136),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,bit stream len=1000,Norm. OBW Diff=0.051042(0.015317),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,bit stream len=10000,Norm. OBW Diff=0.053125(0.014565),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 1.0,bit stream len=100,Norm。OBW Diff=0.05679(0.028744),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,bit stream len=1000,Norm。OBW Diff=0.05679(0.018793),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,bit stream len=10000,Norm。OBW Diff=0.059259(0.018455),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.2,bit stream len=100,Norm。OBW Diff=0.066667(0.026596),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,bit stream len=1000,Norm。OBW Diff=0.052564(0.018851),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,bit stream len=10000,Norm。OBW Diff=0.05641(0.019516),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,bit stream len=100,Norm。OBW Diff=0.011538(0.12822),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,bit stream len=1000,Norm。OBW Diff=-0.029487(0.024079),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,bit stream len=10000,Norm。OBW Diff=-0.038462(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 2.0,bit stream len=100,Norm. OBW Diff=-0.088889(0.1057),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,bit stream len=1000,Norm. OBW Diff=-0.084848(0.025353),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,bit stream len=10000,Norm. OBW Diff=-0.088384(0.01059), HSTSA OBW=6.6e+06

[0069] The (logarithmic) x-axis shows the number of transmitted bits. In Figures 9-15, the y-axis shows the normalized mean difference between SSA and HSTSA OBW for matched conditions, with the standard deviation across 30 trials per condition shown in parentheses. HSTSA values ​​are provided as the last number in each row. The standard deviation is indicated by the error bars.

[0070] Note that while SSA OBW results are provided for 100 to 10,000 simulated transmitted bits, the comparison is with HSTSA OBW, which is based on a 1.5 million bit transmission.

[0071] These data show that the average SSA OBW measurements agree with HSSTA by less than 10% for over 100 simulated transmitted bits. The standard deviation of the difference between the SSA and HSSTA measurements decreases as the number of transmitted bits increases.

[0072] In operation, SSA stores the entire input sequence and analyzes it repeatedly for different frequency bins, while HSTSA analyzes a different portion of the input signal for each frequency bin, which allows SSA to be more compact than HSTSA in terms of the required input sequence length.

[0073] Figure 10 shows a comparison of SSA performance as affected by the number of frequency bins (20, 40, 60, 80, 100, 120, 140). The number of transmitted bits is 10,000, and the SSA IF filter length is 500. The x-axis shows the number of frequency bins. The y-axis shows the normalized average difference between SSA and HSTSA OBW for matched conditions. The standard deviation is shown by the error bars based on the data below. PSW=Gaussian Sigma 0.8,f_num=20,Normalized OBW Diff=0.09375(0),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,f_num=40,Normalized OBW Diff=0.09375(0),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,f_num=60,Normalized OBW Diff=0.041667(0),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,f_num=80,Normalized OBW Diff=0.054688(0),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,f_num=100,Normalized OBW Diff=0.055208(0.013443),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,f_num=120,Normalized OBW Diff=0.059896(0.012138),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,f_num=140,Normalized OBW Diff=0.049107(0),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 1.0,f_num=20,Normalized OBW Diff=0.11111(0),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,f_num=40,Normalized OBW Diff=0.040123(0.039832),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,f_num=60,Normalized OBW Diff=0.049383(0),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,f_num=80,Normalized OBW Diff=0.064815(0),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,f_num=100,Normalized OBW Diff=0.061728(0.017758),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,f_num=120,Normalized OBW Diff=0.049383(0),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,f_num=140,Normalized OBW Diff=0.057319(0.00483),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.2,f_num=20,Normalized OBW Diff=0.15385(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,f_num=40,Normalized OBW Diff=0.057692(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,f_num=60,Normalized OBW Diff=0.089744(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,f_num=80,Normalized OBW Diff=0.057692(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,f_num=100,Normalized OBW Diff=0.05(0.017927),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,f_num=120,Normalized OBW Diff=0.057692(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,f_num=140,Normalized OBW Diff=0.046703(0.0083827),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=20,Normalized OBW Diff=0.15385(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=40,Normalized OBW Diff=-0.038462(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=60,Normalized OBW Diff=-0.038462(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=80,Normalized OBW Diff=-0.038462(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=100,Normalized OBW Diff=-0.038462(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=120,Normalized OBW Diff=-0.038462(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=140,Normalized OBW Diff=-0.039377(0.0050158),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 2.0,f_num=20,Normalized OBW Diff=-0.16667(0),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,f_num=40,Normalized OBW Diff=-0.13005(0.0069157),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,f_num=60,Normalized OBW Diff=-0.12626(0.012583),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,f_num=80,Normalized OBW Diff=-0.12374(0.0085185),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,f_num=100,Normalized OBW Diff=-0.091414(0.010885),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma2.0,f_num=120,Normalized OBW Diff=-0.12037(0.0089793),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma2.0,f_num=140,Normalized OBW Diff=-0.12229(0.0071619),HSTSA OBW=6.6e+06

[0074] Figure 11 shows a comparison between SSA and HSTSA as a function of SSA IF filter length (50, 100, 250, 500, 1000). The bitstream length for the trials reported below is 10,000. The number of frequency bins is 100. The x-axis shows the SSA IF filter length. The y-axis shows the normalized mean difference between SSA and HSTSA OBW for matched conditions. The standard deviation is indicated by the error bars.

[0075] The data used to generate Figure 11 is as follows: PSW=Gaussian Sigma 0.8,filter len=50,Normalized OBW Diff=0.084375(0.014565),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,filter len=100,Normalized OBW Diff=0.0625(0),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,filter len=250,Normalized OBW Diff=0.057292(0.011845),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,filter len=500,Normalized OBW Diff=0.052083(0.014983),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,filter len=1000,Normalized OBW Diff=0.054167(0.014056),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 1.0,filter len=50,Normalized OBW Diff=0.074074(0),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,filter len=100,Normalized OBW Diff=0.074074(0),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,filter len=250,Normalized OBWDiff=0.069136(0.012805),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,filter len=500,Normalized OBW Diff=0.060494(0.018153),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,filter len=1000,Normalized OBW Diff=0.055556(0.018835),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.2,filter len=50,Normalized OBW Diff=0.11538(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,filter len=100,Normalized OBW Diff=0.076923(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,filter len=250,Normalized OBW Diff=0.061538(0.019164),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,filter len=500,Normalized OBW Diff=0.052564(0.018851),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,filter len=1000,Normalized OBW Diff=0.053846(0.019164),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,filter len=50,Normalized OBW Diff=0.038462(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,filter len=100,Normalized OBW Diff=-0.038462(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,filter len=250,Normalized OBWDiff=-0.038462(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,filter len=500,Normalized OBW Diff=-0.038462(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,filter len=1000,Normalized OBW Diff=-0.038462(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 2.0,filter len=50,Normalized OBW Diff=-0.081818(0.009416),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,filter len=100,Normalized OBW Diff=-0.087374(0.0094858),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,filter len=250,Normalized OBW Diff=-0.087374(0.0094858),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,filter len=500,Normalized OBW Diff=-0.087879(0.010066),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,filter len=1000,Normalized OBW Diff=-0.089394(0.010027),HSTSA OBW=6.6e+06

[0076] For the purpose of efficient runtime, especially when used as part of a PSW alphabet design, a key question is how lightweight the SSA parameters can be while maintaining performance. Based on the data used in Figures 9-11, the following parameters were chosen to enable efficient execution: Transmit Bits: 10,000 Number of frequency bins: 100 IF filter length: 100

[0077] FIG. 12 shows the variation as a function of the number of transmitted bits, which are 100, 1,000, and 10,000.

[0078] Figures 13-15 show a comparison similar to that shown in Figures 9-11, but using Method II for calculating OBW instead of Method I. A summary of the comparison is provided in Table 2 below. Table 2: Method II: Comparison of SSA and HSTSA OBW TIFF0007762418000002.tif73165

[0079] In the Method II comparison, the number of transmitted bits was 1.5 million for both SSA and HSTSA. The data show agreement between SSA and HSTSA is within about 20%, which is worse than Method I above. Also, the Method II calculations have a much higher standard deviation across trials. These results may suggest that for these alphabets, Method I is preferred based on the HSTSA comparison, that the time average of HSTSA does not adequately capture the spectrum, or a combination of both.

[0080] Figure 13 shows the SSA performance of Method II as a function of input stream length (in bits). The SSA IF filter length is 500 and the number of frequency bins is 100. This yielded the following data, with the standard deviation over 30 trials for each condition given in parentheses. The HSTSA value is provided as the last number on each line. PSW=Gaussian Sigma 0.8,bit stream len=100,Normalized OBW Diff=0.11042(0.34922),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,bit stream len=1000,Normalized OBW Diff=0.2625(0.39542),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,bit stream len=10000,Normalized OBW Diff=0.28125(0.4235),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 1.0,bit stream len=100,Normalized OBW Diff=0.14815(0.34015),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,bit stream len=1000,Normalized OBW Diff=0.25926(0.36132),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,bit stream len=10000,Normalized OBW Diff=0.34074(0.32918),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.2,bit stream len=100,Normalized OBW Diff=0.15641(0.48038),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,bit stream len=1000,Normalized OBW Diff=0.16154(0.40166),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,bit stream len=10000,Normalized OBW Diff=0.2359(0.1905),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,bit stream len=100,Normalized OBW Diff=-0.012821(0.39744),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,bit stream len=1000,Normalized OBW Diff=0.082051(0.31744),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,bit stream len=10000,Normalized OBW Diff=0.084615(0.23761),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 2.0,bit stream len=100,Normalized OBW Diff=-0.43737(0.24743),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,bit stream len=1000,Normalized OBW Diff=-0.28889(0.20394),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,bit stream len=10000,Normalized OBW Diff=-0.23939(0.17167),HSTSA OBW=6.6e+06

[0081] Figure 14 shows the SSA performance of Method II as a function of frequency bin number (20, 40, 60, 80, 100, 120, 140). The number of transmitted bits is 10,000, and the SSA IF filter length is 500. The x-axis shows the frequency bin number. The y-axis shows the normalized mean difference between SSA and HSTSA OBW for matched conditions. Standard deviations are shown as error bars for the following data. PSW=Gaussian Sigma 0.8,f_num=20,Norm. OBW Diff=0.5(0.15132),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,f_num=40,Norm. OBW Diff=0.25(0.20517),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,f_num=60,Norm. OBW Diff=0.26736(0.27092),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,f_num=80,Norm。OBW Diff=0.24479(0.29868),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,f_num=100,Norm。OBW Diff=0.19792(0.4321),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,f_num=120,Norm。OBW Diff=0.28819(0.29077),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,f_num=140,Norm。OBW Diff=0.15179(0.28583),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 1.0,f_num=20,Norm。OBW Diff=0.48148(0),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,f_num=40,Norm。OBW Diff=0.42593(0.15494),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,f_num=60,Norm。OBW Diff=0.28395(0.23957),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,f_num=80,Norm。OBW Diff=0.2716(0.22802),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,f_num=100,Norm。OBW Diff=0.37531(0.34288),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,f_num=120,Norm。OBW Diff=0.29218(0.21865),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,f_num=140,Norm。OBW Diff=0.18166(0.24946),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.2,f_num=20,Norm。OBW Diff=0.29487(0.18851),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,f_num=40,Norm。OBW Diff=0.26923(0.13923),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,f_num=60,Norm。OBW Diff=0.25641(0.12776),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,f_num=80,Norm。OBW Diff=0.20192(0.21685),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,f_num=100,Norm。OBW Diff=0.28462(0.22418),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,f_num=120,Norm。OBW Diff=0.23504(0.2613),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,f_num=140,Norm。OBW Diff=0.28205(0.31525),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=20,Norm。OBW Diff=0.53846(0),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=40,Norm。OBW Diff=0.13462(0.19786),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=60,Norm。OBW Diff=0.089744(0.23265),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=80,Norm。OBW Diff=0.099359(0.26325),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=100,Norm。OBW Diff=0.010256(0.23361),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=120,Norm。OBW Diff=0.10043(0.23331),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,f_num=140,Norm。OBW Diff=0.043956(0.24741),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma2.0,f_num=20,Norm。OBW Diff=-0.23737(0.027663),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma2.0,f_num=40,Norm。OBW Diff=-0.22727(0.10969),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma2.0,f_num=60,Norm。OBW Diff=-0.23906(0.10438),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma2.0,f_num=80,Norm。OBW Diff=-0.25758(0.13552),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma2.0,f_num=100,Norm。OBW Diff=-0.32828(0.15268),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma2.0,f_num=120,Norm. OBW Diff=-0.26178(0.14972),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma2.0,f_num=140,Norm. OBW Diff=-0.26623(0.15725),HSTSA OBW=6.6e+06

[0082] Figure 15 shows the SSA performance of Method II as a function of SSA IF filter length (50, 100, 250, 500, 1000). The bitstream length for the trials reported below is 10000. The number of frequency bins is 100. PSW=Gaussian Sigma 0.8,filter len=50,Normalized OBW Diff=0.1875(0.46713),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,filter len=100,Normalized OBW Diff=0.1875(0.46713),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,filter len=250,Normalized OBW Diff=0.23958(0.4631),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,filter len=500,Normalized OBW Diff=0.2125(0.42757),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 0.8,filter len=1000,Normalized OBW Diff=0.22083(0.42666),HSTSA OBW=3.2e+06 PSW=Gaussian Sigma 1.0,filter len=50,Normalized OBW Diff=0.27407(0.22043),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,filter len=100,Normalized OBW Diff=0.42963(0.28732),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,filter len=250,Normalized OBW Diff=0.45926(0.28797),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,filter len=500,Normalized OBW Diff=0.43457(0.2934),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.0,filter len=1000,Normalized OBW Diff=0.39506(0.33303),HSTSA OBW=2.7e+06 PSW=Gaussian Sigma 1.2,filter len=50,Normalized OBW Diff=0.30513(0.12368),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,filter len=100,Normalized OBW Diff=0.2641(0.062862),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,filter len=250,Normalized OBW Diff=0.2359(0.12605),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,filter len=500,Normalized OBW Diff=0.24359(0.15128),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.2,filter len=1000,Normalized OBW Diff=0.30769(0.12935),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,filter len=50,Normalized OBW Diff=0.13077(0.21771),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,filter len=100,Normalized OBW Diff=0.0076923(0.25097),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,filter len=250,Normalized OBW Diff=0.04359(0.28186),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,filter len=500,Normalized OBW Diff=0.058974(0.21539),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 1.6,filter len=1000,Normalized OBW Diff=0.028205(0.27881),HSTSA OBW=2.6e+06 PSW=Gaussian Sigma 2.0,filter len=50,Normalized OBW Diff=-0.22525(0.10181),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,filter len=100,Normalized OBW Diff=-0.27778(0.14697),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,filter len=250,Normalized OBW Diff=-0.29394(0.13787),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,filter len=500,Normalized OBW Diff=-0.28788(0.14363),HSTSA OBW=6.6e+06 PSW=Gaussian Sigma 2.0,filter len=1000,Normalized OBW Diff=-0.27172(0.15359),HSTSA OBW=6.6e+06

[0083] Figure 16 shows the eight polynomial symbol waveform (PSW) alphabet corresponding to a root-raised cosine (RRC) filtered 8-phase shift keying (PSK) waveform (8-PSK). Because PSK waveforms are based on sinusoids with constant amplitude, they produce a relatively stationary spectrum within filtering and symbol boundary effects. FT-based spectral analysis techniques can therefore be used to accurately analyze PSK signals. The PSW alphabet corresponding to 8-PSK was constructed by starting with eight sinusoidal polynomials with even phase offsets between them and then convolving them with the polynomial corresponding to RRC for α = 0.1.

[0084] MATLAB® software simulations were performed to estimate the OBW at 99% power for the 8-PSK waveform shown in FIG. 16 using the SSA of the present invention and a MATLAB® FT-based OBW function called obw. As shown in FIGS. 17 and 18, respectively, the OBW calculated using the SSA and FT-based techniques closely matched within a few percent. Close agreement between the SSA of the present invention and the FT analysis is expected because the 8-PSK waveform has a stationary spectrum over time, which is an underlying assumption of the FT analysis.

[0085] In summary, our SSA performed better for signals with stationary spectra when compared to FT-based techniques, and for signals with non-stationary spectra when compared to HSTSA techniques. Thus, SSA provides a robust alternative to HSTSA for use in hardware systems, enabling for the first time the simulation and spectral analysis of signals with non-stationary spectra.

[0086] The foregoing description and accompanying drawings illustrate the principles, preferred embodiments, and modes of operation of the present invention. However, the present invention should not be construed as limited to the particular embodiments discussed above. Additional variations on the above-described embodiments will be apparent to those skilled in the art.

[0087] Accordingly, the above-described embodiments should be considered illustrative rather than restrictive, and it should be understood that modifications to these embodiments could be made by those skilled in the art without departing from the scope of the present invention as defined by the following claims.

Claims

1. 1. A method of signal power analysis, comprising: specifying, via a processor, a frequency spectrum range, the frequency spectrum range having a minimum frequency (f_min) and a maximum frequency (f_max); defining, via the processor, at least one frequency bin within the frequency spectrum range, the frequency bin having a frequency width f_width and a frequency source f_source associated therewith; specifying, via the processor, a target intermediate frequency (f_target) for each frequency bin; Identifying, via said processor, time intervals dt between signal samples; calculating via said processor a sampling frequency Fs=1 / dt and a signal sample length sig_len; defining, via said processor, for each frequency bin, a software intermediate frequency band pass filter ifbpf in terms of Fs, f_target, and f_width; defining, via said processor, an input signal software bandpass filter INbf to remove frequencies outside said frequency spectrum range; receiving, via said processor, an input signal SIGin distributed in amplitude and power over a signal frequency range; applying, via the processor, the input signal software bandpass filter INbf to the input signal SIGin to generate a filtered input signal; normalizing, via the processor, the filtered input signal to generate a normalized signal power SIGin_norm; storing, via the processor, in a memory, the normalized signal power SIGin_norm; calculating, via said processor, for each frequency bin, a mixing frequency f_mix, where f_mix=f_source-f_target; calculating, via said processor, for each frequency bin, a mixing stream cos_mix, cos_mix=cos(0:rads_per_sample:rads_per_signal), rads_per_sample=f_mix*dt*2*π, and rads_per_signal=rads_per_sample*(sig_len-l); generating, via said processor, for each frequency bin an intermediate frequency (IF) signal sig_IF, where sigIF=SIGin_norm*cos_mix; applying, via said processor, for each frequency bin, said band pass filter ifbpf to sig_IF to generate a filtered IF signal sig_IF_filtered; calculating, via the processor, for each frequency bin, a bin power, bin_power, in the filtered IF signal, sig_IF_filtered, by summing the squared amplitudes of sig_IF_filtered and dividing this sum by the time interval, dt, between the signal samples; storing, via the processor, in the memory, the bin power bin_power for each frequency bin; Identifying, via the processor, the frequency bin having a maximum bin_power, max_power_idx; calculating, via the processor, an occupied bandwidth of the input signal by summing the frequency widths of the frequency bins based on the bin power bin_power for each frequency bin; A method comprising:

2. To calculate, 2. The method of claim 1, comprising calculating a two-sided occupied bandwidth of the input signal by starting with the frequency width of the frequency bin having the maximum bin_power and then successively summing the frequency widths of the frequency bins of maximum power adjacent on either side of the maximum bin_power or a previously summed frequency bin until the sum of the powers in the frequency bins equals 99% or some desired percentage of the power of the input signal SIGin.

3. To calculate, The method of claim 2 , comprising calculating the single-sided occupied bandwidth by dividing the double-sided occupied bandwidth by two.

4. The method of claim 1 , wherein the input signal SIGin is received from at least one of a transmitter, a receiver, a monitoring device, and a software signal simulation.

5. The method of claim 1 , wherein the processor and the memory are in at least one of a transmitter, a receiver, a monitoring device, and a computer running a software signal transmission simulation on the processor.

6. adjusting a location of a center frequency of at least one frequency source based on the identified frequency bin having the maximum bin_power. The method of claim 1.

7. defining a software input signal bandpass filter IN-bf; applying the input signal bandpass filter IN-bf to the input signal SIGin to remove frequencies outside the frequency passband from the input signal SIGin; The method of claim 1 further comprising:

8. A transmitter; a receiver in communication with the transmitter; a signal monitoring device in communication with the transmitter to receive as an input signal SIGin at least a portion of a signal being transmitted from the transmitter to the receiver; receiving an input signal having a signal length sig_len, an amplitude, and a power distributed across an input frequency spectrum over a time interval dt; defining at least one frequency bin having a frequency width of the input frequency spectrum; normalizing the power of the input signal to generate a normalized signal power SIGin_norm; For each frequency bin, calculate a mixing frequency f_mix, f_mix=f_source-f_target, f_target = target intermediate frequency of the signal in said frequency bin; f_source = the frequency associated with the frequency bin and the mixed stream cos_mix; cos_mix=cos(0:rads_per_sample:rads_per_signal), rads_per_sample=f_mix*dt*2*π, rads_per_signal=rads_per_sample*(sig_len-l); and generating an intermediate frequency signal sig_IF=SIGin_norm*cos_mix for each of said at least one frequency bin; for each of the at least one frequency bin, applying an intermediate frequency (IF) band pass filter ifbpf to sig_IF to generate a filtered IF signal sig_IF_filtered; calculating a bin power, bin_power, of the filtered IF signal by summing the squares of the amplitudes in the filtered IF signal for each of the at least one frequency bin and dividing this sum by the time interval, dt; providing at least one of the bin power, the input signal power, and the normalized signal power to at least one of the transmitter, the receiver, the signal monitoring device, the management system, and the software simulation; and at least one processor.

9. A non-transitory computer-readable medium storing instructions, comprising: The instruction: When executed by one or more processors, for said one or more processors: receiving an input signal having a signal length sig_len, an amplitude, and a power distributed across an input frequency spectrum over a time interval dt; defining at least one frequency bin having a frequency width of the input frequency spectrum; Normalize the power of the input signal to generate a normalized signal power SIGin_norm, and for each frequency bin, calculate a mixing frequency f_mix, where: f_mix=f_source-f_target, f_target = target intermediate frequency of the signal in said frequency bin, f_source = frequency associated with said frequency bin and mixing stream cos_mix, where cos_mix = cos(0:rads_per_sample:rads_per_signal); rads_per_sample=f_mix*dt*2*π, rads_per_signal=rads_per_sample*(sig_len-1), generating an intermediate frequency signal sig_IF=SIGin_norm*cos_mix for each of said at least one frequency bin; for each of the at least one frequency bin, applying an intermediate frequency (IF) band pass filter ifbpf to sig_IF to generate a filtered IF signal sig_IF_filtered; calculating a bin power, bin_power, of the filtered IF signal by summing the squares of the amplitudes in the filtered IF signal for each of the at least one frequency bin and dividing this sum by the time interval, dt; A non-transitory computer-readable medium comprising one or more instructions.

10. 10. The non-transitory computer-readable medium of claim 9, wherein the one or more processors are located at least proximate to at least one of a transmitter, a receiver, a signal monitoring device, and a management system.

11. Calculating an occupied bandwidth of the input signal by summing the frequency widths of the frequency bins based on the bin power bin_power in each frequency bin. The non-transitory computer-readable medium of claim 9 further comprising:

12. Calculating the occupied bandwidth is identifying said frequency bin with max_power_idx, which is a maximum bin_power; starting from the frequency bin corresponding to max_power_idx, iteratively summing the power in the frequency bins at each iteration; summing the power of frequency bins at successively lower frequencies or successively higher bin indices containing more power; stopping the summation if the summed power is equal to or greater than a predetermined percentage of the power of the input signal; and subtracting a frequency corresponding to the lowest frequency bin used in the summation from a frequency corresponding to the highest frequency bin used in the summation to define an occupied band, obwp, corresponding to the predetermined percentage.

13. 13. The non-transitory computer-readable medium of claim 12, wherein the predetermined percentage is 99%.

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