the embodiments herein and the various features and advantageous details thereof are explained more fully with reference to the non - limiting embodiments that are illustrated in the accompanying drawings and detailed in the following description . descriptions of well - known components and processing techniques are omitted so as to not unnecessarily obscure the embodiments herein . the examples used herein are intended merely to facilitate an understanding of ways in which the embodiments herein may be practiced and to further enable those of skill in the art to practice the embodiments herein . accordingly , the examples should not be construed as limiting the scope of the embodiments herein . the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the fall scope of the invention . as used herein , the singular forms “ a ”, “ an ” and “ the ” intended to include the plural thrills as well , unless the context clearly indicates otherwise . it will be further understood that the terms “ comprises ” and / or “ comprising ,” when used in this specification , specify the presence of stated features , integers , steps , operations , elements , and / or components , but do not preclude the presence or addition of one or more other features , integers , steps , operations , elements , components , and / or groups thereof . it will be understood that , although the terms first , second , etc . may be used herein to describe various elements , components , regions , layers and / or sections , these elements , components , regions , layers and / or sections should not be limited by these terms . for example , when referring first and second photons in a photon pair , these terms are only used to distinguish one element , component , region , layer or section from another region , layer or section . thus , a first element , component , region , layer or section discussed below could be termed a second element , component , region , layer or section without departing from the teachings of the present invention . unless otherwise defined , all terms ( including technical and scientific terms ) used herein have the same meaning as commonly understood by one of ordinary skill , in the art to which this invention belongs , it will be further understood that terms , such as those defined in commonly used dictionaries , should be interpreted as having a meaning that is consistent with their meaning in the context of the relevant art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein . a preferred embodiment of the present invention comprises a sparse - recovery model and reconstruction scheme . the invention allows the reconstruction of a wideband signal that has missing frequency bands . although the invention is demonstrated with the application of this technique to radar applications , the sampling and reconstruction technique of the present invention also works with other communication systems , including television reception and transmission , headphones with noise filters , and other communications devices ; especially ones that employ ultra high frequency modulation schemes . a preferred embodiment of the present invention comprises a sparsity - driven spectral recovery ( ssr ) technique that coherently estimates the spectral information in the frequency bands of the received data that may be either completely corrupted ( due to the presence of interference sources ) or nonexistent ( because of no transmission in the prohibited frequency bands ). a key innovative concept of the invention is that although the information is lost in the received signal due to the contaminated and missing frequency bands , the signal can be modeled as a linear combination of spectrally filtered and time - shifted versions of the transmitted waveform . thus , by constructing a dictionary that comprises spectrally filtered and time - shifted versions of the transmit waveform , the technique solves for a sparse solution of the coefficients that represent the significant components contributing to the received signal . the reconstruction using the preferred embodiment sparsity - driven spectral recovery ssr technique on spectrally corrupted and spectrally notched data is well matched with the original data , while conventional techniques only attempt to reduce the resulting artifacts rather than recover the original information . an advantage of the preferred embodiment technique is that it not only suppresses noise and artifacts from noisy and frequency notched signals , such as uwb data , it also recovers the original complex data ( both amplitude and phase ). although the amplitude information is important for the first stage of signal ( or target ) detection , using this technique the target signature is preserved in the complex data that includes the phase content , which is the key for target discrimination and classification , and coherent change detection . another advantage of this technique is that because the contiguous ( no abrupt changes ) spectral content is recovered , this technique can achieve better performance with higher snr in the suppression of the artifacts in the resulting data and imagery than other techniques that only attempt to reduce the noise and artifacts . in addition , since the resulting data comprises one single substantially contiguous frequency hand instead of multiple non - contiguous subbands , subsequent processing ( image formation , target detection and discrimination . etc .) steps are simple because they do not