Intentional Aliasing Method to Improve Sub-Nyquist Sampling System

1 Abstract —A m odulate d wideband co nverter (M WC) has been int rod uced as a su b - Nyquist sa mpler that exploit s a set of fast al ternati ng pse udo r ando m (PR) signal s. Thr oug h par allel analog chan nels , an MWC compr esses a m ultiband spec trum by mixing it wit h PR s igna ls in t he ti me do ma i n , an d ac quir es i t s sub - Nyquist s amp l es . Previously, t he rati o of co mpressio n was full y de pend ent on t he s peci fic ati ons of PR si gna ls. T hat i s, to further reduce t he sa mpling rate witho ut i nfor mati on los s , faster and l onger - peri od PR sig nals w ere needed. How ever, the i mple ment atio n of s uc h PR si gnal gene rat ors res ults in high pow er consu mption an d large f abricat ion ar ea. In t his pa per, we propose a novel aliased modul ated wi deband converte r ( AM WC ), w hich c an f urther re duce the s ampling rat e of MWC with fixed PR signals . The main i dea is t o ind uce intentional signal aliasing at the anal og -to- dig ital c onvert er ( ADC ). In addition to t he first s pectral compr ession by the signal mixer, the intentio nal aliasing compress es the mix ed spectru m onc e again. We demo n st ra t e that AMWC reduces the number of analog chann els and the rate of ADC for lossless sub - Nyquist sampl ing wi thout nee ding to upgr ad e th e spee d or the perio d of PR signals. Conver sely , for a g iven fixed number of an alog channels and sam plin g ra te, AMWC improves the perf or mance of signal reco nstr ucti on. Index Terms — S ub - Nyqui st sa mpli ng, m odula ted w i deband converter, s a mpling eff iciency, i ntentiona l aliasing, c om pressed sensi ng, r andom filt er. I. I NTRODUCTION pp lications of electronic warfare (EW) s ys te m s , electronic intelligence (ELINT ) systems , or cognit ive radio s are dem anding the observation of a multi ba nd si gnal, i.e. , a collectio n of multiple nar row - band s ign als , each with different center frequencies , scattered across a wi d e freq uency r ange up to tens of g igahertz (GHz) . T he N yqui st sampling rate is t wice t he maxi mum freq uenc y of the wi d e range . When a multiband signal is spa rse , i.e . consist s of a fe w na r r o w b a nd s , the signal can be sam pled without information lo ss at a sub - Nyquist ra te far less tha n the N yquist rate. The theoretical lo wer limi t o f t he rate r equired for lossle ss sub - N yquist sampling is the su m of t he band widt hs, k nown as the Landau rate, when the spectral locations of all t he narrow - band si gnal s are k nown [1] . When spectral locatio ns are unkn ow n , t he lower li mit is dou ble d [2] . The modulated wideband converter (MWC) propose d by Mishali et al. [3] is a lossle ss sub - Nyquist sa mpler tha t aim s at achieving the theoretical lo wer limit o f sampling rate. This wo rk w as suppo rte d by a gra nt - in -a id of HANWHA THALES . J. Jang and the cor re spo ndi ng a utho r H e ung - No Lee are with th e Depa rtm ent of E lectr ical Engin eerin g and Com put er Scien ce, Gwan gju Insti tute of Sc ience and Technology, Gwan gju 61005, Sou th Korea (e - mai l: jjh201 4@gist. ac.kr; heungno@gi st.ac.k r) . S. I m is w ith Ha nw ha Sy s tem s, Se on gna m 1 352 4, So ut h K or ea ( e - m ail: sh.i m@ha nw ha. com ) Similar to o ther s ub - Nyquist sam plers proposed in [4] – [6] , MWC ex ploits ps eudo - random (PR) si gnals, whi ch period ically output pulsed patterns . MW C has multiple analog c hanne ls , each o f w hich co nsist s o f a P R si gnal generator , signa l mi xer , lo w - pass filter (L PF) for anti - aliasing, and lo w - rate a nalog - to - digital converter (ADC) in seq uence . The syst em compresses a multiband spectrum thro ugh the mi xing and LP F pro ced ure s , follo wing which i t samples at a sub - Nyquist rate. The reco nstruc tion o f t he inp ut mult iband spec trum is guarant eed u nder s ome conditi ons of the co mpressed s ensing (CS) theor y [7] – [ 11] . W ith the he lp of CS reco nstruction algor ithms in [2], [12] dev eloped f or the MWC s, it has been proved th at an MW C can ac hie ve the theoretical lo wer limit of the loss le ss sub - Nyqui st sampli ng rate . Ho we v er , to achieve the lower limit of the lossless sub - Nyquist sampli ng rate , the prev ious ly propose d MWC by Mishali et al. reli ed on a hi gh - end P R si gnal ge nerat or, sinc e it was the only spectral compress or . T he ratio of spectral com pression was full y dep end ent o n the oscillation spe ed and length of the puls ed patter ns within a single p eriod o f the PR sig nals. S pecifically , to improv e th e com press ion ra tio for a sparser multiband signal, PR signals with a greater pattern length w ere required . In ad dition, the o scillatio n spee d s hould be faster than the Nyquist rate f or a lossless compress ion. Unfort unate ly , increasi ng the p atte rn length o f a P R signa l generator wit h tens o f GHz - range switchi ng spe ed leads to difficult researc h problem s in the field of chi p engi neer ing , such a s high powe r cons umption a nd large f abrication area due to the high chip speed [13], [14] , whic h hi nder the commercial a vailability of such a PR signal generator chip . Recently, efforts t o reduce the rate for lossless sub - Nyquist sam pling wit h MWC c loser to the theoretical lo wer limit witho ut up gr adin g the PR signal generators have been mad e in [15], [16] . In [15] , the aut hors proposed a m ethod tha t channelize s the multi band spect rum i nto fe w ort hogo nal subb and s b efor e mixi ng wit h the P R signa ls. Si nce t he channelized signals have a lo wer Nyquist rate tha n the original inpu t, for a give n os cil lat ion speed and patter n length of PR sig nals , t he method achieves a high er ratio of spectral com pression . Al tho ugh t he m ethod le d to a further re duc tio n of the lossle ss sub - Nyquist sam pling rate, it requires additional hardware resou rces for the channelization , such as band - pass filters, local oscilla tors, and a greater numbe r of indepe nden t PR signal gene rators propor tion al to th e num ber of subb and s. In [16] , a me t ho d similar to that proposed i n [15] was presented , in whic h the inp ut sig nal was d ivided into in - phase (I) and quadrature (Q) channels before m ixing it with PR si gnals. T he lo ssless sub - Nyquist samplin g rate can be reduced by the s ame principle as in [15] , altho ugh the author s did not me ntio n this point . Ho wever , the s yste m also required additional hardware resources fo r the I - Q divisio n. Jehyuk Jang, Sang hun Im , and Heung-N o Lee*, Senior Memb er, IEEE Intentio nal A liasing Me thod to Improve Sub -Nyquist Sampling System A 2 In this paper, we propose an aliased MWC ( AMWC ), which reduces the lossless su b - Ny quist samp ling r ate for gi ven practical P R signals. T he main idea of AMWC is to break the anti - aliasing rule and induce intentional aliasing at the ADC of ea ch spat ial c ha nnel b y set ting the ban dwidth of th e prior LPF to be greater than the ADC sa mpling rate. I n additio n to the fir st sp ectr al co mpre ssio n by the mixi ng and LPF procedure s , this inte ntional ali asing leads to another spectral compression under a certain relation betw een the ADC sampl ing r ate a nd b and width o f the p rio r LPF. Thr ough t he two spectral compression procedure s , the compression ratio is improv ed w ithou t fa ster or longer P R sig nals . Conseq uen tl y , for a give n and fixed PR sig nal generator , the lossles s sub - Nyquist samplin g rate o f AMWC is closer to the lower limit than t hat o f MW C. The propo sed AMWC achiev es the same eff ect as in previous works [15], [16] , i.e. , r educ tion in the lossl ess sub - N yq ui s t sam pling rate witho ut up gra ding t he PR sig nal ge nerat ors , and requires no additio nal hard ware compo nents . T o our knowle dge , AMWC is novel in tha t no stud y has thus far improv e d the sub - Nyquist samplin g capabilit y of MWC by impr oving t he uti liz atio n effi cienc y of given ha rd war e resources. In [17], [18] , var iations of MWC sim ila r to AMWC that include aliasing at the ADC ha ve been investigat ed for analyzing channel ca pacity . The ir mai n re sults indicate that suppr essin g non - active subba nds before spectral com pression minimizes the lo ss of infor mation rate incurre d by aliasi ng t he n oise spectrum. Inter es tingl y, t he aut hor s o f [18] introduce d a r ule for dete rmini ng th e sampl ing rate of each spatial ch annel similar to that of AMWC (see Section I II - A for details) . H oweve r, t he rule was des ign ed to m ake a fair comparison with other filterbank - based s yste ms by fle xibl y control ling the ban dwidth of subban ds, rather tha n to exploit the aliasi ng at t he ADC to re duce the lo ssless sub - N yq u i st sampling rate . Additionally , ac cor ding to our re sult s , the rul e in [18] is insufficient and ali asing at the ADC may lead to information lo ss. Our main co ntrib utio n is that the ant i - aliasin g rule of MW C is show n to be un necessar y for lossles s s ub - Nyquist sa mpling. We reveal a certain relationship between the AD C sampling rate and bandw idth of the prio r LPF s o that AMWC can avoi d the loss of sign al inf ormation du ring the add itional spectral com pression . We dem onstrate that , for given oscillatio n speed and pattern leng th of PR signa ls, the samplin g rate and analog chan nels of AMWC requ ired for the reconstru ction of a mult iband s ignal are furt her