269500 - Research Topics

Frame Capture in 802.11p vehicularnetworks

Author:Pieter van Wijngaarden

Supervisors:Dr.ir. Geert Heijenk

Martijn van Eenennaam, M.Sc.

August 18, 2010

Contents

1 Introduction 1

2 Vehicular networks and their applications 22.1 Applications and their advantages . . . . . . . . . . . . . . . . . . . . . . 22.2 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 IEEE 802.11 Wireless Networks 43.1 IEEE 802 Networking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 The 802.11 MAC layer: CSMA/CA and the DCF . . . . . . . . . . . . . . 53.3 Orthogonal Frequency Division Multiplexing . . . . . . . . . . . . . . . . 7

3.3.1 Introduction to OFDM . . . . . . . . . . . . . . . . . . . . . . . . 73.3.2 OFDM Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3.3 Guard times, coding and forward error correction . . . . . . . . . . 103.3.4 Strengths and weaknesses of OFDM . . . . . . . . . . . . . . . . . 123.3.5 OFDM Transmission and Reception blocks . . . . . . . . . . . . . 13

3.4 IEEE standard draft: 802.11p . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Frame Capture 154.1 What is FC? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Scenario classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 802.11p scenario predictions . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Potential impact of FC on vehicular networks 23

6 Conclusion 246.1 Master’ thesis lookahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

A Appendix 28A.1 Orthogonality of multiple sinusoidals and their frequency spacing . . . . . 28

1 Introduction

The last few years the IEEE has been working on a new standard in the wireless domainspecifically designed for vehicular environments: 802.11p. This standard is still in thedraft phase, but many universities and companies have already begun exploring possibleapplications. One of these applications is vehicle safety. Allowing wireless communica-tion between vehicles could enable a wide range of safety features in cars such as brakeassist, notification of accidents, and passing emergency vehicles or information aboutthe traffic density on the road ahead and implied speed predictions. Another importantapplication for the Dutch highway network and for our research group is traffic efficiency.We are currently investigating these possible safety and efficiency applications.

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Many challenges lie ahead before these technologies can be seen on the road. One of theimportant challenges is to fully understand how a Vehicular Ad-hoc Network (VANET)behaves at high speeds with extreme mobility. Examples are users/cars entering andexiting networks every second and possible routing information being outdated the verysecond it is generated.

Our main research question is focused on a specific phenomenon at the 802.11 physicallayer called Frame Capture (FC, also referred to as Physical Layer Capture). It is aneffect that comes into play when two frames are being transmitted simultaneously (forexample because of the hidden terminal problem); a receiver might receive one of thetwo colliding frames correctly depending on signal strength and timing variations. Wesuspect that due to the nature of VANET communications with lots of broadcast traffic,FC could have a significant impact on the network throughput. Our goal is to gain moreinsight in the influence FC has on this; if we understand it better we can get a morerealistic model of how network traffic flows in a vehicular network.

This report is structured as follows: in Chapter 2 we look at the possibilities and ad-vantages of vehicular networks and the typical challenges we face when employing one.After that we discuss the standard 802.11 MAC and PHY layers and their properties,where we especially look at 802.11a and p because of their similarity. We also look at thedetails of OFDM and the modulations used in both standards. Then, in Chapter 4 wefurther explain FC and perform a literature study to get a more detailed view on whenand under which circumstances FC occurs and what the various signal requirements are.In Chapter 5 we discuss why we suspect FC to be so important for vehicular networks,and finally in Chapter 6 we look at various simulation options and scenarios we willperform later, using this literature study as a basis.

2 Vehicular networks and their applications

This chapter discusses some of the basics about VANETs; why do we want them, whatcan they do for us and what do we need to take into consideration when designing sucha network?

2.1 Applications and their advantages

Vehicular networks are networks between vehicles which can be used for a wide varietyof applications. If the network involves an infrastructure with nodes or access pointsalongside of the road (connected to the Internet), these applications could be any user-centered application which are already available on a normal personal computer or mobiledevice, such as instant messaging, automatically updating maps for use with GPS orGalileo, streaming videos for the kids in the back of the car or even multi-player gaming.These networks however can also be made in a complete ad-hoc manner, creating a

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network between adjacent vehicles without access points or central organization. Thisdoes not enable many connectivity features for the user or driver of the car, but canmainly be useful in safety- and efficiency-oriented applications. This last scenario iswhat we focus on, we talk about a vehicular network as if it were a Mobile Ad-HocNetwork (or MANET) and call it just like many others a Vehicular Ad-hoc Network (orVANET) [1, 2].

To give some examples of safety-improving applications: cars could send notificationsabout events such as accidents to all cars which are about to pass a crash site. Theycould then take appropriate action like slowing down or switching lanes in a coordinatedmanner. Emergency vehicles could transmit messages requesting to free up a lane toadvance quickly to the crash site. Or in a more preventive way: cars could send amessage to the cars behind them if they need to brake suddenly, implementing someform of communication-aided brake assist.

Apart from increased vehicle safety VANETs also have possible efficiency success factors.The traffic density on the Dutch highways is enormous, not just around the big cities,and on average working days there are over 200 km of traffic jams. This is about 4% ofthe entire Dutch highways network. Peaks in congestion due to weather conditions canrise up to 800 kilometers. The Dutch Ministry of Transport, Public Works and WaterManagement calculated that the annual economical damage caused by traffic jams wasbetween 2,8 and 3,6 billion euros in 2008. This was an increase of 7 percent comparedto 2007, and even a 78 percent increase compared to 2000 [3]. These costs includeboth direct and indirect economical damage; things like extra fuel usage, the impliedenvironmental footprint, and commuters arriving late at work which causes companiesto lose time and money. This doesn’t only apply to the Dutch highways however, mostbig cities around the globe suffer from heavy traffic congestion at peak hours.

If vehicles on the highway would be able to coordinate their speed (and possibly theirroute/destination) with all other vehicles around them using V2V communication andmake intelligent decisions regarding their speed, not just based on the 10 vehicles infront of the driver (what the driver can see), but based on all vehicles in the coming 50kilometers, the result would be a lot more even flow of traffic, resulting in fewer trafficjams and major decrease of economical damage because of traffic congestion.

2.2 Challenges

In a vehicular networking environment, two typical scenarios need to be considered:urban and highway environments. In both cases the ’ad-hoc’ characteristic of the networkis quite extreme. On the highway vehicles usually do not stay close to each other for along time, and all vehicles travel in the same direction but at every fork or exit nodescan enter or leave the network. It is thus incorrect to assume that a node in the networkis still there if it was 60 seconds ago. This is even more true in an urban environment,where the vehicles have an even higher degree of freedom and mobility [4, 5].

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In such an environment, it is very impractical and in some cases impossible to use astandard 802.11a, b or g wireless (ad-hoc) network. Things like authentication, updatingrouting tables and sending packets to other nodes is very unreliable because the networktopology changes too much in too short periods of time. Typical AP (access point)scanning costs between 70 and 600 ms, authentication between a few ms up to morethan a second, depending on the type of authentication, and association costs around 15ms but is quite vendor specific. Also the association time needed increases if the user’ssessions need to be handed over from another AP to the new one the user is connectingto in the case of user mobility [6]. If for example a vehicle moving at a typical highwayspeed such as 100 km/h passes a road-side access point, it is hardly authenticated tothat AP the moment it is already leaving its range.

Apart from the limitations from the MAC layer such as authentication, the physicallayers from standard WLAN are also not designed to handle that much movement. Oneof the normal WLAN varieties, 802.11a, uses OFDM (Orthogonal Frequency DivisionMultiplexing) which is very good in dispersive, noisy environments but also very sensitiveto Doppler-shift. If no effective countermeasures are taken against it, Doppler shift willsignificantly influence the efficiency and throughput of the vehicular network [7, 8].Altogether many factors have to be considered in a vehicular network that are not thatrelevant in normal 802.11 infrastructure or ad-hoc networks.

3 IEEE 802.11 Wireless Networks

In this chapter we will discuss various aspects of the 802.11 wireless networking familyand the required specifics about the MAC and physical layers. Hereby we focus on themodulation and multiplexing techniques of 802.11a and p (OFDM). Readers alreadyfamiliar with the 802.11 MAC layer could probably skip Sections 3.1 and 3.2 about theMAC layer, CSMA/CA and the hidden terminal problem.

3.1 IEEE 802 Networking

802.11 is the family of wireless networking standards created and published by the IEEE.Figure 1 shows how 802.11 relates to the other IEEE 802 networking standards, itbasically consists of a new Medium Access Control layer and new physical layers, andoffers an Ethernet-like network over a wireless medium to the higher layers. Note that802.11 has quite a few different physical layers, they use different modulation techniquesand various frequency bands. The MAC layer is roughly the same for all variations,however for 802.11p this is also changed to counter some of the issues mentioned in 2.2related to authentication and association.

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802

Overviewand

architecture

802.1

Management802.2 Logical Link Control (LLC)

802.3

MAC

802.5

MAC

802.3 802.5

PHY PHY

802.11 MAC

802.11

FHSS PHY

802.11

DSSS PHY

802.11

OFDM PHY

802.11

HR/DSSS

802.11

ERP PHY

Figure 1: The IEEE 802 family (Source: Figure 2-1 from [9]).

3.2 The 802.11 MAC layer: CSMA/CA and the DCF

Because the wireless medium is a lot different from the wired medium used in for example802.3 (Ethernet), a new MAC layer needs to be defined to successfully mitigate problemssuch as interference, collisions and the increased security vulnerability. Therefore, anadapted access scheme is used in this MAC layer: CSMA/CA which stands for CarrierSense Multiple Access with Collision Avoidance. This access scheme counters at leastsome of these problems. It contains a few different coordination functions, among themthe DCF (Distributed Coordination Function). This is the most common one, and isthe one relevant to discuss in this report.

The DCF organizes the access to the medium according to a set of basic rules.

• If the station wants to send a frame, it first senses the channel to see if somebodyelse is sending (this is called carrier sensing). If the medium is busy, the stationdefers from sending and waits until the medium becomes available again. After themedium has become available, it waits for a predefined period of time, the DIFSor Distributed Inter-Frame Space.

• After this DIFS time the station enters a so-called contention window or backoffwindow. This is a time window divided in slots. The window size (in numberof slots) depends on previous transmissions, but has a minimum and maximumdefined by the standard. In 802.11p the contention window size is between 15and 1023 slots and always increments in powers of 2. The station then randomlychooses a number of slots (within the contention window), that is the number oftime slots it will wait before it actually starts trying to transmit. This means thatif two stations were both waiting for a transmission to end, they will not bothstart transmitting directly after the DIFS (and generate a collision). After pickinga random number of slots a backoff timer starts counting down to zero, when itreaches zero the station can transmit.

• If, while waiting a random number of slots, the station senses another transmission,because another station was also waiting and picked a random number smaller thanits own, it freezes the backoff timer. Then, after that transmission and a DIFS ofidle time on the medium, it restarts the backoff timer but does not pick a random

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Data

Defer access

SIFS

ACK

DIFS

Data

Contention window

Backoff after defer

Sending station

Receiving station

Other stations

Figure 2: The IEEE 802 DCF. (Source: Fig 3-7 from [10])

number again. This means that a station which had to wait the first round, has ahigher probability of gaining access first to the medium in the subsequent rounds.

• If the backoff timer reaches zero, the station transmits and the transmission fails(i.e. no ACK is received), it doubles the contention window size and picks a randomnumber again. This means that when the number of nodes in the network increasesand the number of collisions increases, stations automatically start waiting longerand the algorithm remains stable.

• If a station needs to transmit a frame that, in the algorithm, logically follows thejust transmitted frame (such as an ACK frame to indicate correct reception, ora new data frame if the entire frame is being fragmented), the station does nothave to wait for a DIFS period. Instead, it can start transmitting after a SIFSperiod (Short Inter-Frame Space), this guarantees that no other station that is notinvolved in the current transmission can seize the medium before it is completed.

In Figure 2 part of the algorithm is shown in action. A sending station starts transmittinga data packet after waiting a DIFS, the receiving station replies after waiting a SIFSwith an ACK, and after a DIFS and a random backoff period another station that alsowanted to send a frame can start transmitting.

A common problem in the wireless medium is the hidden-terminal problem, shown inFigure 3. This happens when two nodes both want to transmit but they cannot heareach other’s transmission. This might happen for many reasons; for example becausethere is an obstacle in between, or because they are simply too far apart. It does create asevere problem though, because in network situations like the one in Figure 3 the carriersensing mechanism alone is not enough. If either Node A or C starts transmitting to B(because it thinks the channel is idle) while the other node is also transmitting, either toB or to another node in the network, their transmissions will overlap spatially at nodeB, causing B to perceive a collision of two signals.

To cope with this hidden terminal problem, the 802.11 DCF includes yet another fea-ture: so-called RTS/CTS frames or Request-To-Send / Clear-To-Send frames. Using the

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A CB

Figure 3: The hidden terminal problem.

standard waiting protocol described above, when a station obtains the medium it doesn’tstart sending its data but instead it first sends a Request-To-Send, a short frame askingfor permission to send, containing basic information like sender, receiver and the lengthof the data frame the station wants to transmit. The receiving station then has to replywith a Clear-To-Send frame, containing the same information. In this case (looking atFigure 3 again), when A wants to send something to B and sends an RTS and B replieswith a CTS, station C and any other station within transmission range of either A or Bwill be notified that the medium will be occupied for the duration defined in the RTSor CTS frame. Of course, A and C could still send and RTS at the same time thatwould collide at B, but then neither A nor C would receive a CTS and they would bothknow somebody else is trying to transmit, starting a random backoff procedure. Thismechanism greatly decreases the collision-related throughput loss, especially for largedata frames [9].

3.3 Orthogonal Frequency Division Multiplexing

3.3.1 Introduction to OFDM

The 802.11 variation dedicated to vehicular networking, 802.11p, uses a physical layersimilar to 802.11a with OFDM as the primary multiplexing technique. In this sectionwe will look into OFDM and describe how it works, so that the concept is clear in thelater sections about 802.11p and Frame Capture (Section 3.4 and Chapter 4).

Normal (single-carrier) modulation techniques use the whole channel and modulate dataonto a signal at a high rate, with one symbol occupying the entire bandwidth for a veryshort time. OFDM however divides the channel in many small subcarriers and everysubcarrier is modulated at a much lower rate. In order not to waste too much bandwidthon guard bands between these subcarriers, their frequencies are chosen in a really smartway to create orthogonality; without spacing between the carriers they do not interferewith each other. This of course greatly improves the channel efficiency. In Figure 4 we

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Figure 4: Orthogonality of subcarriers in the frequency domain. (Source: [11])

clearly see that at the peak of every subcarrier all other subcarriers are zero, i.e. atthat frequency the other subcarriers do not contribute any energy to the signal. Theorthogonality is crucial to OFDM, if subcarriers are not chosen orthogonal to each other,they will overlap and interfere significantly.

Apart from the ’multiplexing’ part of OFDM, the individual subcarriers are modulatedto contain the data. Depending on the chosen modulation both amplitude and phasecan vary in an OFDM subcarrier. The frequency however has to remain constant topreserve the orthogonality.

3.3.2 OFDM Mathematics

In order to explain how these orthogonal subcarriers are exactly generated, we need tolook a little bit into the mathematics behind OFDM. Let’s first look at OFDM in thetime domain:

An OFDM subcarrier is a normal sine wave at a certain frequency, modulated with thedata. As said before the frequency does not change, only the phase and amplitude can bemodulated. Typical modulation schemes that achieve this are Binary and QuaternaryPhase Shift Keying (BPSK & QPSK) or Quadrature Amplitude Modulation (QAM).These modulations can be described in the complex field as ’constellations’ of points,where each point is the same sine wave but at a different phase and/or amplitude, andeach point encodes a number of bits. This is illustrated in Figure 5. If the numberof points in the constellation increases, more bits can be mapped onto each point. InFigure 5(a) for example, there are 2 points; only a 0 and a 1 can be mapped on bothpoints. In Figure 5(c) there are 16 points and there can be mapped 4 bits onto 1 point(24 = 16). In the complex field, a vector can be drawn from the origin to the point: theangle that the vector makes with the x-axis is the phase (ranging from 0 to 360◦), and

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I

Q

(a) BPSK constellationdiagram

(b) Example of BPSK modula-tion

I

Q

A

φ

(c) Explanation of the con-stellation

Figure 5: Modulation of OFDM signals

∫∞−∞ rect(t) · e−i2πft dt = sin(πf)

πf = sinc(f)

(a) Fourier-transform of rectangular window (b) The resulting spectrum

Figure 6: Fourier

the length of the vector is the amplitude.

The desired orthogonality is created by doing two things: carefully choosing the subcar-rier frequencies and, after summing all the subcarriers, convoluting it with a so-calledrectangular window. A rectangular window is quite a simple signal: it is 1 between −T

2and T

2 , 12 at the borders and 0 otherwise. The rectangular window used in OFDM is 1

between 0 and T (shown in Equation 1).

rect(t) =

0 if t < 0 or t > T1 if t > 0 and t < T12 ift = 0 or t = T

(1)

This rectangular pulse in the time domain is quite obvious. When Fourier-transformingthis rectangular pulse, we see that the spectrum of this rectangular is a sinc function (asin Figure 6(a)).

A sinc function looks exactly like one of the carriers in Figure 4, it has a peak atone specific frequency and is 0 at the center frequencies of the adjacent subcarriers, orin the time domain, at the nonzero integer values of t. This means that rectangularpulse in the time domain looks like a sinc function in the frequency domain. Hence, amodulated signal convoluted with a rectangular pulse of the right width looks exactly

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like a subcarrier as in Figure 4. This is valid with only one subcarrier, but by choosingthe other subcarriers just the right way (explanation in Appendix A.1), summing themand after that convoluting them with the rectangular window creates an OFDM signal(convolution in the time domain is equal to multiplication in the frequency domain). Thefrequency difference between the subcarriers needs to be fc = 1

T . As an example, T in802.11a is 3.2µs, this corresponds to 1/3.2 · 10−6 = 312.5 kHz subcarrier spacing.

All the above would result in a spectrum (for 802.11) that looks like Figure 7. At the gapsin the spectrum are special pilot carriers, used for time and frequency synchronizationpurposes.

Carriernumber

Centerfrequency

-25 -20 -15 -10 -5 0 5 10 15 20 25

Figure 7: 802.11 OFDM channel structure (Source: Figure 13-8 from [9]).

So in short: all subcarriers of an OFDM signal simultaneously transfer a number of bits.The number of bits per symbol depends on the subcarrier modulation. The symbol rateof the subcarriers themselves is low, but because OFDM uses many subcarriers the totaldata rate is equal to a single-carrier system.

3.3.3 Guard times, coding and forward error correction

There are many types of interference and disturbances which can disrupt or destroy asignal in the wireless medium. Two of them are ISI and ICI, or Inter-Symbol- and Inter-Carrier-Interference. ISI is the phenomenon that two adjacent symbols interfere witheach other; this is caused by multipath propagation or frequency delays which disrupt thereceiver’s timing synchronization. ICI is caused when (perhaps in a narrow part of thefrequency band) the carrier frequency shifts a little bit. This disrupts the orthogonalityof the subcarriers and causes energy of one subcarrier to leak into another. This can becaused for example by Doppler shift, which can be a serious problem in OFDM.

To counter ISI, every OFDM symbol starts with a guard time; this is a small portion ofthe symbol time. The symbol time is divided in the guard time and the FFT integrationtime (also shown in Figure 8). This is important because multipath delay can causevarious ’copies’ of the original symbol to arrive at the receiver, also called differentsignal components. These components are generated by reflection of the signal againstbuildings, metal structures and other big objects. The Line-Of-Sight (LOS) componentis the part of the signal that arrives directly at the receiver, and usually at variousdelays one or more multipath components show up, at a lower amplitude than the LOScomponent. If because of multipath delay a part of the previous symbol arrives duringthe guard time, this doesn’t interfere with the symbol itself. This is only true however

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Guard time FFT integration time

Subcarrier 1

Subcarrier 2

Previoussymbol

Delay

Figure 8: Cyclic prefix extension (Source: Figure 13-6 from [9])

if the guard time is chosen such that it is bigger than the biggest reasonably expectedmultipath delay. The multipath components do not cause variations in the frequency,so the orthogonality is still preserved. However to preserve the orthogonality of thesignal the receiver must have an integer number of cycles in the FFT integration time.This integer number is needed because the Fast Fourier Transform (FFT) works withdiscrete samples drawn from the signal, not the original signal itself. If in the FFTintegration time there is not an entire cycle (360◦ in the complex field) or an integermultiple of that, the sampling stops at some signal value and the next sampled value(belonging to the next symbol) will differ greatly. This results in discontinuities in thesignal, and this changes the perceived frequency of the subcarrier. This in turn destroysthe orthogonality [9].

To prevent destroying the orthogonality, whatever is transmitted during the guard timeshould have the same frequency as the symbol itself. In that case when the signal shiftsa bit over time, the FFT integration time still contains a nice (integer multiple of cycles)part of the signal. To achieve this the transmitter sends the last part of the symbol itselfduring the guard time (see Figure 8). This is called cyclic prefix extension. Also, insituations where there is no LOS component, the OFDM receiver could in the best casestill figure out based on a multipath component what the original symbol was.

In order to further increase robustness of the signal against severe channel conditions,a typical OFDM transmitter also adds coding to the data. This is strictly not part ofOFDM, but very common to use. The coding enables Forward Error Correction (FEC)at the receiver; with FEC a receiver can detect and correct errors without requiringretransmission. The encoder expands the original data bitstream, thereby adding re-dundancy. How much redundancy is added, is given by the coding rate, for 802.11 thisvaries between 1/2 and 3/4. A coding rate of 1/2 means that every original data bitis replaced with 2 coded bits. There are many different coding algorithms which varyin complexity, the one that is used in 802.11 is a so-called convolutional encoder. Aconvolutional encoder works on a continuous stream of bits; it takes m bits as input and

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produces n output bits, where the coding rate is m/n. It uses a transformation functionthat has a shift register with the last 7 input bits. On every step it computes n outputbits based on the 7 bits in the register, and m bits in the register are replaced withnew input bits. An example of a small convolutional encoder (with only 3 bits in theregister) is given in Figure 9. In this encoder the values in the registers are summed tocreate the various output bits. The way the register bits are combined (or the generatorpolynomials for the various output bits) is critical to the error-correction properties ofthe encoder, they vary depending on the coding rate and the number of registers (orconstraint length).

Figure 9: An example of a convolutional encoder with 1 input bit, 3 output bits and 3bits in the register.

One can understand that using such an encoder, an input bit kind of ’spreads’ itself overtime, as it is still present in the shift register a few computation rounds later. Dependingon the generator polynomials, a certain string of input bits of arbitrary length will alwaysgenerate the same output bits. Therefore when decoding, depending on the incomingbitstream, a decoder can find out what the original data bits were. 802.11 recommendsa so-called decoder which uses the Viterbi-algorithm, which is a maximum-likelihooddecoder. If bit errors occur, the Viterbi algorithm outputs the most likely sequence ofdata bits, depending on the encoding and the occurrences of the bit errors. [9, 12]

One last smart thing that the OFDM transmitter does is interleaving: it spreads thecoded bits over all subcarriers (1 bit at carrier 1, next bit at carrier 2, next at carrier3 and so on), so that in the case of narrowband interference the bit errors are spreadover the input bit stream. The probability of multiple bit errors close together (after de-interleaving) decreases, and if the decoded bits have errors, they can be more easily becorrected by CRC checks at the MAC layer. The receiver thus has a higher probabilityof finding the right data bits based on the encoded bits.

3.3.4 Strengths and weaknesses of OFDM

The multiplexing scheme used in OFDM has a few significant advantages over the single-carrier schemes used for instance in 802.11b. All the small subcarriers together generate

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Transmitter

FEC (Convolution

encoder)

Interleaving,mapping, andpilot insertion

IFFTGuardinterval

insertion

I-Qmodulation HPA

Pilot remove,deinterleaving,and demapping

FFTGuardintervalremoval

I-QdemodulationLNA AGC FEC

decoder

Receiver

Figure 10: An OFDM Transceiver block diagram (Source: Figure 13-17 of [9])

a very flat, evenly distributed spectrum. The spectral efficiency is high; subcarriers areorthogonal to each other and placed very closely together. The various protection mech-anisms make the signal very robust to narrowband interference and multipath fading,and ISI is effectively countered by the cyclic prefix extension (given that the delays arenot much longer than the guard time), while the efficiency loss caused by this cyclicprefix is acceptable because of the low symbol rate. Another advantage is that OFDMcan be implemented in transmitters and receivers using relatively simple components,no advanced equalization circuits are needed.

There are also disadvantages however; because of the dependence on orthogonality anOFDM channel is very sensitive to carrier- and Doppler shifts. Also, summing of all theindependent subcarriers may create a signal with some high (amplitude) peaks, whichcauses a high Peak-to-Average-Power-Ratio or PAPR. This increases the complexity atthe transmitter, since practical power amplifiers (PA’s) have a range at which they arelinear (i.e. linearly amplifying the signal), and another part (near saturation) where theyare non-linear. If the amplification is non-linear, this changes the form of the receivedsignal at the receiver and thus increases bit errors [13, 14].

3.3.5 OFDM Transmission and Reception blocks

This section shows transmitter and receiver blocks for typical OFDM chipsets, andexplains some of the steps that a transmitter goes through when transmitting data.

When looking at Figure 10, an OFDM transmitter does the following steps when trans-mitting (note that some of these steps are specific to 802.11) [9]:

1. Transmission rate selection. This depends on the channel conditions but is chipset-implementation dependent. The rate does dictate however which modulation isused and which coding rate. Together the modulation and coding rate determine

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how many bits are transmitted per symbol. Table 1 contains an overview of theused combinations.

2. Transmission of the PLCP preamble (at a fixed rate of 1 Mbps), these are a fewlong and short symbols to train the receiver and enable frequency synchronization.

3. Transmission of the PLCP header, also at 1 Mbps, which contains info about theframe that is about to be transmitted (frame length, encoding, transmission ratethat will be used).

4. Creation of the data packet itself: some protocol-specific fields are added, the datais scrambled to prevent long sequences of zeros or ones and some padding bits areadded. This data is encoded using the convolutional encoder.

5. Division of the coded bits into blocks (depending on the number of bits per OFDMsymbol) to perform the interleaving process.

6. Pilot subcarrier insertion and using the IFFT or Inverse Fast Fourier Transformto generate the data signals.

7. Modulation of the subcarriers with the data signals using I-Q modulation. Thismodulation type is how QAM and QPSK signals are generated, for more informa-tion readers are referred to Chapters 9-5 and 9-6 of [15].

8. Amplification of the entire signal in an High-Power Amplifier (HPA) for transmis-sion over the antenna.

When a frame is being received, the signal is first amplified using a Low-Noise Amplifier(LNA), and Automatic Gain Control (AGC) is applied to get the signal to the (standard)power levels needed for demodulation. The rest of the reception process is basically thereverse of the sending process, as can be seen in Figure 10.

Table 1: 801.11 data rates and modulations / coding rates [9]

Transmissionrate (Mbps)

Modulation Codingrate

Coded bitsper carrier

Coded bitsper symbol

Data bits persymbol

6 BPSK 1/2 1 48 249 BPSK 3/4 1 48 3612 QPSK 1/2 2 96 4818 QPSK 3/4 2 96 7224 16-QAM 1/2 4 192 9636 16-QAM 3/4 4 192 14448 64-QAM 2/3 6 288 19254 64-QAM 3/4 6 288 216

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3.4 IEEE standard draft: 802.11p

In an attempt to solve the problems described in Chapter 2, an IEEE working groupis currently creating an amendment to the 802.11 standard. It will specify many of theneeded adaptations, one of the main new concepts they introduced is called WAVE (orWireless Access in Vehicular Environments). The amendment carries the name 802.11p.The standard is still in the draft phase, but has been under development since 2006 andshould be completed in the fourth quarter of 2010. The next passage provides a shortoverview of the new functionality and behavioral changes:

An STA (Station) can operate in WAVE mode, and when it does it can send messagesto any other STA in WAVE mode with a valid MAC address, including group addresses,without first joining a BSS. If communication with a DS (Distribution System, a fixedroad-side network) is necessary, stations can also join a WBSS (WAVE Basic ServiceSet). Authentication or association is not required in WAVE mode, only the first step(joining the BSS) has to be performed. After that, data is sent data directly to the DS[16].

The physical layer is also adapted for increased movement in 802.11p. The frequencyband at which it will operate is probably in the 5.9 GHz ITS channel in the US, this bandhas been designated by the FCC for use by Intelligent Transport Systems (ITS). TheEuropean telecommunications authority, the CEPT, has designated a similar frequencyspectrum after extensive studies [17, 18].

To counter the increased vulnerability of 802.11p (also see Section 3.3.4), such as theincreased amount of Doppler shift due to movement, some physical layer parameters arechanged in the 802.11p standard. One 802.11a OFDM channel uses 52 subcarriers outof which 48 are used to transmit data and 4 are pilot carriers. 802.11p uses the samenumber of subcarriers, but uses a smaller bandwidth per channel; 10 MHz opposed to 20MHz in 802.11a. This also means that all parameters in the time domain (guard time,symbol time) are doubled compared to 802.11a. This has as a result that Doppler spreaddecreases (due to the smaller frequency bandwidth), inter-symbol interference is alsodecreased due to the higher guard times. These doubled parameters in the time domainhalve the effective data rate (3 to 27 Mbps against 6 to 54 in 802.11a) [19]. The standardalso specifies more stringent ACR (Adjacent Channel Rejection) requirements, becausethe guard bands between channels are smaller. Table 2 contains a small comparison ofvarious 802.11a and p PHY parameters.

4 Frame Capture

The phenomenon that this report focuses on is the so-called capture effect. It is alsoreferred to as Physical Layer Capture (or PLC), or Frame Capture (FC). This chapterwill further elaborate on it: what is it exactly, in which situations does it occur and

15

Table 2: 802.11a and 802.11p PHY values [2]

Parameter 802.11p 802.11a

Channel bandwidth 10 MHz 20 MHzData rates 3 to 27 Mbps 6 to 54 MbpsSlot time 16 µs 9 µsSIFS time 32 µs 16 µsPreamble length 32 µs 20 µsPLCP header length 8 µs 4 µsAir propagation time < 4 µs << 1 µsCWmin 15 15CWmax 1023 1023

what happens exactly at the hardware level?

4.1 What is FC?

Frame Capture occurs at the physical layer of 802.11 networks. Depending on chipsetdesigns it might occur in other types of networks as well, but this was no part of ourinvestigation. It has already been observed and investigated in many different papers[20, 21, 22, 23, 24], mostly for 802.11a. [20] discusses the implied unfairness and thatcloser nodes can effectively ’capture’ the channel if their signal is sufficiently stronger.[21] describes various ways to model the problem. [23] performs real experiments with802.11a testbeds and [22] only simulations. However, none of these were specific to802.11p or vehicular networking, they mostly focus on the basic phenomenon.

Frame Capture occurs (under some conditions) when two transmissions overlap spatially.Normally the 802.11 coordination function (CSMA/CA, also see Section 3.2) preventsmost collisions by implementing RTS/CTS and random backoff mechanisms. However,in safety-oriented VANET applications most traffic will very likely be broadcast traf-fic, which is sent without a prior RTS/CTS handshake and without acknowledgments.Therefore, collisions are much more likely to occur.

If at a given node two signals arrive simultaneously, this would normally be regardedas a collision and both packets would be assumed lost. The last few years however itappeared that this is not the way most 802.11 chipsets handle a collision: under somecircumstances one of the packets is being received correctly even if another transmissionis occurring simultaneously [25, 26]. This phenomenon of being able to receive a frameeven in the presence of another frame is called Frame Capture. It is quite dependent onchipset design; ultimately the chipset makes the decision to try and start demodulating adifferent (stronger) signal when the interference on the current signal becomes too strong.However, currently most network simulators do not take this behavior into account (see

16

also Chapter 5); but experiments point out that under some network circumstances thedifference in throughput can be substantial [27].

FC can occur in many different situations, and there are many variables which influencethe behavior (and possible outcome). If two frames are being transmitted simultaneously,the following factors are important:

• Exact time of arrivalThe exact time difference between the arrivals of two colliding frames is an impor-tant factor. The timing difference we talk about here is in the order of microsec-onds. If the medium is busy the CSMA/CA mechanism implies a slot synchroniza-tion which will cause the frames to arrive more or less at the same moment. How-ever in the hidden terminal situation a colliding frame could arrive at any momentduring reception, because a hidden node that doesn’t sense ongoing transmissionsis allowed to start sending at any moment.

• Signal strengthApart from the timing, one of the two signals has to be sufficiently stronger thanthe other for the receiver to be able to decode it even in presence of the interferenceof the weaker signal. If both signals are equal in RSS (Receiver Signal Strength),it really depends on the chipset whether it is able to decode either one of them.Also, the required difference in RSS depends on the timing relation. If the strongerframe arrives before the weaker frame, the stronger frame will probably be receivednormally, and the required SIR difference is quite small. If one of the two framesinterferes with the other one’s preamble (which is an important phase of the re-ception process: during the preamble the receiver tries to lock onto the signal),the required difference in RSS is a lot higher because it is harder for the receiverto correctly lock onto one of the signals.

• Chipset manufacturerIt appears that even chipsets from different manufacturers behave differently underthe above mentioned timing relation. In a Prism chipset [28] for example, the effectonly occurs if the second, stronger frame arrives before or during the first frame’spreamble. However in Atheros chipsets [29] capturing a stronger frame will occuralso after the reception of the weaker frame’s preamble (i.e. almost independentof the timing relation between the two arriving frames [30, 26]).

4.2 Scenario classifications

Based on literature [21, 23, 25, 27], we have distinguished the following relevant scenar-ios where 2 frames arrive almost simultaneously at the receiver. In this classificationoverview we assume that when we have two frames (F1 and F2) and that F1 is alwaysthe strongest. All scenarios where F2 would be stronger always have their equivalents inTable 3 below.

17

In advance it is impossible to say which of the two frames is the desired one, so we justassume that the stronger frame is the one we want to receive. We cannot make thisdistinction based on the content of the frames, but receiving the stronger frame is alsodesirable because a stronger SIR means a lower bit-error rate (BER), which increasescorrect packet reception probability. In a vehicular networking context, receiving thestrongest frame is strongly related to receiving the frame of the nearest vehicle. Thisis also a desirable feature, because the nearest neighboring vehicles pose the greatest’threat’ to the vehicle itself; safety messages like emergency braking and other problemsshould really be received first. It is thus quite reasonable to assume that we want toreceive the strongest frame.

Table 3 illustrates the FC scenarios with different timing and SIR situations. ∆t is theexact difference between the arrival times (in µs), Lpreamble is the length of the frame’spreamble (32µs in 802.11p), and the SIR (Signal to Interference Ratio) is RSSF1 −RSSF2. As said, in all cases RSSF1 > RSSF2.

Table 3: Frame Capture scenarios (data rate of 6 Mbps). These timing relations andresults are for Atheros chipsets [23]. The experiments were performed with 802.11a.

Timing relation Result

1. ∆t > LpreambleP Frame 1

P Frame 2

Frame 1 is captured ifSIR > ∼ 0 dB

2. ∆t < LpreambleP Frame 1

P Frame 2

Frame 1 is captured ifSIR > ∼ 12 dB

3. ∆t < LpreambleP Frame 1

P Frame 2

Frame 1 is captured ifSIR > ∼ 12 dB

4.∆t > Lpreamble, receiver

locked on to Frame 2

P Frame 1

P Frame 2

Frame 1 is captured ifSIR > ∼ 10 dB

5.∆t > Lpreamble, receiver

NOT locked on toFrame 2

P Frame 1

P Frame 2

Frame 1 might becaptured, but SIR

should be at least ∼20dB. Probability

increases linearly asSIR increases.

Basically Table 3 tells us the following: a stronger frame can always be captured (andthus received correctly if no other interferers arrive) in the presence of another frame,if the SIR of the stronger frame is high enough. The required SIR is higher if ∆t isless than Lpreamble, because the frame’s preamble is an important phase of the carrier

18

detection and lock-on process. If the receiver is locked on to the weaker frame, it is abit easier to receive a stronger frame if it arrives, because the ability to receive a signalmeans that the receiver is better able to suppress it as well [24].

Figure 11 from [23] shows the required SIR in more detail. It shows the FRR (FrameReception Ratio) on the y-axis for various SIR values and for various situations. Inthese experiments a testbed was created with a receiver, two senders (one normal senderand an interferer) who cannot hear each other, and two extra receivers to monitor thetwo senders and get exact timings on the sent frames (see also Figure 1 in [23]). Whenexperimenting with the induced collisions at the receiver, the authors make three dis-tinctions; Sender First (SF) capture, Sender Last&Interferer Clear capture (SLC) andSender Last&Interferer Garbled (SLG) capture. The Sender in this case is the sender ofthe desired frame, the other sender acts as an interferer. In the SF case the strongestframe arrives first (corresponding to Scenario 1 and 2 in Table 3), in the SLC case thestrongest frame arrives second while the receiver is decoding the weaker frame (Scenario3 and 4), and in the SLG case the strongest frame arrives second while the receiver is,for any reason, not decoding the weaker frame (Scenario 5). The capture probabilitydepends a lot on the data rate of the stronger frame, but at 6 Mbps in the SF case almostall frames can be captured even if the SIR is around 0 dB (both frames equal in signalstrength). For higher data rates the required SIR also increases. It is clear to see thatin the SLC case the required SIR is a lot higher, but does not change much if the datarate increases - only at the highest data rates does the required SIR also increase. If thereceiver is not locked on to the weaker frame (or ’interference’), the data rate becomeseven less important - in the presence of the interfering signal energy, the desired signalmust simply be strong enough to be received. For a reasonable Frame Reception Ratio(80 %) the SIR needs to be around 20-22 dB.

It is also important to note that there are two different thresholds:

• The capture threshold. The SIR of an arriving frame that determines if the receiverdrops the current frame and starts locking on the new frame

• The required SIR during the entire transmission. This one is different from thefirst one because it is bitrate-dependent.

The decision to capture a frame is made based on the signal power of the newly arrivingframe. If it is high enough, the receiver might start preamble detection and all mightgo well, but then during the PLCP header reception the bitrate of the rest of the framebecomes known. It might thus happen that a receiver correctly locks onto the frame butafter receiving the PLCP header the bitrate at which the frame is sent appears to be toohigh and the MAC CRC check at the end of the frame reception process fails. