IEEE P Wireless Personal Area Networks

Size: px

Start display at page:

Download "IEEE P Wireless Personal Area Networks"

Kristian Garrett
6 years ago
Views:

1 November 6 IEEE P c IEEE P8.5 Wireless Persoal Area Networks Project Title IEEE P8.5 Workig Group for Wireless Persoal Area Networks (WPANs) Statistical 6 GHz Idoor Chael Model Usig Circular Polarized Ateas Date Submitted Source Re: [] [ November 6] [Zhiguo Lai, Uiversity of Massachusetts] [Abbie Mathew, NewLANS] [Salvador Rivera, NewLANS] [Dev Gupta, NewLANS] [zhlai@ece.umass.edu] [amathew@ewlas.com] [srivera@ewlas.com] [dgupta@ewlas.com] Abstract [This paper describes a statistical 6 GHz idoor chael model with circular polarized ateas, which icludes time-of-arrival (ToA) ad agle-of-arrival (AoA) characteristics. A time-domai measuremet system that was used to simultaeously collect the temporal ad spatial data is described. Data processig methods are outlied, ad results obtaied from data take i office ad residetial eviromets are preseted.] Purpose [] Notice Release This documet has bee prepared to assist the IEEE P8.5. It is offered as a basis for discussio ad is ot bidig o the cotributig idividual(s) or orgaizatio(s). The material i this documet is subject to chage i form ad cotet after further study. The cotributor(s) reserve(s) the right to add, amed or withdraw material cotaied herei. The cotributor ackowledges ad accepts that this cotributio becomes the property of IEEE ad may be made publicly available by P8.5. Submissio Page Zhiguo Lai, UMass

2 November 6 IEEE P c Statistical 6 GHz Idoor Chael Model Usig Circular Polarized Ateas Zhiguo Lai, Abbie Mathew, Salvador Rivera, ad Dev Gupta Uiversity of Massachusetts, Amherst, MA 3, USA NewLANS Compay Ic., Acto, MA 7, USA s: zhlai@ece.umass.edu; {amathew, srivera, dgupta}@ewlas.com Abstract: I this paper, a statistical 6 GHz idoor chael model, which icludes time-of-arrival (ToA) ad agle-of-arrival (AoA) characteristics, is proposed based o a measuremet campaig usig circular polarized ateas. A time-domai measuremet system that was used to simultaeously collect the temporal ad spatial data is described. Data processig methods are outlied, ad results obtaied from data take i two differet eviromets are preseted.. Itroductio I high speed idoor radio commuicatios systems, delay distortio due to multipath compoets (MPCs) is a serious cause of chael degradatio. Circular polarized sigals have the property of reducig MPCs. This has bee reported i the work doe by a umber of researchers i this area [-3]. However almost all the existig statistical models are based o liear polarized ateas. Specifically, the work doe by Maabe ad Sato [] provides iterestig results. Their work showed that the root-mea-square (RMS) delay spread with right had polarizatio is about half that of vertical ad horizotal polarizatios. They also reported that circular polarizatio reduced MPCs up to 3 db with respect to the liear polarizatios. They the coducted experimets to determie a relatioship betwee polarizatio ad bit error rate (BER). For a 5 Mbps BPSK trasmissio with a SNR of 5 db, the BER with circular polarizatio was sigificatly lower tha that with liear polarizatios. The authors of this paper are ot aware of papers describig 6 GHz chael model with circular polarized sigals. Motivated by this, Uiversity of Massachusetts ad NewLANS udertook a 6 GHz idoor measuremet campaig to support the IEEE 8.5.3c task group i the developmet of a wireless persoal area etwork (WPAN) stadard. The objective of the campaig was to develop a statistical chael model for circular polarized pulse trasmissio i support of applicatios i office ad residetial eviromets. The data acquisitio system developed for this measuremet campaig is capable of resolvig MPC arrivals simultaeously i time ad space domais. The system was used to collect a umber of data sets from these two eviromets. This paper reports the results of these measuremets, ad the proposes a statistical model which icludes both time-of-arrival (ToA) ad agle-of-arrival (AoA) characteristics of the idoor chael. Most of the temporal-spatial chael models proposed so far made the assumptio that the statistics for the ToA ad the AoA are idepedet. To the best of the authors kowledge, the first thorough ivestigatio (ad probably the oly oe available i the literature) o the temporal-spatial correlatio i idoor propagatio eviromets based o real measuremet data was that of Chog, et. al. [4], although Specer et. al. [5] cotaied a brief discussio o this Submissio Page Zhiguo Lai, UMass

