On Detecting White Space Spectra for Spectral Scavenging in Cognitive Radios fred harris

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1 On Deecing Whie Space Specra or Specral Scavenging in Cogniive Radios red harris San Diego Sae Universiy, San Diego, CA 92182, USA Absrac: A primary ask perormed by a Cogniive Radio is ha o specral esimaion o locae he segmens in a specral span ha conain hie zones, spans ha conains noise only, or grey zones, spans ha conain signals ih signiican inervals o o-ime. The ideniicaion o specral regions conaining noise only specra, as opposed o regions conaining signals ih lo specral densiy, is surprising diicul. The esimaor mus deal ih quesions o ransorm lengh, indo selecion, indo lengh, indo overlap, and ensemble average opions. This paper describes he impac o each selecion opion and presens he archiecure o he opimal specral esimaor. This is sor o annoying since e are emped o use larger ransorm o obain improved resoluion o periodic signal componens bu e do so a he cos o increased variance o he noise like componens. Bear in mind ha modulaed signals are noise like so he o signals o ineres o us are noise like and variance consideraions ill dominae our signal processing opions. The response o he inconsisency o he DFT specral esimae is o selec a ransorm lengh required o obain he desired specral resoluion acceping he variance associaed ih ransorm lengh and hen reduce he variance by orming an ensemble average over many realizaions. Inroducion: Cogniive Radios (CR), aare o channel condiions and aciviy modiy is operaing parameers o enable reliable, inererence ree, communicaions. The aareness is obained by combinaions o monioring and probing aciviies. The monioring and probing aciviy resides close o he physical layer o he radio. This process includes mehods o rapidly obain specral and spaial aareness as ell as rapid acquisiion o channel esimaes. The response includes adapive modulaion ormas o enable lexible specral allocaion and uilizaion, as AGC seings, as carrier and iming acquisiion, and as signal-o-noise raio esimaes. These asks are DSP inensive and every one o hem is a challenge o implemen in he operaing region o mos ineres o he cogniive radio: he lo signal o noise raio regime! A primary signal processing ask perormed by a CR is specral esimaion o locae hie and grey zone specral regions in a moniored specral span. A hie zone is one ha conains noise only and a grey zone is one ha conains signals ih signiican and resolvable inervals o o-ime. The deecion o specral regions conaining signals ih high specral densiy levels is very simple bu only addresses hal he guidance rules. We pos he high specral densiy regions ih no respassing signs. We sill have o pos he noise-only specral densiy regions ih you can go here signs. We mus be careul o disinguish he noise-only specral regions ih specral regions conaining signals ih lo specral densiy levels. This proves o be a surprising diicul ask. As described by Haykin [1], he analysis o non-saionary signals is no as ell disciplined as he analysis o saionary signals. He poins ou ha he periodogram is a badly biased inconsisen esimaor o he poer specrum. He suggess he use o indos o suppress bias eecs and addiional processes o purchase back he increase in variance due o he use o indo. We remind he reader ha a consisen esimaor reduces is variance as he daa lengh increases. By comparison, a DFT based specral esimaor exhibis increased variance as he ransorm size increases. As shon in igure 1, he lengh normalized DFT o a real or complex hie noise sequence has a mean and variance ha increase ih ransorm size. Figure 1. Sample Mean and Variance o -poin FFT o Complex AWG Sequences Specral Esimaes: Robus specral esimaors incorporae overlapped indoed DFT. The DFT o course is implemened by he FFT. An eecive pos process is he ensemble average o he indoed specra. The signal processing lo o his esimaor is shon in igure 2. The design parameers o be opimized in his process include ransorm size, indo selecion, indo overlap, and pos deecion averager srucure and bandidh. Figure 3 illusraes he imporance o ensemble averaging o obain reliable specral esimaes by presening specral esimaes ormed rom a succession o overlapped indoed ransorms. The inpu signal o he specral esimaor conains a sine ave, a modulaed QAM signal and hie noise. The o signal componens have a +5 db SR bu neiher can be reliably deeced ill he pos deecion averager has processed 16 or more specra. The reason he ensemble averaging is required o obain reliable specral esimaes can be clearly seen in he densiy uncions o measuremens ormed by he FFT. Each specral measuremen is Chi-square ih 2-degrees o reedom obained by summing he squares o he quadraure componens o he FFT.

