Suppression of Co-Channel Interference in High Duty Ratio Pulsed Radar Receivers

Transcription

1 Suppresson of Co-Channel Interference n Hgh Duty Rato Pulsed Radar Recevers C M Alabaster* *Whte Horse Radar Lmted, K, emal: clve@whradar.com Keywords: Co-Channel Interference, Pulsed Radar, Recever, Hgh duty rato. Abstract Ths paper descrbes a technque for the suppresson of nterference, partcularly co-channel components arsng from nterference on neghbourng channels, wthn a pulsed radar recever operatng on a hgh duty rato. The technque employs two RF swtches wth a band pass flter between them and nestng the gatng control of the second swtch to le wthn the gatng control of the frst swtch. Smulatons ndcate that rejecton of n-band nterference components arsng from nterference that s offset by 60MHz from the channel centre frequency some 0dB greater than that of a conventonal flterng s possble. The technque s most effectve at hgh duty ratos. Introducton One of the most damagng forms of nterference for any radar s co-channel nterference from another, smlar radar, snce the nterference sgnal closely resembles that expected by the vctm radar. Interference on a neghbourng channel frequency but wthn the front-end bandwdth of the radar can be accepted nto the radar causng saturaton of front-end components and ntermodulaton products wthn the channel bandwdth. Furthermore, the recever protecton swtchng can modulate the nterference resultng n addtonal frequency components wthn the channel bandwdth. Consequently, the nterference can cause a flood of false alarms, saturate the processng and corrupt the operaton of the constant false alarm rate (CFAR) detector. To combat ths problem, a radar may cancel nterference sgnals wthn ts sgnal processor or use auxlary antenna cancellaton technques, or steerable antenna nulls, n the drecton of the nterference sources; orthogonal waveforms may be used by lke radars operatng wthn range of each other n order to suppress mutual nterference. It would, however, be preferable to reject nterference wthn the recever before t reaches the processor snce ths reduces the processng burden, does not dstort antenna beams f steered close to nulls n ther radaton patterns and does not requre coordnaton of waveforms between smlar radars. A smple technque s descrbed here that can provde hgh levels of nterference rejecton n pulsed radars operatng on a hgh duty rato. Hgh duty ratos are often assocated wth hgh PRF waveforms but ths need not always be the case. The soluton uses standard components wthn the rado frequency (RF) radar recever; namely two RF swtches and a band pass flter (BPF) between the two swtches. The frst swtch pulse modulates the nterference at the pulse repetton frequency (PRF) of the vctm radar and the BPF passes only hgh order PRF components that are manfest as mpulses of n-band nterference at the swtchng edges of the frst swtch. The gatng control of the second swtch s nested wthn the gatng control of the frst swtch such that the mpulses of nterference are gated out. The crtcal and novel elements descrbed here are the relatve tmngs of the swtch gatng sgnals and the flter requrements. Secton descrbes the technque n more detal and secton 3 provdes an analytcal bass. The fourth secton descrbes detals of a smulaton, ncludng some smulaton results. Fnally, secton 5 draws some conclusons. Descrpton of the Technque. The Problem Consder a stuaton whereby a pulsed radar s recevng nterference from another pulsed radar. The stuaton s portrayed n Fgure for a source of pulsed nterference and a vctm recever both operatng on nomnally the same PRF wth a 50% duty rato. The top trace (a) n Fgure shows the transmtted pulses from an nterference source. The mddle trace (b) of Fgure s the gatng functon of a vctm recever that, when low, solates the recever durng the transmsson of ts own pulses and, when hgh, permts the recepton of sgnals. The bottom trace (c) of Fgure shows the noneclpsed fragments of the nterference pulses that are accepted nto the vctm recever. Snce the tmngs of the transmtted pulses and the gatng of the vctm recever wll, n general, be asynchronous, n addton to an unknown range delay between the two, there wll be a random tmng offset between the two. Ths means that only fragments of the transmtted pulses wll be receved correspondng to perods when the transmtted pulses are present durng the recevng tme of the vctm radar. For 50% duty ratos, one edge of the fragment of nterference wll be defned by the transmtted modulaton (the tralng edge, as shown n Fgure ) and the other edge wll be defned by the recever gatng (the leadng edge, as shown n Fgure ). Although the two radars may both be runnng on notonally the same PRF, n practce, there

