Abstract:
A method of and apparatus removing of a plurality of relatively narrow banded signals in a relatively wide banded input signal. The method involves and the apparatus provides for compressively sensing one relatively narrow banded signal in the relatively wide banded input signal and removing one relatively narrow banded signal from the relatively wide banded input signal before detecting and removing another relatively narrow banded signal in the relatively wide banded input signal, the step of and apparatus for compressing sensing occurring with respect to both (i) the input signal with the previously detected narrow banded signals removed therefrom and (ii) a frequency shifted version of (i).

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This invention is an improvement over the invention described in U.S. patent application Ser. No. 13/091,020, filed Apr. 20, 2011 and entitled “Adaptive Interference Removal for Compressive Signal Detection and Reconstruction in Real Time”, the disclosure of which is hereby incorporated herein by reference. 
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     None. 
     TECHNICAL FIELD 
     Disclosed is an improved scheme for sampling, detecting and reconstructing analog signals residing in a wide spectrum through compressive sensing in real time. By applying compressive sensing techniques, this scheme is able to detect signals quickly over a very wide bandwidth. Unlike existing compressive sensing approaches which carry out the sampling and reconstruction procedures separately, in this new scheme, the sampling process and the detection/reconstruction process closely interact with one another. In this way, this new scheme offers real-time detection and reconstruction of signals, which is not available using known compressive sensing techniques. 
     BACKGROUND 
     Since first introduced by Candes (Candes and T. Tao. Near optimal signal recovery from random projection: universal encoding strategies?  IEEE Trans. on Information Theory , 52:5406-5425, 2006) and Donoho (D. L. Donoho. Compressed sensing.  IEEE Trans. on Information Theory , 52:1289-1306, 2006), compressive sensing has been adopted in many applications to save sampling resources. Besides signal reconstruction, this technique has also been used in many different signal processing fields. Signal detection is also an important area for applying this technique and has shown strong advantage for detecting signals over very wide frequency band (See Xiangming Kong and Mohin Ahmed, “Quick signal detection and dynamic resource allocation scheme for ultra-wideband radar”, Proc. SPIE 8021, 802111 (2011)). 
     A detection scheme we previously developed (see U.S. patent application Ser. No. 13/091,020 referred to above) has been shown to be able to track input signal S well when the input SNR is large and the signal dynamic range is moderate. However, when these conditions are not met, detections are missed and false alarm rate increases quickly. To overcome these shortcomings, a better detection scheme is proposed herein. 
     BRIEF DESCRIPTION OF THE INVENTION 
     In one aspect the present invention provides a method for detecting of a plurality of relatively narrowbanded signals in a relatively widebanded input signal, the method comprising: 
     (i) compressively sensing the strongest narrowbanded signal in said input signal and determining a center frequency of said first narrowbanded signal; 
     (ii) utilizing the center frequency and the bandwidth of said strongest narrowbanded signal to remove the strongest narrowbanded signal from said input signal and to produce a version of the input signal which version corresponds to said input signal without said strongest narrowbanded signal; 
     (iii) repeatedly compressively sensing additional narrowbanded signals in the input signal and repeatedly utilizing a center frequency and a bandwidth of each additional narrowbanded signal to remove the each additional narrowbanded signal from said input signal and thereby to produce said version of the input signal which version corresponds to said input signal less said strongest narrowbanded signal and less each additional narrowbanded signal as the additional narrowbanded signals are repeatedly compressively sensed and removed from said input signal; 
     (iv) assigning each compressively sensed narrowbanded signal which is removed from the input signal to a separate narrowband receiver for signal analysis; and 
     wherein the compressive sensing which occurs for said strongest narrowbanded signal and with respect to each additional narrowbanded occurs with respect to both said input signal less all previously detected narrowbanded signals and a frequency shifted version of said input signal less all previously detected narrowbanded signals. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1 a  and 1 b    are schematic diagrams setting forth the architecture of the disclosed system having a single compressive sensing unit; 
         FIG. 2  is schematic diagrams setting forth the architecture of the disclosed system having two compressive sensing units; 
         FIGS. 2 a  and 2 b    depict possible embodiment of the narrowbanded receivers shown in  FIG. 1   a;    
         FIG. 2 c    depicts an embodiment of the compressive sensing units shown in  FIG. 2 ; 
         FIG. 3  shows two frequency spectrums extending from 0 (DC) to 100 GHz each with two signals in it, one frequency spectrum being shifted by preferably a half bandwidth of a subband; 
         FIG. 4  shows a plot of the detection rate at different SNR based on simulation studies. 
         FIGS. 4 a  and 4 b    present flow diagrams of the steps which preferably occur at the signal detectors and receivers, respectively. 
         FIG. 4 c    shows a general purpose computer which may be utilized in practicing the present invention. 
         FIG. 4 d    shows two embodiments of non-volatile computer technology readable medium which may be utilized to store computer technology readable instruction means. 
         FIGS. 5 a  and 5 b    depict the results of simulations done showing the improvements provided by the new design disclosed herein. 
     