have to perform on multiple data sets . in addition , more information ( wideband without gaps ) would translate to better performance . a third advantage of this technique is that it can also be employed to extrapolate the spectral information outside the included band . thus , the technique effectively increases the bandwidth and improves the resolution of the signal . this will result in better performance in the detection and discrimination of targets . although a preferred embodiment of the invention discloses application of this technique for radar applications only , it is within the skill of the art to use the spectral recovery techniques of the present invention apply to such applications as communications systems , including uwb communication systems . in accordance with the principles of the present invention , the influence of noise signals is effectively removed from radar signals via sparse recovery . the preferred embodiment technique estimates and subtracts noise signals from the contaminated radar signals in the time domain . therefore , it does not suffer from high sidelobe and reduced target - amplitude effects as in existing notch - filtering approaches . additionally , the technique is completely adaptive with the changing environment and does net assume any knowledge ( amplitude , frequency band , modulation scheme , how many interference sources are present , etc .) of the interference sources . the preferred embodiment technique is based on a sparse recovery approach that simultaneously solves for ( i ) radar signals embedded in noise with large amplitudes and ( ii ) interference noise signals . fig1 illustrates overall schematic system block diagram of a preferred embodiment . at each pulse repetition interval ( pri ), the radar transmitter ( 101 ) transmits radar signals to the area of interest and receives return radar signals that correspond to the physical objects from the area . the transmitter 101 transmits an ultra wide band signal ( uwb ), however , the use of the preferred embodiment is not limited to uwb signals . the receiver ( 102 ) receives the return radar signals , or backscattered pulses that are reflected from objects in the scene . the backscattered pulses are generally contaminated by interference sources ( radio , tv , cellular phone , etc .). thus , the receiver 102 receives the signal with a spectral content which 1 ) lacks the “ notched ” content from non - transmission frequency bands , and 2 ) has the corrupted frequency bands due to interference sources . a preferred embodiment of the present invention essentially extracts or “ subtracts out ” the noise . at regular intervals { k | k mod l = 0 }, the transmitter is turned off so that the receiver only collects pure interference noise signal . this data collection scheme is illustrated in fig2 . in order to determine the background noise , the preferred embodiment system incorporates noise - only intervals that occur when the transmitter 101 is turned off and the data switch 109 sends the noise signals to the memory buffer in which dictionary 105 resides via channel 111 to construct a “ noise ” dictionary . during the standard radar transmit and receive cycle a pulse repetition frequency ( prf ) is established where the signal is first transmitted by the transmitter . depending upon the range or distance in which targets are to be detected , the receiver waits a time interval t = 2r / c where r is the range and c is the speed of flight . during the operation of the preferred embodiment however , after a certain number of prf cycles , transmission is stopped and the noise is observed whereupon the data switch 109 is programmed to allow noise data to enter the dictionary 105 . noise data is preferably entered into the dictionary 105 using a shifted window mechanism or approach . that is , data for the first column of the dictionary 105 is entered from the noise signal ( i . e ., the return signal occurring when no transmission from the transmitter 101 is taking place ). to enter the second column , the window is effectively shifted a minute amount via the sliding window , and the second column is data realized from the shifted interval of the noise signal . similarly , columns of noise data are entered into the dictionary 105 during the time interval when the switch 109 diverts the data from the signal recovery processor 103 into the dictionary 105 . noise data using the sliding window approach may be entered by an additional processor or processor or such operations may be performed by the signal recovery processor . note further that the switch 109 may be embedded or realized in software or may comprise a demultiplexer configuration , in a manner known to those skilled in the art . moreover , the channel 111 needs not connect through the switch 109 but instead may be connected to the receiver 102 . moreover , receiver 102 may comprise a plurality of receivers , one of which is tasked with capturing noise from the received signal . the noise in the return signal is detected during the intervals during which the transmitter does not transmit . hence , there is no signal component present in the return signal in this case . it is noted that the signal may be transmitted and received in a matter of nanoseconds thus the interval between the prf cycles and the reception of the noise only signal for the purposes of entering data into the dictionary 105 may occur within a matter of nanoseconds or tens of nanoseconds . the channel 111 