2 d(n) d(n) d (n) D (k) P(k) P (k) AVG Inpu Daa Buer Windo Processing Buer FFT Algorihm Figure 2. Classical Specral Esimaor 2 Pos Deecion Average Buers Deecion Algorihms Figure 3. Specral Esimaes ormed Wih Dieren umber o Ensemble Averaged FFTs The chi-square densiy is exponenially disribued ih variance equal o is squared mean. We have he sense ha he measuremen o noise poer, he average squared value, is noisy and he SR o he measuremen is 0 db. Oering an esimae ih his saisic is equivalen o saying he esimae has an error bound o plus or minus 100%. A no very useul measuremen! As an ensembles average is generaed ih successively a greaer number o erms, he densiy is sloly modiied rom 2 degrees o reedom o 2M degrees o reedom. The primary change is ha he ensemble average changes is shape, morphing rom an exponenial densiy uncion moving oards a Gaussian densiy uncion. This is seen in igure 4. The irs 16-o 32 averages primarily aec he shape o he densiy uncion hile addiional erms in he averaging process aec he variance o he morphed and no approximaely Gaussian densiy uncion. This leads o an ineresing relaionship beeen he mean and variance o a specral densiy o a noise only specral region and o a random modulaed signal specral region. Since he sample variance is equal o he squared mean energy e ind ha specral regions ih larger signal componens also have large variance. And in ac he sandard deviaion o he specra is equal o specra o he random modulaed signal. We see his in igure 5 hich presens he specral esimae ormed rom an ensemble o 32 indoed DFTs and he requency dependen sandard deviaion ormed rom he same 32 DFTs. We see he sandard deviaion has he same srucure and levels as he signal specra. oe ha on a log scale, he larger variance regions have he same displacemen as he small variance regions. The lesson e learn here is ha deecion mus be perormed on a log magniude scale as opposed o a magniude scale. Figure 4. Densiy Funcions or Chi-Square M-degrees o Freedom Random Variable

3 The eecs o he indo on he specrum o a signal can be readily seen. We recall ha he Fourier ransorm o he consan envelope sinusoid originally has zero idh and he Fourier ransorm o he T ide recangle indo is he ubiquious sin(πt)/(πt)). We noe o eecs o he indo on he indoed specrum. Firs, he indo idens he signal s specral idh rom ininiesimally small o he main lobe o he sin(πt)/(πt)), and second, specral leakage smears or leaks he specrum hrough he sin(πt)/(πt) side-lobes o remoe specral regions ar removed rom he specral locaion o he unindoed specrum. This leakage decays quie sloly ih an Figure 5. Specra and Sandard Deviaion o Specra ormed rom Ensemble o 32 DFTs The specra shon in igure 6 illusrae a number o imporan conceps in specral analysis. The upper subplo as ormed ih recangle (someimes called he deaul) indoed ransorms hile he loer subplo as ormed ih Kaiser-Bessel indoed ransorms. Shon in each subplo is he ransorm o a single realizaion o he composie signal, he average o 256 ransorms, and he peak value o he ransorm values over he 256 ransorms. We irs noe ha he recangle indoed ransorm exhibis specral leakage rom he srong signal region ha spills ino he neighboring specral region and nearly covers he lo-level specral mass o is le. This specral leakage does no appear in he specrum ormed ih he Kaiser-Bessel indo. The specral leakage is observed as a requency and signal dependen bias. The lesson here is ha indoing is required i e are o deec lo level signals in he presence o nearby high level signals. The peak specrum is ineresing because i is a soring process requiring a compare and replace operaion a each FFT sample poin raher han a scaled sum a each poin. When he signal specrum is random, as a ell designed modulaion signal should be, he peak value has an expeced value o approximaely 8-dB above he