    
    
     DETAILED DESCRIPTION 
     The Earlier Solution 
     Before discussing the improved detection scheme proposed herein, it might be useful to again consider the technology (the earlier solution) disclosed in U.S. patent application Ser. No. 13/091,020. The earlier system and the present system are designed to cover very wide bandwidth S simultaneously while sampling at below the Nyquist rate. Compressive sensing is applied to the sampling process to reach simultaneous coverage of the entire supported band. By adding an interference removal procedure in the sampling process that closely interacts with the signal detection procedure, our system becomes an adaptive sampling system. 
     Let W be the entire bandwidth covered by the system. This bandwidth is divided into N frequency subbands. Signals in each subband will be treated as a different signal. Hence, the input is typically composed of a mixture of multiple narrowband signals. We further assume that at most M out of N bands will be simultaneously occupied at any moment (U.S. patent application Ser. No. 13/091,020, filed Apr. 20, 2011 discusses embodiments for which this assumption does not have to be made). The input signal can be approximated (with this assumption) as 
     
       
         
           
             
               
                 
                   
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     At the front end of the system, the input is divided into M+1 paths by a power divider  10  as shown by  FIG. 1 a   . The first path is denoted as the main path P 0 , whose major functionality is monitoring the entire spectrum and performing signal detection. In this path, measurements are taken and processed through a signal detector  30 . The signal detection process follows an iterative manner Measurements are taken in rounds. In each round, a set of measurements is taken. From these measurements, preferably only one signal (the strongest) is detected during each iteration. Through each set of measurements, the signal detector  30  can quickly determine the center frequency of the strongest signal (found during a given iteration) in the input on path P 0  downstream of summing node  60 . Then this information is sent to one of M narrowband receivers  40  which will preferably (i) analyze that signal and (2) cooperate with summing node  60  to remove that signal from path P 0  so that the signal detector can go on to detect the next strongest signal still coming through the summing node  60 . In the narrowband receivers  40 , the input is filtered to isolate this signal which is being analyzed and removed from P 0 . Besides processing the removed signal to obtain other information about the signal, the narrowband receiver  40  also sends the isolated signal to the main path P 0  where it is subtracted (removed) from the input as interference by summing node  60 . Then another round of measurements is taken from the “clean” input and the detection process will be repeated to detect another signal. This system architecture is shown in  FIG. 1 a    herein and in  FIG. 1 b    herein which respectively generally correspond to  FIGS. 3 a  and 3 b    of U.S. patent application Ser. No. 13/091,020. The iterative process described about is described in even greater detail in U.S. patent application Ser. No. 13/091,020. 
     In detecting one (the strongest) signal in the input, the signal detector  30  processes the input (with interference removed, i.e., downstream of summing node  60 ) in a compressed sensing fashion using compressed sensing measurements block  20 . The input in the main path with interference removed is first modulated by a pseudorandom signal p(t) which has the format 
     
