Measuring Pulsed Interference in 802.11 Links

1 Measuring Pulsed Interfe rence in 8 02.11 Links Brad W . Zarik off , Douglas J. Leith Hamilton Institute, NUI Maynooth Abstract —Wireless 802.11 links opera te in unlicensed spectrum and so must accommodate oth er unli censed transmitters which generate pulsed interference. W e propose a new approa ch for detecting the prese nce of pulsed interfere nce affecting 802.11 links, and fo r estimating temporal statistics of this interfer - ence. This approa ch builds on recent work on disti nguishing collision losses from noise losses in 802.11 li nks. When the interva ls between interfer ence pulses ar e i.i. d., th e ap proach is n ot confin ed to estimating the mean and variance of these interva ls but can reco ver the complete probability distribution. The appro ach is a transmitter -side technique that provides per- link informa tion and is compatible with standard hardware. W e demonstrate the effectiv eness of the p roposed approach usin g extensiv e experimental measureme nts. In addition to a pplications to monitoring, managem ent and d iagnostics, the fun damental informa tion provided b y our approach can potentially be used to adapt the frame durations used in a network so as to in crease capacity in the p resence of pul sed interference. I . I N T RO D U C T I O N W ir eless 8 02.11 link s operate in unlicensed spectr um and so m ust acco mmodate other unlicensed transmitters. The se transmitters include n ot only other 80 2.11 WLANs but also Bluetooth devices, Zigbee devices, domestic applian ces etc . Impor tantly , the resulting interfer ence is o ften pulsed in nature. That is, the interf erence that consists of a seque nce of “on ” periods (or p ulses) during which the interference power is high, interspersed by “off ” p eriods where the interfer ence power is lower, illustrated schematically in Fig. 1. The former might be thought of a s corre sponding to a p acket transmission by a h idden ter minal and th e latter as the idle times betwe en these transmissions. For this type of interferer, RSSI/SINR measuremen ts are of limited assistance since the SI NR mea- sured fo r one packet may b ear little relation to th e SINR experienced by other packets. A furth er complicating factor is that in 80 2.11 link s f rame loss due to co llisions is a featur e of normal op eration in 802.11 WLANs, and thus we need to be car eful to distingu ish losses due to collisions an d losses due to chan nel impairmen t. In this p aper we p ropose a new appr oach for detectin g the presence of pulsed interferen ce affecting 8 02.11 link s and for estimating temp oral statistics of this inter ference und er mild assumptions. Our appro ach is a transmitter-side techniqu e that provides per-link info rmation an d is comp atible with stand ard hardware. This significantly extend s recent work in [1], [ 2] which establishe s a MAC/PHY cross-layer tech nique c apable of classifying lost tr ansmission oppor tunities into noise-related losses, collision induced losses, hidden-nod e losses and of distinguishing these losses from the unfairness caused b y exposed nodes and cap ture effects. Supported by Scienc e Foundati on Ireland grants 07/IN.1/I901 and 08/SR- C/I1403. W ireless LAN P ul s e d T ra ns m i t t e r t i m e Pulsed T ransmitter time Fig. 1. Illustra ting a WLAN with interfering pulsed transmitter (e.g. 802.11 hidden terminal, Blueto oth devic e, microw av e ove n, baby monitor , etc ) inducing pack et loss. Detection and measuremen t of pu lsed interferen ce is par- ticularly topical in v iew of the trend towards increasingly dense wireless deployments. In addition to bein g o f interest in th eir own rig ht for network monitor ing, managemen t and diagnostics, our temporal statistic measuremen ts can b e u sed to ad apt network p arameters so as to significantly increase network capacity in the presence of p ulsed interference. This is illustrated in Fig. 2, which shows experimental measurem ents of packet error rate (PER) versus modulatio n and cod ing scheme (MCS) f or an 802 .11 network in the presence o f a pulsed microwa ve oven (MWO) in terferer . T wo curves are shown, one for each f ragment of a two packet TXOP burst (below we discuss in m ore detail ou r interest in using packet pairs). Observe that the PER is lowest at a PHY r ate of 18-24 Mbps – importantly , the PER rises no t on ly f or h igher PHY rates, as is to be expected due to the lower resilien ce to noise at higher rates, but also rises for lower PHY rates. Th e in crease in PER at lower PHY rates is d ue to the p ulsed nature of the interferen ce – since the frame size in ou r experimen t is fixed, the tim e taken to transmit a frame incr eases as the PHY ra te is lo wered, increasing th e likelihood that a frame “collides” with an interference b urst. At a PHY rate of 1Mbps, the frame dura tion is lo nger than the maximu m interval betwe en interferen ce pu lses an d, as a r esult, the PER is close to 100%. W e discu ss this example in more detail in Section IV -B, but it is clear the approp riate choice of PHY rate can lead to significant throug hput gains in such situation s. W e briefly n ote that this ty pe of MAC layer ad aptation comp lements p roposed PHY layer interfer ence a v oidance techniqu es such as cognitiv e radio [3]. 2 I I . R E L AT E D W O R K Previous work on estimating 8 02.11 channel conditions can b e classified into three categories. First, P HY link-level approa ches using SINR and bit-error rate (BER). Second, MAC appr oaches relyin g on throughpu t and delay s tatistics, or frame loss statistics derived fro m tr ansmitted frames which are not A CK ed and/o r from signa ling messages. Fin ally cr oss-layer MAC /PHY appr oaches that c ombine in formation at b oth MAC and PHY layers. Most work on PHY layer appro aches is based on SINR measuremen ts, e.g. [4] –[6]. The basic idea is to a priori m ap SINR measur es in to link qu ality estimates. Howe ver , it is well known th at the c orrelation between SINR and actual packet de- li very rate can be weak due to time- varying channel conditions [7], pulsed in terferenc e being one such example o f a time- varying chan nel. [8 ] con siders lo ss diagnosis by examining the erro r pattern with in a p hysical-layer symbo l, with the aim of exposin g statistical differences between collision and weak signal based losses, but does not co nsider p ulsed interfer ence. The co gnitive rad io literature co nsiders PHY layer techniqu es for op timising perf ormance in the p resence of interfer ence v ia joint spectral an d tempo ral analysis [ 9]. Th ere are some solu- tions tailo red to the ISM band [ 3], wher e customised h ardware has b een devised with the aim of providing a synchr onisation signal based on periodic interference. Ho we ver , cogniti ve r adio technique s are largely g eared tow ards interference a voidance and ma ke u se of non- standard har dware. MA C appr oaches make up some of the most p opular and earliest rate co ntrol algo rithms. T ech niques such as ARF [ 10], RB AR [11] an d RRAA [12] attempt to use frame tra nsmission successes and failures as a mea ns to indirectly me asure chan - nel condition s. Howev er , these technique s cann ot disting uish between noise, c ollision, or hidden noise so urces of error . In [13], rate contro l via