2 wll always be a slght dfference n ther PRFs. Furthermore, the range between the transmtter of the nterference and the recever may constantly be changng. Both the PRF dfference and changng range wll ensure that the tmngs of the transmtted pulses and recever gatng drft n and out of phase wth each other and, as a result, the wdth of the fragment of nterference s constantly changng. Addtonally, the defnton of the leadng/tralng edges of ths fragment swaps perodcally between the transmtted modulaton and recever gatng. The bandwdth of the nterference s gven by the recprocal of the pulse fragment wdth and wll therefore constantly be changng. The rate at whch the spectrum of the nterference decays at hgher offset frequences from ts carrer frequency depends on the rse and fall tmes of the transmtted modulaton and recever gatng. Snce both these sgnals tend to have sharp edges, there wll be consderable spectral spread nto neghbourng channels. As the duty rato of the transmtter and recever reduce, or the wdth of the transmtted pulses becomes much less than the vctm recever s open tme, both edges of the nterference pulses become more lkely to be defned by the transmtted modulaton. Notwthstandng ths, there s always some lkelhood that fragments of nterference wll have one edge defned by the recever gatng. A complete soluton to mutual nterference must therefore address both the transmsson of pulses and the recever. If, however, only one edge of the pulse has the 0ns cosne profle and the other edge has a ns rse tme, then the mprovement s only 4.5dB at a 60MHz offset frequency. Clearly, t s mportant that both edges of the nterference pulse fragment be shaped n order to reduce co-channel nterference. Whlst t s possble to shape the recever gatng functon n much the same way that transmtted pulses may be shaped, t s not always practcal to do so. Frstly, f the nterference s powerful enough to drve any subsequent amplfer stages nto saturaton, the profled edges wll be re-sharpened. Secondly, the extended rse and fall tmes on the recever gatng functon ncrease eclpsng losses and the mnmum range []. The reducton n spectral spread can be mparted wthn the recever usng a combnaton of two swtches wth a flter between them. A block dagram of a typcal superheterodyne recever front-end s shown n Fgure. The crucal and novel elements are the two swtches, and specfcally, the tmng controls to these swtches, wth the band pass flter (BPF) sandwched between them and these components are hghlghted usng the lght grey shadng n Fgure. Any pulsed radar recever requres a protecton swtch before the low nose amplfer (LNA) n order to solate t from damagng levels of transmtter leakage power durng ts transmtted pulses; ths forms the frst RF swtch. Receved sgnals are then amplfed n the LNA and down-converted to an ntermedate frequency (IF) usng a mxer. The BPF s desgned to flter out nterference fallng outsde the channel bandwdth of the radar and so may typcally have a bandwdth of a several MHz; t s not ntended to be the matched flter, whch would typcally come further along the processng chan. The BPF s more easly mplemented n the IF sgnal path, rather than at the ncomng (mcrowave) RF secton, snce t requres a more modest Q-factor here. The solaton provded by the protecton swtch may typcally be around 60dB and, although adequate to protect the LNA from saturaton, stll results n a detectable level of transmtter leakage power wthn the recever. Ths should be removed usng an addtonal hgh solaton swtch before the hgh gan amplfer chan; ths s the second swtch shown on the rght of Fgure. Fgure : Interference from pulsed radar operatng on a 50% duty rato. (a) Pulsed nterference, (b) Gatng control of vctm recever (hgh = recever open, low = recever closed), (c) Fragment of nterference pulse accepted nto recever.. The Soluton The classc soluton to lmt the spectral spread of transmtted pulses s to shape the pulses wth a weghtng functon []. Ths extends the rse and fall tmes of the transmtted pulses. Even the smple expedent of mpartng a cosne functon rse and fall (voltage) profle over the frst and last 0ns of a.5μs pulse can reduce the spectral envelope by 5dB at an offset frequency of 60MHz compared wth ns rse and fall tmes. Fgure : Radar superheterodyne recever front-end block dagram. In order to solate the recever effects from those of shapng the transmtted pulses, t s most convenent to consder contnuous wave (CW) nterference. If CW nterference s receved on a neghbourng channel t should be subject to a