       
         
           
             
               
                 
                   
                     
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     where p n  is a sequence of random numbers that takes values±1 randomly and N is an integer number, which is the number of point frequencies used to approximate the signal. The samples y[n] can be expressed in matrix format as: 
               y   =       HDFs   ⁢           ⁢     where   ⁢   H       =     [         1       1       1       1                                                                                                                                                           1       1       1       1                                                                                                                                                           1       1       1       1         ]         ⁢                         approximates   ⁢           ⁢   the   ⁢           ⁢   integration   ⁢           ⁢   process     ,         
D=diag(p n ) and F=[e −jπn(2k+1)/N ] n,k  for n=0, 1, . . . , N−1 and k=−N/2, −N/2+1, . . . , N/2−1.
 
     Since in each set of measurements, we only need to determine the most significant coefficient in s, we can directly apply the first step of Orthogonal Matching Pursuit (OMP) which guarantees to find the coefficients in the support of the signal if the number of nonzero coefficients in the signal satisfies the condition 
                              x   0          0     &lt;       1   2     ⁢     (           μ   2     ⁡     (   Á   )         -   1       +   1     )         ,           Eqn   ⁢           ⁢     (   3   )                 
where μ 2 (Á)=           |a i *,a j | is the coherence of the measurement matrix. OMP is discussed in Tropp (J. A. Tropp and A. C. Gilbert, Signal Recovery from Random Measurement via Orthonal Matching Pursuit,  Information Theory, IEEE Transaction on , vol. 53, no. 12, pp. 4655-4666, 2007).
 
The Improved System

     Processing the samples in rounds (iteratively) as discussed above is equivalent to multiplying the signal by a square window. Although each narrowband signal only occupies one frequency subband, after windowing, the bandwidth of the narrowband signal is enlarged. Hence, when we project the signal onto the N center frequencies (of the N subbands), the coefficient for the frequencies of the subbands not occupied by the signal can also be large and resulting in a false alarm. Although increasing the window size reduces the bandwidth enlargement, as long as the window exists, the problem cannot be completely overcome. 
     To reduce this problem, we obtain preferably two sets of the measurements, one set being directly from the input signal (downstream of summing node  60 —see  FIG. 2 ), and the other set from the input signal frequency shifted by preferably a half bandwidth of a subband. 
     Before considering  FIG. 2  in greater detail, consider  FIG. 3 .  FIG. 3  depicts two frequency spectrums extending from 0 (DC) to 100 GHz each with two signals in it, one frequency spectrum being shifted by preferably a half bandwidth of a subband (see the shift noted by B/2 in the upper most depicted spectrum) as noted above. Each frequency spectrum is divided into N subbands numbered 1 through N, each with width B=W/N. We arbitrarily and preferably assume that at most M out of N subbands will be simultaneously occupied at any given moment (this assumption reduces the amount of equipment needed to practice this invention—so the assumption need not be made by using additional equipment). 
     Signals in each subband will be treated as a different signal. Hence, the input signal x(t) is composed of a mixture of multiple narrowband signals. In  FIG. 3  only three subbands are shown as being occupied by a signal. The signal at subbands n2 and n2+1 is wider than W/N and therefore multiple (in this case two) subbands are assigned to it and it is treated as being made up of multiple (two in this case) different signals in the frequency spectrum. The signals in subbands n, n2 and n2+1 comprise an input signal x(t) to the disclosed system. The number of subbands signals in the input signal x(t) may occupy as many as M subbands. When the shifted signal is processed, we take into consideration the half bandwidth shift, therefore, each original narrowband signal is preferably still centered at the relative axis. 
     In the detection process, we obtain two sets of frequency coefficients, one from each measurement set associated with each of the two frequency spectrums shown in  FIG. 3 . These two coefficient sets are power combined (see Eqn. (5) below). Correspondingly, the system architecture changes into the system architecture of  FIG. 2  which has two compressive sensing measurement blocks  20  and  20   fs . Compressive sensing measurement block  20   fs  is shifted frequency-wise by one half the bandwidth of a subband (B/2 in  FIG. 3 ). Unlike using two sets of direct measurements which may still leave large power out of the desired band whereas the in-band power is small, the frequency shift between the two sets of measurements ensures that at least one set of coefficients catch large in-band power. This is equivalent to using two point frequencies to approximate each band instead of one as with the original disclosure of U.S. patent application Ser. No. 13/091,020. In this way, the contrast between the coefficients for the occupied band and those for the empty band is increased. Hence, the system is more robust. This process can be expressed in the following equations: 
     