average value independen o he average value. While he peak specrum can be used as a deecion saisic e have o keep in mind ha he variance o he peak specra is greaer han he variance o he average specrum. This dierence in variance levels can be seen in igure 6. Windos: The DFT perorms a inie sum and as such processes daa sequences colleced over inie inervals radiionally describe as being o lengh -samples. The collecion o samples rom he ime (or index) line has he eec o applying a gaing or indoing uncion o he colleced daa. The gaing operaion is equivalen o a recangle indo and is oen reerred o as he deaul indo. The produc beeen he daa rom he ime line and he indo hich deines and exracs he inie inerval is seen o be equivalen o a convoluion o heir respecive specra. For he sampled daa domain his convoluion is a circular convoluion. The ime and requency domain eecs o he indo operaion are seen in igure 7. Figure 6. Specra o Single, Peak, and Average Recangle and Kaiser Windoed Transorms s() () s() S() W() S() Figure 7. Time and Specral Represenaion o Signal, Windo, and Windoed Signal

4 envelope decay rae o 1/. This specral leakage hrough he indo s specral side lobes bias he measuremens o lo level specral componens and ill ill in deep specral nulls. The side-lobe srucure o he indoed ransorm limis he abiliy o he ransorm o deec specral componens o signiicanly loer ampliude in he presence o a large ampliude componen hile he mainlobe idh o he indoed ransorm limis he abiliy o he ransorm o resolve or separae nearby specral componens. The irs o hese limiaions is demonsraed in igure 8 here a sylized poer specrum o o sinusoids o ininie exen and o inie exen is presened. For his example, he relaive ampliude o he lo-level signal a requency 2 is 60 db belo he high level signal a 1. oe ha he side-lobe srucure o he high level signal is greaer han he main lobe level o he lo-level signal, hence masks he presence o he lo-level signal. I he lo-level signal is o be deeced in he presence o he nearby high level signal, he indo applied o he daa mus be modiied. Windos mus be seleced ih side-lobe srucure signiicanly loer han he sidelobe srucure o he recangle indo. he indo lengh. The red lines overlaid on he specra are spaced s/ and indicae he locaion o adjacen DFT bin ceners. db db Poer specrum, unindoed Poer specrum, indoed Figure 8. Specral Represenaion o Un-indoed and o a Recangle Windoed Sinusoids o Signiicanly Dieren Ampliudes. The high side lobe levels o he recangle indo are due o he abrup disconinuiy a he boundary edges. We reduce side lobes by reducing he severiy o he boundary disconinuiies. A good indo is an even symmeric eighing uncion ha smoohly and genly ransiions rom near zero levels a he boundaries o uniy ampliude a is cener [2], The modulaion o he indo s envelope increases is specral main lobe idh. Since he indo is even symmeric ih a narro main lobe bandidh i can oen be approximaed by a shor cosine ransorm. The successive specral erms in he ransorm are locaed on he zero crossings o he earlier erms and heir conribuion o he specra is o old. Firs heir ose sinc(πt) ransorms conribue side lobes ha desrucively cancel he side lobes rom he earlier specral erms and second heir separae main lobes increase he composie main lobe idh. Figure 9 presens he specra o our indos hich illusrae he opimal exchange beeen main lobe idh and side lobe levels ha cab be achieved ih shor cosine ransorm based indos. The main lobe specral idh, he peak-o-irs zero crossing, o hese indos is seen o be 1,2,3 and 4 FFT bin idhs respecively, here a bin idh is 1/T or s/, ih s=1/t and being Figure 9. Specra o Recangle and oher Good Shor Cosine Transorm Windos Windo Overlap: Our specral esimaion ask requires us o collec a sequence o realizaions o be indoed, ransormed and averaged. The indoing operaion impacs subsequen