reduce d. F or given sa mpli ng rat e and num ber of analo g ch annel s, we s ho w t hat t he reconstruction performan ce of AMWC for a multi band s ignal with a g ive n spar sit y is imp rove d . Additionall y , we s ho w that the bene fit s fr om intentio nal aliasing can be further stre ngt hened usin g a no n - flat LPF . The non - flat frequ ency respons e of LPF r esult s in a different input - output relat ionshi p for eac h frequen cy com ponent of the s ub - N yquist sa mple s of AMWC . S imulation results sho w that t he re ducti on o f lossle ss sub - Nyquist samplin g rate is boosted when t he fil ter resp onse is sample s of a random distribution as the inp ut - output re lati onshi ps of diff erent frequency components become independent . The remainder of t his paper is organiz ed as fol lows. In Section II , we br iefly i ntroduce MWC with the anti - alia sing rule and the n de fine the goal of this paper . I n Sectio n III , we propose AMWC a nd de rive i ts input - output relatio nship. The relationship b etween the s ampling rate of A DC and bandw idth of LPF to avoi d inform ation loss is als o provide d. In Sec tio n IV , a revised i nput - output relatio nship of AMWC correspon din g to the use of a non - ideal LPF is provi ded. S imulation result s are provid ed in Section V . Section VI concludes the paper. II. B ACKGROUND AND P ROBLEM F ORMULATION The modulated wideband converter (MWC) is a sub - Nyquist sampling syste m f or multiband signals. A signal ( ) xt is a multiband si gnal if its spec trum ( ) Xf is com posed of B K disj oint conti nuo us bands of maximum ban dwidth B [2], [3] . We assume that the m aximum frequency of a target mult iband s ignal doe s not exceed max f , i.e. , ( ) 0 Xf = for NYQ f ∈   , whe re [ ) max max , NYQ ff −   , and NYQ   is the complementar y set of NYQ  . W e denote the Nyqu ist rate by max 2 NYQ ff  . A. System C onstitution and P aramete rs MWC consist s of M analog channels in parallel (s ee F ig. 1- (a) ) . E ach channel consists of a PR signal generator, a mixe r, a n LPF , and a n AD C in seq ue nce. Each PR si gna l ( ) i pt for channel ind ex i is p T - periodi c and outputs chi p s of an odd length L withi n a s ingle p e riod p T . Each chip lasts for a chip duration 1 cp T TL − = . We denote the chip speed by 1 cc fT −  and the re petition rate o f the PR signal by 1 pp fT −  . The LPF has a cut - of f fre quenc y 2 LPF W , whe r e LPF W denotes the bandwidth of the filter incl udi ng the ne gati ve Fig . 1 . S amplin g system of AMWC . Th e system is eq ui valen t to cMWC whe n and . In AMWC , the sa mplin g rate is - ti mes lower t han t he filte r ba ndwid th with to inten tionally ind uce alia sing. 3 freq uenc y. T he L PF b and widt h is se t to , LPF p W qf = wh e re q is the channel - tr adin g param eter , an odd positive integer . Finall y, we deno te the sa mpli ng rat e , which i s equal at ever y channel , by s f . T he total s ampling r ate is the su m of sampling rates of all channels, defined by , s total s f Mf  . MWC fi rst compresses the input multiband spectrum us ing PR si gnals . After that , nonze ro subba nds of the multiband spectrum are recovered b y CS recovery algorithms. For the successful CS recovery , all spectral component s with in th e Nyqui st ra nge NYQ  of each PR signa l are needed to be independent, which requires a fast chip speed c NYQ ff ≥ [3] . Thr oughou t thi s pap er, we set c N YQ ff = . B. Convent ional Mod ulate d Wi deband Co nvert ers In the origi nal p ape r [3] by Mishali et a l. , for lossle ss sub - Nyquist samplin g , the ADC follo wed the anti - aliasing rule , i.e. , s LPF fW ≥ . T his co nventi ona l rul e has suffi ced fo r lossles s sub - N yquist sampling . W e refer to MWC t hat follow s t he anti - ali asing rule as conve ntional M WC ( cMWC ). T he input - ou tput relatio nship o f cMWC is gi ven i n [3] . T he input ( ) xt at the i -th cha nne l is fir st mix ed with t he p T - periodi c PR si gnal ( ) i pt that period ically outputs a sequence of L mixin g chi p s . B y the per iodicity, the Fo urier tra nsfo rm (FT ) of ( ) i pt is an impulse train . T he F T of the mixed signa l ( ) ( ) ( ) ii s t xt p t = is the convol ution ∗ of the two spectra : ( ) ( ) ( ) ( ) ( ) 2 , , j ft i i il p l S f s t e dt Pf X f c X f lf π ∞ − −∞ ∞ =−∞ = ∗ = − ∫ ∑  (1) where , il c for ,, l = −∞ ∞  are the Fourier series coefficients of ( ) i pt . The mixed signal ( ) i st and ( ) p X f lf − in (1) a re filtered b y the LP F ( ) Hf . We let ( ) 1 Hf = for LPF f ∈  , and otherwise, ( ) 0 Hf = , where [ ) 2, 2 LPF LPF LPF WW −   . S ince ( ) Xf is ba nd - limited by NYQ  , the infinite - order sum mation i n (1) is reduced to a fin ite order as follow s: ( ) ( ) ( ) ( ) ( ) 00 00 , , for , ii Lq i l p LPF l Lq Y f S fH f c X f lf f + = −+ = = −∈ ∑  (2) w here 0 L is com puted by ( ) 0 12 LL = − [3] , and ( ) 0 1 2. qq −  Next , the ADC of rate 1 ss fT − = takes sa mp l es of ( ) i yt , i.e. , [ ] ( ) s ii t nT yn y t = = . B y th e c onve ntio nal anti - aliasing rule , we s et s LPF fW = . T hen, the discrete - time FT (D TFT) o f [ ] i yn preserves the spectrum of (2). I n (2 ), every sub band ( ) p X f lf − is spectrally correlated with nea rb y 1 q − subb and s , sinc e t he band wid th LPF W is wider than t he shi fti ng inte rva l p f . To make them spectrally ort hogona l , the sample s [ ] i yn are mod ulated and lo w - p ass filtered in p ara llel t hro ugh q digital c hanne ls by  [ ] ( ) [ ]  2 , ps p j sf T n is i f n nq z n y ne h n π − =   = ∗   (3) for 00 ,, sq q = −  , whe re [ ] p f hn is a digital LPF with t he cut - o ff fre q uenc y of 2 p f and a flat pass band response . T he DTF T of (3) is ( ) ( ) 0 0 2 ,, for , s L j fqT is il s p p lL Z e c X f lf f π + =− = −∈ ∑  (4) where ) 2, 2 p pp ff  −    . T he subbands ( ) p X f lf − in (4 ) are spectrally orthogonal to e ach other , sinc e the ba ndwid th equals the shif ting interval . As ( ) Xf is a multiba nd sig nal, o nl y a fe w subb and s in (4) have no nzero value s. If p fB ≥ , t he u pper boun d on t he spa rsity K of th e sub band s is 2 B KK ≤ , sinc e the u nifo rm grid of inte rval p f splits each band into two pieces at most . Conseq uen tl y, e ac h analo g chan nel o utp uts q diffe re nt seque nce s, a nd ther efo re , cMWC obtains tota lly Mq equa tions for i nput reco nstr uct ion. Depending on the number of eq uatio ns, it was sho wn in [3] that the input spectru m can be perfectly reconstru cted. Pre viousl y, t o obtai n mo r e equa tions f or a fix ed number of ch annels M and fo r a gi ven specificatio n p f for P R signa l gene ratio n, cMWC ha s to r el y on the increased sampli ng r ate sp f qf = by cont roll ing the channel - trading parameter q . In th is paper, we ai m t o s ho w there is anot her wa y to obtain mor e equat ions and improve the input reconstruction performance, witho ut the cost inte nsive ways o f increasing the total s ampling rate , s total s f Mf = or red ucing p f , or both. C. Sampling E fficiency I n (4) , MWC splits the input spect rum into many subband s along a u nifor m grid of a splitting interval , and it then t ake s sam ples of th e weight ed sum of subb and s. We denote the splitting inte rva l b y I f . N ote that the splitt ing interval o f cMWC , I cMWC f equals p f . Fro m the sa mpl es, a CS recover y Fig . 2 . Illus trat ion of the samplin g e ffic iency i n the re la tion be tw een the maxim um bandwidt h a nd s plitti ng i nterval . (a) When , M WC wastes a portion o f the tot al s ampl ing rate becau se of the unuse d ba nd in t he non zero subban ds . (b) T he r eg ula ted i mpr oves th e samp ling effic ienc y. 4 algorithm ( e.g. , [11], [12], [19 ], [20] ) fina ll y recovers the K nonze ro subba nd s co ntai ning the split pieces of the B K multi bands . Co nseq uent ly , the to tal samplin g rate is cons umed to t ake sa mple s of K nonze ro sub band s of band widt h I f . T his in dicates that the total sa mpling rate req ui red for lossless sa mplin g by an MWC wou ld be at least , 2 s total I f Kf ≥ , where th e factor of 2 arise s fr o m t he unknow n suppor ts of the nonz ero subba nd s. I n contr ast , a resul t in [2] stat es that, for a general sub - Nyquist sa mpling system, the minimum req uirement for lossless sam pling of a multiband signal is , 2 s total B f KB ≥ , where B KB is the uppe r bound of the ac tual spe ctral occupancy of a mult iba nd si gnal . That is, when I f is far greater than B , MWC co nsumes a portion of the total sampling r ate inefficiently . Specifically, I f greater than B yields a higher probability for the K nonze ro subb and s to be com pri sed of unus ed band s , i.e. , zero s . The inefficient use o f total sa mpling rate is il lustrate d in Fig. 2. Ideally, w he n the splitting inter val I f becomes finer and closer to B while satisf ying I fB ≥ , the sampl ing effic ienc y is im proved , as sho wn in Fi g. 2. The efficiency is maximized whe n IB Kf K B = . Base d on th is obser vation , we define the sampling efficienc y α of MWC as the ratio between the actual spectral occupancy of t he multiband sig nal a nd the to tal band widt h of t he recovered subband s , i.e. , . B I KB Kf α  (5) Note that , b y the definition o f K , 1 α ≤ al ways hold s . In su m ma r y, i mpr ovi ng α has t wo ad vanta ges. Fi rst, for the lossle ss sa mpling o f a given multib and si gnal, it w ould reduce the req uired total sampling rate , s total f closer to the theoretical minimum requirement , 2 s total B f KB ≥ . B y t he defini tio n, the hi ghe r α