       
         
           
             
               
                 
                   
                     
                       
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     The cost paid to get this improvement is that we need to add one more measurement path (to accommodate  20   fs ), which increases the hardware complexity somewhat. Hence there is a trade-off between the detection accuracy and hardware complexity. In our simulation, we found that two branches for the main path (Po downstream of summing node  60 ) are good enough for SNR around 0 dB, whereas the previous approach has good performance only for SNR larger than 20 dB. Since the increase in the hardware cost is not that high, this new architecture is a better choice for low SNR applications. 
     Schematic diagrams of two possible embodiments of the narrow band receivers  40  are depicted by  FIGS. 2 a  and 2 b    which show two different embodiments of one of the narrowband receivers  40  in greater detail. Within a given narrowband receiver  40 , the input is preferably first filtered to isolate the signal. To do this, there are two options. First, the signal can directly pass through a bandpass filter  49  to remove all other signals and noise and then be demodulated to baseband by mixing it with a carrier e −j2πf     m     t  (where f m  is the center frequency of band m, m=1, 2, . . . N) at a mixer  41  before applying it to a signal processor  44 . This embodiment is shown by  FIG. 2 b   . However, since a receiver  40  may later be dedicated to process another signal with a different center frequency, it means the bandpass filter  49  should have an adjustable center frequency. This can be difficult to implement. Hence, a second option shown by  FIG. 2 a    is preferably utilized. In the embodiment of  FIG. 2 a    the input is first demodulated to baseband by mixing it with a carrier e −j2πf     m     t  at a first mixer  41 . The demodulated signal then passes through a lowpass filter  42  to remove all other signals and noise. The bandwidth of the lowpass filter  42  should preferably be equal to the width of a subband W/N. Then the signal is then modulated back to its original passband through another mixer  43  by a carrier e j2πf     m     t  before being directed to summing node  60  via multiplexer  50 . The advantage of this embodiment of  FIG. 2 a    is that all narrowband receivers  40  have the same components except the center frequency of the oscillators that generate the carriers. Usually the center frequency is easy to adjust. The modulated signal output from mixer  43  is the same as that in the input to mixer  41 , except that it has effectively been bandpass filtered using this technique. Hence, it is fed to the main path where it is subtracted as interference at summing node  60 . Therefore, the input to the measurement block x 1 (t) no longer contains the isolated signal and hence it will not be detected again. This interfering signal removal technique is preferably carried out as described with reference to the embodiment of  FIG. 2   a.    
     Tuning of the narrowband receivers  40  is controlled by the carrier frequency of the two mixers  41 ,  43  in the narrowband receiver embodiment of  FIG. 2 a    or the bandpass filter  49  in the embodiment of  FIG. 2 b   . In  FIGS. 2 a  and 2 b   , all the signal processors  44  handle the signal at baseband and therefore the signal processors  44  may be separate entities in each receiver  40  or a common signal processor  44  may be used for a number or all receivers  40 , if desired. 
     Compressive sampling is now explained with reference to  FIGS. 2 and 2   c . The input in the main path P 0 , occurs after summing node  60  where the output of the summing node is divided into two paths, one going to CS measurement block  20  and the other going CS measurement block  20   fs . The input in the main path with interfering signals removed (at the summing node  60 ) is first modulated by a pseudorandom sequence p(t) in CS measurement block  20 , which pseudorandom sequence has the form: 
     