loss differentiation is suggested via a modified ARF algorithm; it was shown to greatly improve perfor mance via the in clusion of a N AK signal, but this requires a m odification to the 802 .11 MAC . Use of R TS/CTS signals h as been prop osed for distinguishing collision s fro m channel noise loss es, e.g. [14], [15]. However , such approach es can p erform poor ly in the p resence of pulsed in terference such as hid den terminals [ 1]. W ith regard to combine d MAC/PHY app roaches, the present paper builds upo n the packet p air approach pro posed in [ 1], [2] fo r estimating the frame erro r r ates due to collisions, noise and hidden termin als. See also the c losely r elated work in [16]. [ 1], [2], [ 16] foc us on time- in variant ch annels and do not con sider estimation of temporal statistics. [ 17] consider s a similar proble m to [1], but uses channel b usy/idle time informa tion. Some work has been done on packet length ad aptation as a me ans of exp loiting a time-varying chann el. [18] mo difies the Gilber t-Elliott chann el model to model bursty ch annels; howe ver , they do no t consider th e MA C lay er . Ther e are m any examples that use MA C fram e er ror in formatio n [19]–[2 3], but they lac k th e ability to disting uish betwee n noise and collisions. There h as been some recen t interesting work on a cross-layer mo del f or packet len gth adaptatio n in [24], which 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p k t 1 p k t 2 P S f r a g r e p l a c e m e n t s PHY rate (Mbps) Packet Error Rate Fig. 2. Experimenta l measurements of packet error rate (PER) versus modulati on and coding scheme (MCS) for an 802.11 network operating on channe l 9 and physicall y located near an operational micro wa ve ov en (MWO). See Section IV -B for further details of the experi mental setup. T wo curves are sho wn, one for ea ch fragment of a t wo pack et TXOP burst. Observ e that the PER is mini mised around 1 8-24 Mbps and rise s at both lo wer and higher MCS rates due to the pulsed nature of the interfere nce. relies on separ ation between no ise er rors and collision errors as a means of tuning the packet length and optimising throughput. I I I . P U L S E D I N T E R F E R E N C E T E M P O R A L S TA T I S T I C S : N O N - PAR A M E T R I C E S T I M A T I O N A. Basic Idea W e start with the ob servation that packet transmissions over a time-varying wire less link can be tho ught o f as sampling the channel conditions. Each sample cov ers an extended in terval of time, equal to the d uration T D of the pa cket transmission, see Fig. 3 . On a chan nel with pulsed interfer ence, the frequency with wh ich packet tran smissions overlap with inte rference pulses (and so the le vel of packet loss) depend s on the duration of the p acket tran smissions relative to the interv als betwe en pulses, and on th e duratio ns of the pulses. For example, it is easy to see th at when the packet du ration T D is lar ger than the max imum time between inter ference pulses, then every packet transmission ov erlaps with at least one interf erence pulse and we can expect to observe a high rate of packet loss. Con versely , wh en the packet duration T D is much smaller than the time between interf erence pulses, most o f th e p acket transmissions will not encounter an interference pulse and we can expec t a much lower rate of packet loss. Hence, by varying the pac ket transmit du ration and ob serving the correspo nding change in packet loss rate, we can hope to infer infor mation ab out the timing of th e inter ference p ulses. W e ca n make this intuiti ve in sight more precise as follows. Assume that the in tervals betwee n pulses are i.i. d. so that they are char acterised by a probab ility distribution fu nction. Then, w e will sho rtly show that the infor mation co ntained in such packet lo ss inf ormation is sufficient to fully rec onstruct this distribution function. This, somewhat surprising, result has im portant practica l implications. Namely , that even wh en the in terference pulses are no t dire ctly o bservable (whic h we expect to u sually be the case), we are nevertheless still able to reconstruc t key tem poral statistics of the inter ference pro cess from easily measured p acket loss statistics. 3 time Interference pulse Data packet Fig. 3. Schemati c illustrat ing “samplin g” of a t ime-v arying channe l by data pack et t ransmissions. Since the data transmissions occup y an interv al of time, the s ampling is of the channel conditions over that interv al, rather than at a single point in time. As the durati on of the data transmissions increase s, the chance that a data transmission ove rlaps with an interference pulse also tends to increase . B. Mathematical Ana lysis W e now for malise the se claims. Consider a sequenc e of interferen ce pulses indexed by k = 0 , 1 , 2 , ... and let T k denote the start time of the k th interferen ce pu lse with T 0 = 0 , S k > 0 denote the du ration o f the k th pulse and ∆ k = T k +1 − ( T k + S k ) > 0 be the interval between the end of k th pulse and the start of the ( k + 1) th pulse. Defining state vector X t := ( t, T k ( t ) , S k ( t ) , ∆ k ( t ) ) , t ∈ R + , the sequence { X t } forms a stocha stic p rocess with T k +1 = T k + S k + ∆ k , T 0 = 0 , k ( t ) = sup { k : T k < t } . W e assum e that the rando m variables ∆ k , k = 1 , 2 , ... are i.i.d. with finite mea n. Then ∆ k d = ∆ , where d = denotes eq uality in distribution, and let Prob[∆ ≤ x ] = F ( x ) . Similarly , we assume that th e pulse duration s { S k } ar e i.i.d. with fin ite mean and S k d = S . Pick a sampling inter val [ t − T D , t ] . Th is samplin g in terval can be tho ught of as a p acket transmission ending at time t . Define indicator fu nction U T D ( X t ) = 1 if interval [ t − T D , t ] does not overlap with a ny interfe rence pulse, an d U T D ( X t ) = 0 o therwise. That is, U T D ( X t ) =  1 t ∈ [ T k + S k + T D , T k +1 ) for some k 0 otherwise . (1) Suppose we transmit a seq uence of packets and let { t j } deno te the seq uence o f times when transmissions finish. Assum e fo r the mom ent that (i) a packet is lost whenever it overlap s with an interf erence pulse and (ii) the intervals between packet transmissions are exponentially r andomly d istributed and are indepen dent o f the interferen ce pr ocess. W e will sho rtly relax these assumptio ns. By assumption (i), U T D ( X t j ) equals 1 if the p acket tran smitted at time t j is received successfu lly an d 0 o therwise. Hen ce, th e emp irical estimate of the packet loss rate is ˆ P t ( T D ) = 1 − 1 N ( t ) N ( t ) X j =1 U T D ( X t j ) , (2) where N ( t ) is the number of packets transmitted in interval [0 , t ] . Provided the packet du ration T D is sufficiently small relativ e to the mean time between packets, b y assum ption (ii) the transm it times { t j } effecti vely possess the Lac k of Anticipation prope rty (the nu mber o f pa cket transmission s in any interval [ t, t + u ] , u ≥ 0 , is indep endent o f { X s } , s ≤ t [25]). When this property holds, by [25, Theorem 1] we almost surely have lim t →∞ ˆ P t ( T D ) = lim t →∞ P t ( T D ) = : p ( T D ) where P t ( T D ) = 1 − 1 t Z t 0 U T D ( X s ) ds. That is, the packet loss rate estimato r (2) provides an asy mp- totically un biased estimate o f the mean value of U T D . Assumption (i) can be replaced by the wea ker requirem ent that th e p acket loss rate is hig her wh en a pa cket transmission overlaps with an interf erence pulse than when it does not. W e consider this in m ore d etail later, in Section V. Assumption (ii) can be relaxed to any sampling approa ch that satisfies the Arriv als See Time A verages (AST A) pro perty , see for example [26], [2 7]. It remains to show that statistic p ( T D ) con tains useful inf or- mation about the interf erence pr ocess. W e begin by ob serving that Y t = sup { k : T k ≤ t } is a renewal proce ss – since the ∆ k and S k are i.i.d., the start times { T k } of th e interfer ence pulses are renewal times. The mean time between renewals is E [ S + ∆] . On ea ch renewal interval t ∈ [ T k , T k +1 ] we have that U T D ( X t ) = 1 for du ration [∆ k − T D ] + , where [ x ] + equals x when x ≥ 0 and 0 othe rwise. The m ean value o f U T D ( X t ) over a renewal interval is there fore R ∞ T D ( x − T D ) dF ( x ) and, by the strong law of large nu mbers, p ( T D ) = 1 − 1 E [ S + ∆] Z ∞ T D ( x − T D ) dF ( x ) . Since F ( • ) is a d istribution fun ction it is differentiable alm ost ev erywhere, and thus so is p ( • ) . At every point T D where p ( • ) is differentiable we h av e dp dT D ( T D ) = 1 E [ S + ∆] Z ∞ T D dF ( x ) = 1 E [ S + ∆] Prob[∆ > T D ] . Provided p ( • ) is differentiable at T D = 0 , then E [ S + ∆] = 1 dp (0) /dT D since Pro b[∆ > 0 ] = 1 , an d so Prob[∆ > T D ] = 1 dp (0) /dT D dp dT D ( T D ) . (3) Hence, k nowledge of statistic p ( T D ) as a fu nction of T D is sufficient to allo w us to c alculate not on ly the mean time between interference p ulses E [ S + ∆] , but also the entire distribution function F ( x ) = 1 − Prob[∆ > x ] of the interferen ce pulse inter-arriv al times. Note that while we can formally d ifferentiate p ( T D ) , its estimate ˆ p ( T D ) will be noisy and so differentiating ˆ p ( T D ) is 4 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 Empirical packet loss rate Theoretical packet loss rate P S f r a g r e p l a c e m e n t s T D (ms) p ( T D ) (a) Periodic in terfere nce, period T ∆ = 100 m s 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 P S f r a g r e p l a c e m e n t s T D (ms) 1 − F ( T D ) (b) ccdf of ∆ for periodic interfer- ence 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 Empirical packet loss rate Theoretical packet loss rate P S f r a g r e p l a c e m e n t s T D (ms) p ( T D ) (c) Poisson interfe rence , mean inter -arri val time 1 /λ ∆ = 10 ms 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 P S f r a g r e p l a c e m e n t s T D (ms) 1 − F ( T D ) (d) ccdf of ∆ for P oisson interfer- ence Fig. 4. Theory and simulation for periodic and Poisson interferenc e. Pack et transmissions are Poisson with mean rate λ = 0 . 01 . not advisable. The f ormal differentiation step is merely used to gain insight in to the statistical inform ation conta ined within p ( T D ) and th ere is no need to actu ally differentiate ˆ p ( T D ) in order to infer characteristics of the interferen ce proc ess ( e.g. see the examples in the n ext section). C. T wo Simple Examples W e pr esent tw o simple examples illustratin g the use of statistic p ( T D ) a nd f or which explicit calcu lations are straight- forward. 1) P erio dic impulses: The first example is where the in- terference consists of p eriodic impulses with period T ∆ (so Prob(∆ = T ∆ ) = 1 ) and packets are always lost when they overlap with an interferen ce pulse. In this case, p ( T D ) = 1 − 1 E [ S + ∆] Z ∞ T D ( x − T D ) dF ( x ) =  T D T ∆ T D ≤ T ∆ 1 T D > T ∆ . That is, p ( T D ) is a trunc ated line with slope T ∆ . Fig. 4( a) plots this theory line, alon g with th e measured packet lo ss rate obtained from simulation s. Th e in terference p eriod T ∆ can be directly estimated from the slope of th e me asured line of pac ket loss versus T D . The ccdf 1 − F ( T D ) shown in Fig. 4(b ) can be calculated using ( 3) or deduced based o n the in terference period . 2) P o isson inte rfer enc e: The second simp le example is where the inter ference pu lses are Poisson impulses, with rate λ ∆ . In this case, p ( T D ) = 1 − 1 E [ S + ∆] Z ∞ T D ( x − T D ) dF ( x ) = 1 − λ ∆ Z ∞ T D ( x − T D ) λ ∆ e − λ ∆ x dx = 1 − e − λ ∆ T D . Fig. 4(b ) shows the correspo nding measured pa cket loss r ate obtained from simulatio ns. Onc e again , the rate parame ter λ ∆ can be d irectly estimated from the mea sured curve of pa cket loss versu s T D (namely from the slope wh en p ( T D ) is plotted on a log scale versus T D ). The ccdf is also shown in Fig. 4(d), and calcu lated as 1 − F ( T D ) = e − λ ∆ T D . D. Distinguishing Collision and Interfer ence Lo sses in 802 .11 The foregoing analy sis f ocuses o n packet loss du e to inter- ference and ignores other sources of packet loss. As alre ady noted, packet loss d ue to co llisions is p art of th e prop er operation of the 802.1 1 MAC. In even qu ite small wireless LANs, the loss rate due to collisions can be significant ( e.g. in a system with o nly tw o users, the collision probability can appro ach 5% [28]) and so it is essential to distinguish between packet loss d ue to collisions and packet loss due to no ise/inteference. T o ach iev e this we bo rrow the packet- pair bursting idea first proposed in [1]. W e ma ke u se of the following p roperties of th e 802. 11 MA C: 1) Time is slotted , with well-define d b oundar ies a t which frame transmissions by a statio n are p ermitted. 2) Th e standa rd data-ACK hand shake mean s th at a sender- side an alysis can reveal any frame loss. 3) Transmissions occurring befo re a DIFS are protected from collision s. This is used , for example, to protect A CK tran smissions, wh ich ar e transmitted after a SIFS interval. Using pro perty 3, when two frames are sent in a b urst with a SIFS between them, th e first frame is subje ct to b oth collision and no ise losses but the seco nd frame is pro tected from collisions and only suffers from noise/interference losses. Such packet-p air bursts can be gen erated in a n umber of ways ( e.g. u sing the TXOP f unctiona lity in 80 2.11e/n, or the pa cket fragmen tation functionality a vailable in all flav ours of 802 .11). For 802.1 1 link s, we therefo re consider sampling the chan - nel u sing p acket pair bursts ra ther th an using single packets. For simplicity we will assume that the duration of both packets is the same an d equa l to T D / 2 , alth ough this can b e relaxed. In the rema inder of th is p aper we will often r efer to the first packet in a burst as pkt1 , an d the second p acket as pkt2 . It is imp ortant to no te that the 80 2.11 MA C only sends pk t 2 when an A CK is successfully received for pk t 1 . T o re tain the Lack of Anticipation pro perty , when no A CK is rece i ved for the first p acket we intr oduce a virtual transm ission of th e second packet i. e. no actual packet is tra nsmitted b u t t he sender still pauses for the time that it would have taken to send the second packet. In practice this is straig htforward to implement by simply adding T D / 2 to the interval betwee n pa cket p airs when an A CK for the first packet is not received. W ith this proced ure, whe n the in tervals between the comp letion of on e packet p air an d the start of the next packet pair form a Poisson