closer to 1 indicates that a porti on of , s total f inefficiently co nsumed for taki ng sampl es of the unus ed ban ds in Fi g. 2 is reduced. By the reduced , s total f , th e numbe r of c ha nnels M or the sampling rate s f of AD C at each channel is reduced . S econ dly, for gi ven a nd fi xed , s total f , we will show through out the rest of paper that impro ving α yiel ds more in depende nt equati ons for signal reconst ruction , and thus , mor e compl ex mult iba nd signa ls wit h highe r B K can be recovered perfectly . D. Limitations i n C onventional MW C For cMWC , the sampling eff iciency depends entirely on the hardware capabilities of PR signal g enerators, which may resul t in severe impl ementation problem s. T he sa mpling efficiency of cMWC depends on the sp ecification s o f PR signal generators s ince , I cMWC f is fixed to p f . By t he definition, t he only way to improve the sa mplin g effi cie ncy cMWC α of cMWC has been to mak e the repe tition rate p f of the P R signa ls c loser to B . As di scuss ed, th e chip s peed c f of PR s igna ls shou ld not be l ess than the Ny quist rate, i.e. , c NYQ ff ≥ . Thus , from the relation 1 pc f fL − = , the chip lengt h L is the only free parameter to con trol p f . Since B is usual ly far smaller than NYQ f , to fit p f closer to B , a ver y long L is needed. However, in applications wh ere NYQ f reaches t ens o f gi gahe rtz, due to th e ex tr em ely high chip speed c f , implementing P R sig nal generators having a hig h chip l engt h L poses probl em s in te rms of p o wer consumption and fabrication area [13], [14 ] . Hence, other means to im prove α witho ut r elyin g on the c hip lengt h L o f the P R signa ls are ve ry i mpor tant. For ex a mp le , s uppose one is obse rving on - ai r radar signals of ban dwidth up to 30 B = [MHz] over an extre m ely wide observ ation frequenc y scope max 40 f = [GHz]. This setting is reasonable in rad ar systems [21], [22] . We discusse d that the chip s peed sh oul d not be less than the Nyquist rate , i.e. , c NYQ ff ≥ , wh ere 80 NYQ f = [GHz]. In this example, to achieve p fB ≈ , the c hip le ngth needs to be 11 21 L = − . Altho ugh ha rd ware i mple mentat ion s of suc h PR sig nal generator s havi ng 80 c f = [GHz] and chip lengt h greater than 11 21 L = − we r e proposed i n the literature [23], [ 24] , the y require very large f abrication area s a nd high po we r cons umpti on , whic h has hinder ed practical use s thus fa r . E. Probl em form ulation The goal of th is paper is to i ntrodu ce th e prop osed sa mpl ing system, Section III, whi ch aim s to im prove the sam pling efficiency α with gi ve n and fi xed sp ecifi catio ns p f , c f , and L for PR signal generation . T hro ughout thi s pap er, we assume small L and B and a large NYQ c ff = , which implies p f large enough compared to B and ma k es room f or impr oving α . That is , p f pB ≥ for a natu ral number 1 p > . The n, i mprovi ng α can be made without upgrading the PR signa l gene rato rs and caus ing t he said imple me ntatio n iss ues such a s higher power con sum ption an d larger fabrication area discu ssed in the prev ious su bsection . T h us, ver y wide band signals can be lossl essly sampled u sing commercially available PR s ignal ge nerato rs and ADCs, wh ile this wa s not possible in the pas t with the co nventional cMWC s yste m. III. A LIAS ED MWC W e propose Aliased modul ated w ideband c onvert ers ( AMWC ) . AMWC renders the ant i - aliasing rule s LPF fW ≥ used in cMWC unnecessary , as revealed later . Instead, AMWC inte ntiona ll y induc es a nd exploit s aliasing at the ADC t o regula te t he splitting i nter val I f and im prove α witho ut relying on the specificat ion of PR signals. In thi s section, we first discuss o ur meth od to induc e controlled aliasing at the ADC a nd de rive r evi sed input - output relatio nship s of AMWC . We the n investigate ho w to control the alia sing fo r lossle ss sa mpling . F inall y, we compare the sampling efficiency of AMWC with tha t of cMWC . A. Intent ional Al iasi ng Method The AMWC sys te m is depicted in Fig. 1 . As me ntio ned already, c om pared to cMWC , AMWC i s d esigned to not sa tisfy the ant i - al iasi ng rul e at the AD C ; rather , it i s designed t o induce intentional a liasing by setting the b and widt h of LPF greater than the sa mpli ng ra te . In fact, in bot h cMWC a nd AMWC , an aliasin g is introduced first by the mixer . T he effect of this firs t alia sing is sh own i n (2) , wh e re the mixe r s hift s , 5 gives weig ht s, and has t he signa l spectrum ( ) Xf overlapped with shifted versions of itsel f at intervals of p f . By t he second aliasing at the ADC, the overlapped spectrum is aliased again at in tervals of new sampling rate of AMWC s f ′ , which is s maller tha n the filter band width . B y adj ustin g the relationship between p f and s f ′ , the splitting in terval I f , which is the interval at which ( ) Xf is split in t he outputs of AMWC , is regulated . Specifically , we set t he n e w samp ling rate s f ′ of AMWC : , sp q ff p ′ ′ = (6) where q ′ is the new channel trading parameter for AMWC and an odd num ber . T he ba ndw i dth of LPF is LPF p W qf ′ = ,and therefore, LPF s W pf ′ = for the inte ger aliasing paramete r 1 p > . We will show that copri me p and q ′ wi t h qp ′ > is necessary for no information loss o f ( ) Xf . T he ne w sampling rate induces additio nal aliasing and r egulates the splitting i nterva l I f to i mpro ve the samp ling e ffi cienc y. W e let p p f f p ′  (7) denote the least common shifting inte rval (LCS), whic h will become the splitti ng interval o f AMWC , i.e., , I AMWC p ff ′ = . With the intro duction of new sa mpling rate s f ′ in (6 ) , it becomes easier to compare AMWC with cMWC . Specifically , with the samp ling r ate fi xed , the n umber of equ ation s f or the input reconstructio n obtained by cMWC and tha t b y AMWC can be compared ; w ith th e num ber of equa tions f ixed, the sampling rates for the two can be compared. For a give n sampling rate sp f qf p ′′ = , we will show in this section, the numbe r of equ ati ons obt ain ed by AMWC is Mq ′ . Fo r a given sampling rate sp f qf = , from Secti on II -B , the numbe r of equat ions obtaine d by cMWC i s Mq . With the sa mpling rate fixe d th e same , i.e., ss ff ′ = , we note that q qp ′ = . T his implies that AMWC has p - times more equations than t hat of cMWC . T AB LE I presents an example of the increase in the numbe r of equ ati ons of AMWC . Wit h th e num ber of equat ions fixed, i.e., Mq Mq ′ = , on the othe r han d, AMWC requires p - times smaller sampl ing r ate t han cMWC does . In [18] , a variation of MWC using a sampling rate similar to (6) w as considered , to analyze the noise factor incurred by the alia sing o f subban ds . There appear coprim e relatio ns between p and q ′ similar to t hat in t his paper . Ho we v er , the purpose of us ing coprime p and q ′ in [18] was completely diffe re nt fro m that of this paper, i. e., t he y reg ulated the splitting i nter val of the subba nds to make a fair comparison with other filterb ank - based sa mpling syste ms wit h regar d t o the effect of noise . No relation between p and q ′ for loss les s sa mp ling and impro vin g sa mplin g e fficie ncy was studied in [18] . To suppor t intentional aliasing, AMWC requi res an ADC wit h a n operating bandw idth wide r tha n it s sa mpling rate . Such a n ADC can be im plemente d by using a wi deband track - and - hold amplifi er (THA ) develope d by Hi ttit e Corp. for the applica tions of EW and ELLINT in [25 ] . Th is THA has an 18 GHz bandwidth and can be integrated at the front end of commercial ly available A DC s of sam plin g rate up to 4 g iga - s a mp les per second. To sho w that the AMWC obtai ns Mq ′ equati ons , we observ e the inp ut - o utput relatio nship s of the aliased sam ples  [ ] i yn in Fig. 1 . Without loss of generalit y, we assume qq ′ = and ss f pf ′ = . By the sa mpli ng theo re m, t he DTF T of  [ ] i yn is the sum o f shi ft s of ( ) i Yf :  ( ) ( ) ( ) ( ) 2 , , s j fT i is r il s p s rl Y e Y f rf c X f rf lf H f rf π ∞ ′ =−∞ ∞∞ =−∞ = −∞ ′ = − ′′ = −− − ∑ ∑∑ ( 8) where ( ) 1 ss Tf − ′′  and ( ) i Yf give n in (2) is the spectrum of the output o f the LPF ( ) Hf . Within on l y a singl e period of  ( ) 2 s j fT i Ye π ′ in (8), i .e. , ( ) [ ) 0 00 , ss f ff f ′′ +   for any 0 f ∈  , because t he band wid th o f ( ) i Yf is limited b y t he L P F ( ) Hf , mo st o f the shi fts ( ) is Y f rf ′ − for sufficiently l arge r are zeros . In other words , there exist ( ) 012 ,, f RR such t hat t he infi nite order of the outer su mmati on in ( 8) is reduced to a finite orde r , i.e. ,  ( ) ( ) ( ) 2 1 2 , s R j fT i il s p s r Rl Y e c X f rf lf H f rf π ∞ ′ = =−∞ ′′ = −− − ∑∑ (9) for ( ) 0 s ff ′ ∈  . A ssuming ( ) 1 Hf = for LPF f ∈  , if 0 f , 1 R , and 2 R satisf y the c ondit io ns of Le mma 1 , the LP F resp onse s in (9) are replaced with ( ) 1 s H f rf ′ −= for ( ) 0 s ff ′ ∈  . N ote that , wh en 1 p = , i.e. , no aliasing exi sts at the ADC, 12 RR = , whi ch is equivale nt to cMWC . TABL E I P ARAMETER C OMPARISONS BETWEEN AMWC AND C MWC Multiband model System speci ficat ion Param eters cMWC AMWC (w ith ) Chann el - trad ing p aram eter Sam plin g ra te [M Hz] Spli tting in terva l [MHz] Spa rsity N umber of rows of To tal num ber of equa tio ns 6 Le mma 1 . E q uatio n (9) is eq uiva lent to (8) if 0 f , 1 R , and 2 R wi t h 12 RR <∈  satisf y 21 1, RR p −= − ( 10 ) and 02 . 