       
         
           
             
               
                 
                   
                     
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     where p n  is a random sequence of ±1 and N is the number of subbands. The modulated signal is lowpass filtered at block  23  and sampled by switch  24  at a rate R. This rate is lower than the Nyquist rate 2 W and was empirically chosen. The bandwidth of the lowpass filter h(t) should be roughly equal to the bandwidth of a subband. See  FIG. 2 c    which shows a preferred embodiment of both the compressive sensing (or measurement) block  20  as well as compressive sensing (or measurement) block  20   fs . Compressive sensing (or measurement) block  20   fs  is identical to compressive sensing (or measurement) block  20  except that the output from the summing node  60  is frequency shifted in block  25  before being modulated at a mixer  22  by a pseudorandom sequence in block  21  as explained above. 
     The same (or a different) pseudorandom sequence can be used in the compressive sensing (or measurement) blocks  20  and  20   fs . 
     In the embodiments of the compressive sensing (or measurement) blocks  20  and  20   fs  the pseudorandom sequence p(t) is generated by a pseudorandom number generator  21  and modulated with the signals appearing on P main  by the mixer  22 . The output of the mixer  21  is applied to a filter  23  having a bandwidth h(t) and thence to a sampling gate or sampler  24 . The outputs y 1 [n] and y 2 [n] in  FIG. 2 c    are applied to signal detector  30  as shown in  FIG. 2 . 
     The signal detection process follows in an iterative manner. See  FIG. 4 a    for a flow diagram of the steps SD 1 -SD 10  which preferably occur at the signal detector  30  and  FIG. 4 b    for the steps R 1 -R 9  which preferably occur at a receiver  40 . Measurements are taken in rounds and the detection process is preferably accomplished using a suitably programmed processor in signal detecter  30 . In each round, only one signal (in a given subband) is detected by the signal detector  30 , but Nc samples are taken. That is why  FIG. 4 a    mentions that “a set of measurements are taken” at step SD 1 . Then the signal detector  30  reports the center frequency of the subband occupied by the detected signal by a control message to a given one of the narrowband receivers  40  of the set of receivers  40   1 - 40   M  which is dedicated to isolate this signal. The narrowband receiver  40  first tests whether there is truly a signal in this subband preferably by energy detection (is the energy above some threshold (Thr). The test result is sent back to the signal detector  30  as a feedback message. If there is a signal in the subband, the narrowband receiver assigned to it would preferably process it further. Otherwise, the narrowband receiver in question would preferably wait for further control message from the signal detector. So, after the first round, receiver  401  isolates the first detected interfering signal and, after the second round, receiver  40   2  isolates the second detected interfering signal and so forth. The isolated signal(s) also occur in the main path P 0  (along with all the other signals in x(t)) and they are iteratively removed by a summing node  60 . In the next round of measurements, the previously detected signal is subtracted from the input as interference and a new signal will be detected. The subtraction of signals received by the narrowband receivers  40  from the signals (x(t)) on P 0  occurs at a subtracting input of the summing node  60  after the signals from the narrowband receivers  40  are combined by a multiplexer  50 . The signal x 1 ( t ), after having the signals received by the narrowband receivers  40  subtracted therefrom, occurs on path P main . The signal on P main  is applied to compressive sensing measurements block  20  which is described in greater detail with reference to  FIG. 2 c   . This process continues until either all signals are removed or the system runs out of receivers  40  since the number of interfering signals must then greater than the number M of receivers  40 . Each interfering signal (appearing as signals s 1 ( t )-sM(t)) can be subjected to analysis by a signal processor  44  associated with each receiver. The signals s 1 ( t )-sM(t) may be digitized and applied to a digital signal processor (DSP) for such analysis and a single DSP can be used with multiple receivers  40 , if desired, using well known multiplexing techniques. 
     The flow diagram of the steps SD 1 -SD 10  of  FIG. 4 a    which preferably occur at the signal detector  30  are preferably implemented by software means (such as a computer program product comprising computer-readable instruction means preferably stored on a non-transitory computer-readable medium, which could be a non-volatile computer-readable medium such as that shown in  FIG. 4 d   , that are executable by a CPU of a computer) running on a CPU in or cooperating with signal detector  30 . The a flow diagram of the steps R 1 -R 9  of  FIG. 4 b    which preferably occur at receiver  40  are preferably implemented by software means (such as a computer program product comprising computer technology readable instruction means preferably stored on a non-volatile computer technology readable medium that are executable by a CPU of a computer) running on a CPU in or cooperating with receiver  40 . So each receiver  40  preferably has a CPU  41  associated therewith and signal detector  30  likewise preferably has a CPU  31  associated therewith. The CPUs  31 ,  41  are preferably located in their respective receivers  40  and in the signal processor  30 , although the CPUs  31 ,  41  could alternatively be located remotely and, moreover, a single CPU (or multiple CPUs) could control multiple receivers  40  and also the signal processor  30 , if desired. The non-volatile computer technology readable medium may implemented by a hard disk or other non-volatile memory means such as flash memory or read only memory. The non-volatile computer technology readable medium may be disposed locally at the CPUs  31 ,  41  or remotely, if desired, 
     The term “CPU” as used in the preceding paragraph is intended to refer, in one embodiment, to a Central Processing Unit which typically comprises the hardware within a computer that carries out the instructions of a computer program (the software means noted above) by performing the basic arithmetical, logical, and input/output operations of the computer utilizing the CPU. The CPU may be implemented as an integrated circuit which itself may comprise more than one CPU to allow the computer with which the CPU is associated to multitask. So the term “CPU” as used in the preceding paragraph is also intended to refer, in another embodiment, to a CPU in a multitasking computer where the CPU resources may be shared among the signal detectors  30  and/or the receivers  40  mentioned above. The term “CPU” as used in the preceding paragraph is intended to refer, in still yet another embodiment, to a computational system where multiple parallel processing units are used where no one of the multiple parallel processing unit is necessarily “central” to the others. The term “CPU” as used in the preceding paragraph, in still another embodiment, may be implemented as a field programmable gate array. 
     The term “computer” is intended to refer to the CPU and the other elements with which the CPU communicates directly, such as communication ports for transferring data for the signals detectors  30  and/or receivers  40  noted above between those elements and the CPU, as well as communication data and/or address lines connected to such communication ports and/or the computer technology readable instruction means noted above. The term computer can also refer other units traditionally associated with CPUs, such a displays, data input devices, disk drives and the like, but it is anticipated that in most embodiments such units would preferably not be employed, except perhaps during initial testing and design of the overall system disclosed herein, since the CPUs are preferably located in their respective receivers  40  and in the signal processor  30 , as noted above. 
     A computer may be implemented as a application specific, dedicated or imbedded processor on one hand and be implemented in their respective receivers  40  and in the signal processor  30 , as noted above, or it may be implemented as a more general purpose computer  1200  such as that shown in  FIG. 4 c    on the other hand, particularly during initial design and/or testing. Some practicing the present invention may choose to use a more general purpose computer  1200  at all times and not just during initial design and/or testing. An exemplary computer system  1200  in accordance with an embodiment is shown in  FIG. 4 c   . Exemplary computer system  1200  is configured to perform calculations, processes, operations, and/or functions associated with a program or algorithm. In one embodiment, certain processes and steps discussed herein are realized as a series of instructions (e.g., software program) that reside within computer readable memory units and are executed by one or more processors of exemplary computer system  1200 . When executed, the instructions cause exemplary computer system  1200  to perform specific actions and exhibit specific behavior, such as described herein. 