3 hgh degree of rejecton by the BPF, however, the acton of the protecton swtch pulse modulates the nterference at the recevng system s PRF and causes sgnfcant spectral spread nto the pass band of the BPF, especally f the protecton swtch mparts fast rse and fall tmes. Ironcally, the protecton swtch has exacerbated the problem. The BPF passes hgh order PRF harmoncs that result n an output sgnal comprsng short bursts (mpulses) of nterference occurrng at the rsng and fallng edges of the pulses of nterference,.e. at the rsng and fallng edges of the gatng functon of the protecton swtch. The BPF has dfferentated the pulses of nterference. The duraton of these mpulses s proportonal to the rse and fall tmes of the protecton swtch and nversely proportonal to the bandwdth of the BPF and the frequency of the nterference; they may typcally last for a few tens of nano-seconds for a flter bandwdth of 5-30MHz. The hgh solaton swtch that follows the BPF must now have a gatng control nested wthn that of the protecton swtch such that these mpulses are gated out. The gatng tmngs are llustrated n Fgure 3 and a close-up vew of the rsng and fallng edges wth 30ns nested tmngs s shown n Fgure 4. In gatng out the mpulses, the n-band nterference s removed. Should ths nterference not be removed t could cause false target ndcatons, mask smaller genune target responses fallng concdent n range and velocty as the nterference, sap effort from the tracker whch may attempt to track these responses and corrupt the operaton of the constant false alarm rate (CFAR) detecton; n short, the nterference could have smlar effects as would false target jammng. The only n-band nterference that remans are hgh order PRF harmoncs assocated wth the fltered nterference. Consequently, the n-band nterference s reduced by approxmately the level of the flter rejecton at the centre frequency of the nterference and wll be sgnfcantly lower than that seen on the BPF output, or f the gatng functon to the hgh solaton swtch was not properly nested wthn the gatng functon of the protecton swtch. The exact tmngs n a practcal system would have to be adjusted to account for the group delay between the two swtches. The (small) penalty pad for the hgh degree of nterference rejecton s the slghtly ncreased dead tme due to the nestng and subsequent ncreases n eclpsng losses. It s therefore advantageous to ensure that these mpulses be as short-lved as possble, such that the dead tme due to swtch nestng may be mnmsed, and so ths demands rapd swtchng of the protecton swtch, a wde pass band of the BPF and mnmal dsperson n all the crcuts between the two swtches. To ths end, the BPF should have a lnear phase flter response across ts (wde) pass band, whch s most readly acheved f ts pass band s centred at a hgh frequency,.e. a hgh value of IF s used (VHF (30-300MHz) or HF ( MHz) beng better than HF (3-30MHz) [3]). The desgn of the BPF s crucal to the success of the nterference rejecton. The mpulses represent the flter s transent response to the rsng and fallng edges of the pulse of out-of-band nterference. The duraton of the mpulses s reduced for a large offset between the flter pass band and the nterference and for a wde pass band. The pass band should not be so wde, however, that t compromses the rejecton of out-of-band nterference. Resstve flter losses ntroduce dampng whch lengthens the mpulses; dsperson n the flter, and other components before the solaton swtch, also stretches the mpulses and should be avoded. Fgure 3: Sgnal waveforms. (a) protecton swtch gatng, (b) BPF output sgnal, (c) solaton swtch gatng, (d) sgnal at solaton swtch output. Fgure 4: Close-up of waveforms at swtchng rse/fall tmes. (a) protecton swtch gatng, (b) BPF output sgnal, (c) solaton swtch gatng, (d) sgnal at solaton swtch output. 3 Theory The processng of a receved sgnal by the analogue recever components hghlghted n grey n Fgure entals swtchng between the tme and frequency domans. The gatng functon provded by the two swtches s most convenently appled n the tme doman whereas the flterng functon of the BPF s most convenently appled n the frequency doman. The Fourer transform, together wth the nverse Fourer transform, enables one to transform from the tme (t) to frequency (ω) domans and back agan [4]. Let the CW nterference sgnal be gven by: s0t cos t () where ω s the angular frequency of the nterference. 3