2 s p fR f  ′ = −   ( 11 ) Proof: See Appe ndix A. We represen t the s hift ing indi ces sp rf lf ′ + in (9) in terms of the LCS p f ′ . T hen ,  ( ) ( ) ( ) 1 1 1 2 , s Rp j fT i il p rR l Y e c X f rq lp f π +− ∞ ′ = =−∞ ′′ = −+ ∑∑ ( 12 ) for ( ) 0 s ff ′ ∈  . To merge the inner and outer summations i n ( 12 ) , we u se Le mma 2. Le mma 2 . I f p and q ′ are co prime, the linear combi natio n rq lp ′ + for { } 11 ,, 1 r R Rp ∈ +−   and l ∈  spans e ver y inte ger . Proof: We consider the fol lowing c ongr uen t relatio nship ( ) mod . k rq p ′ ≡ ( 13 ) By modular arithmetic, if p and q ′ are coprime , there alwa ys exis t s one - to - one correspondence betw een r and k in the least residue s ystem modulo p . Since p =  , ( ) mod rq p ′ for r ∈  in ( 13 ) spans eve ry nu mb er in the least residue syste m of modul o p . Hence, for r ∈  and l ∈  , mod rq lp k p lp ′ += + spa ns ever y int eger. ■ B y de noti ng k rq lp ′ = + in ( 12 ) , we ha ve the equi vale nt relationship  ( ) ( ) ( ) 2 ,1 ,, s j fT i ik p k Y e d R p q X f kf π ∞ ′ =−∞ ′′ = − ∑ ( 14 ) for ( ) 0 s ff ′ ∈  , wh er e ( ) ,1 ,, ik d R pq ′ are the ne w sensin g coe ffici ent s o f AMWC . P ropos ition 3 p rovid es the rule t o obtain the coefficients , ik d fr o m t he Fourier coeffi cients , il c of PR sig nals . Fig . 3 . Principle of improv ing the sam pling e ffic ie ncy by AMWC at a sin gle ana log channel is ill ustra ted, w ith setti ng , , , and . At the firs t sta ge, the input s pectr um is aliased b y mix ing it w it h the P R si gnal an d lo w - pas s f ilte ring it . T h is aliased - version of is d epict ed a s . In (a), th e main di fferen ce b etween cMWC and AMWC is how to take t ime - samples of . cMWC pr event s the spe ctr um fr om bei ng al iase d in tak ing time - samples. AMWC , on t he co ntrary , a ims to make th e sp ectru m inten tion ally alia sed once a gain , a s depi cted as i n ( b). I n (c ), as a res ult, the sp litti ng - in terval of cMWC is , w her eas in (d) , t hat o f A MW C is halv ed to . Thus , the sam pl ing e ffic iency of AMW C be com es doubl e d ( as ). 7 Proposi tion 3 . For coprim e p and q ′ , let us d efi ne ( ) ( ) ( ) { } 1 1 11 1 ; , , mod , I k R pq k q q k R p R p −  ′ ′′ − −+   ( 15 ) where ( ) ( ) 1 mod qp − ′ is the multip licative inverse of q ′ modul o p . E qua tion ( 14 ) is equivalent to ( 12 ) if ( ) ( ) 1 ,1 , ; ,, ,, . ik iI k R pq d R pq c ′ ′ = ( 16 ) Proof: See Appen dix B. In ( 14 ) , the band widt h of t he s ubb and s ( ) p X f kf ′ − for ( ) 0 s ff ′ ∈  equals s f ′ and is q ′ ti mes wid er tha n the ir sh ift ing interval p f ′ . Therefore, eve r y sub band is co rrelated wit h the closest 1 q ′ − subb and s . B y ma kin g these subb and s spectrally orthogonal, the M relations hip s fo r 1, , iM =  are expan ded to Mq ′ equa tions t o enha nce the input reconstructio n performance . A s imi lar wor k wa s done fo r cMWC thro ugh (3 ) to (4) , whic h furt her d ivide s the observi ng freq uenc y d o mai n ( ) 0 s f ′  ( 14 ) into q ′ tiny d o mai n s. Specifically , fo r 0, , 1 uq ′ = −  , the u - th tiny freq uenc y doma in is defined b y ( ) 0 pp f uf ′′ +  , whe re ( ) ) 0 00 ,. pp f ff f ′′  +    ( 17 ) The n, the c orr espo nding divided outpu t s have r e lationship s  ( ) ( ) ( ) ( ) ( ) 2 ,1 0 , , for , p u j fT i ik p p p k Ye d R p q X f kf f f uf π ′ ∞ =−∞ ′ ′ ′′ = − ∈+ ∑  ( 18 ) for 0, , 1 uq ′ = −  . Finally , we define t he outp ut  ( ) 2 , p j fT iu Ze π ′ of AMWC as follo w s:  ( )  ( ) ( ) ( ) ( ) 22 , ,1 ,, pp p u j fT j fT iu i f f uf ik u p k Ze Ye d R p q X f kf ππ ′′ ′ = + ∞ + =−∞ ′′ = − ∑  ( 19 ) for ( ) 0 p ff ′ ∈  . T he fi nal o utp ut  , iu zn    in the d iscrete - time doma in can be obt ain ed by perform ing digi tal fre que ncy mod u latio n and lo w - pass filte ring on  [ ] i yn , a s similar ly done fo r cMWC in ( 3). The s pecific design of the digital pro cessing s yste m is s hown in Fig. 1- (b) . Conseq uen tl y, in ( 19 ), the input ( ) Xf is split int o spectrall y orthogonal sub band s at interval s of p f ′ . T he refore, the splitt ing int erval of AMWC equals the LCS p f ′ : , , p I AMWC p f ff p ′ =  ( 20 ) which is p ti mes lo wer tha n , I cMWC f . B y reduc ing t he splitting inte rva l b y controllin g the aliasing parameter p , the sampl ing e ffici ency o f AMWC i n (5 ) is im proved. Fi g. 3 illustrates ho w AMWC re gulates the splitti ng interval and impr oves t he sa mpli ng e fficie nc y. In c ontr ast , as disc ussed i n Section II - D , r egula tin g t he splitting inter val of cMWC require s a v ery cos tly soluti on of advanced PR si gna l generator s wit h a lar ger c hip le ngth . Co nse que ntly , both cMWC and AMWC obtain Mq Mq ′ = equa tions for i nput rec onstruc tio n, al thoug h AMWC consu mes a p - times lo wer total sampling rate (6) . In Se ctio n III -C , we sho w that the Mq ′ equati ons of AMWC are independent . B. Matrix F orm of I nput – Output R elation ship For conveni ence of analyzing and so lvi ng li near simult ane ous Mq ′ equa tions ( 19 ) , we cast them as a matri x equa tion . To thi s end , we f irst reduce the inf inite summation in ( 19 ) to be finite. We then discr etize the c onti nuous spec tra to fo r m a ma t ri x wi th a finite number of c olumns . S ince ( ) Xf is ba nd - li mited to NYQ f ∈  , within the limited freq uenc y range ( ) 0 p ff ′ ∈  , the i nfinite summation order in ( 19 ) is reduced to a finite order as f ollow :  ( ) ( ) ( ) ( ) 2 1 2 , ,1 , 0 , , for , p j fT iu N i k u I AMWC p kN Ze d R p q X f kf f f π ′ + = ′′ = −∈ ∑  ( 21 ) where 1 N and 2 N a re , respectively, t he smallest and la rge st ind ex k of th e subban ds ( ) , I AMWC X f kf − that c ontai n so me active value of ( ) Xf wi thin NYQ f ∈  . Na mely , these indices 1 N and 2 N indicate ( ) , 0 I AMWC X f kf −= for 1 kN < and 2 kN > , and thus h elp us obtain a matri x equation of ( 21 ) with finite dimens ions . To m athematicall y d efi ne 1 N and 2 N , TABL E II S UMMARY OF AMWC P ARAMETERS ( C MWC WHEN AND ) maxim um frequenc y of multiban d signal Nyquis t rate of mu ltiband si gnal, , maximum bandw idth a nd n umbe r o f t he n arr ow band s in a multi ba nd sig nal numbe r of nonz er o sub ba nds ( spa rsity ) , if . numb er of analog cha nnels length of PR chips wit hin a single p eriod chip sp eed of PR signals, repe titio n rate of P R si gnals , , ch annel - trading parame ter, al iasing parame ter bandw id th o f L PF, le ast com mon s hif ting interval, sampli ng ra te of an AD C, total sam pli ng rate , spli ttin g interv al, sampli ng ef ficien cy, 8 n ote that the k - th subba nd ( ) , I AMWC X f kf − in ( 21 ) obse rves the fre que ncy r ange ) 0, 0, , k I AMWC I AMWC p f kf f kf f ′  − −+    ( 22 ) of ( ) Xf . The n, the indice s 1 N and 2 N are defined by { } { } 1 0 , max min : min : k N YQ I AMWC Nk k f kf f ∈ ∩ ≠∅ = ∈− <    ( 23 ) and { } { } 2 0 , max max : max : , k NYQ I AMWC p Nk k f kf f f ∈ ∩ ≠ ∅ ′ = ∈ − + >−    ( 24 ) respectively. Usi ng the p ara meter s and r elati ons gi ven i n T ABLE II and Lemma 1 , th e two probl ems ( 23 ) and ( 24 ) turn into ( ) 12 min : 2 q Lp N k Rq k ′ +   ′ = ∈− <     ( 25 ) and ( ) 22 max : 1 2 q Lp N k Rq k ′ −   ′ = ∈ − +>     ( 26 ) respectively. As both q ′ and L are odd posit ive i nteger s , the s oluti ons of tw o problem s ( 25 ) and ( 26 ) are determ ined as follow: ( ) 12 1, 2 q Lp N Rq ′ + ′ = −+ ( 27 ) and ( ) 22 . 2 q Lp N Rq ′ − ′ = − ( 28 ) Finall y, t he output spect rum  ( ) 2 , p j fT iu Ze π ′ in ( 21 ) turns i nto a lin ear com bina tion of unknown subb and s ( ) , I AMWC X f kf − for ( ) 0 p ff ′ ∈  . The matrix - multiplicati on form Z = DX of ( 21 ) is provided in ( 30 ). We denote the numb er o f subbands , i.e., the di mension of matrix X , b y N , whi c h equals 21 1 . NN N Lp = −+ = ( 29 ) Since ( ) Xf consist s of B K narr ow ba nds o ver t he wide Nyqui st ra nge, o nly a fe w of its subba nd s ( ) , I AMWC X f kf − for ( ) 0 p ff ′ ∈  have nonzero v alues. Therefore, t he matrix X in ( 30 ) is r ow - wi se sp arse with a sp arsit y K related to B K . T o draw a relationship between the analytic result ( 30 ) and actually acquired samples  , iu zn    , we c onver t the DT FT ( 30 ) to the DFT o f  , iu zn    by t akin g the fr equenc y sa mples o f the infinite columns of Z and X . W hen the inpu t is ob served for a finite duration o T , taking samples of th e spect ru m ( 21 ) at fre quenc y inter val s of 1 o fT − ∆= does not cause a ny information loss . T he samp les o f spectrum  ( ) 2 , p j fT iu Ze π ′ is obt ained by taki ng t he DF T of the actually acquired ti me - sa mple s  , iu zn   . Co nseque ntl y, for a finite obser vation ti me 2 op T WT ′ = for a sample len gth 2 W , we r ewrite the matr ix - multip lication for m ( 30 ) as 22 , WW Z = DX ( 31 ) where column s of 2 2 Mq W W ′ × ∈ Z  and 2 2 NW W × ∈ X  are sub - colum ns of Z and X , respectively, at freq uenc y intervals of f ∆ . This concept will be exploite d in Section IV to derive a rev ised i nput - out put relationship of AMWC for usin g LPF wit h a no n - fla t fr eq uency r esp onse . C. Choosing the Aliasing P arameter For a given total sa mpling ra te, AMWC obta ins more equa tions used for input rec onstruction t ha n cMWC does . What remains is to check if t he extended equations provide indep end ent info r mation. We reveal a conditi on on the aliasing parameter p that necessit at es the li nea r syst em ( 30 ) to be well - pos ed for e very K - sparse signal matrix X . Proposi tion 4 . There e xists the unique solution of (30) for every K - sparse si gnal X only if p and q ′ are coprime