process, the packet loss statistics wil l satisfy the AST A proper ty . Assuming that packet collisions occur in depend ently of interferen ce pulses, the packet loss r ate f or the first packet in the pair ˆ p 1 ( T D / 2) is then an estimato r for p 1 ( T D / 2) = 1 − 1 − p c E [ S + ∆] Z ∞ T D 2 ( x − T D 2 ) dF ( x ) , 5 where p c is the packet collision prob ability . Note that it is difficult to separate out the contr ibution p c due to collisions from measurements o f p 1 ( T D / 2) , as already discussed. Th e second p acket in a pa ir is only transmitted if the first packet was received successfully (per the standard 802.11 TXOP and fragmen tation semantics) and so th e second packet measur e- ment data is censored . W e there fore h av e that the packet loss rate f or th e secon d packet in th e pair ˆ p 1 ( T D / 2) is an estimato r for p 2 ( T D / 2) = 1 − 1 (1 − p 1 ( T D / 2)) E [ S +∆] R ∞ T D ( x − T D ) dF ( x ) . Combining the loss statistics p 1 ( T D / 2) and p 2 ( T D / 2) for the first and second packets, we can re cover o ur desired loss statistic p ( T D ) fro m p ( T D ) = 1 − (1 − p 2 ( T D / 2)) (1 − p 1 ( T D / 2)) , (4) and in this way separ ate ou t the co ntribution to packet loss from interf erence from th e con tribution due to co llisions. E. Carrier S ense The 802.1 1 MA C uses carr ier sense to distinguish betwee n busy and idle slots on the wireless medium . If the energy on the channel is sensed ab ove the carrier-sense th reshold, then the PHY CCA.indicate(BUSY) signal will be issued by the PHY to indicate to the MA C layer th at the channel is busy . Consequently , when an interfer ence p ulse is above th e carrie r- sense threshold at t he transmitter, packet transmissions will not start. Instead, a packet waiting to be transmitted will be queued until the chann el is sensed idle (PHY CCA.indicate(IDLE) ), and th en transmitted. This mean s tha t the packet transmission times are n o longer independen t of the inte rference process and the AST A p roperty is gen erally lost. I n particu lar , the packet loss rate is biased an d tends to be un derestimated sin ce packet transmissions that should hav e started during an interferenc e pulse (and so likely to have led to a p acket loss) are deferre d until after the p ulse finished (and so m uch less likely to be lost since the time to the next interferenc e pulse is then maximal). When the dur ation of the interferen ce pulses is short relati ve to the time between pulses, then the mag nitude of this bias can be expected to be small. When the interfe rence pu lse dura tion is larger , an approx imate compe nsation for the bias can be carried o ut as follows. Con sider the in dicator fun ction ˜ U T D ( X t ) =  1 t ∈ [ T k + T D , T k +1 ) for some k 0 other wise . This mod ifies ( 1) b y lump ing the tim e when th e interfer ence pulse is active into the goo d window , roughly ca pturing the fact th at packet transmissions schedu led durin g a pulse will b e deferred until the p ulse finishe s. When the inter ference p ulse on and off time s a re i.i. d., this modified lo ss statistic is equal with p robability one to ˜ p ( T D ) = 1 − 1 E [ S + ∆] Z ∞ 0 dG ( y ) Z ∞ T D ( y + x − T D ) dF ( x ) = 1 − 1 E [ S + ∆]  E [ S ] F c ( T D ) − Z ∞ T D ( x − T D ) dF c ( x )  = p ( T D ) − ǫ, (5) 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 Carrier Sense No Carrier Sense P S f r a g r e p l a c e m e n t s T D (ms) Packet Loss Rate (a) Pack et loss rate versus packet duration T D with and without carrier sense 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 ˆ F ( x ) F ( x ) P S f r a g r e p l a c e m e n t s T D (ms) 1 − ˆ F ( T D ) (b) Estimate ˆ F ( T D ) of distributi on functio n F ( T D ) Fig. 5. Simulation example illustrati ng how the estimation bias introduced by carrie r sense can be largely remov ed using (6). Periodic interferenc e, similar to the micro wa ve oven interferenc e expe rimental ly measured in Section IV -B (period ∆ = 11 ms, pulse duration S = 9 ms). where F c ( x ) = 1 − F ( x ) is the ccdf, G ( y ) = Pro b[ S > y ] and ǫ = E [ S ] E [ S +∆] F ( T D ) is an a pprox imation to the estimation bias. Using integratio n b y parts and that ˜ p (0) = 1 − E [ S ]+ E [ ∆] E [ S +∆] , (5) can be r e written as ˜ p ( T D ) = 1 − E [ S ] E [ S ] + E [∆] F c ( T D ) + 1 E [ S ] + E [∆] Z ∞ T D F c ( x ) dx. (6) Assuming that the m easured packet loss rate app roximates ˜ p ( T D ) , the n g i ven measurem ents of loss rate for a rang e of T D values we can solve e quation (6) to obtain an estimate for F ( T D ) and E [ S ] . Th is can be carried ou t in a n umber of ways – one simp le appro ach is to write F ( x ) as a weighted sum of P K i =1 w i g i ( T D ) of orthogo nal b asis fun ctions { g i ( T D ) } , and select the weights { w i } an d E [ S ] to minimise the square error between the RHS of (6) and the mea surement of the LHS. W e illustrate use of this approach in Fig. 5, which presen ts d ata generated using a simulation with carrier sense an d periodic interferen ce. T he on- time o f the inter ference p ulses is S = 9 ms and th e time between pulses is ∆ = 11 m s. Fig. 5(a) plots the measu red packet loss rate versus T D which is assumed to appr oximate ˜ p ( T D ) . Also shown is the loss rate p ( T D ) when carrier sense is disabled. The bias ǫ between ˜ p ( T D ) and p ( T D ) is clearly evident. Using this biased d ata for ˜ p ( T D ) and re ctangular basis functions { g i ( T D ) } , so lving (6) yields the estimate ˆ F ( T D ) sho wn in Fig. 5(b). It can be seen that ˆ F ( x ) accu rately es timates the true distribution fu nction F ( T D ) (also marked in Fig. 5(b) ) i.e. th at we hav e successfully 6 T ABLE I S P E C T RU M A NA L Y S E R D E TA I L S A N D S E T U P F O R Z E RO S PA N M E A S U R E M E N T S . Model Rohde & Schwarz FSL6 with optional pre-amp V ideo BW 10 MHz Resoluti on BW 10 & 20 MHz Sweep time 20 ms Antenna LM T echnologies LM254 2.4 GHz dipole compen sated fo r the carrier sense b ias. I n p articular, the sharp transition at 11 ms is accur ately e stimated. I V . E X P E R I M E N TA L M E A S U R E M E N T S In this section, we presen t experimental measuremen ts demonstra ting the power and practical utility of the propo sed non-p arametric estimation ap proach . W e collecte d data in two sep arate measurement c ampaigns. The first con sists of measuremen ts on a n 802. 11 lin k affected by interferen ce fr om a do mestic microwa ve oven (MWO). Such interf erence is common , and so of considerable practical importance. The second shows me asurements from an 80 2.11 lab testbed, with two tr ansmitting n odes a nd a numb er of hidd en no des actin g as the pu lsed interfer ence sour ce. A. Har dwar e and Software Asus 700 laptop s equ ipped with Atheros 802.11 a/b/g chipsets ( radio 1 4.2, MAC 8.0, PHY 10 .2) were used a s client station s, ru nning Debian Lenny 2.6.2 6 and using a modified Linux Mad wifi driver based on 10 .5.6 HAL and 0.9.4 dr i ver . A Fujitsu Lifeb ook P7010 equipped with a Belkin W ir eless G card using an Atheros 802.1 1 a/b/g chipset (AR2417, MAC 1 5.0, PHY 7.0) was used as th e access point, runnin g FreeBSD 8.0 with the RELEASE kernel and using th e standard FreeBSD A TH d riv er . The b eacon period is set to the maximum value of 1 s. W e disab led the Ather os’ Amb ient Noise I mmunity feature which has b een reported to cause unwanted side effects [29]. T r ansmission power of the lap tops is fixed and antenna diversity is disabled . In previous work we h av e taken c onsiderable care to confirm th at with this hardware/software setup the wireless stations accurately follow the IEEE 802. 11 standard and the packet pair measurement approa ch is correctly implemented (see [ 1], [29], [ 30] for further de tails). A Rohde & Schwarz FSL-6 spectrum an alyser is used to verify that the test channe ls are unoccup ied and also to captu re the tim e-domain traces (see T able I for details). B. Micr owave Oven Interfer ence 1) Experimenta l Setu p: The experimental setup co nsisted of one c lient station, the AP an d a 7 00 W microw av e oven. During the exper iments, the MW O is o perated at maximum power to heat a 2 L bowl of water , and is located approxi- mately 1 m a way f rom the client station a nd AP; the e xact geometry of the setup is not importan t since the MWO is close en ough to the laptops to disrupt communications. The antenna co nnected to the spe ctrum a nalyser is located su ch that the en ergy from each RF sour ce is o f similar ma gnitude. 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p k t 1 p k t 2 P S f r a g r e p l a c e m e n t s T D 2 (ms) p ( T D ) (a) Measured packe t loss rate v ersus pack et duration T D . Confidence interv als based on the Clopper-Pea rson method are displayed, but are s mall enough to be par- tiall y obscured by the point markers. 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 P S f r a g r e p l a c e m e n t s T D (ms) 1 − ˆ F ( T D ) (b) Inter -arri val distrib ution of interferenc e pulses Fig. 6. Experimenta l measurements with micro wave oven (MWO) inter- ference . Data frames are transmitted at a PHY rate of 1 Mbps rate and the duratio n T D is vari ed by adjusting the packe t size. Both pk t 1 and p k t 2 are equal length T D 2 . The MWO o perates in the 2.4 GHz I SM ban d, with signif- icant overlap ( > 50 % ) with the W iFi 20 MHz channels 6 to 13; this was verified using the spectrum analyser . Our 80 2.11 experiments used ch annels 7 and 9 and took p lace in a roo m that was cleared fo r co-chann el interference before, durin g and after each experimen t. The client station transmits packets to the AP with the MTU, FRA G and packet size set to values th at ensur e that both p k t 1 and pk t 2 are of nearly identical d uration (th e deviation o f T D / 2 is kept to below 1%). Th e packet dur ation is adjusted by varying the p acket size betwee n 3 0 and 211 0 byte s ( yielding T D from 1 .4 ms to 18 m s). Th ese p ackets are gene rated using the stand ard pin g comm and in a b ash scr ipt. The interval between each set o f packet pairs is expo nentially distributed with r ate λ = 30 pac kets pe r seco nd, an d th e modu lation and coding rate is fixed at 1 Mbp s. 2) Inferring I nterfer ence S tatistics F r om P a c ket Loss Mea- sur ements: Fig. 6(a) pre sents the measured packet loss rate between the client station and the AP versus the packet duration T D / 2 . Each point is averaged over more th an 1 0 4 ob- served packets. Using this p acket lo ss data, Fig. 6(b ) plo ts the estimated distrib ution fu nction ˆ F ( T D ) for interfere nce pulse inter-arriv al times. W e use the ap proach described in Section III-E to comp ensate for the bias intro duced by carrier sense at the c lient station. It can be seen that ˆ F ( T D ) exhibits a sharp 7 transition arou nd 11 ms, a long with som e residual proba bility mass between 11 and 1 5 ms. This indicates that th e MWO interferen ce is estimated to be ap proxima tely periodic with period ∆ = 11 ms. W e confirm the accuracy of this inference indepen dently using direct spectrum analyser measurem ents of the M WO in terferenc e in the next section , see Fig. 7. Before pr oceeding howev er , it is worth com paring the experimentally measure d 802.1 1 loss data in Fig. 6(a) with the simulation data in Fig. 4(a). This comparison highlights the ad ditional comp lexity introduced by carrier sen se and the censoring of second packet loss data. Nevertheless, our approa ch is able to successfully disen tangle these effects in a principled way an d there by estimate F ( T D ) . 3) V alidation: Fig. 7(a) presents spectru m analyser d ata showing two interference pulses generated b y the MW O. A packet pair transmission by the client station can also be seen, lying between the interference bursts (this particular packet pair transmission is successfully received by the AP , verified b y n oting th e presence of MA C A CKs at the end of each pac ket). From this and other d ata, we find that the MWO interfer ence is appr oximately periodic, with pe riod T = 1 /f = 20 ms i.e. a freque ncy o f 5 0 Hz , as expected due to the A C circuitry that is driving the MWO. Th e pro file of the interferen ce bursts is, h owe ver, not unifor m. Fig . 7( b) sh ows a measured in terference burst of where the inter ference p ower is rough ly constant over the dur ation (app roximately 9 ms) of the pulse. Fig. 7(c ) shows an interfere nce pulse wher e the interferen ce power dips during the middle of the pulse, so as to effecti vely create two n arrower pulses sp aced appr oximately 4 ms apart. Th is variation in burst energy p rofile is attributed to frequen cy in stability of the MWO cavity magn etron, a known effect in MWOs [3 1]. Ou r measurem ents indicate that the MWO interference consists of pulses with me an in terval 11 ms between pulses, w ith som e d eviation (Fig. 6(b)). These direct measurements ar e therefore in g ood ag reement with th e estimated distribution function , which was deri ved ind irectly using pa cket lo ss measur ements. C. 802. 11 Network W ith Hidden Nod es 1) Experimenta l Setup: This test be d co nsists of a WLAN formed fr om two clien t stations and an access poin t, plus three additional stations configur ed as hidden nod es. These hid den nodes ( HNs) are cr eated by modifying the Madwifi dri ver such that the carrier sense is disabled (using the techn ique as detailed in [32]) and setting the NA V to zero for all packets – this effecti vely makes the HNs unrespo nsiv e to any packets that they deco de from the client, or energy th at may trig ger a ph ysical carrier sense. A scr ipt gener ates p ing traffic on the h idden no des having exponentially distributed intervals between packet transmissions, with a mean interval of 50 ms. Th e ping packets sent are o f duratio n 4.5 ms (verified via the spec trum analyser) . Since the transm issions by each HN are Poisson with intensity λ = 20 packets/s, the agg regate interference is also Poisson a nd with intensity λ = 60 packets/s. The experim ents used chann el 13 of the ISM b and, and took place in a ro om that was cleared for co- channel inter ference bef ore, durin g an d after the experiments. (a) Pack et pair transmitted between two MWO bursts. The y-axis grid is in 2 ms increments. T he pack et pair is encoded at the 1 MBps 802.11 rate, with both packets havi ng duration 4.36 ms. (b) Second pack et in a pair suf fering a colli sion with a MWO burst; after the MWO burst has finished and carrie r sense indica tes the channel is idle, the packet is retransmit ted. The y-axis grid is in 2 ms increments. (c) Packet pair and a MWO burst. The y-axis grid is in 2 ms increments. T he resoluti on bandwidth is set to 20 MHz, and thus captur es about 99% of the WLAN signal. The MWO burst has a dip in the middle, which is attrib uted to frequenc y instability in the MWO cavity magnetron. Fig. 7. Sp ectrum analyser measureme nts of micro wa ve oven (MWO) interfe rence. 8 0 2 4 6 8 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 p k t 1 p k t 2 P S f r a g r e p l a c e m e n t s T D 2 (ms) p ( T D ) (a) Measured packet loss rate v ersus packe t duration T D . Confidence interv als based on the Clopper-Pear son method are displayed, but are s mall enough to be par- tiall y obscured by the point markers. 0 2 4 6 8 10 0.5 0.6 0.7 0.8 0.9 1 ˆ F ( x ) F ( x ) P S f r a g r e p l a c e m e n t s T D (ms) 1 − ˆ F ( T D ) (b) Inter -arri val distrib ution of interferenc e pulses. Fig. 8. Experimental measurements; primary netw ork has two nodes transmitt ing to AP , interference netw ork has three hidden nodes. 2) Inferring I nterfer ence Statistics F r om P a ck et Loss Mea- sur ements: Fig. 8(a) p lots the measured p acket loss rate in the WLAN versus the p acket duration. Note tha t this loss rate includ es a contribution due to collisions between the two client stations in th e WLAN a nd a contribution due to interferen ce fr om the hidden nodes. Nevertheless, using ou r packet p air ap proach we are ab le to disen tangle these two sources of packet loss. Fig . 8(b) plots th e resultin g distribution of in terference pulse inter-arriv al times estimated using this packet loss data. The data plotted in Fig. 8(b ) is the estimate of 1 − F ( T D ) , and is displayed using a logarith mic y -axis. Also plotted in Fig. 8( b) is th e theory line 1 − F ( T D ) = e − λT D correspo nding to Poisson distributed interference with rate λ = 60 packets/s. It can b e seen that the estimated data is approx imately linear on this log scale, as expected for a Poisson distribution, and that the slope is close to the expected value of λ = 60 . The offset between the Poisson theory line and the e stimated line is explained b y the presence of a b aseline packet loss rate of approx imately 5% in our experimental setu p – this baseline lo ss rate is confirmed b y separate mea surements (not shown h ere). V . P U L S E D I N T E R F E R E N C E T E M P O R A L S T A T I S T I C S : P A R A M E T R I C E S T I M A T I O N Thus far we ha ve con sidered estimating the interference distribution fu nction in a no n-param etric m anner . By m aking stronger, structural assumptions about the interfer ence process, we can alterna ti vely parameterise the distribution function and our task then b ecomes one o f estimating these mo del parameters. A fairly direct trade-o ff in effort is inv olved here, which is why it is important to consider both non - parametric an d param etric approac hes. Nam ely , we ha ve the bias-variance tra de-off whereby no n-param etric appr oaches make on ly wea k assumptions about the interferen ce proce ss, but req uire mo re m easurement data, wher eas parametric ap- proach es make stron g assumptions, but req uire less measure- ment data for the same estimation accur acy (assumin g that the model structur e is accu rate). In this section we pre sent a p arametric estimation appr oach for o ne c lass o f mo del. Th e mo del is related to the tw o- state Gilbert-Elliot chann el model [33], which is po pular for analy sing commun ication ch annels with bursty losses, extended to inc orporate carr ier sen sing and the packet tran s- mission p rocess. Alth ough simple, this mo del is usef ul and we demonstra te its effecti veness for estimating h idden terminal interferen ce. A number of extensions are possible, in cluding to a m ulti-state inte rference mod el [3 4], cor related losses [35], fast fading [36] and so on, but we leave consider ation of th ese extensions to f uture work. A. P arametric P ack et Loss Model 1) Interfer en ce: W e mod el pulsed interfer ence as switching random ly between two states, “g ood” ( G ) and “bad” ( B ), with exponentially distributed d well times in each state. Formally , let S = { G, B } d enote th e set of in terference states, Q =  − λ B λ B λ G − λ G  , (7) and Π =  0 1 1 0  . (8) Let Y = { Y n , n = 0 , 1 , 2 , ... } be a sequ ence of ran dom variables tak ing v alues in S and r epresenting the ev olving state, with Prob[ Y n +1 = j | Y n = i ] = Π ij . (9) W ith our the choice of Π , the Y n flip bac k and fo rth between th e G and B states so that Y is of the form { ..., G, B , G, B , ... } . Let { k } index the sub-sequ ence of B states in Y . Let S k denote the dwell tim e in the k th B state an d ∆ k the dwell time in the following G state. The dwell times S k and ∆ k are indepen dent exponential ran dom variables having, resp ectiv ely , mean 1 /λ B and 1 /λ G . The sequence T k +1 = T k + S k + ∆ k is the sequence of ju mp times at which the inter ference enters state B . 2) P acket T ransmissions: The wireless station perfor ming measuremen ts transmits a seq uence of packets to a destination station, with expo nentially distributed pauses between trans- missions. Similar to the foregoin g in terference model, we le t { T x, I dl e } be the two transmitter states, where T x corr e- sponds to transmission of a p acket. Let { V m , m = 0 , 1 , 2 , ... } denote a sequence o f r andom v ariables wh ich flip b ack and forth between the T x and I dl e states. The dwell time in th e T x state is a constant T D , the dwell times in the I dl e state are 9 indepen dent exponential ran dom variables with mean 1 /λ D . W e index the su b-sequenc e of T x states b y p acket n umbers in { n } , and let t n denote the tim e when transmission of p acket n starts. 3) Carrier Sen se: The interfere nce state at the p acket transmit time t n is Y k ( n ) where k ( n ) = sup { k : T k ≤ t n } . Let p cs = P rob[ t n ∈ [ T k ( n ) , T k ( n ) + ∆ k ( n ) ]] := α λ G λ G + λ B , where 0 ≤ α ≤ 1 and λ G λ G + λ B is the proba bility than the interferen ce is in state B . In the f ollowing, we consider two limiting situations. Firstly , wher e the carrier sense threshold lies above the noise level in both interfere nce states, in which case the p acket transm ission times are decoupled fr om the interferen ce state and α = 1 . Seco ndly , where the carrie r sense threshold lies above th e noise le vel in in terference state G but below the noise level in state B , in which ca se α = 0 . 4) P a ck et Loss: Packets ar e discar ded whe n th ey fail a checksum test at the receiver . Henc e, we treat the channel as an era sure channel. Let δ n denote a rand om v ariable that takes value 1 when packet n is er ased an d value 0 otherwise. Let ˜ S n denote the time that the ch annel spend s in state B during the transmission o f packet n . In g eneral, we expect that the p robab ility Prob[ δ n = 1] that packet n is erased depend s on ˜ S n . Nevertheless, to streamline the presentation we m ake the simplifying assum ption that P rob[ δ n = 1] = p B whenever ˜ S n > 0 and Pro b [ δ n = 1] = p G otherwise, wh ere p B and p G are c hannel p acket loss rate par ameters in the B and G states r espectiv ely . W e also assume that pa cket erasures occur indepen dently , i.e. the random variables δ n , δ m are indepen dent fo r n 6 = m . 5) P a ck et Err o r Rate Analysis: T o deter mine the packet error rate as a fu nction of the packet transmit dura tion, we need to analyse two coup led stochastic pro cesses, namely the channel and transmission proc esses. The joint process takes state values in { G, B } × { I dl e, T x } . Since our interest is in counting the frequ ency o f packet losses, observe th at we can lump th e ( I dl e, G ) and ( I dl e , B ) states together, since we know that no packet lo ss can occur in these ( I dl e, • ) states. Also, when the system first enter s state ( T x, B ) , the n a p acket loss occu rs and we d o not n eed to keep count o f the nu mber of subsequen t transition s between ( T x, G ) an d ( T x, B ) . W e can therefor e partition time in to slots, with each slot being of thre e possible types: I dl e (corr esponding to the lum ped ( I dl e, • ) states), Loss (corr esponding to lumping of states ( T x, G ) an d ( T x, B ) after th e first tr ansition from ( T x, G ) to ( T x, B ) ) and T r ansmitti n g (cor respondin g to a dwell time in state ( T x, G ) ). Th e transitions between th ese slots ar e as shown in Fig. 9 and T ab le II. The transition matrix P of this slotted time Mar kov chain is: P =   0 1 − p cs p cs 1 − p i ( T D ) 0 p i ( T D ) 1 0 0   , (10) where 1 − p i ( T D ) = exp( − λ B T D ) . The stationary state distribution satisfies π = π P , where π 1 = Prob[ I dle ] , P S f r a g r e p l a c e m e n t s I dle T ransm i tting Loss 1 − p cs 1 − p i p cs p i 1 Fig. 9. Slotted time Marko v chain. π 2 = Pr o b[ T r ansmitting ] , and π 3 = P rob[ Loss ] . Solving yields, π T = 1 2 + p i ( T D ) (1 − p cs )   1 1 − p cs (1 − p cs ) p i ( T D ) + p cs   . The packet er ror probability f or the first packet in a pair is p 1 ( T D ) = (1 − p i ( T D )) π 2 p G + p i ( T D ) π 2 p B + p cs π 1 p B (1 − p i ( T D )) π 2 + p i ( T D ) π 2 + p cs π 1 = (1 − p i ( T D ))(1 − p cs ) p G + ( p i ( T D ) (1 − p cs ) + p cs ) p B =: G 1 ( T D , λ B , p B , p G , p cs ) . (11) The first term in th e expr ession for p 1 ( T D ) correspo nds to the event wh ere the interf erence stays in state G throug hout a packet tr ansmission and a packet loss occurs. The seco nd term c orrespon ds to the event that a packet transmission starts with the inter ference in state G , but the in terference change s to state B durin g the course of the tr ansmission and a pac ket loss o ccurs. Th e third term corr esponds to the event that a packet tran smission starts with the inter ference in state B and a pa cket lo ss occu rs. Conditioned on the first p acket tr ansmission bein g success- ful, the p acket erro r pr obability f or the seco nd p acket in a p air is p 2 ( T D ) = (1 − p i ( T D )) λ B λ B + λ G p G +  1 − (1 − p i ( T D )) λ B λ B + λ G  p B =: G 2 ( T D , λ B , p B , p G , p cs ) , (12) where th e λ B λ B + λ G factor accounts for the event that the interferen ce is in the B state up on starting tr ansmission of pk t 2 . B. Model pa rameters Equation s (1 1) and (12) together form a parametric mod el of the p acket pair loss process, which is described b y p arameters λ B , p B , p G and p cs . Before proceedin g, we briefly illustrate how the m odel parameters λ B , p B , p G and p cs affect the o bserved p acket 10 T ABLE II M A R KO V M O D E L S TA T E T R A N S I T I O N S . I dle → T r ansmitting (start Tx, interfer ence in state G) : 1 − p cs I dle → Loss (start Tx, interfer ence in state B) : p cs T r ansmitting → I dle (interfe r ence in state G thr oughout Tx) : 1 − p i = exp( − λ B T D ) T r ansmitting → Loss (interfer ence enters state B during Tx) : p i = 1 − exp( − λ B T D ) Loss → I dl e (Tx of damage d packet ends) : 1 10 −1 10 0 10 1 10 2 0 0.2 0.4 0.6 0.8 1 p k t 1 p k t 2 P S f r a g r e p l a c e m e n t s T D (ms) p ( T D ) λ B = 100 λ B = 1000 Fig. 10. Pa cke t error rate versus pack et duration T D ; λ D = 30 , v ariabl e λ B , p G = 0 , p B = 1 , p cs = 0 , T S I F S = 10 µ s. 10 −1 10 0 10 1 10 2 0 0.2 0.4 0.6 0.8 1 p k t 1 p k t 2 P S f r a g r e p l a c e m e n t s T D (ms) p ( T D ) p B = 0 . 9 p B = 0 . 7 Fig. 11. Pa cke t error rate versus packe t duration T D ; λ D = 30 , λ B = 100 , p G = 0 . 1 , vari able p B , p cs = 0 , T S I F S = 10 µ s. loss versus T D curves. Our aim is to (i) illustra te the typ es of loss c urves that th e mo del is able to captur e and (ii) gain som e intuitive insigh t into the role of the various model paramete rs. Fig. 1 0 shows the impa ct of λ B , wh ich pro duces a horizon tal shift in the loss cur ves. Fig. 11 shows the im pact o f p B , whic h determines the right- hand asymptote of the loss curves. Fig. 12 shows the impact of the carr ier sense parameter p cs (by v arying α ), which produ ces a vertical shift in the left-hand asymptote. Although no t shown, the impact o f p G also pr oduces a vertica l shift in the left-han d asy mptote. C. Maximum Likelihoo d P arameter Estimation Our ob jectiv e is to estimate the model par ameters λ B , p B , p G and p cs from measur ements of pa cket loss. Th e empirical estimators for loss p robabilities p 1 ( T D ) and p 2 ( T D ) are ˆ p 1 ( T D ) = 1 N 1 N 1 X n =1 δ 1 n ˆ p 2 ( T D ) = 1 N 2 N 2 X n =1 δ 2 n , where N 1 is the numb er of first packets, N 2 the n umber o f second p ackets, δ 1 n is the indicator functio n that equ als 1 wh en 10 −1 10 0 10 1 10 2 0 0.2 0.4 0.6 0.8 1 p k t 1 p k t 2 P S f r a g r e p l a c e m e n t s T D (ms) p ( T D ) α = 0 α = 0 . 2 α = 0 . 8 α = 1 Fig. 12. Pa cke t error rate versus packe t duration T D ; λ D = 30 , λ B = 100 , p G = 0 , p B = 0 , variab le p cs (by var ying α ), T S I F S = 10 µ s. the n th first packet is lost and 0 oth erwise, and similarly δ 2 n for second p ackets. Collec ting packet loss measu rements for a sequ ence of pa cket dur ations T D 1 , T D 2 , ... and stacking the correspo nding loss prob ability estimates we have        ˆ p 1 ( T D 1 ) ˆ p 2 ( T D 1 ) ˆ p 1 ( T D 2 ) ˆ p 2 ( T D 2 ) . . .        =        G 1 ( T D 1 , λ B , p B , p G , p cs ) G 2 ( T D 1 , λ B , p B , p G , p cs ) G 1 ( T D 2 , λ B , p B , p G , p cs ) G 2 ( T D 2 , λ B , p B , p G , p cs ) . . .        + η , (1 3) where η denotes the estimation erro r in the p acket lo ss esti- mates. For N 1 , N 2 sufficiently large, the estimation erro r η is close to b eing Gau ssian distributed. The max imum likelihood estimates fo r p arameters λ B , p B , p G and p cs are then the values that minimise the square error between the LHS and RHS in (13). D. Experimental Measurements 1) Experimental Setup: W e revisit the WLAN exp erimental setup discu ssed in Sectio n IV - C, but now chan ge the setup slightly so that only a single wireless client (rather than two clients) transmits in the WLAN. This ch ange is intro duced because, for simplicity , we h av e not included p acket c ollisions in our parame tric m odel. 2) P acket Lo ss Measur ements: Fig. 13 shows the measured packet lo ss rate versus the packet d uration T D . Note th at the range of packet durations that we can use is con strained by the maxim um 802.1 1 frame size of 2272 B to lie in the interval 1 .4 m s to 1 8 m s. T wo sets of results ar e shown, fo r one and for three hidden nodes active. Each experimental point is calculated as the av erage of mor e than 6 × 10 5 packet transmissions. Also shown are the maximu m likelihood fits to this data u sing param etric mod el (11) an d (12); the 11 10 −1 10 0 10 1 10 2 10 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 ˆ P ( 1 ) e ˆ P ( 2 ) e p k t 1 p k t 2 P S f r a g r e p l a c e m e n t s T D (ms) p ( T D ) 3 int 1 int Fig. 13. Experimenta l measurements and model fit for WLAN with hidden node interferen ce. D ata points are for experiment s using 1 and 3 interfere rs, with each interferer having a packet transmission rate of λ B = 20 . Initial v alues for the paramete r estimator are ˆ λ B = 20 , ˆ p cs = 0 , p G = 0 , and p B = 0 . 5 . Model parameters are give n in T able III . Number of interferers ˆ λ B ˆ p cs ˆ p G ˆ p B 1 19.9932 0.0286 0 .0080 0.267 8 3 54.7173 0.1011 0 .0055 0.405 5 T ABLE III D E T A I L S O F T H E M A X I M U M L I K E L I H O O D PA R A M E T E R E S T I M ATE S F O R M E A S U R E M E N T D AT A I N F I G . 1 3 correspo nding m odel pa rameter estimates are given in T able III, o btained using an inter ior-point solver . 3) V alidation: The hidden node interfer ers each make transmissions with exponen tially distributed idle time b etween packets so that the mean tran smit rate is 2 0 packets/s. When one in terferer is active, we exp ect λ B = 20 and wh en three interferers ar e active we expect λ B = 60 . It can be seen f rom T able III that the mo del estimates are clo se to these predictions. Measurements taken with no hidd en n ode interferer s a cti ve indica te that th e b aseline p acket loss rate is less than 1% and it can be seen fr om T ab le III that the model estimate for p G is in good agreement with this. While it is difficult to similarly validate the estimates for p arameters p cs and p B , we n ote tha t th e estimated values are very r easonable. 4) P a rametric vs No nparametric Estimation: A pa rametric model m akes strong structural assum ptions that allow the loss cur ves to be parameter ised using a small n umber of parameters. Sin ce th ere are fewer par ameters, we expect to be able to estimate the ir values with less data, but at th e co st of introdu cing a bias if the structu ral assumptions turn o ut to b e inco rrect. Fig. 14 p lots max x | ˆ F ∞ ( x ) − ˆ F N ( x ) | versus the numbe r of o bserved packets N for both the parametric and no n-para metric app roaches, where ˆ F N ( x ) is the estimate of F ( x ) obtain ed u sing N observations and ˆ F ∞ ( x ) is the estimate using all 6 × 10 5 observations. For the parametric model, the parameter estimates are fed back into the mo del equations (11) and (12), and the resulting parame terised P e curves are used to calculate ˆ F N ( x ) . This p rovides a rou gh indication of how estimates conv erge as the amou nt of d ata is increased. It can be seen that the param etric solution conv erges to within 5% of the asymptotic estimate after N = 90 0 packets and to within 2 .5% af ter N = 4 000 packets, while the non - parametric solution req uires N = 6000 and N = 20 000 100 1,000 10,000 100,000 0 0.025 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Non- para m e tric P ar am e tric P S f r a g r e p l a c e m e n t s N max x | ˆ F ∞ ( x ) − ˆ F N ( x ) | Fig. 14. Con vergen ce of estimates of F ( x ) versus the number of packets observe d. ˆ F N ( x ) denot es the estimat e using N pac ket observ ation s and ˆ F ∞ ( x ) denotes the estimate obtained using the full measurement trace . For each N , we take 100 random subsamples of N packe ts from the full measurement trace, calcula te max x | ˆ F ∞ ( x ) − ˆ F N ( x ) | for each subsample, and avera ge this value over the 100 subsamples to obtain the curves sho wn. Data is s ho wn for both parametric and non-parametric estimates. The data set used is from the three interferer experiment, see Fig. 13. Fig. 15. Spectru m analyse r s napshot of hidden terminal interferers in time. The y-axis grid is in 2 ms increments. Interferer burst durations are fixed at 4.5 ms, with arri v als at 10, 19, 80, 83 and 89 ms. Since each interferer has a differe nt path to the spectrum analyser antenna, the pulses are at dif ferent po wer le vels. T he th ird and fourth p ulses colli de, resulting in a steppe d feature. packets, respectively , to a chieve the same level of estimation accuracy . 5) Discussion: It is interesting to no te that, despite its simplicity , th e param etric mo del used here is remarkab ly effecti ve at ca pturing the behaviour in a complex phy sical en vironme nt. For example, the model ign ores the fact that the in terference power will depen d o n th e nu mber of h idden node transmission s taking p lace at the same time. T his effect can be seen in the spec trum analyser measur ements in Fig. 15, where overlapping tr ansmissions b y interfe rers leads to a stepped interference pulse profile. T he model also assum es t hat the d uration o f inte rference pulses is expon entially distributed, but this will n ot b e the case in o ur exp erimental setup . Mo re complex parametr ic models are also p ossible, an d in par ticular can leverage the wealth of r esearch on bursty com munication s channels, but we leav e this to fu ture work. V I . C O N C L U S I O N S In this paper we propose a new approach for de tecting the presence o f pulsed interference af fecting 802 .11 links, and for estimating tempo ral statistics of th is interf erence. Our 12 approa ch is a transmitter-side technique th at provide s per-link informa tion and is compa tible with standar d h ardware. This significantly extends r ecent work in [1], [2] wh ich establishes a MAC/PHY cross-layer techniqu e capable o f classifying lost transmission o pportu nities into noise-related losses, collision induced losses, hidden- node losses and o f distinguishin g the se losses fr om the un fairness caused by exposed nod es a nd capture effects. V I I . A C K N O W L E D G E M E N T S Many thank s to Ken Duffy and Giu seppe Bianchi for th eir helpful com ments. 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T o wsle y, “Measurement and Modeling of the T emporal Dependenc e in Packe t Loss, ” in 18th Annual IEEE Int. Conf . on Computer Commun. (INFOCOMM) , vol. 1, Mar . 1999, pp. 345 – 352. [36] H. S. W ang and N. Moayeri, “Fini te-stat e Marko v cha nnel-a useful model for radio communication channels, ” IEEE T ransact ions on V e- hicular T ec hnolo gy , vol. 44, no. 1, pp. 163–171, Feb . 1995. 13 PLA CE PHO TO HERE Brad Zarikoff rece i ve d the B.Eng. degree with distinc tion in electrical engineeri ng from the Uni- versi ty of V ict oria, V ictoria, Canada, in 2002 and the M.A. Sc. and Ph.D. degre es from Simon Fraser Uni ve rsity , Burnaby , Canada, in 2004 and 2008, respect i vel y . He is currently a research fellow at the Hamilton Insti tute, Nat ional Uni ve rsity of Ireland Maynooth. His current research intere sts include interfe rence mitigati on, power line communicat ion netw orks, and synchronisation for network MIMO systems. PLA CE PHO TO HERE Doug Leith graduated from the Uni ver sity of Glas- go w in 1986 and was awa rded his PhD, also from the Uni versi ty of Glasgo w, in 1989. In 2001, Prof. Leith moved to the Nationa l U ni ve rsity of Ireland, Maynooth to assume the position of SFI Principal In vestigator and to establish the Hamilton Institute (www .hamilton.ie) of which he is Dir ector . His current resear ch interests include the analysis and design of netwo rk congesti on control and resource alloc ation in wireless networks.