     Exemplary computer system  1200  may include an address/data bus  1210  that is configured to communicate information. Additionally, one or more data processing unit, such as processor  1220 , are coupled with address/data bus  1210 . Processor  1220  is configured to process information and instructions. In an embodiment, processor  1220  is a microprocessor. Alternatively, processor  1220  may be a different type of processor such as a parallel processor, or a field programmable gate array. 
     Exemplary computer system  1200  is configured to utilize one or more data storage units. Exemplary computer system  1200  may include a volatile memory unit  1230  (e.g., random access memory (“RAM”), static RAM, dynamic RAM, etc.) coupled with address/data bus  1210 , wherein volatile memory unit  1230  is configured to store information and instructions for processor  1220 . Exemplary computer system  1200  further may include a non-volatile memory unit  1240  (e.g., read-only memory (“ROM”), programmable ROM (“PROM”), erasable programmable ROM (“EPROM”), electrically erasable programmable ROM “EEPROM”), flash memory, etc.) coupled with address/data bus  1210 , wherein non-volatile memory unit  1240  is configured to store static information and instructions for processor  1220 . Alternatively exemplary computer system  1200  may execute instructions retrieved from an online data storage unit such as in “Cloud” computing. In an embodiment, exemplary computer system  1200  also may include one or more interfaces, such as interface  1250 , coupled with address/data bus  1210 . The one or more interfaces are configured to enable exemplary computer system  1200  to interface with other electronic devices and computer systems. The communication interfaces implemented by the one or more interfaces may include wireline (e.g., serial cables, modems, network adaptors, etc.) and/or wireless (e.g., wireless modems, wireless network adaptors, etc.) communication technology. 
     In one embodiment, exemplary computer system  1200  may include an input device  1260  coupled with address/data bus  1210 , wherein input device  1260  is configured to communicate information and command selections to processor  1220 . In accordance with one embodiment, input device  1260  is an alphanumeric input device, such as a keyboard, that may include alphanumeric and/or function keys. Alternatively, input device  1260  may be an input device other than an alphanumeric input device. In an embodiment, exemplar computer system  1200  may include a cursor control device  1270  coupled with address/data bus  1210 , wherein cursor control device  1270  is configured to communicate user input information and/or command selections to processor  1220 . In an embodiment, cursor control device  1270  is implemented using a device such as a mouse, a track-ball, a track-pad, an optical tracking device, or a touch screen. The foregoing notwithstanding, in an embodiment, cursor control device  1270  is directed and/or activated via input from input device  1260 , such as in response to the use of special keys and key sequence commands associated with input device  1260 . In an alternative embodiment, cursor control device  1270  is configured to be directed or guided by voice commands. 
     In an embodiment, exemplary computer system  1200  further may include one or more optional computer usable data storage devices, such as storage device  1280 , coupled with address/data bus  1210 . Storage device  1280  is configured to store information and/or computer executable instructions. In one embodiment, storage device  1280  is a storage device such as a magnetic or optical disk drive (e.g., hard disk drive (“HDD”), floppy diskette, compact disk read only memory (“CD-ROM”), digital versatile disk (“DVD”)). Pursuant to one embodiment, a display device  1290  is coupled with address/data bus  1210 , wherein display device  1290  is configured to display video and/or graphics. In an embodiment, display device  1290  may include a cathode ray tube (“CRT”), liquid crystal display (“LCD”), field emission display (“FED”), plasma display or any other display device suitable for displaying video and/or graphic images and alphanumeric characters recognizable to a user. 
     Exemplary computer system  1200  is presented herein as an exemplary computing environment in accordance with an embodiment. However, exemplary computer system  1200  is not strictly limited to being a computer system. For example, an embodiment provides that exemplary computer system  1200  represents a type of data processing analysis that may be used in accordance with various embodiments described herein. Moreover, other computing systems may also be implemented. Indeed, the spirit and scope of the present technology is not limited to any single data processing environment. Thus, in an embodiment, one or more operations of various embodiments of the present technology are controlled or implemented using computer-executable instructions, such as program modules, being executed by a computer. In one exemplary implementation, such program modules include routines, programs, objects, components and/or data structures that are configured to perform particular tasks or implement particular abstract data types. In addition, an embodiment provides that one or more aspects of the present technology are implemented by utilizing one or more distributed computing environments, such as where tasks are performed by remote processing devices that are linked through a communications network, or such as where various program modules are located in both local and remote computer-storage media including memory-storage devices. 