3 November 6 IEEE P c issue. I this paper, a joit probability distributio fuctio (PDF) for the ToA ad the AoA is preseted followig the approach described i [4]. The orgaizatio of the paper is as follows. Sectio describes the measuremet setup ad eviromets. Sectio 3 presets the techique used to extract MPC ToA ad AoA iformatio from the measured data. A ew statistical chael model is proposed i Sectio 4, followed by a discussio of parameters i Sectio 5. Fially, appropriate coclusios are draw i Sectio 6.. Measuremet Setup ad Eviromets. Measuremet Setup The measuremets were made i time-domai such that reflectios ad impulse resposes are available almost istataeously. A block diagram of the system is show i Figure. The pulse geerator modulates the 6.6 GHz from the dielectric resoator oscillator (DRO) to create a s pulsed sigal. The receiver dow coverts the received sigal to a itermediate frequecy (IF) of 5.4 GHz. The pulsed data is recorded usig a Agilet 86A digital storage oscilloscope coected to a laptop computer, which captures the iformatio displayed by the scope ad saves the data i a Excel spreadsheet. At the start of each measuremet, the trasmitter ad the receiver were bore sighted. The basic techique used to collect the data ivolved poitig the receiver i a certai directio, measurig the time-domai impulse respose (from the IF output) of the chael i that directio, rotatig the receiver a small amout (about ) with the trasmitter fixed, ad repeatig the procedure util full 36 coverage was obtaied. Measuremets were take at each icremet. The trasmitter ad receiver ateas, both right had polarized, have a half power beam width (HPBW) of 35 ad 3 respectively. Figure. Measuremet setup Submissio Page 3 Zhiguo Lai, UMass

November 6 IEEE P8.5-6-398-3-3c. Eviromets Office ad residetial eviromets were cosidered i the measuremet campaig. The office eviromet cosisted of cubicles, coferece rooms, ad hallway/corridor.

cubicles, metal shelves, ad walls The residetial measuremets were made i US homes ad they cosisted of livig/family rooms, diig rooms, ad kitche.

4 November 6 IEEE P c. Eviromets Office ad residetial eviromets were cosidered i the measuremet campaig. The office eviromet cosisted of cubicles, coferece rooms, ad hallway/corridor. Each of them comprised of followig articles: Office cubicles: metal shelves, white boards, ad office widows Coferece rooms: metal shelves, white boards, ad office widows Hallway/corridor: office cubicles, metal shelves, ad walls The residetial measuremets were made i US homes ad they cosisted of livig/family rooms, diig rooms, ad kitche. The eviromets cosisted of widows, doors, picture frames, woode furiture, ad fire place. Table summarizes the umber of measuremets take i each eviromet ad Figures shows the layouts of some locatios cosidered i the campaig. Table. Number of measuremet poits i the campaig Eviromet Number of Locatios Number of Measuremets Office 33 6,6 Residetial 3 5,46 63,466 (a) Office cubicle (b) Coferece room (c) Kitche (d) Livig room Figure. Sketch layouts of some locatios cosidered i the measuremet campaig Submissio Page 4 Zhiguo Lai, UMass

November 6 IEEE P8.5-6-398-3-3c 3. Data Processig ad Aalysis 3. Basebad Pulse Recovery As metioed i the previous sectios, time-domai impulse resposes were measured as the receiver rotated through 36.