and qp ′ > . Proof : See Appe ndix C. Propo sition 4 gives a co ndition qp ′ < for copri me p and q ′ that make s AMWC an ill - pose d syste m. T his indicates that , within the set of coprime qp ′ > , there may be a subset t hat make s AMWC guarantees the e xiste nce o f uniq ue sol utio n of ( 30 ) for every K - spars e signa l mat ri x X . In [11] , a CS resu lt stat es there exist the u nique solutio n of a multiple measure ment vector ( MMV ) CS equa tion Z = DX for every K - sparse sig nal X if  ( )  ( )  ( )  ( ) ( ) ( ) ( ) ( ) ( ) 11 2 11 2 11 2 11 2 2 1,0 1, 1, 1 1, 2 1, 1 1, 1 1, 1 1 1, 1 2 2, 2, 1 2, 2 ,0 2 ,1 ,1 1 , ,1 p p p p j fT NN N j fT q Nq Nq N q j fT NN N j fT MN q MN q MN Mq Mq Ze dd d Ze dd d dd d Ze dd d Ze π π π π ′ + ′ ′ − ′′ ′ +− +− + +− ′ + ′ ′′ ′ +− +− + + ′ − ×∞ ′ ∈      =       Z               ( ) ( ) ( ) ( ) ( ) 1 1 2 1 1 p p p q N Mq N X f Nf Xf N f X f Nf − ×∞ × ′ ∈ ∈    ′ −    ′ −+        ′ −    X D             ( 30 ) 9 ( ) ( ) 2 spark 1 ran k , K < −+ DX ( 32 ) where sp ark is the minim um num ber of l inearly depen dent colu mns in D . Mean wh ile, the spark of a n Mq ′ - by - N matr ix is u pper bou nded to 1 Mq ′ + by the S ingle to n bound [26] . Based on th ese results, we find a sufficient condition on p and q ′ fr o m Monte Carlo exper iments in Se cti on V-A ( Fig. 4 ) that maxi mize s the spar k of . Main R e sult 5. Let 2 Mq K ′ ≥ . F or every K - sparse sig nal X , t here e xists the u nique sol utio n of (30) , and therefore, AMWC does not lose an y informati on of K - sparse si gnal X , if p and q ′ are coprime and qp ′ > . Meanwhile , we ch oose p to mi n i mi ze the maxi mum o f the sparsi ty K , wh i c h i s the numb er o f nonzer o sub band s of ( ) Xf at splitting interval s , I AMWC p ff ′ = . T he spar sity K is dependent on the center f requencies of B K multiband s and their max imu m band widt h B . When , I AMWC fB ≥ , ever y multiband occupies at most two subba nd s, which implies 2 B KK ≤ . On the o ther hand , whe n , I AMWC fB < , so me multiband s may o ccup y mor e t han t wo sub band s, whic h provides an oppor tun ity to increas e K b eyo n d 2 B K . Hence, we recommend choos ing the aliasing parameter p as . p f p B  ≤   ( 33 ) D. Sampling E fficiency A nalysis We compare the sampling effici encies of AMWC , AMWC α , and cMWC , cMWC α , d efined in ( 5) . The sa mplin g effic ie nc ies are funct ion s of the sparsi ty K , which is a r andom variable in general. We denote the sparsity o f cMWC a nd AMWC b y cMWC K and AMWC K , respectively. To make them deter ministic, we put assumption s on a nd AMWC K that i n both cMWC and AMWC , the B K band s in ( ) Xf respectively occupie s exact ly o ne s ubba nd, i .e., cMWC AMWC B KK K = = . This oc curs with high proba bility when 1 p fp B −  and the center frequ encies of mul tiban ds are far enough apart from each other with a small . Unde r the assu mption above , the samplin g efficiencies o f cMWC and AMWC are obtained by , , B cMWC cMWC I cMWC p KB B Kf f α = = ( 34 ) and , , B AMWC AMWC I AMWC p KB pB Kf f α = = ( 35 ) respectively. Note that i f 1 p = , AMWC and cMWC ar e completel y identical, and t herefore AMWC cMWC αα = . W hen 1 p > , the inte ntional aliasi ng of AMWC takes effect and improv es the sa mpli ng ef ficie ncy proportionally to p . IV. N ON - IDEAL L OW - PAS S F ILTER S The input - o utput relation ship in the p revious section is based on the ideal LP F ( ) Hf havi ng a flat p ass - ba nd response. However, in real applications , t he pass - band respo nse of an LP F s ign ifica ntly fl uctua tes . In the c ase of cMWC , a po st digital - pro cessing tec hniq ue to equalize the effec ts o f non - flat filte r resp onse s was propose d in [27] . Unfort unate ly , o wi ng to the aliasing a t ADC, t he equalization s cannot be a pplied to AMWC . In this section, we inst ead provide a revised input - outp ut relation ship of AMWC based on the fluctuated LPF ( ) Gf . W ithout loss of gener ality, we assume all analog channels use the same LPF. We assume that the res ponse ( ) Gf is nonzer o and known within the pass - band LPF f ∈  and i s zero for C LPF f ∈  . We derive a revis ed inp ut - o utput relation ship reflecting the e ffect of ( ) Gf . P aradoxically , o ur e mpirical results in Sec tion V conc lude tha t , fo r a gi ven samp ling e fficie nc y , an irregularly fluctuated filter res ponse is helpful to further decrease the total samplin g rate required for l ossl ess sub - Nyquist sam pling . The derivatio n start s fr om substit utin g ( ) Hf in the input - output relatio n s of (8)-( 12 ) with ( ) Gf . Without los s of generality, we assume qq ′ = and ss f pf ′ = . E quati on (9) the n turns into  ( ) ( ) ( ) ( ) 2 1 2 , s R j fT i il p p r Rl Y e c X f rq lp f G f rq f π ∞ ′ = =−∞ ′ ′ ′′ = −+ − ∑∑ ( 36 ) for ( ) 0 s ff ′ ∈  , where 1 R and 2 R are chosen fr o m Lem ma 1 . By Lemma 2, we substitute rq lp k ′ += and merge the outer and in ner s umma tio ns:  ( ) ( ) ( ) ( ) ( ) 2 1 2 ,1 ,, s j fT i N ik p p p kN Ye d R p q X f kf G f k f π γ ′ = ′′ ′ = −− ∑ ( 37 ) for ( ) 0 s ff ′ ∈  , where the sensing coefficients ( ) ,1 ,, ik d R pq ′ , 1 N , and 2 N are , respectively , computed fro m Proposit ion 3, ( 27 ) , and ( 28 ) . We define the funct ion p γ of k that ma p s k in ( 37 ) to the corres pondin g rq ′ in ( 36 ) so that the t wo equations are equivalent . Le mma 6 r eveals the mapping rule for ( ) p k γ . Le mma 6. Un der th e condi tion s of Lemma 1 a nd Lemma 2 , (36) a nd (37 ) are equivalent if the m apping ru le of p γ is assi gned b y ( ) ( ) 1 ; ,, , p k k pI k R p q γ ′ = − ( 38 ) where the p icki ng reg ular ity ( ) 1 ; ,, I k R pq ′ is defined in ( 15 ). Proof: See Appe ndix B. As don e in ( 14 ) to ( 19 ), the final output s  , iu zn    for 0, , 1 uq ′ = −  are obtained b y pr ocess ing the time - sa mples  [ ] i yn of the spectrum ( 37 ) using the digital system given in D cMWC K B K 10 Fig. 1- (b). Then, th ose spectra  ( ) 2 , p j fT iu Ze π ′ have the following i nput - outp ut relatio nship s:  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 2 , ,1 ,1 ,, ,, , p j fT iu N ik u p p p p kN N ik u p p p kN Ze d R p q X f kf G f uf k u f d R p q G f k u f X f kf π γ γ ′ + = + = ′ ′′ ′ = − +− + ′ ′′ ′ = −− ∑ ∑ ( 39 ) for ( ) 0 p ff ′ ∈  , whe re ( ) ( ) , pp ku k u u γγ ′ +−  . Consequently, the linear coefficients on the subb and s ( ) p X f kf ′ − in ( 39 ) become fre quency - sel ective. To numer icall y sol ve ( 39 ) , we discretize th e continuous freq uenc y , as d iscu ssed i n Secti on III -B . We assu m e t hat the sign al is observ ed for the fi nite duration 2 op T WT ′ = , wh ere 2 W is the length of the discretized si gnal . Then, the sa mp le s of spe ctrum are defined by  [ ]  ( ) ( ) ( ) ( ) ( ) [ ] ( ) 1 2 1 1 2 1 1 2 ,, , ,, , p o o o j fT iu iu f wT N ik u p p f wT kN N p iu k f wT kN Z wZ e d G f k u f X f kf b w X f kf π γ − − − ′ = + = = = =  ′′ ′ = −−  ′ = − ∑ ∑  ( 40 ) for ( ) { } 00 ,, 1 o po w fT f f T ′ ∈ +−   , where the freq uenc y - selective se nsing coe fficients ( ) [ ] ,, iu k bw ar e defined as ( ) [ ] ( ) ( ) 1 , ,, , o ik u p iu k f wT b w d G f ku f γ − + = ′′ −  ( 41 ) for w ∈  . No te that , by the relation between DFT and DTF T, th e s pectrum samples ( 40 ) ar e ob tained by taking the DFT as follo w s:  [ ]    ( ) 21 2 mo d 2 2 ,, 0 for , n W j wW W iu iu n Z w z ne w π − =  = ∈  ∑   ( 42 ) where  , iu zn    are the out put seque nc es of AMWC . For convenience, w e represent the inp ut - outp ut relation of ( 40 ) for w ∈  in a vector fo rm as  [ ] [ ] [ ] , w ww = Z BX ( 43 ) whe re t he elements of the output colum n vector  [ ] Mq w ′ ∈ Z  are  [ ] , iu Zw for row indices 1, , iM =  and 0, , 1 uq ′ = −  . T he unkno wn column vector [ ] N w ∈ X  consist s of ( ) 1 o p f wT X f kf − = ′ − for ro w indices 12 ,, kN N =  . T he freq uency - selective sensi ng m atri x [ ] Mq N w ′ × ∈ B  consists of ( ) [ ] ,, iu k bw wi t h row indices i and u and column i nd ex k . The CS model ( 43 ) is called MMV w ith differe nt se nsin g matrices, fo r whic h ma n y numerical solvers have been devel ope d [9], [28] . The exi sten ce of uniqu e sol ution of ( 43 ) depen ds on t he spark of sensi ng m atrix [ ] w B . Note that from ( 41 ), the elements o f [ ] w B are multiplicatio ns of the ele ments of D and the samples o f the low pass filter ( ) Gf . In [ 29] , Davies et al. proved that the spark of a matrix from an independent continuous d istribution ac hieves the S ingleton boun d with probabil ity one. Wh en the f ilter response ( ) Gf is de signed to b e irregular, i.e., its samples are drawn from an indepe nden t ran dom dis tribut ion, the spark o f [ ] w B after multiplicatio n w ith the sa mples o f ( ) Gf sho uld gr ow clo ser to ac hievi ng the Sin gleto n bound . Whe n the spar k o f [ ] w B indeed achieves the Singleton bou nd and t he co ndit ion ( 32 ) holds , fo r e ve r y K - sparse si gnal X the uniq ue sol utio n to ( 43 ) always e xists. V. S IMULATION A. Spark of Sens ing Mat rix To suppor t Main R esult 5 , t he sufficienc y of lossless sub - Nyq u is t sa m pl i ng b y AM WC , w e demons trate that the sensin g matri x D with coprim e parameters qp ′ > achieves the S inglet on boun d. Monte Carlo experiments were performed under various setti ngs of p and q ′ . W ith 127 L = , we us ed the maxi mum lengt h seq uence s of le ngt h L as the chip va lues of PR signal for each channel 1, , iM =  . We set t he num ber of an alog channels to 3 M = . For 5 5 10 × independ ent trials, we rand oml y selected Mq ′ column s of D and counted the rate for whic h the selected colu mns are lin early in depende nt. Fig. 4 sho ws how the linear independency of columns in D varies a s p and q ′ cha nge . The white poi nts in the plot indicate the pairs of p and q ′ where every sel ection of Mq ′ column s of D is line arl y inde pe ndent . The dark poi nts indicate th at at least one selection of Mq ′ columns ha s linear dependency . The up pe r triangular area indicates th e region of ( ) , pq ′ wi t h qp ′ > whe re all points e xcept for the points t hat p and q ′ are not coprim e belong to t he white set . T hat is , for copri m e qp ′ > , all the selections of Mq ′ columns are linea rly i ndep ende nt , and t hus the sp ark o f D achieve s the Fig . 