4 It s necessary to consder just a sngle cycle of the pulsng operaton of the protecton swtch whch can therefore be represented by the dealsed tme gatng functon: gt 0 t () 0 elsewhere Ths swtch modulates the nterference sgnal wth zero nserton loss when the swtch s closed and nfnte solaton when open. It results n a rectangular pulse of nterference havng a pulse wdth, τ, of nfntely fast rse and fall tmes. The pulse modulated nterference sgnal on the swtch output s therefore: st s0t gt cos t 0 t (3) 0 elsewhere (The amplfcaton and down-converson process s nconsequental to the rejecton of nterference and so the same nterference carrer frequency has been retaned throughout.) The s (t) sgnal s a sngle pulse of the orgnal nterference from tme, t = 0 to t = τ. The spectrum of ths sgnal s gven by ts Fourer transform, whch s defned as: jt s t s e dt S F. (4) In ths case, the functon s (t) s gven by the product of two functons, so: s t g t S S F 0. 0 G (5) where S 0 (ω) represents the Fourer transform of s 0 (t), G (ω) represents the Fourer transform of g (t) and * represents the convoluton of the two. In practce, t s not necessary to use convoluton snce the Fourer transform of s (t) may be derved drectly from the defnton of the Fourer transform gven by Equaton (4) over three tme ntervals; t < 0, 0 t τ and t > τ. The resultng spectrum s gven by: j.. e.sn. j j j. e.cos S (6) Ths s the classc snc functon spectrum centred at ω wth an angular frequency offset to the nulls at multples of. Should the analyss be extended to capture at least two perods of the gatng functon, g (t), ts perodcty would be evdent and would result n a set of dscrete spectral responses offset by harmoncs of the PRF and whose envelope conforms to the snc functon gven by Equaton (6). Let the BPF have an dealsed transfer functon gven by: j F cos jsn e L (7) 0 elsewhere where ω L and ω are the lower and upper frequency lmts of the pass band, respectvely, and β s the phase constant of the flter. Wthn ts pass band, the flter has a magntude response of and a phase response of β. Thus, the flter has a rectangular response of zero pass band nserton loss and nfnte stop band solaton and has a lnear phase response wthn ts pass band; the flter s therefore non-dspersve. It may be assumed that ω s well outsde the pass band, ω L to ω, and so the flter passes only low-level, hgh order sdelobes of the snc functon that are offset well away from ω represented by Equaton (6). The flter output therefore has a spectrum gven by: S S F (8) Substtuton of Equatons (6) and (7) nto Equaton (8) gves: S j j e.cos.. e.sn. j j. e L j (9) S 0 elsewhere The tme doman representaton of ths sgnal may be obtaned va an nverse Fourer transform,.e.: j s t S S e t F. d (0) In ths case, the upper expresson for S (ω) gven by Equaton (9) can be substtuted n Equaton (0) and the lmts of ntegraton can be reduced to ω L and ω. Note also that: () v p where v p s the phase velocty through the flter. The tme doman representaton of the flter output s therefore gven by: s t j L j. e j t v p j t v p j. e.cos.... t vp.sn. d.... e () Ths sgnal s ncdent on the solaton swtch. Agan, t s necessary to consder just a sngle cycle of the pulsng operaton of the solaton swtch whch can therefore be represented by the dealsed gatng functon: gt tn t tn (3) 0 elsewhere where t n s the nestng margn of the solaton swtch wthn the tmngs of the protecton swtch, g (t). The gatng functon to the solaton swtch s a pulse of wdth τ-t n delayed by t n wth respect to g (t). As for the protecton swtch, the solaton swtch exhbts nfntely fast rse and fall tmes, zero nserton loss when the swtch s closed and nfnte solaton when open and t provdes further modulaton of the nterference sgnal. The waveform on the output of the solaton swtch s therefore gven by: t s t g t (4) s3 The spectrum of the output sgnal s gven by the Fourer transform of s 3(t),.e.: S3 F s 3t S G (5) where G (ω) represents the Fourer transform of g (t). 4