may include a change detector to detect changes or fluctuations in the noise such that when the noise signal changes or fluctuates substantially , the interval during which noise data is captured is more frequent . likewise , if the noise does not change rapidly and remains relatively constant , then the interval between entering the noise data into the dictionary 105 is greater , allowing more time to be devoted to the standard prf cycles of transmitting and receiving the wide - band signals . at other intervals when the transmitter are turned on , the data switch 109 allows the received signals that include noisy radar signals to pass to the signal recovery processor 103 for processing . in addition to the noise dictionary 105 , the signal recovery processor 103 uses dictionary 104 that includes many time - shifted versions of the transmitted waveform . the upper or first dictionary 104 includes the time shifted versions of the original transmit waveform . this data may be stored in column beginning with data at the beginning of the range of interest . for example , if the range of interest is between 1000 meters to 5000 meters , the first column of data in the dictionary 104 may be a pulse representing , the appearance of a target at 1000 meters . the second column of data may then represent a pulse returned from a target at a short increment of distance from 1000 meters , the increment depending upon the number of column entries of data and the range of interest . the signal recovery processor 103 utilizes the noise dictionary 105 to estimate and extract the noise signal components from the noisy receive signals to generate clean radar signals . specifically , the lower or second dictionary 105 includes the column of noise data which are to be extracted out of the processed , signal in combination with the processing of the time - shifted versions of the transmitted waveform included in dictionary 104 . the output of the signal recovery processor 103 is the receive signal with contiguous spectral content . in a typical communications system , the output signals from the signal recovery processor 103 may be processed by a detector ( not shown in fig1 ). in radar block diagram of fig1 , the output signals from the signal recovery processor 103 are sent to the image processor ( 106 ), and subsequently , to the target detection and discrimination ( 107 ). the resulting synthetic aperture radar ( sar ) image and detection information are then visualized by the display ( 108 ). a further description of the image formation is found in u . s . patent application no . 2010 / 0141508 entitled “ method and system for forming an image with enhanced contrast and / or reduced noise , by lam h . nguyen and jeffrey sichina , published jun . 10 , 2010 , herein incorporated by reference . referring now to the dictionary 104 , this dictionary is constructed from time - shifted versions of the transmit signal s ( t ). in the absence of any spectral notches and interference sources , the received signal y ( t ) would be simply the summation of reflections of all targets within the radar &# 39 ; s range swath and beamwidth , i . e ., ideally the received signal would be composed of a linear combination of delayed and weighted replicas of the transmitted pulse s ( t ) as follows : y ⁡ ( t ) = ∑ i = 1 m ⁢ α i ⁢ s ⁡ ( t - τ i ) , ( 1 ) where the weighting coefficients α i represent the target amplitudes and the time - shifting parameters τ i model the phase change that corresponds to the delay time it takes the transmit signal pulse s ( t ) to travel the distance between the transmitter / receiver and the point target of interest . in other words , the received signal may comprise the output of a linear time invariant ( lti ) system modeled by the reflection coefficients and its sparsity is directly related to the complexity level of the scene . with the addition of the rfi noise r i ( t ) and the system noise w i ( t ), the complete signal model is shown in the top of fig9 . it is important to note that although the observed scene might be complex with many objects , the complexity m of the receive data record is significantly reduced since the reflections from all targets that have the same range distance from the radar transmitter / receiver would be represented by a single reflection coefficient and phase . the publication by lam nguyen and trac tran , “ robust recovery of synthetic aperture radar data from uniformly under - sampled measurements ,” ieee igarss , ( jun . 28 , 2011 ) ( hereby incorporated by reference ), describes how the model can robustly represent real sar data , and is herein incorporated by reference . fig9 illustrates the received signal in digital format 220 , the signal dictionary 104 , the rfi noise dictionary 105 , and the sparse coefficient vector memory portion 250 . the signal dictionary 104 of fig9 is the same as the dictionary 104 of fig1 in u . s . application ser . no . 13 / 477 , 282 . the columns 231 , 232 , 233 , 234 are selected by the nonzero entries in the solution α i , in the top part of vector memory portion 250 . vector memory portion 250 contains two sections ; one selection is the sparse coefficients 231 c , 232 c , 233 c , 234 c of the solution and the other portion is the rfi noise coefficients e i represented by 241 c , 242 c , 243 c , which will determine the location of the nonzero entries in noise dictionary 105 . the sparse coefficients 231 c , 232 c , 233 c , and 234 c are associated with the columns 231 , 232 , 233 , 234 and the rfi noise coefficients 241 c , 242 c , 243 c are associated with the columns 241 , 242 , 243 . the vector coefficients in vector memory portion 250 , which includes both α i and e i , is obtained from the l 0 norm or l 1 norm minimization . as shown in fig9 , the sparse - signal sparse - noise model for the received radar data record y i ( where i can be considered as the pri index ) may be represented as : y i = x i + r i + w i = d i x α i + d i rfi e i + w i ( 5 ) using this model , the original sar signal x i is assumed to be sparse with respect to d i x — the time - shifted dictionary in box 104 of fig9 constructed from our transmitted signal s ( t ). with the data collection strategy as illustrated in fig2 and using the assumption that the interference noise signal r i is relatively stationary within a small temporal window , the interference noise signal r i also has its own sparse representation : r i = d i rfi e i , where d i rfi is the time - adaptive rfi noise sparsifying , dictionary ( box 105 ) constructed from the either or both nearby pure - noise data records : y p ( past ) and / or y f ( future ). more specifically , if y p , n and y f , n represents the n - point noise sequences collected at time n from data record p and f , respectively , y p , n and y f , n can be represented as two column vectors as follows y p , n ≡[ y p [ n ] y p [ n + 1 ] y p [ n + 2 ] . . . y p [ n + n − 1 ]] t y f , n ≡[ y f [ n ] y f [ n + 1 ] y f [ n + 2 ] . . . y f [ n + n − 1 ]] t ( 6 , 6a ) next , the noise dictionary ( 105 ) can be constructed from a collection of such vectors at various times n ranging from 0 to n − 1 as shown in the matrix below d i rfi =[ y p , 0 y p , 1 . . . y p , n − 1 | y f , 0 y f , 1 . . . y f , n − 1 ] ( 7 ) in other words , columns of the noise dictionary 105 as represented in equation ( 7 ) are n - point noise segments of different phase shifts extracted directly from y p ( past ) and y f ( future or anticipated ) via a sliding window mechanism ( moving a window of size n to various different locations in the rfi data record to extract various rfi noise dictionary atoms ). note that more entries can be added to the dictionary since a few past noise records can be exploited rather than just a single noise record y p . in fact , a sophisticated learning dictionary can be considered to further improve the dictionary and hence the resulting sparse representation . the trade - off is generally in the computational complexity of the learning ( or loading data into dictionary 105 ) as well as the recovery algorithm . in practice , construction of a dictionary 105 using equation ( 7 ) is efficient and little performance can be gained from venturing far away temporally . referring now to fig2 , a flow diagram of the methodology of a preferred embodiment is set forth . in box 201 , a time - shifted dictionary 104 is generated . this may occur on a one time basis at the start of operations or may be updated as desired to reflect changes in the target area or target range . the circle 202 represents the step of inputting the time - shifted data into the dictionary 104 . in box 203 the receiver is operational to receive either waveforms reflected from targets ( in combination with noise ) or , when the transmitter 101 is shut off during certain predetermined intervals , noise signal data . diamond 204 reflects the operation of switch 109 by which the noise data is directed into the rfi noise dictionary 105 , as described above . box 205 represents the input of the noise data into the dictionary 105 . continuing in the flow chart of fig2 , in box 206 , using the data from both dictionaries 104 and 105 , the system solves for the sparse solution that represents both noise and data as represented mathematically by equation ( 5 ). continuing in the description of the flow chart of fig2 , in box 207 , the noise contents of the inputted signal y i is estimated as r i = d i rfi e i and in box 208 the signal is reconstructed by extracting the estimated noise contents as x i = y i − r i = y i − d i rfi e i where the dense noise w i in equation ( 5 ) has been ignored . the steps 203 through 208 are then repeated with the acquisition of new data in another aperture in order to form a complete image . box 210 represents the image formulation and box 211 represents the display of the final sar image . reference is made to u . s . patent application publication no . 2010 / 0141508 , herein incorporated by reference , for the further information relating to boxes 210 and 211 . referring hack again in equation ( 5 ), the w i is the system &# 39 ; s dense noise with small bounded energy ∥ w i ∥ 2 ≦ σ . the main difference between the two noise sources w i and e i is that e i is sparse and can be captured with only a few significant entries , but each can be large in magnitude , whereas w i is dense and generally insignificant , in magnitude . this leads to the following optimization problem , which can be solved approximately with orthogonal matching pursuit ( omp ): { α i ′ , e i ′ } = arg ⁢ ⁢ min α i , e i ⁢ {  α i  0 +  e i  0 } ⁢ ⁢ ⁢ ⁢ subject ⁢ ⁢ to ⁢ ⁢ y i = [ d i x d i rfi ] ⁡ [ α i e i ] + w i . ( 8 ) it has been well - known in the compressed sensing literature that the following relaxed convex optimization problem can be implemented as an alternative to matching pursuit type algorithms in equation ( 8 ) where λ and τ are tuning parameters that control the trade - offs between the sparsity priors and the data consistency constraint { α i ′ , e i ′ } = arg ⁢ ⁢ min α i , ɛ i ⁢  y i - d i x ⁢ α i - d i rfi ⁢ e i  2 + λ ⁢  α i  1 + τ ⁢  e i  1 . ( 9 ) note that in both formulations of equations ( 8 )-( 9 ), the entries in both dictionaries 104 and 105 should be normalized . hence , the optimization is not dependent on the noise energy level if the parameters λ and τ are pre - determined appropriately . the resulting noise - suppressed signal can then be computed as x i = y i − d i rfi e i . each data record that is expected to contain the sar signal of interest is recovered independently . all are then supplied to the image processor ( 106 ) to produce the final sar image . a number of recovery techniques in the current compressed sensing ( cs ) literature can be employed to solve the optimization problem in equation ( 8 ). in a preferred embodiment , orthogonal matching pursuit ( omp ) was used due to its simplicity , recovery robustness , and fast computation . omp is also very simple to set up : the only required parameter is the sparsity level k . once the sparsest coefficient vector α is found , the raw data record is recovered as in equation ( 9 ). all of the recovered data records are then supplied to the back - projection image formation algorithm to produce the final sar image . find gγ n ε d with maximum inner product rf n , gγ n ; an extension of matching pursuit ( mp ) is its orthogonal version : orthogonal matching pursuit ( omp ). the main difference with mp is that coefficients are the orthogonal projection of the signal f on the dictionary d . in fact , this algorithm solves the sparse problem : min x ⁢  f - dx  2 2 ⁢ ⁢ such ⁢ ⁢ that ⁢ ⁢  x  0 ≤ n , with ∥ χ ∥ 0 the l 0 pseudo - norm equal to the nonzero elements of x . fig3 is a description of a generic orthogonal matching pursuit algorithm . in box 220 α , which represents the coefficients , is initialized . in relation to the preferred embodiment , the coefficients relate to what columns are selected as “ best ” matches , in box 221 the parameter r , referred to as the residual , is set initially to y , the incoming received signal . in box 222 , the operation is conducted to find out what is in the dictionary d i what is the column that most looks like signal being examined . in box 223 , the objective is to try to find the index of columns with the best match between r and the dictionary d i , which is the solution in box 224 , the chosen index or indices are appended to the solution vector . in box 225 , the residue r is updated where d pi the pseudo inverse of the matrix ( dictionary ) d . the subtrahend represents the pseudo inverse matrix ( dictionary ) multiplied by the matrix d multiplied by the signal y . the process stops when all indices corresponding to the solution have been determined . if all indices have not been determined , there is a loop back to box 222 ; as once one best match is found , the system loops back to find the second best match and so forth . the term α a represents all indices of columns corresponding to the approximated solution , which is set forth in box 226 . the noise extraction technique of the present invention was tested and evaluated using the data from the u . s . army research laboratory ( arl ) uwb low - frequency sar that transmits radar signals occupying the frequency spectrum from 50 to 1150 mhz . interference noise data were collected from the real environment with the antenna pointing toward washington . d . c ., where a typical example is shown in fig4 . for each aperture location i where radar data are collected , the noise record was randomly selected and added to the raw radar data record . the noise - only records used to construct the corresponding noise dictionary were chosen such that the time interval between when the noise records and the radar records are collected are at least 1 ms apart . this delay amount yields a typical radar pulse repetition frequency of 1 khz . fig5 is an illustration of the frequency spectrum showing frequency ranges in which fm , digital tv and cellular phones operate . fig6 illustrates a comparison of the time - domain plots of a typical original radar record , its interference noise corrupted version , and its rfi extracted version . the interference corrupted version shows that the target responses are completely obscured by the strong interference noise signals . although a small level of residual noise still remains in the rfi extracted radar record , responses from the targets are essentially recovered and very well matched with the original data . fig7 illustrates a visual depiction of the resulting sar images after interference noise suppression at various interference levels ( where the interference noise signal strength in db scale is 7 , 13 , and 20 db stronger than the radar signal , respectively , when measured in the raw data domain ). the figures in the left column of fig7 show the original side - looking sar images formed using data from the arl radar without any interference . this sar image shows a few targets ( vehicles ) hiding behind foliage and tree area . the figures in the middle column depict the sar images formed using radar data contaminated with noise signals without any processing . finally , the figures in the right column of fig7 illustrate the resulting sar images after the proposed rfi extraction technique has been applied to the contaminated radar data . for the three cases shown in fig7 , the level of rfi suppression ( which is different than the strength of the noise interference ) turns out to be 11 . 3 , 13 . 5 , and 14 . 7 db , respectively . in all cases , as illustrated in the follow - up fig8 ( where the three data points associated with fig7 are in square boxes ), an average of more than 10 db of rfi energy level is consistently suppressed after the image formation stage ( i . e ., a signal - to - noise ratio ( snr ) is measured in the sar image domain ). when the noise level increases , the suppression level can get up to 15 db and levels off at around 15 . 5 db . in addition to radar , the invented noise extraction technique works with any communications systems that suffer from interference noise . examples of such practical systems and applications include : radars operating in combative environments with frequent enemy jamming ; audio recordings with background noise ; noise - cancellation headphones in noisy environments such as airplanes , airports , construction sites , etc . the practical applications of the invention in relation to radar include wide area mapping , surveillance , target detection and discrimination , concealed target detection , through - the - wall building mapping , tunnel detection , ground penetration radar , collision and obstacle , imaging , radar . also , the present invention may be used for radar jamming avoidance ( the noise being the jamming signal ). the present invention may be used with high speed data links and networks . another practical application is usage in noise cancelling systems such as headphones , car and recording technology . an advantage of the preferred embodiment described , is that it is adaptive with the changing environment . it does not assume any knowledge ( amplitude , frequency band , modulation scheme , number of noise sources , etc .) of the interference sources . a preferred embodiment technique directly estimates and subtracts interference noise signals from the radar signals in the time domain . therefore , it does not suffer from : 1 ) high sidelobes , and 2 ) reduced target - amplitude effects as in existing notch - filtering approaches . the preferred embodiment technique simultaneously solves for ( i ) the radar signals embedded in interference noise with large amplitudes and ( ii ) the interference noise signals within one unified sparsity - driven optimization framework . such techniques operate in the low - frequency spectrum that spans from under 100 mhz to several ghz in order to penetrate foliage , walls , etc . and employ wide - bandwidth signals to achieve the desired resolution . other approaches , such as notched filters result in ( i ) large sidelobes in the time domain of the received signal and ( ii ) reduced target amplitudes . in addition to radar , the invented noise extraction technique works with any communications systems that suffer from interference noise . the techniques provided by the embodiments herein may be implemented on an integrated circuit chip ( not shown ). the chip design is created in a graphical computer programming language , and stored in a computer storage medium ( such as a disk , tape , physical hard drive , or virtual hard drive such as in a storage access network ). if the designer does not fabricate chips or the photolithographic masks used to fabricate chips , the designer transmits the resulting design by physical means ( e . g ., by providing a copy of the storage medium storing the design ) or electronically ( e . g ., through the internet ) to such entities , directly or indirectly . the stored design is then converted into the appropriate format ( e . g ., gdsii ) for the fabrication of photolithographic masks , which typically include multiple copies of the chip design in question that are to be formed on a wafer . the photolithographic masks are utilized to define areas of the wafer ( and / or the layers thereon ) to be etched or otherwise processed . the resulting integrated circuit chips can be distributed by the fabricator in raw wafer form ( that is , as a single wafer that has multiple unpackaged chips ), as a bare die , or in a packaged form . in the latter case the chip is mounted in a single chip package ( such as a plastic carrier , with leads that are affixed to a motherboard or other higher level carrier ) or in a multichip package ( such as a ceramic carrier that has either or both surface interconnections or buried interconnections ). in any case the chip is then integrated with other chips , discrete circuit elements , and / or other signal processing devices as part of either ( a ) an intermediate product , such as a motherboard , or ( b ) an end product . the end product can be any product that includes integrated circuit chips , ranging from toys and other low - end applications to advanced computer products having a display , a keyboard or other input device , and a central processor . the embodiments herein may comprise hardware and software embodiments . the embodiments that are implemented in software include but are not limited to , firmware , resident software , microcode , etc . furthermore , the embodiments herein can take the form of a computer program product accessible from a computer - usable or computer - readable medium providing program code for use by or in connection with a computer or any instruction execution system . for the purposes of this description , a computer - usable or computer readable medium can be any apparatus that can comprise , store , communicate , propagate , or transport the program for use by or in connection with the instruction execution system , apparatus , or device . the medium can be an electronic , magnetic , optical , electromagnetic , infrared , or semiconductor system for apparatus or device ) or a propagation medium . examples of a computer - readable medium include a semiconductor or solid state memory , magnetic tape , a removable computer diskette , a random access memory ( ram ), a read - only memory ( rom ), a rigid magnetic disk and an optical disk . current examples of optical disks include compact disk — read only memory ( cd - rom ), compact disk — read / write ( cd - r / w ) and dvd . a data processing system suitable for storing and / or executing program code will include at least one processor coupled directly or indirectly to memory elements through a system bus . the memory elements can include local memory employed during actual execution of the program code , bulk storage , and cache memories that provide temporary storage of at least some program code in order to reduce the number of times code must be retrieved from bulk storage during execution . input / output ( i / o ) devices ( including but not limited to receivers , etc .) can be coupled to the system either directly or through interverting i / o controllers . network adapters may also be coupled to the system to enable the data processing system to become coupled to other data processing systems or remote printers or storage devices through intervening private or public networks . modems , cable modem and ethernet cards are just a few of the currently available types of network adapters . the compressed sensing theory asserts that under several mild conditions of the matrix s n , the above l 0 - norm minimization problem ( l 0 - norm is simply its number of significant ( non - zero ) elements ) can be efficiently solved by recasting it as the following l 1 - norm convex optimization : and the original signal can be recovered as x *= s n a *. all of the recovered data records are then supplied to the time - domain projection image formation algorithm to produce the final sar image . there are various techniques for solving the inverse linear system of equations for the sparsest a * in the compressed sensing community , which can be classified into two categories : basis pursuit via linear programming ( as described in e . candès and t . tao , “ decoding by linear programming ,” ieee trans . on information theory , vol . 51 , no . 12 , pp . 4203 - 4215 ( december 2005 )( hereby incorporated by reference ) or gradient projection ( as described in m . a . t . figueiredo , r . d . nowak , and s . j . wright . “ gradient projection for sparse reconstruction : application to compressed sensing and other inverse problems ,” ieee journal of selected topics in signal processing : special issue on convex optimization methods for signal processing , vol . 1 , no . 4 , pp . 586 - 598 ( april 2007 )( hereby incorporated by reference ); matching pursuit via orthogonal matching pursuit ( as described in j . tropp and a . gilbert , “ signal recovery from random measurements via orthogonal matching pursuit ,” ieee trans . on information theory , vol . 53 , no . 12 , pp . 4655 - 4666 ( december 2007 )( hereby incorporated by reference )) regularized orthogonal matching pursuit as described in d . needell and r . vershynin , “ signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit ,” ieee journal of selected topics in signal processing , vol . 4 , pp . 310 - 316 ( april 2010 ) ( hereby incorporated by reference )), subspace pursuit ( as described further in w . dai and o . milenkovic , “ subspace pursuit for compressive sensing : closing the gap between performance and complexity ,” ieee trans . on information theory , vol . 55 , pp . 2230 - 2249 ( may 2009 ) ( hereby incorporated by reference )), sparsity adaptive matching pursuit ( as described further in t . t . do , l . gan , n . h . nguyen , and t . d . tran , “ sparsity adaptive matching pursuit algorithm for practical compressed sensing ” in proc . asilomar conf . on signals , systems , and computers , pp . 581 - 587 , pacific grove ( october 2008 )( hereby incorporated by reference ), etc . in this implementation the orthogonal matching pursuit technique may be employed as described in j . tropp and a . gilbert , “ signal recovery from random measurements via orthogonal matching pursuit ,” ieee trans , on information theory , vol . 53 , no . 12 , pp . 4655 - 4666 ( december 2007 ) ( hereby incorporated by reference ) to solve for a * due to its simplicity , recovery robustness , and fast computation . this algorithm is graphically illustrated in fig3 . in the vector space c n of n - dimensional signals , vector x can be represented as x = ψa , where ψ is called the sparsifying matrix which in other words , the representation through ψ can be ( and in many cases , should be ) close to the original signal x . the sparsifying matrix is said to be complete if its columns span the entire n - dimensional space . the signal x is said to be strictly k - sparse when there are only k non - zero components in a . when the sorted magnitudes of ( a i ) decay very quickly and x can be well approximated with only k components , then x is said to be k - compressible . the effectiveness of the recovery algorithm of a class of signal x heavily depends on the sparsest representation of the signal class . in standard compressed sensing , fixed linear transform bases such as the dct , fft and the discrete wavelet transform ( dwt ) or a combination of all three are often employed to obtain sparsity . in the uwb sar system , the transmitted pulse is a monocycle impulse with an approximated bandwidth range of 300 - 3000 mhz . current collected raw data captured in the 8 receivers do not exhibit any common sparse pattern . in other words , a quick spectrum analysis reveals that the raw data is not time - sparse or frequency sparse or even wavelet sparse . hence , a naïve direct application of compressed sensing ( cs ) via random projection with fourier , cosine , or wavelet bases yields disappointing results . as used herein “ processor ” may include but is not limited to a computer , central processing unit ( cpu ), microprocessor , multiprocessor , main frame computer , personal computer , or laptop computer . as used herein the terminology “ matching pursuit ” means a technique for finding the “ best matching ” projections of multidimensional data onto an over - complete dictionary . as stated in wikipedia , the basic idea is to represent a signal from hilbert space as a weighted sum of functions ( called atoms ) taken from called atoms ) taken from d : f ⁡ ( t ) = ∑ n = 0 + ∞ ⁢ a n ⁢ g γ a ⁡ ( t ) where indexes the atoms that have been chosen , and a weighting factor ( an amplitude ) for each atom . given a fixed dictionary , matching pursuit will first find the one atom that has the biggest inner product with the signal , then subtract the contribution due to that atom , and repeat the process until the signal is satisfactorily decomposed . by taking an extremely redundant dictionary one can look in it for functions that best match a signal . finding a representation where most of the coefficients in the sum are close to 0 ( sparse representation ) is desirable for signal coding and compression . as used herein , the terminology “ sparsity driven ” of “ sparsity - driven ” is a relative term relating to the finding of a compressible solution which is intended to be treated broadly . for example , a sparse matrix is a matrix with enough zeros that it pays to take advantage of them ; commonly interpreted from an economics view point in that if one can save time and memory by exploiting the zeros , then a matrix is sparse . the terminology sparsity refers to the selection of a model , within a hierarchy of model classes , that yields a compact representation ; i . e . a model that depends on only a few of the observations , selecting a small subset of features for classification or visualization . selection of an optimal representation which is sufficiently sparse enables efficient computation by optimization techniques and alleviates the extreme difficulty encountered without sufficient sparsity . as used herein , the terminology “ target ” area means area of interest , which may be , for example , a scene , an animal or human body or portion thereof , face ( as in face recognition ), object , ground region , field , landscape , aerial environment , or a combination thereof . the term “ noise ” as used herein relates to observation noise . there are many sources that cause noise in the resulting observed signal . noise can be divided into two categories : additive noise and multiplicative noise . system noise , thermal noise , quantization noise , self - interference noise , radio frequency interference ( rfi ) noise are some examples of the additive noise . multiplicative noise is much more difficult to deal with since it is data dependent . some sources that cause multiplicative noise include : timing jitter in data sampling , small aperture size compared to image area , the under - sampling of aperture samples , the non - uniform spacing between aperture samples , errors in position measurement system , etc . multiplicative noise results in undesired sidelobes that create high noise floor in the image and thus limit the ability to detect targets . as used herein , the terminology “ dictionary ” means an organized listing of data stored in machine - readable form for reference . the dictionary may be located in any type of storage or memory for storage of data . as an example , the “ dictionary ” may comprise time shifted versions of the transmitted probing signal s ( t ). as used herein the terminology “ spectral ” means of or relating to a spectrum . the terminology “ spectrum ” refers to a band , portion or region of the electromagnetic spectrum which may or may not be continuous , for example , the radio spectrum is the region of the electromagnetic spectrum spanning the radio frequency range . the terminology “ spectra ” is the plural of spectrum . the foregoing description of the specific embodiments are intended to reveal the general nature of the embodiments herein that others can , by applying current knowledge , readily modify and / or adapt for various applications such specific embodiments without departing from the generic concept , and , therefore , such adaptations and modifications should and are intended to be comprehended within the meaning and range of equivalents of the disclosed embodiments . it is to be understood that the phraseology or terminology employed herein is for the purpose of description and not of limitation . therefore , while the embodiments herein have been described in terms of preferred embodiments , those skilled in the art will recognize that the embodiments herein can be practiced with modification within the spirit and scope of the appended claims .