4 . Inde pende ncy rates unde r v ari ous and for which rand omly selec ted co lum ns o f t he s ensi ng m atrix of AMWC are ind epend ent . Whe n a nd are coprim e and , every se lection of col umns is l ine arl y indepe nde nt . 11 Sing leton bou nd. T his re sul t is co nsis tent wit h Proposition 4 and supports Main R esult 5 . B. Reducti on of T otal S ampling R ate W e demonstrate that , wi th the improv ed sampl ing efficiency , AMWC indeed reduces the total sam pling rate required f or lossless sub - N yquist sampling for give n specificatio ns of PR signals . Add itionall y , when t he frequ ency response of low - pass fi lters is dr a wn at r a n do m, the reduction of total sa mpling r ate is boosted. The re ducti on of total samplin g rate reduce s the num ber of channels as well as the s amp ling rate of each c hannel. For simulation , we generated real - value d m ultiband input s ( ) xt as the su m of B K narrow band s ign als of bandw idth [ ] 5 MHz B = . The energies of narrow band s are equal . T he center frequen cies of narrow band si gnals we r e dra wn at ra ndo m , whi le those spectra w ere not overlapped wit h eac h othe r. T he maxi mum fr eque nc y of ( ) xt does not exceed [ ] max 10 GHz f = . The signals last fo r the duration 2 op T WT ′ = seconds wit h 15 W = . The parameters of P R signa ls were 127 L = , 1 max 2 p f fL − = [ ] 157.48 MHz .  We used maxi mum le ngth se que nce s with different initial seed s as the ch ip va lues of PR sig nals fo r chan nel ind ice s 1, , . iM =  We expressed the continuous sig nals in simulation on a dense discrete - time g ri d with int er vals o f ( ) 1 2 NYQ qf − ′ second s. The band widt h of lo w - pass filte rs and the sampling rate followed the param eter relations of AMWC , i.e., LPF p W qf ′ = and 1 s LPF f pW − ′ = . We considered the ideal LPF ( ) Hf with a flat passban d response and the non - ideal Fig . 5 . Rate of succ essful su pport recover y of cMWC a nd AMWC as a functio n of to tal sam pling rate f or v arious al iasi ng param ete rs and m ultiban ds . The num ber of c han nel s wa s fixed to . Ideal ((a) - (b)) and random ((c) - (d)) low - pass filters were used. Fig . 6 . Rate of succes sful supp ort recovery of cMWC and AMWC as a fun cti on of t otal s amp ling rat e when SNR=3 [ dB] . The num ber of chan ne ls wa s fixe d to , and the num ber of mul tib ands in is fixed t o . Id ea l ( a ) and r an dom ( b ) low - pa ss filters were us ed . 12 LPF ( ) Gf wi t h a n irr egular pass band re spons e. In simula tio n, t he i mpulse r es ponse o f ( ) Gf w as d ra wn initially fr o m the n ormal d istribution, windowed to li mit the filter band wid th, a nd t hen hel d fixe d through out th e whole simulation . W e call ( ) Gf the r ando m LPF w ith this irregular passban d response . Und er va rio us settin gs of p , q ′ , and B K wi t h coprim e qp ′ > , we m ea sured th e rate of succe ss ful recov ery of the su pports of X b y t he distributed CS or thogo nal matchi ng pur suit ( DCS - SOMP ) a lgorithm [28] . For s ingle s upports esti mation, DCS - SOMP wa s run for 2 B K iterations. It ai m ed to find one distinct s upport per each iteration out o f K supp ort s, given 2 B KK ≤ . O nce the support s are found , x can be reconstru cted by the least square s. T he success ful supp ort recovery was declared if  ⊆  , where  and   are , respectively , t he true and foun d supports . The support recovery rate in simulat ions was defined as the number of successful support recovery divided b y total 500 trials with randomly rege nerated ( ) xt . Fig. 5 sho w s the support r ecov ery ra te of AMWC as a function of to tal sampli ng ra te w hen 3. M = We set { } 10, 20 B K = . P lots (a) and (b) are r esults o f using t he ideal LPF ( ) Hf . I t is demonstra te that compared to cMWC , AMWC reduces the total s ampling rate required for reconstructio n of given multiba nd signals. Inve rse ly, for a given total samplin g rate, AMWC takes sub - Nyqui st sampl es of more multiband s than cMWC does , wit hout information lo ss. Ho wever, when p increases , alt hough the samp li ng effic ienc y is imp ro ved p ropor tiona l ly to p fr o m ( 35 ) , the total sampling rate does not decrease anymore. This is cause d by t he la ck o f de gree s of free do m in the sens ing matri x D . The elements of D are made of the Fourier coeffi cients , il c of the PR sign als, and most elem ents are repeatedly reused. Altho ugh i t was d emo nstr ated in the p revio us s ub - sectio n that D has the maximum spark and well preserves the sparse signal X , re cover ing X by n on - optimal CS algor ithms requires D to have a large degrees of freedo m [10] . T h is limitation is overco me by usi ng the ra n d o m LPF ( ) Gf . Plots (c) and (d) are the results of using t he r ando m LPF ( ) Gf . I t is sho wn that AMWC further reduces the total sampling rate required for successful support recovery as the sampling efficie ncy i mp ro v e s . Cons equen tly , t he ra ndo m respon se of ( ) Gf enha nce s the de gree s of f reedom of sensin g matrice s [ ] w B for d iffe re nt fre que ncy ind ice s w and i mproves the recovery performance by the non - optim al algorithm DCS - SOMP . T his enha nceme nt c annot b e ap plie d for cMWC , since the effect of random response becomes removable by equalization [27] . In Fig. 6 , ad ditive white Ga ussi an noise ( ) nt of SN R=3 [dB] was considered , where t he si gnal - to - ratio noi se (SNR) in decibel is de fined as ( ) 22 10 SNR 10 log xn  . We fixed 10 B K = . Plots (a) and (b) are the results for using the ideal LPF and the ran dom LPF, respectively. Despite the additive noise , t he res ults sho w that AMWC still re duces the to tal sampling rate or improves the recovery performance. In clud i ng the results in Fi g. 6 , we conducted more simu lation s under v arious { } [ ] S N R = 6 , 3, 0 , 3,1 2 d B −− but omitted to repeat the plots as the gr aphs ex hibit the similar pattern. Instead, we summarized the mi nima l sampl ing point resul ts in T AB LE III , where th e minimal s ampli ng point is defined as the minimal total sa mpling rate which achieves t he support recov ery rate of 90% . In the r esult s, a s p and/or SNR increase, the m inimal sam pling p oint gets smaller , which is expected. Fig. 7 de monstrate s tha t AM WC red uces the num ber of channels required for the support recovery . We set 10 B K = and co mpared the support recovery rates of cMWC and AMWC for vari ous M and given sam pling rate of each channel. In plot (a), the sup por t rec over y rate of AMWC slightly TABL E III T HE T OTAL S AMPLING R ATE R EQUIRED FOR 90% S UPPORT R ECOVERY R ATE WITH V AR IOU S SNR AND V ALUES OF SNR [dB] LPF p =1 ( cMWC ) p =2 ( AMWC ) p =3 ( AMWC ) p =4 ( AMWC ) -6 Ide al 6.14 2 4.01 6 3.62 2 3.89 8 Random 6.14 2 3.54 3 2.67 7 2.24 4 -3 Ide al 6.14 2 3.54 3 3.62 2 3.42 5 Random 6.14 2 3.07 1 2.04 7 1.53 5 0 Ide al 5.19 7 3.07 1 2.67 7 2.95 3 Random 5.19 7 2.59 8 1.73 2 1.53 5 3 Ide al 5.19 7 3.07 1 2.67 7 2.48 0 Random 5.19 7 2.59 8 1.73 2 1.29 9 12 Ide al 5.19 7 3.07 1 2.67 7 2.24 4 Random 5.19 7 2.12 6 1.73 2 1.06 3 The flo ating numbe rs in ce ll s ind icate the m inimal total sa mpl ing rate in G Hz which a chieves the supp ort rat e recovery of 90%. The number of analog ch annels a nd mu ltib ands were set to and , resp ecti vely. Fig . 7 . Rat e of successful suppor t recovery of cMWC and AMWC a s a funct ion of sa mpli ng rat e of each ch annel for vari ous al iasin g param eters and th e numbe r o f c hanne l s . The nu mbe r o f mul ti ban ds wa s fixed to . Ideal (a) a nd random ( b ) low - pass filters were us ed 13 outperf orm s cMWC , a ltho ugh AMWC uses fe wer cha nnels with a lower sampling rate of each channel t han cMWC . Additionall y , in plot (b), when the ran dom l ow - pass fil ter is used, AMWC usin g a si ngle c ha nnel outpe rfor ms cMWC usi ng six c hannel s . As the increase in the number of rows in Z in ( 30 ) o r in ( 43 ) by p - times , t he performance of AMWC is improv ed but the computatio nal complexit y (C C) f or th e su pport recov ery with AMWC ine vitab ly i ncrea ses a s wel l . T he CC of a co mpre ssed sensing algorithm depends on the sizes of matrices in th e linear inverse problem Z = DX . Let equation Q , sample Q , a nd subband Q denote t he num ber of rows and c olumns of Z and the num ber of rows of X for cMWC problem , respectively. We make note of t he report that the CC of D CS - SOMP with cMWC is ( ) 2 equation subband sample OQ Q Q [28] . W hen the t wo to tal sampli ng rat es , s total f of cMWC and , s total f ′ of AMWC are eq ual to each other , the number of rows of Z of AMWC beco me s equation pQ and that o f X becomes subband pQ , respectively , as discussed in Sec tion III -A . I n add ition, since the ba ndwid th of the subban ds of AMWC is p - times na rrower th an t hat of cMWC , the number of column s of Z becomes 1 sample pQ − . Thus , the C C of DCS - SOMP wi t h AMWC is ( ) 22 equation subband sample O pQ Q Q . VI. C ONCLUS ION We proposed a new MWC system called AMWC which impr oves t he sa mplin g effi cie ncy by int entio nall y ind uci ng an aliasing at the ADC. W e sh owed that th e impr oved sampling effic iency leads to redu ction on the s ampling rate and number of ch annels required for obtaining a certain numbe r of equ ati ons fo r s igna l reconstruction. We provided conditions that the sensing matrix of the equations o btained by AMWC a chie ves t he Sin gleto n bo und , and thus no l oss fro m sam pling is gu arante ed. In summ ary, the improv ed sampl ing efficiency of AMWC r educes the total sam pli ng rate requi re d for lo ssless sampl ing. In other words , wit h fewer channe ls and less sampling rate of each channel t han t hose o f t he conve ntio nal MW Cs , a multiband signal can be captured without infor mation loss by AMWC . Conver se ly, fo r gi ven hardware r esourc es, the i nput r econs truc tion with AMW C outperf orm s the c onventi onal MWC s. Extensi ve si mula tio n demonstrated that AMWC indeed reduces the total sampling rate or im proves the recon struction perf ormanc e signi fica ntl y . Additionall y, it was demonstra ted t hat the ben efit s of AMWC are maintained in various SNRs. Moreover, us e of LPF wi t h rand om pa ssband res ponse , it was sho wn, furt her impro ves the sa mpli ng e fficie nc y. A PPEND IX A P ROOF OF L EMMA 1 With the re lationship LPF s f pf ′ = , the pass - band f requen cy of ( ) s H f rf ′ − in (8) is g iven b y , 22 ss ss pf pf f rf rf ′′  ′′ ∈− +    . When we ob serve (8) o nly for a single perio d ( ) 0 s f ′  , since LPF s Wf ′ > , s om e of ( ) s H f rf ′ − , the pass band s of w hich incl ude the fre que ncy d omai n ( ) 0 s f ′  , can be replace d b y t he const ant freq uenc y re sponse . Without loss of generalit y, we set the pass - ban d respon se to one , i.e. , ( ) 1 Hf = for LPF f ∈  . The n , for r ∈  satisfying 0 2 s s pf rf f ′ ′ −≤ ( 44 ) and 0 , 2 s ss pf rf f f ′ ′′ + ≥+ ( 45 ) the shift s of filter respon se s in (8) are replaced wit h ( ) 1 s H f rf ′ −= wi thin ( ) 0 s ff ′ ∈  . L et 1 R and 2 R be the mini mum a nd maxi mum i nte ger s r satisfying ( 44 ) and ( 45 ) , respectively . Add itionally , for (8) and (9) to be equivalent, w e add s ome condi tions on 1 R and 2 R such t hat the pass bands of ( ) s H f rf ′ − for r smalle r t han 1 R and gr eate r tha n 2 R have no interse cti on with ( ) 0 C s ff ′ ∈  . In other w ords, we have follo wi ng conditi ons on 1 R and 2 R : ( ) 20 1 2 s ss pf R f ff ′ ′′ + − ≥+ ( 46 ) and ( ) 10 1 2 s s pf Rf f ′ ′ − +≤ ( 47 ) so that ( ) 0 s H f rf ′ −= withi n ( ) 0 s ff ′ ∈  for 1 rR < or 2 rR > . By co mbini ng ( 44 ) and ( 46 ) , we ha ve a co ndit io n on 2 R that 20 , 2 s s pf Rf f ′ ′ −= ( 48 ) an d fr o m ( 45 ) a nd ( 47 ), we ha ve a c ondit ion o n 1 R that 10 . 2 s ss pf Rf f f ′ ′′ += + ( 49 ) Finall y, c ombi ning ( 48 ) and ( 49 ) provides th e con dition s of Lem ma 1 . ■ A PPEND IX B P ROOF S OF P R OPOSITION 3 AND L EMMA 6 A. Proof of Propos ition 3 We t rack t he inp ut - outpu t relation starti ng from ( 12 ):  ( ) ( ) ( ) 2 1 2 , s R j fT i il p r Rl Y e c X f lp rq f π ∞ ′ = =−∞ ′′ = −+ ∑∑ for ( ) 0 s ff ′ ∈  , wh ere 1 R , 2 R , and 0 f satisfy L emma 1. Alter nati vel y, b y usi ng 1 r rR ′ = − , we have  ( ) ( ) ( ) ( ) ( ) ( ) ( ) 21 2 ,1 0 1 ,1 0 s RR j fT i il p rl p il p rl Y e c X f lp r R q f c X f lp r R q f π − ∞ ′ ′ = = −∞ − ∞ ′ = =−∞ ′ ′′ = − ++ ′ ′′ = − ++ ∑∑ ∑∑ ( 50 ) for ( ) 0 s ff ′ ∈  , wh ere 21 1 RR p −= − by Le mma 1 . W e replace the ter m ( ) 1 r Rq ′′ + in ( 50 ) b y a combinatio n of its quot ient ( ) 1 ;, p rqR µ ′′ and remainder ( ) 1 ;, p rqR ρ ′′ by divisor p , whic h are , respectively , define d by ( ) ( ) 1 1 ;, p r Rq rqR p µ ′′ +  ′′    ( 51 ) and ( ) ( ) ( ) 11 ; , mod . p rqR r R q p ρ ′′ ′ ′ +  ( 52 ) 14 By substituting ( ) ( ) ( ) 1 11 ;, ;, pp r R q p rqR rqR µρ ′ ′ ′′ ′′ += ⋅ + into ( 50 ), we have  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 , 0 1 , 0 s j fT i p il p lr p p il r lr Ye c X f lp p r r f c X f lp r f π µ µρ ρ ′ − ∞ ′ =−∞ = − ∞ ′ − ′ =−∞ = ′ ′′ = − +⋅ + ′′ = −+ ∑∑ ∑∑ ( 53 ) for ( ) 0 s ff ′ ∈  , where the notations ( ) 1 ;, p rqR µ ′′ and ( ) 1 ;, p rqR ρ ′′ are simplified to ( ) r µ ′ and ( ) r ρ ′ , respectively . When p and q ′ are coprime , by m odular ar ithmetic , ther e exist s one - to - one correspondence between ( ) r ρ ′ and r ′ modul o p . We arr ange the ord er of inne r s ummat io n of ( 53 ) by intr od ucing a utilit y variable ( ) { } 0, , 1 vr p ρ ′ ∈−  :  ( ) ( ) ( ) ( ) ( ) 1 1 2 1 , ;, 0 s p j fT i p p il vq R lv Ye c X f lp v f π µρ − ′ − ∞ ′ − =−∞ = ′ = −+ ∑∑ ( 54 ) for ( ) 0 s ff ′ ∈  , where the invers e ( ) 1 1 ;, p vq R ρ − ′ of th e remainder ( ) 1 ;, p rq R ρ ′ modu lo p is com puted by ( ) ( ) ( ) 1 1 11 ; , mod , p vq R v q R p ρ − − ′′ −  ( 55 ) where ( ) 1 mod qp − ′ is the multiplicative inverse of q ′ modul o p . We simplify the e xpressio n ( ) 1 1 ;, p vq R ρ − ′ to ( ) 1 v ρ − . F r o m Lem ma 2, we can me r ge the inner a nd out er summa tions of ( 54 ) as follo w s:  ( ) ( ) ( ) ( ) 1 2 , mod s j fT i p k i kp k p Y e c X f kf π µρ − ∞ ′  −  =−∞  ′ = − ∑ ( 56 ) for ( ) 0 s ff ′ ∈  . We now simpl ify the p ic king re gulari ty of the coefficients ( ) , iJ c ⋅ in ( 56 ) , wh ic h is defined by ( ) ( ) ( ) ( ) ( ) 1 1 1 ; , , mod . k J k R pq k p p k k p µρ µρ − −  ′ −    = −    ( 57 ) Meanwhile, b y the definitio ns of the quotie nt ( ) µ ⋅ and remainder ( ) ρ ⋅ , we ha ve ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 11 11 11 1 1 1 1 mod 1 1 mod . k Rq k p k Rq k Rq p p k Rq k p k Rq k p p ρ µρ ρρ ρ ρρ ρ − − −− −− −  ′ +  =   ′′ = +− + ′ = +− ′ = +− ( 58 ) By substituting ( 58 ) into ( 57 ) , ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) 1 1 1 1 1 1 11 1 mod ; ,, 1 mod ; ,, . k Rq kk p J k R pq pp p k Rq k pp k q kq R p R p I k R pq ρ ρ − − − ′ +  ′ = +−   ′ + = −  ′′ = −⋅ − +  ′ = ( 59 ) Th us, t he proof is completed. ■ B. Proof o f Lemma 6 We t rack t he inp ut - outpu t relation starti ng from ( 36 ):  ( ) ( ) ( ) ( ) 2 1 2 , s R j fT i il p p r Rl Y e c X f rq lp f G f rq f π ∞ ′ = =−∞ ′ ′ ′′ = −+ − ∑∑ for ( ) 0 s ff ′ ∈  . Und er the cond itio ns o f Le mma 1 and Lem ma 2 , b y usi ng 1 r rR ′ −  , we have  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 21 2 ,1 1 0 1 ,1 1 0 s j fT i RR il p p rl p il p p rl Ye cX f l p r Rq f G f r Rq f cX f l p r Rq f G f r Rq f π ′ − ∞ ′ = = −∞ − ∞ ′ = =−∞ ′′ ′ ′′ ′ = − ++ −+ ′′ ′ ′′ ′ = − + + −+ ∑∑ ∑∑ ( 60 ) for ( ) 0 s ff ′ ∈  . As done in ( 50 ) to ( 54 ) , we intro duce a utility variable ( ) vr ρ ′  and substitu te ( ) ( ) ( ) 1 1 r Rq p v v µρ − ′′ += ⋅ + into t he inp uts o f X and G in ( 60 ) . I t then follows  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 , ; ,, 2 1 0 s p p i J k R pq j fT i lv p c X f lp v f Ye Gf p v vf π µρ − ∞ ′ ′ − =−∞ =  ′ −+  = ′  ⋅− +   ∑∑ ( 61 ) for ( ) 0 s ff ′ ∈  . Aft er mer gin g the i nner a nd out er summa tions ba sed on Lem ma 2 , we obtain ( 37 )  ( ) ( ) ( ) ( ) ( ) 2 ,1 ,, s j fT i ik p p p k Y e d R p q X f kf G f k f π γ ∞ ′ =−∞ ′′ ′ = −− ∑ for ( ) 0 s ff ′ ∈  , w here ( ) p k γ is def ined by ( ) ( ) ( ) ( ) ( ) 1 1 mod mod mod . p k p kp kp p kk p γ µρ µρ − − + = +  ( 62 ) B y ( 58 ) and the de finiti on o f ( ) 1 k ρ − in ( 55 ) , ( 62 ) t urns into ( ) ( ) ( ) ( ) 1 1 1 11 mod . p k k Rq q kq R p R γρ − − ′ = +  ′ = −+  ( 63 ) By the defi nitio n of ( ) 1 ; ,, I k R pq ′ in ( 15 ) , we final ly ha ve ( ) ( ) 1 ; ,, . p k k pI k R p q γ ′ = − ( 64 ) Th us, th e proof is completed. ■ A PPEND IX C P ROOF OF P ROPOSITION 4 We fi rst s ho w that i f pq ′ > for copri me p and q ′ , at least two c olumns of D are identical. Then, fr o m a result in [11] , this violates a nec essary condi tion for the uniq ue existence o f a K - sparse so lutio n. We fir st mathematically formulate the m eaning o f t wo column s of D bei ng identica l. From Proposition 3 and ( 30 ) , the entrie s ( ) ,1 ,, ik u d R pq + ′ of D are picked fro m ( ) 1 , ; ,, iI k R pq c ′ , 15 where k and u in , ik u d + rep rese nt the co lumn a nd r o w position, re spectively. To search for identical co lumns in D , w e inve sti gate the existence of pairs ( ) , k ω ∗∗ of a colum n ind ex k ∗ and shi ft in dex ω ∗ such tha t ,, ik u ik u dd ω ∗ ∗∗ + ++ = for every ro w i nde x { } 0, , 1 uq ′ ∈−   . In othe r words, we fi nd pairs ( ) , k ω ∗∗ satisfying ( ) ( ) 11 ; ,, ; ,, . I k u R pq I k u R pq ω ∗∗ ∗ ′′ ++ = + ( 65 ) for every u ∈  , where the functi on I is de fined in ( 15 ) . W e use a co mputatio n result o f ( ) ( ) 1 ; ,, Ik Ik R p q ′  in the second line of ( 59 ): ( ) ( ) ( ) 1 1 , k Rq k Ik pp ρ − ′ + = − ( 66 ) whe re ( ) ( ) 11 1 ;, p k kq R ρρ −− ′  is a fun ction modu lo p defi ned in ( 55 ) by ( ) ( ) ( ) 1 1 11 ; , mod . p kq R k q R p ρ − − ′′ −  B y sub stit utin g ( 66 ) into ( 65 ) , we re w rite ( 65 ) as ( ) ( ) ( ) ( ) 11 . Ik u Ik u k u ku q ω ω ρωρ ∗∗ ∗ ∗ −∗ ∗ −∗ + += + ⇔ + + = ++ ′ ( 67 ) W e sho w that , if pq ′ > and copri me , there exis t s at least one pair ( ) , k ω ∗∗ of th e column in dex k ∗ and shi fting index ω ∗ that satisf y ( 67 ) for every r o w i nd e x u ∈  . Be fore proceeding, we check a computation of ( ) 1 kq u ρ − ′ ++ for every u ∈  . By the definition, it follo ws ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 = mod mod 1 mo d 1 mod . kq u kq u q R p k uq R p p ku p ρ ρ − − − − ′ ′′ ++ ++ − ′ = +− + = ++ ( 68 ) Note that ( 68 ) indicates when ω ∗ is cho sen to q ′ , it satisfies ( 67 ) , fo r k ∗ ∈  suc h that ( ) 1 1 ku p ρ −∗ + <− . What task re mains is to s how t he e xiste nce k ∗ satisfies ( ) 1 1 ku p ρ −∗ + <− for every row i ndex u ∈  , which i mplies the existence o f identical col umns in D and completes the proof. To this end, w e fin d a set of ( ) mod kp such tha t ( ) 1 1 ku p ρ − += − . F rom the defi nitio n, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 m o d 1 m o d 1 mod 1 mo d . ku p p kuq R p p k p Rq u p k R qu p ρ − − + ≡− ′ + − ≡− ′ ≡ −+ − ′ ≡− − ( 69 ) Note that ( ) 1 1 Rq ′ − is a c onsta nt . S i nce the ri ght - han d side of ( 69 ) varies by u ∈  , the cardinalit y of set of ( ) mod kp suc h that ( ) 1 1 ku p ρ − += − is q ′ =  . Since pq ′ > , this implies there exists ( ) { } m o d 0 , 1, , 1 kp p ∗ ∈−  suc h that ( ) 1 1 ku p ρ −∗ + <− , and k ∗ ∈  such that ( ) 1 1 ku p ρ −∗ + <− exist s as well. Conseq uen tl y, if copri me pq ′ > , there must ex ist at lea st one pair of identical columns in D . T he exis tence of identical colu mns in D implies ( ) spark 2 = D . Theorem 2 in [11] states that there exis t the unique solutio n of a linear equa tion Z = DX for every K - sparse so lutio n X only if ( ) ( ) spark 1 rank , 2 K −+ < DX ( 70 ) where sp ark is the minim um num ber of l inearly depen dent colu mns in D . If ( ) spark 2 = D , for sign als X wi t h ( ) rank 2 1 K ≤− X , the condition pq ′ > violates ( 70 ) . ■ R EFERENCES [1] H. J. L anda u, “N ec ess ary dens ity cond itio ns f or sam pling an d interp olati on of certain entire fu nctions, ” Act a Mat h. , vol. 117, no. 1, pp. 37 – 52, 1967 . [2] M. Misha li and Y. C. Eldar, “Blind Mu ltiban d Signal Recon struction: Compr es se d Se nsing f or A nalog Si gnals, ” IEEE Trans . Signal Process. , vol. 57, no. 3, pp. 993 – 10 09, Mar . 2009 . [3] M. Mishali a nd Y. C . Eld ar, “F rom Theory to Practice : Su b - Ny quist Samp ling of Sparse Wideb and Analog Si gnals,” IE EE J. Sel . Top. Sign al Pr ocess. , vol. 4, no. 2, p p. 3 75 – 391, Apr. 20 10. [4] Y. Chen , H. Chi, T. Jin , S. Zhen g, X. Ji n, and X. Zhan g, “Sub - Ny quist S am ple d A nalo g - to - Digita l Conversi on Ba sed on Photo nic Time Str etc h an d Com pre ssiv e S ensing Wit h O ptic al Random Mix ing,” J. Lig ht. Tech nol. , vol. 31, no. 21, p p. 3395 – 3401, Nov. 2013 . [5] Y. Zhao, Y. H. Hu, and J . Liu, “Rand om Triggering - Ba sed Sub - Nyquis t Sampl ing Sy ste m f or S pars e Mul ti ba nd Sig na l,” IEEE Trans . Ins tr um. Me as. , vol. 66, no. 7 , pp. 1789 – 179 7, Jul . 201 7. [6] J. A. Tr opp, J. N. L aska, M. F . Duar te, J. K. Romb erg, and R. G. Bara niu k, “ Bey ond Ny quist: Eff icie nt S am pling of Spa rse Bandlim ited Sig nals ,” I EEE Tr ans . I nf. T he ory , vol. 56, n o. 1, pp. 520 – 544 , Jan. 2010. [7] D. L . Do noho , “Co m pre sse d se nsing , ” IEEE Trans. Inf . Theory , vol. 52, n o. 4, pp. 12 89 – 1306, Apr. 2 006. [8] E. J. Candès , J. Rombe rg, an d T. Tao, “ Robust uncer tainty principl es: ex act sig nal re con struc tio n fr om highl y incom pl ete fre quenc y infor mation,” IEEE Tran s. Inf. Theo ry , vol. 5 2, n o. 2, pp. 489 – 509, Feb. 200 6. [9] S. Park , N. Y. Yu, and H. - N. Lee, “ An Informat ion - Theoret ic Study for Joint S pars ity Patte rn Reco ve ry w ith Diff ere nt Se nsing Matr ice s,” IEE E T ran s. I nf. The ory , vol. 63, no. 9, pp. 55 59 – 5571, 2 017. [10] M . F. Duart e and Y. C. E ldar, “Stru ctu red Co mpres sed S ensin g: Fro m Th eo ry to A pplica tio ns,” I EE E T rans . Sig nal Proc ess . , vol. 59, no. 9 , pp. 4053 – 408 5, Sep. 2011. [11] M . E. D avies a nd Y. C. Eldar , “R ank A war enes s in Joint S parse Recov ery ,” IE EE Tran s. Inf. The ory , vol. 58, no. 2 , pp. 1135 – 1146, Feb. 201 2. [12] S. A. Varma an d K. M. M. Pr abhu, “ A new appro ach to nea r - theo reti cal sa mplin g rat e for mod ulat ed wi deban d con verter, ” Sign al Pr ocess. C omm un. SP CO M 20 14 I nt. Conf . O n , pp. 1 – 5, 2014. [13] M . Sak are, “ A Power and Ar ea Ef ficient A rchitecture o f a PR BS Genera tor W ith Mult iple Out puts, ” IEEE Trans. C ircuits Syst. I I Express Briefs , vol. 64, n o. 8, pp. 92 7 – 931, Aug. 2 017. [14] L. Vera and J . R. L ong, “A 40 - G b/s 2 11 - 1 PRBS Wit h D istri bute d Clo cking and a T rig ge r Co untdow n Outpu t,” IEEE Tra ns. Cir cui ts Syst. II Express Briefs , vol. 63 , no. 8 , pp. 7 58 – 762, Aug. 2 016. [15] T. Chen, M. G uo, Z . Ya ng, an d W. Zhang, “ A nove l and ef ficien t comp ress ive mu ltip lexer f or mu lti - chann el com pres sive s ensi ng bas ed on modulated wid eband convert er,” 2016 IEEE Inf. Techno l. Netw. Electr on. Auto m. Cont rol Con f. , pp. 347 – 35 1, Ma y 2016. [16] T. Haque, R. T . Yazi cigil , K. J. - L . Pan, J . Wrig ht, and P . R. K inget, “T heory and Desig n of a Q uadr ature A nalog - to - Informa tion Converter for E nergy - Effici ent W ide band Spe ctr um S ensi ng, ” IEEE Trans . C ircu its Sys t. Reg ul. P ap. , vol. 62, no. 2, pp. 527 – 535, Feb. 2015. [17] Y. Chen, A. J . Go ldsmith , and Y . C. El dar, “ On the Minim ax Capac ity L oss Unde r S ub - Nyquist Uni versal S amp ling,” I EEE Trans . In f. The ory , vol. 63, n o. 6, pp. 3 348 – 3367 , Jun. 2017. [1 8] Y. Chen, Y . C . Eld ar, an d A . J. G ol dsmit h, “S han non Me ets N y quist: Capac ity o f Sam ple d G aussi an Cha nne ls ,” IEEE Tra ns. Inf. Theor y , vol. 59 , no. 8, pp. 488 9 – 4914, Aug. 20 13. 16 [19] J. Chen a nd X. H uo, “T heo retic al Re sul ts o n Spa rse Rep res enta tio ns of Mu lt iple - M easu remen t Vect ors, ” IE EE Tr ans. Si gna l Pr oces s. , vol. 5 4, n o. 12, pp. 46 34 – 4643, Dec. 2 006. [20] Y. J in and B. D. Ra o, “Supp ort Recovery of Spa rse Sign als in th e Presen ce of M ulti ple Mea surem ent Vecto rs,” I EEE Tran s. Inf. The ory , vol. 59 , no. 5, pp . 3139 – 3157, May 201 3. [21] IEEE stand ard for letter design ations for radar - freque ncy ba nds, IEEE Standard 521 - 200 2, Ja n. 200 3. [22] F. E. Nathans on, P. J. O’Re illy , and M. N. Cohen , Radar design princ ip les: si gna l pr ocess ing an d th e env ironmen t . Ra lei g h, NC : Scite ch P ubl., 20 04. [23] T. O. Dickso n et al. , “A n 80 - Gb/s 2/s up 31/ - 1 pse udorando m bin ary sequ enc e gener ator in SiGe B iCMO S techn ology, ” IEEE J. Sol id - State Circ uits , vol. 4 0, no. 12, pp . 2735 – 2745, Dec. 2005 . [24] A. Gharib, A . Tal ai, R . We ig el, and D. Kissin ger, “A 1.16 pJ/bit 80 Gb/ s 2 11 - 1 PRB S generator in SiGe b ipola r technology,” Eu r. Mic row. Inte gr. Ci rcuit Co nf. EuM IC 2014 9th , pp. 27 7 – 280, 2014. [25] H ittite Mic row ave, “H ittete ’s 1 8 G Hz Ul tra W ideban d Track - and - Hold Ampli fier En hance s Hig h S pee d A DC Per formance,” [ online] . Avai lable: http:/ /www .analog .com/me dia/e n /te chnical - documenta tion/t echnica l- articl es /trac k -n- hold_0411 .pdf [26] R. Sing le ton, “M axim um dis tance q - nary codes ,” IEEE Tra ns. Inf . The ory , vol. 10, no. 2, pp . 116 –1 18, 1964 . [27] Y. Chen, M. M ishal i, Y. C. Eldar, an d A. O . H ero III , “Modulated Wideba nd Converter with Non - ideal L owpas s Filte rs,” 2010 IEEE Int. Conf. Ac oust. Spee ch Sig nal Process . ICASS P , pp. 3 630 – 3633, Jun. 2010. [28] D. Baron, M. F . Duarte , M. B. Wa kin, S. S arv otham, a nd R . G. Bar aniuk , “Di strib uted comp ressi ve sensi ng,” ArX iv P rep r. ArXi v09013403 , 2009 . [29] T. Blume nsa th a nd M. E. Davi es, “Sa mpli ng T heore ms f or Sign als From th e Union of Finite - Dimensio nal L inear Subsp aces,” IEEE Trans . In f. The ory , vol. 55, no. 4 , pp . 1872 – 18 82, Apr. 200 9. Jehyuk Jang receiv ed his B.S degree in e lectronic e ngi neer ing fr om the Kumo h National Institu te of Technolog y, Gumi , Korea in 2014 and M.S in Information and Co mmuni cat ion E nginee ri ng fro m GIS T. Cur rentl y, he i s pursui ng a P h.D. degree at the School of E l ectrical Engi neeri ng a nd Co mpute r Sc ience in GIST, Kor ea. His research in terests incl ude sub - Nyquist s ampling , com pressed s ensing . Sanghun I m received the B.S. degree in electronics engineering f rom Soongsil Unive rsit y, Se oul, Kor ea, i n 2 00 9, a nd the M.S. degree and Ph.D. degree in electrical eng ineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2011 an d 2016, res pectiv ely . He is curr entl y wor king at H anwh a Syst ems, Korea. His current research interests include communic ation theories and signal processing for wireless communications and physical layer secu rity. Heung - No Lee (SM’13) receiv ed the B.S., M.S., and Ph.D. degrees from the Unive rsit y of Ca lifo rnia , Los Angele s, CA, USA , in 1993, 1 994, a nd 199 9, resp ectively, al l in electrical engine eri ng. He t hen worke d at HRL Laboratories, LLC, Malibu, CA, USA, as a Research S taff Member from 1999 to 2002. F rom 2002 to 200 8, he work ed as an Assis tant Profess or at t he Univ ersi ty of Pitts burgh , PA, USA. In 2009, he th en moved to the School of Elect rical Engineering and Computer Science, GIST, Korea, wh ere he is currently affiliated. His areas of research include information theory, sign al proces sing th eory, com mun icati ons/netw orking theory, and their ap plication to wir eless commun ications and netwo rki ng, c ompr essi ve se nsi ng, f uture inte rnet, and bra in – computer interface. He h as received several prestig ious nati onal awards , includi ng the T op 100 Nat iona l Res earch and De velopme nt Award in 2012, the Top 50 A chie vements of Fundamental Researches Award in 2013, and the Science/Engineer of the Month (Janu ary 2014) .