5 Ascertanng the spectra and waveforms of the sgnals on the nput and output of the solaton swtch entals an evaluaton of the ntegral of Equaton (). The precse nature of these sgnals depends on the values of ω and τ wth respect to ω L and ω. Of partcular nterest here, s the case of both ω L and ω beng offset from ω by many tmes π/τ,.e. typcally,, 00. L Evaluaton of Equaton (9) and the ntegral of Equaton () s awkward but some general trends may be observed for the partcular choce of parameter values of nterest here that ad understandng. Equaton (9) s of the general form: j a j e (6) j a c j The e term n Equaton (9) merely ntroduces a phase offset to the spectrum or a tme shft equvalent to the delay through the flter nto the waveform, Equaton (0), and may be gnored here. The nverse Fourer transform of Equaton (6) represents the waveform of an exponentally decayng cosne wave startng at tme t = 0 followed by an exponentally decayng negatve cosne wave startng at tme t = τ. The frequency of the cosne waves s gven by ω c and the factor a s the tme constant of the exponental decay and wll be large and postve for a wde pass band flter centred well away from the carrer frequency of the nterference.e.: s large and, 00 L. L The exponentally decayng cosne waves are the underdamped transent responses of a resonant crcut,.e. the response of the deal, loss-less flter to the rsng and fallng edges of the nput pulse. The frequency, ω c, corresponds to the resonant frequency of the crcut,.e. the centre frequency of the flter pass band. The duraton of a flter s response s gven by the nverse of ts bandwdth, therefore a. As the flter bandwdth ncreases, a becomes L ncreasngly large and the exponental decay becomes more rapd wth the result that the waveform tends towards a postve mpulse at t = 0 and a negatve mpulse at t = τ. Ths waveform represents the dfferental of the nput pulse whose j spectrum s of the form: e. It may therefore be understood that, for the partcular choce of parameters descrbed here, the waveform, s (t) approxmates the dfferental of the s (t) pulse and resembles the mpulses shown n Fgure 4, trace (b). The exponentally decayng cosne mpulses reduce to 37%. Settng the of ther ntal ampltude after a tme L nestng offset to the nverse of the BPF bandwdth, t, enables waveform g (t) to gate out most of n L the mpulses. Increasng t n to allows the L mpulses to decay to 4% of ther ntal ampltude, and so more of the mpulses are gated out but the dead tme and eclpsng losses are doubled. The waveform on the solaton swtch output s now very low: s t 0 for all t. In ths way, 3 sgnfcant rejecton of the nterference wthn the pass band can be acheved. 4 Smulaton Results The sgnals wthn the IF secton of a hypothetcal recever have been smulated n accordance wth the technques and theory descrbed n the earler sectons. An 8 pole Butterworth flter desgn has been modelled. A hgh PRF wth a hgh duty rato s assumed and would be typcal parameters for a hgh PRF ar-to-ar velocty search mode of an arborne ntercept or fre control radar [5]. The followng parameters have been assumed: PRF of recever 50 khz Duty rato 37.5% (recever open tme.5 μs) IF centre frequency 00 MHz BPF nserton loss at 00MHz.6 db BPF 3dB bandwdth 7.5 MHz Interference centre frequency n IF 60 MHz (.e. 60MHz above IF centre) BPF nserton loss at 60MHz 6.0 db (Rejecton wrt to pass band 4.4 db) It was further assumed that the protecton swtch had a rse and fall (voltage) profle of a cosne functon over 0ns. The solaton swtch had a ns rse and fall tme that was nested 30ns wthn the 50% ponts of the rse and fall of the protecton swtch edges (Fgure 4). The fast rse and fall tmes of the solaton swtch was ntended to gve rse to the slowest spectral decay n the nterference and hence the most pessmstc result for nterference rejecton; slower rse/fall tmes would result n reduced n-band nterference. The extra 30ns nestng margn at the begnnng and end of each recevng perod ncurs a dead tme totallng 60ns. Ths dead tme reduces the recever open tme to.5μs 60ns =.44μs that results n an extra eclpsng loss of: 0.log 0(.44/.50) = 0.dB. A smaller ncrease n eclpsng loss would be ncurred at a lower PRF. The spectra of the varous sgnals are shown n Fgure 5. These spectra were obtaned va a unformly weghted FFT over two complete cycles of the recever pulsed waveform (8μs). The nterference, pulse modulated by the protecton swtch, and down-converted by the mxer, s shown n the dark blue and ts peak level at 60MHz s normalsed to 0dB. Its envelope at 00MHz s at -56.5dB and s due entrely to the pulse modulaton mparted by the protecton swtch. The flter characterstc n decbels s shown n red. The flter provdes 4.4dB rejecton of nterference at +60MHz offset from ts centre frequency. The green spectrum corresponds to the nterference sgnal seen on the output of the BPF. Its peak at 60MHz s at -6.04dB and corresponds to the flter nserton loss at ths frequency. Its envelope at 00MHz s at -58.dB whch s only.6db lower than the nterference level at the IF centre frequency seen at the mxer output (due to the.6db nserton loss of the BPF at 00MHz). Ths llustrates that the flter wll prevent the out-of-band nterference from saturatng further amplfcaton and processng stages but provdes almost no rejecton of n-band nterference components ncurred by the pulsng of the protecton swtch. The nterference seen on the solaton swtch output s shown 5

6 n cyan and has a peak level of -6.6dB at 60MHz. The small decrease wth respect to the BPF output (green) of 0.dB s due to the slght reducton n the wdth of the nterference pulse, totallng 60ns, due to the 30ns of nestng at each edge, and s consstent wth the extra eclpsng loss. From the lower trace of Fgure 4, one may observe how the mpulses of n-band nterference have been removed, leavng the (suppressed) pulse modulated 60MHz sgnal. Its envelope at 00MHz s at -79.3dB, whch shows a.db decrease wth respect to the nterference at 00MHz on the flter output (green). The.dB mprovement s somewhat less than the 4.4dB rejecton provded by the flter at 60MHz offset; the dscrepancy beng due to the slower decay of the spectrum due to the faster edges of the solaton swtch (plus a mnor dscrepancy due to the slght change n ts pulse wdth, as noted earler). +00MHz MHz Table : Improvement of n-band rejecton of nterference versus offset frequency. () the greater degree of rejecton of n-band nterference results from extendng the nestng to 50ns. Very hgh levels of nterference rejecton are acheved n these cases due to the hgh flter rejecton close to zero Hertz. These results are gven for smple, unmodulated pulses. If ntra-pulse modulatons are used for the purposes of pulse compresson, there wll be addtonal spectral components wthn the IF pass band, however, these would also be rejected by the levels gven n Table. 5 Conclusons Fgure 5: Sgnal spectra. Blue = nterference at mxer output, Red = flter characterstc, Green = nterference at flter output, Cyan = nterference at solaton swtch output. The mprovements for nterference at other frequency offsets are tabulated n Table. The maxmum mprovement wll be lmted to ether the flter rejecton at the centre frequency of the nterference or the solaton of the solaton swtch, whchever s the least. Snce several hgh solaton swtches can be ganged n seres, t s the flter rejecton that sets the practcal lmt on nterference rejecton. Frequency offset of Interference Flter rejecton wrt pass band [db] Improvement of n-band nterference rejecton [db] - 80MHz , 6 () -60MHz , 53 () -40MHz MHz MHz MHz MHz MHz A full soluton to the rejecton of mutual nterference entals spectral contanment of the transmtted pulses through pulse shapng and rejecton wthn the recever. The use of two swtches wth a band pass flter between them as part of a typcal superheterodyne recever archtecture can provde a hgh degree of rejecton of neghbourng-channel nterference. The rejecton of nterference s dependent on the nondspersve nature of the crcutry between the two swtches, the characterstcs of the flter and the relatve tmngs of the gatng functons to the swtches; the gatng of the second swtch beng nested wthn the gatng tmngs of the frst. Rejecton of n-band, co-channel, spectral components of mutual nterference up to the level of the flter rejecton at the centre frequency of the nterference s possble. Ths offers consderable mprovements over flterng on ts own and would probably be benefcal n suppressng all forms of nterference. A small ncrease n eclpsng losses s ncurred due to the nestng margns between the two gatng sgnals. Acknowledgements I would lke to acknowledge the nvaluable contrbuton to ths work of the late Chrs Bourne, an ex-colleague and frend. References [] D. Emery. The Dgtal Synthess of HF Surfacewave Radar Waveforms, MSC thess, Cranfeld nversty, Shrvenham, K, (004). [] C. M. Alabaster. Pulse Doppler Radar, Sctech (now IET), (0). [3] IEEE Std Standard Letter Desgnatons for Radar-Frequency Bands. (976, revsed 984, 00). [4] James W. Nlsson, Electrc Crcuts, 4 th Ed., Addson- Wesley, ISBN (993). [5] Davd Lynch, Jr and Carlo Kopp, Multfunctonal Radar Systems for Fghter Arcraft, chapter 5 n Radar Handbook 3 rd Edton, Ed M. I. Skolnk, McGraw-Hll, ISBN (008). 6