processing in an imporan ay, namely he amoun o overlap. The eec o he indo can be described ih he aid o he subplos o igure 10. The irs pair o subplos presens he ime series and he specrum o he deaul recangle indo o lengh. oe he main lobe idh o he ransorm, peak o irs zero, is s/. Due o he high specral side lobe, 13 db belo he main lobe, his is an inappropriae indo or use in a robus specrum analysis ask. The second pair o subplos presens he ime series and specrum o he 4-erm Blackman-harris indo ih same lengh. The maximum side lobe or his indo is 92 belo he main lobe ih a main lobe idh o 4s/, idened by a acor o our. The hird pair o subplos illusraes one mechanism ha e can employ o rerieve he original main lobe idh o s/. Wha e have done here is lenghen he ime exen o he indo o samples hich ses he specral main lobe idh o 4s/() or s/. This lengh indo has he same 92 db side lobe level as he lengh indo bu i has he same main lobe idh as he recangle indo o lengh. While he main lobe idh o he lengh indo oers he same main lobe idh as he lengh recangle indo he specral main lobes separaed by s/ cross a heir -10 db poin raher han a he -3.9 db poin o he recangle indo s ransorm. The ourh pair o subplos shos a lengh 6 indo designed by he Remez algorihm ih he same main lobe idh and same side lobe levels o he 4-erm Blackman-harris indo. This main lobe has a laer pass band han sandard indos due o he ime domain side lobes. The laened specral main lobe no cross a here 1-dB poins and hus exhibis smaller scalloping loss. We no choose he lenghened indo suggesed in igure 10, o say lengh or 6, o achieve he argeed main lobe idh o s/ ih he desired 90 db side lobe levels. The problem is he lenghened indo appears o require a ransorm o commensuraely longer lenghs or 6. These longer ransorms ill oer specral spacing o s/() or s/(6) respecively. We have no desire or need or he denser specral spacing, in ac e an o preserve boh he specral idh and he specral spacing o he

5 Figure 10. Time Domain and Specral Response: Windos ih -90 db Side Lobes and Main Lobe Widh s/ recangle indoed -poin FFT. We can achieve his desired goal by simply sub-sampling he poin FFT by a acor o 4-o-1 or he 6 poin FFT poin FFT by a acor o 6-o-1. This resampling is shon in (1) or lengh indo. We no choose he lenghened indo suggesed in igure 10, o say lengh or 6, o achieve he argeed main lobe idh o s/ ih he desired 90 db side lobe levels. The problem is he lenghened indo appears o require a ransorm o commensuraely longer lenghs or 6. These longer ransorms ill oer specral spacing o s/() or s/(6) respecively. We have no desire or need or he denser specral spacing, in ac e an o preserve boh he specral idh and he specral spacing o he recangle indoed -poin FFT. We can achieve his desired goal by simply sub-sampling he poin FFT by a acor o 4-o-1 or he 6 poin FFT poin FFT by a acor o 6-o-1. This resampling is shon in (1) or lengh indo H (k) = (n)h(n)e -1 = (n)h(n)e (n+)h(n+)e (n+2)h(n+2)e (n+3)h(n+3)e -j (n+)k -j (n+2)k -j (n+3)k (2) -1 H (k) = (n)h(n) e -1 H (4k) = (n)h(n)e -1 = (n)h(n)e -j n(4k) Under he 4-o-1 don sampling, he kernel o he poin DFT is no he same as he kernel o he poin DFT. We sill have he problem o packing a poin sequence ino an -poin DFT. We accomplish by pariioning (1) as shon in (2). Since he kernel is periodic in, he erms n+r can be replaced by n hich leads o he double sum shon in (3). The inner sum o (3) is he 4-old aliasing ha occurs in he ime domain due o he 4-o-1 don sampling in he requency domain ha e used o conver he poin kernel o he poin kernel. (1) -1 H (k) = (n)h(n)e -1 3 = (n + r)h(n + r) e r=0 The sequence o operaions e have described in his secion is compacly illusraed in igure 11. Here e irs see a ime line spanned by adjacen inervals o lengh covered by -poin recangle indos. The recangle proved o be an unaccepable indo and as replaced ih a 4-erm Blackman-harris indo. Hence he nex ime line spanned by adjacen inervals o lengh covered by -poin B-h indos. This covering obviously leaves gaps in he ime line ha are no colleced and processed by he indoed ransorms. This suggess ha e should overlap he inervals. Ho much overlap [3] is appropriae? The yquis crierion direcs us o have an oupu sample rae mached o he signal bandidh. The main lobe idh o he indo s specrum is s/ (3)

6 hich means ha he oupu sample rae should also be s/. We inerpre his o mean e ha e mus deliver inpu samples and remove 1-oupu sample rom each DFT bin. Saed oherise, he poin inervals mus shi -Samples beeen ransorms. Addiional overlap ill over saisy he yquis crierion and oers no improvemen in variance reducion and lesser overlap under saisies he yquis crierion and leads o sub-opimum, higher variance specral esimaes. This 25% shi is equivalen o 75% overlap. The 75% overlapped indos o lengh is shon on he hird ime line spanned by overlapped inervals on lengh. The inal presenaion in igure 11 is he 4-o-1 aliasing or olding relaed o he 4-o-1 don sampling o he -poin indoed rans orm. The olding is implemened in he ime domain o aec he equivalen 4-o-1 don sampling so ha he -poin sequence can be processed in he original -poin ransorm. The olding has anoher name in he signal processing communiy. The olded indo is described as a polyphase iler pariion. The srucure o he (or 6) polyphase pariion and is associaed -poin FFT, recognized as he sandard -Channel polyphase iler bank [4, 5], is shon in igure 12. oe ha he ransorm rae is 1-ransorm per -inpu samples, he same as or he non overlapped recangle indo processing. The specral resoluion or his process is ha o he 4-h subplo pair o igure 10. -Poin Windos, Folded 4-o-1 -Poin Windos, 75% Overlap Figure 11. Processed Time Segmens: -Poin Recangle Adjacen Windos, -poin B-h Adjacen Windos, -Poin B-h 75% Overlapped Windos, and -Poin B-h 75% Overlapped and 4- o-1 Folded o -Poin Folded Windo d() s d(). Polyphase Pariion h(n) 0 h(n) 1 h(n) 2 h(n) 3 h -2(n) h -1(n). h(n)=h(r+nm) r PT FFT D() P() P () AVG. Figure 12. Polyphase Channelizer Wih Folded Windo in Polyphase -Pah Filer Concluding Commens: We have revieed some o he classic signal processing ools and imporan consideraions in radiional specral analysis direced o he speciic ask o ideniying hie specra in a span o moniored bandidh. These consideraions include he need or indos o suppress srong signal masking o nearby lo level signals, indo specral occupancy as a uncion o side lobe deph, indo overlap o saisy The yquis crierion or he indo s expanded bandidh, and specral don sampling o avoid high correlaion o adjacen specral esimaes. We also addressed he diiculy making specral measuremen o noise like signals due o poor saisics o he sum o square erms obained rom he oupu o he DFT. This brough our aenion o he need or ensemble averaging and he dieren saisics available rom boxcar averages and exponenial averages. These opions enable he monioring o ime varying saisics, an imporan consideraion hen he specrum occupancy is ime varying. Reerences: 1. Simon Haykin, Cogniive Radio: Brain-Empoered Wireless Communicaions, IEEE Journal on Seleced Areas in Communicaion, Vol. 23, o. 2, February 2005, pp red harris, On he use o Windos or Harmonic Analysis ih he Discree Fourier Transorm, pp , Proceedings o he IEEE, Vol. 66, o. 1, January red harris, On Overlapped Fas Fourier Transorms, Inernaional Telemeering Conerence (ITC-78), pp , Los Angeles, CA ov Doug Ellio, Handbook o Digial Signal processing; Engineering Applicaions, Chaper 8. Time Domain Signal processing ih he DFT. Academic Press, 1987, 5. red harris, Mulirae Signal Processing or Communicaion Sysems, Prenice-Hall, Magniude Square. Ensemble Average. Logarihm P. db () Deecors.