     We tested our improved system through simulation. The detection rate at different SNR values is plotted in  FIG. 5 a   . Three different types of input are tested. Each input is made of 9 signals. In the first type of input, all 9 signals have random amplitudes. In the second type, one signal s amplitude is 10 times larger than all other signals. In the third type, all signals have equal amplitude. The result shows that the detection rate is lowest for the second type and highest for the first type. Since the noise power is determined by the entire input, in the second type of input, when the noise is strong, only one signal has relatively large SNR. Other signals are all too weak compared to the noise. Hence, we can only detect the dominant signal. While in other types of signals, multiple signals are strong enough to be detected. Hence, the detection rate would increase. In the third type of input, since all signals have equal power, the interference from band expansion is larger than other types of signals. Hence, the first type of input has the best detection rate. As a comparison, the results are also given for the system where only one set of measurements is taken (marked as original algorithm). We can see that with one extra set of measurements, the detection rate obviously improves a lot. With the new system, the detection rate is close to 100% for SNR as low as 0 dB. 
     To check the system s performance on detecting and tracking frequency hopping signals, which is the main type of signal this disclosed invention intended to cover, we also simulated nine frequency hopping signals of random power. The detecting and tracking result at 20 dB SNR is given in  FIG. 5 b   . Notice that the 20 dB SNR is calculated from entire signal power. The maximum signal dynamic range is around 35 dB. Hence, for weak signals, the individual SNR drops below −10 dB. The time-frequency distribution of the input is plotted in  FIG. 5 a    while  FIG. 5 b    depicts the detection result. In this plot, the gray dots represents the time-frequency of the signals correctly detected. The circles represent the time-frequency of the undetected signals. The dark circle corresponds to the signals not detected due to lack of narrowband receiver or other signals being detected in the same time slot. The two circles marked with a “TM” correspond to true misses. That is, no signal is detected in that time slot although there is at least one narrowband receiver available for processing the signal. The stars correspond to the false alarm. Dark stars are false alarms due to late detection of a signal s disappearance. The single stars marked with a “FA” is a true false alarm when the detected frequency does not correspond to any signal. From the simulation, we can see the system performs pretty well at 20 dB SNR. 
     It is possible that adding even more compressive sensing measurement blocks could provide additional benefits. Thus instead of adding only one compressive sensing measurement block  20   f , (and shifting the input frequency by one half a subband bandwidth), two compressive sensing measurement blocks  20   f , could be used instead with a first one of the two added compressive sensing measurement blocks  20   fs  would be shifted by ⅓ subband bandwidth and the second one of the two added compressive sensing measurement blocks  20   f , would be shifted by ⅔ subband bandwidth with reference to the input frequency. So this improvement envisions adding one or more compressive sensing measurement blocks  20   fi  each of which would shift the input frequency by a different fraction of a subband bandwidth depending on the total number of compressive sensing measurement blocks  20  and  20   fs  utilized. 
     This concludes the description including preferred embodiments of the present invention. The foregoing description including preferred embodiments of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise forms disclosed. Many modifications and variations are possible within the scope of the foregoing teachings. Additional variations of the present invention may be devised without departing from the inventive concept as set forth in the following claims.