5 November 6 IEEE P c 3. Data Processig ad Aalysis 3. Basebad Pulse Recovery As metioed i the previous sectios, time-domai impulse resposes were measured as the receiver rotated through 36. The respose at each agle is a IF sigal cetered at about 5.4 GHz. A evelope detector was digitally implemeted i Matlab to recover the basebad pulses: The respose was first badpass filtered at 5.4±. GHz. The evelope of the output was obtaied through Hilbert trasform [6]. Basebad pulses were recovered after lowpassig the evelope (cutoff of. GHz). 3. Image Processig Data produced by this procedure ca best be viewed i the form of a image plot of the relative received power (with respect to the LOS sigal) as a fuctio of time ad agle. A example of such plots is show i Figure 3a. This data set was take from cubicle measuremets (Figure a), where the trasmitter ad receiver separatio is about.8 m. Note that i his figure, the vertical axis represets the absolute delay from the trasmitter. Also ote that a -3 db threshold has bee applied. From this image plot, oe ca clearly see the sigal due to LOS path (zero agle) as well as two reflectios at large agles. ormalized power (db) delay (s) - - Measuremet7a θ (deg) 5 LOS θ (deg) (a) Image plot (b) Reflectio plot Figure 3. Example of a processed data set. Computer algorithms, such as CLEAN ad KDE, are available to extract MPC iformatio (power, ToA, ad AoA). However these algorithms are usually time-cosumig ad they ted to pick up false detectios which may eed to be maually removed. We had made accurate layout diagrams for each locatio durig the measuremet campaig. Due to the simplicity of the measured data, MPC iformatio i our data sets was maually collected from processed data alog with the help of eviromet layouts. Submissio Page 5 Zhiguo Lai, UMass

6 November 6 IEEE P c Cosider the example give i Figure 3a, which is the reflectio measuremet of the office eviromet i Figure a. To accurately determie the MPC iformatio, the peak value ad its correspodig delay of the received power at each agle were plotted i Figure 3b. Lookig at the layout diagram ad the reflectio plot, oe ca determie that the true reflectios, ot the aomalies of the atea itegratig reflected power, were captured. Oe ca further determie that ad are sigle ray reflectios (from the metal shelf ad the wall behid the receiver respectively). The image plot ad the reflectio plot i Figure 3, cofirmed the selectio. This procedure was used for all the measuremets. 3.3 Atea Decovolutio The atea decovolutio process was carried out by a process explaied i the previous paragraph. This was made possible as oly a few scattered MPCs were observed i each of the measuremets. This was facilitated by the -3 db threshold. The 3 HPBW receiver atea ad circular polarizatio further reduced multipath reflectios. 4. Proposed Statistical Idoor Chael Model I this sectio we propose a statistical model for the idoor multipath chael that icludes a sigle-cluster versio of the S V model [7] ad icorporates spatial iformatio. Model parameters are derived from the data collected durig the measuremet campaig described i Sectio. By assumig a fiite umber of MPCs (sice a proper threshold has bee applied), the chael impulse respose (CIR) ca be expressed as L h( t, θ ) = β δ ( t tlos, θ ) + α lδ ( t tlos t l, θ θl ) () l= where δ( ) is the Dirac delta fuctio, β is a ormalizatio factor that ca be determied from the measuremet setup (e.g., the free-space path loss, trasmitter ad receiver gais, etc.), t LOS is the absolute delay correspodig to the LOS path, L is the umber of rays, α l, t l, ad θ l are the amplitude, the ToA, ad the AoA of the l-th ray, respectively. Notice that the last three parameters are all relative values with respect to the LOS sigal. 5. Extractio of Chael Parameters This sectio presets the statistical characterizatio of chael parameters defied i the previous sectio. The variatios of these parameters may be characterized statistically by fittig the measuremet data agaist the proposed theoretical distributios. 5. Joit Distributio of the ToA ad the AoA Figure 4 shows the scatter plots of the ToA versus the AoA for office ad residetial eviromets. It ca be see that for both eviromets the ToA ad the AoA are strog related. Roughly speakig, rays arrivig at the receiver with shorter (or loger) delays ted to have relatively smaller (or larger) AoAs. Also it appears that there are o arrivals withi ± of the LOS directio for the office eviromet ad ± for the residetial eviromet. Submissio Page 6 Zhiguo Lai, UMass

7 November 6 IEEE P c office eviromet 5 ToA (s) AoA (deg) residetial eviromet 5 ToA (s) AoA (deg) Figure 4. Scatter plots of the ToA versus the AoA for office ad residetial eviromets. To better model the temporal-spatial property of the multipath chael, it is desirable to study the joit PDF betwee the ToA ad the AoA, f(t l, θ l ), which, followig the approach described i [4], ca be expressed as f ( tl, θ l ) = f ( θl tl ) f ( tl ) () where f(θ l t l ) is the coditioal AoA PDF ad f(t l ) is the margial ToA PDF. The margial ToA PDF, f(t l ), was estimated by accumulatig all rays AoAs ad is usually modeled by a Poisso process [7]: f ( tl tl ) = λ exp[ λ( tl tl ) ], tl < tl < (3) where λ is the mea ray arrival rate. Figure 5 shows the margial ToA histograms for both office ad residetial eviromets with their estimated PDFs plotted as solid curves. The estimated parameter from the measured data was /λ =. s for the office eviromet ad.9 s for the residetial eviromet. Figure 6 shows the joit histograms for the ToA ad the AoA. Note that the agular axis i this figure spas from to 36 istead of from -8 to 8 cosiderig the fact that the majority of arrivals are from the viciity of ±8 except for extremely short delays. I [4], the coditioal AoA PDF, f(θ l t l ), is associated with a set of so-called partial coditioal AoA PDFs, {f(θ l ), =,,, N-}, as follows: N {,, L, N } ( ) = f θ l tl f ( θl τ ), (4) = where = τ with τ beig the delay step size ad N beig the umber of delay steps. Here τ = s was chose i order to esure that the umber of rays withi each delay step,, was Submissio Page 7 Zhiguo Lai, UMass

November 6 IEEE P8.5-6-398-3-3c sufficiet (e.g., at least ) for further statistical aalysis of f(θ l ).

8 November 6 IEEE P c sufficiet (e.g., at least ) for further statistical aalysis of f(θ l ). By observig the shape of f(θ l ), a kow PDF is proposed to represet the varyig shape of f(θ l ) by a chage of its parameter values (usually the mea ad stadard deviatio). 5 office eviromet occurrece (%) occurrece (%) t (s) residetial eviromet t (s) Figure 5. Margial ToA PDFs for office ad residetial eviromets. Figure 6. Joit histograms of the ToA ad the AoA for office ad residetial eviromets. Cosiderig the similarity betwee office ad residetial eviromets, i the followig aalysis, we combie all the measured data for a sufficietly large sample space. Figure 7 shows a series of histograms, i.e., graphs of partial coditioal AoA PDFs, {f(θ l ), =,,, N-}. The value of to geerate each of these graphs are also show o the plots. Note that at large Submissio Page 8 Zhiguo Lai, UMass

9 November 6 IEEE P c delay values ( > s), fewer arrivals were observed. I order to obtai a good statistical represetatio, τ was chose to be s to esure a sufficietly large umber of samples. Also ote that graph (a) was geerated differetly: it was observed that rays with extremely short delays ted to arrive withi the viciity of ; so -8 to 8 otatio was used ad histogram of absolute AoA values was plotted (cosiderig the symmetry). It ca be see that a Gaussia PDF cetered at 8 with various stadard deviatios ca provide a reasoably good match to the rest graphs. Note that although Figure 7b is more or less uiform, it ca still be treated as a Gaussia distributio (still cetered at 8 ) with very large stadard deviatio. Therefore a modified Gaussia PDF is proposed to model f(θ l ): ( θ l θ ) exp < τ < τ π σ σ f ( θl τ ) = (5) o ( θl 8 ) exp otherwise π σ σ θ θl l where τ, θ, ad σ are properly chose costats (τ = s, θ = 3, ad σ = i our case), ad σ θ l is the stadard deviatio coditioed upo ad its variatio ca be approximated by the followig fittig: σ θ ( τ ) a ( b τ ) l τ = exp (6) where a ad b are costats that ca be estimated usig a oliear least square regressio method. Figure 8 show the extracted σ from measured data ad its least square fittig. θ l Much more measuremets are eeded for more accurate modelig of the joit PDF. At the time of desigig this measuremet campaig, o prior iformatio was available o the multipath characteristics (AoA ad ToA) at 6 GHz ad with circular polarizatio. 5 4 (a) = [:] s = 34 5 (b) = [:4] s = 35 5 = [4:6] s = 46 (c) occurrece (%) = [6:8] s = 46 (d) (e) = [8:] s = (f) = [:] s = AoA (deg) Figure 7. Partial coditioal AoA PDFs for each delay step. Submissio Page 9 Zhiguo Lai, UMass

10 November 6 IEEE P c stadard deviatio (deg) ( τ ) = 64.5 exp(. τ ) σ θ 9 l curve fittig measured data delay step, (s) Figure 8. Stadard deviatio of the coditioal AoA PDF as a fuctio of delay step. 5. Power Delay Profile The power delay profile (PDP) cotais iformatio about how much power arrives at the receiver with a delay betwee [τ, τ+dτ] ad has bee ivestigated i umerous studies (theoretically ad experimetally). It is therefore oly shortly discussed here. I accordace with [4], [5], ad [8-], it is foud that the average PDP is reasoably well modeled by a oe-sided expoetial decayig fuctio, i.e., P exp( τ / γ ) τ > P ( τ ) (7) otherwise where γ is the ray decay factor. Figure 9 shows the scatter plots of the ormalized power versus the ToA for both office ad residetial eviromets with their estimated curves superimposed. The estimated parameters from the measured data was γ = 3.8 s for the office eviromet ad.56 s for the residetial eviromet ad P = -.84 db for the office eviromet ad -. db for the residetial eviromet. Oe of the key compoets for small-scale fadig is the amplitude statistics which is ofte assumed to follow Rayleigh distributios with their parameters determied from the average PDP, i.e., σ ( τ ) = P( τ ). The Rayleigh distributio with parameter σ has a PDF give by α α exp > α f ( α) = σ σ (8) otherwise If we defie a ormalized parameter ˆ α = α / σ, it is easy to show that αˆ also follows a Rayleigh distributio with parameter. Obviously it is much simpler to work with a ormalized variable. Submissio Page Zhiguo Lai, UMass

11 November 6 IEEE P c Let the measured istataeous amplitude be α (τ ). Defie the ormalized amplitude as ˆ α( τ ) = α( τ ) / P( τ ). To obtai f (αˆ ), the PDF of ˆα, we follow the approach described i the previous sectio, i.e., we associate it with a set of partial ormalized amplitude PDFs, { f ( ˆ α τ ), =,, L, N }, as follows: N {,,, N } ( ˆ) = f α f ( ˆ α ), L (9) = where = τ with τ beig the delay step size ad N beig the umber of delay steps just like before. Agai the office ad residetial data were combied to esure a sufficietly large umber of samples. Figure shows a series of histograms, i.e., graphs of partial ormalized amplitude PDFs, { f ( ˆ α τ ), =,, L, N }, all superimposed by a Rayleigh distributio with parameter (the red curves). We ca see a reasoably good agreemet betwee the measured data ad the aalytical expressio. Aother two parameters of practical iterest are the mea delay, τ, ad the RMS delay spread, τ RMS, defied as [] P( τ ) ( τ ) d ( ) ( ) τ dτ τ = ad τ τ P τ dτ τ =. () RMS P τ P( τ ) dτ Note that both parameters are defied from a sigle PDP. Figure shows the computed τ ad τ RMS as well as the cumulative probability of the RMS delay for both office ad residetial eviromets. The results are cosistet with those reported i []. ormalized power (db) ormalized power (db) office eviromet ToA (s) residetial eviromet ToA (s) Figure 9. Scatter plots of the power versus the ToA for office ad residetial eviromets. Submissio Page Zhiguo Lai, UMass

12 November 6 IEEE P c.5 (a) = [:] s = 34.5 (b) = [:4] s = 35.5 (c) = [4:6] s = PDF (d) = [6:8] s = 46.5 (e) = [8:] s = 34.5 (f) = [:] s = ormalized amplitude 4 6 Figure. Partial ormalized amplitude PDFs for each delay step. 5 office eviromet 5 residetial eviromet mea delay (s) measuremet umber 6 mea delay (s) measuremet umber 4 rms delay (s) 4 rms delay (s) 3 cumulative probability measuremet umber rms delay spread (s) cumulative probability measuremet umber rms delay spread (s) Figure. The mea delay ad the RMS delay for office ad residetial eviromets. Submissio Page Zhiguo Lai, UMass

13 November 6 IEEE P c 5.3 Extracted Parameters Chael parameters of all the proposed PDFs ad PDP related parameters extracted from the measuremet data for both office ad residetial eviromets are give i Table. Table. Statistical Chael Parameters Parameters Values Office Residetial Uit Ray Arrival Rate, /λ..9 s Ray Decay Factor, γ s τ s θ 3 deg Coditioal AoA PDF σ deg Mea 8 deg Stadard a 64.5 deg Deviatio b.9 s -/ Mea Delay, τ s RMS Delay Spread, τ RMS.8.4 s 6. Summary ad Coclusios A ew statistical 6 GHz idoor chael model, which icludes both ToA ad AoA characteristics, has bee proposed based o the measuremet data collected i two differet eviromets usig circular polarized ateas. A time-domai measuremet system that was used to simultaeously collect the temporal ad spatial data was described. Data processig methods were outlied ad statistical characteristics of the model parameters were preseted. Oe of the major cotributios of this paper is the thorough study of the joit PDF betwee the ToA ad the AoA, which fully describes the temporal-spatial correlatio properties of the multipath chael. To the best of the authors kowledge, this paper is the first 6 GHz chael model based o circular polarized sigals. I geeral, the data preseted here exhibited some geeral characteristics as the sigle-cluster S-V model ad other modified S-V models. The ToAs i the measured data closely follow a sigle Poisso process, the mea amplitude of each arrival approximately follows a patter of expoetial decay, ad the istataeous amplitude follows a Rayleigh distributio. However there were some differeces observed i the measured model parameters: About 5% (8%) of the arrivals have a relative power of -5 db (- db) or less compared to the LOS sigal for both office ad residetial eviromets. No arrivals were observed withi ± of the LOS directio for the office eviromet ad ± for the residetial eviromet. For both eviromets the ToA ad the AoA are strog related. Rays arrivig at the receiver with shorter (or loger) delays ted to have relatively smaller (or larger) AoAs. Submissio Page 3 Zhiguo Lai, UMass

14 November 6 IEEE P c The coditioal AoA PDFs are well described by a series of Gaussia distributios cetered at 8 with various stadard deviatios except for those with extremely short delays. 7. Ackowledgemet The authors wish to thak Dr. Su-Khiog Yog ad Dr. Toy Pollock for their helpful commets. 8. Refereces [] T. Maabe, et. al., Multipath Measuremet at 6 GHz for Idoor Wireless Commuicatio Systems, IEEE 44th Vehicular Techology Coferece, 95-99, Jue 994. [] T. Maabe, et. al., Polarizatio Depedece of Multipath Propagatio ad High Speed Trasmissio Characteristics of Idoor Millimeter Chael at 6 GHz, IEEE Trasactio o Vehicular Techology, Vol. 44, No., 68-74, May 995. [3] K. Sato, et. al., Measuremets of Reflectio ad Trasmissio of Office Buildig i the 6 GHz Bad, IEEE Trasactio o Ateas ad Propagatio, Vol. 45, No., , December 997. [4] C. C. Chog, et. al., A New Statistical Widebad Spatio-Temporal Chael Model for 5 GHz Bad WLAN Systems, IEEE Joural o Selected Areas i Commuicatios, Vol., No., 39-5, February 3. [5] Q. H. Specer, et. al., Modelig the Statistical Time ad Agle of Arrival Characteristics of a Idoor Multipath Chael, IEEE Joural o Selected Areas i Commuicatios, Vol. 8, No. 3, , March. [6] S. Hayki, Commuicatios Systems (4 th Editio), Joh Wiley & Sos, Ic., May. [7] A. A. M. Saleh ad R. A. Valezuela, A Statistical Model for Idoor Multipath Propagatio, IEEE Joural o Selected Areas i Commuicatios, Vol. 5, No., 8-37, February 987. [8] A. Davydov, et. al., Saleh-Valezuela Chael Model Parameters for Library Eviromet, IEEE doc.: c. [9] T. Pollock, et. al., Office 6 GHz Chael Measuremets ad Model, IEEE doc.: c. [] T. Pollock, et. al., Residetial 6 GHz Chael Measuremets ad Model, IEEE doc.: c. [] H. Sawada, et. al., LOS Office Chael Model Based o TSV Model, IEEE doc.: c. [] T. S. Rappaport, Wireless Commuicatios: Priciples ad Practice ( d Editio), Pretice Hall Ptr., December. Submissio Page 4 Zhiguo Lai, UMass

IEEE P Wireless Personal Area Networks

IEEE P Wireless Personal Area Networks September 6 IEEE P8.-6-398--3c IEEE P8. Wireless Personal Area Networks Project Title IEEE P8. Working Group for Wireless Personal Area Networks (WPANs) Statistical 6 GHz Indoor Channel Model Using Circular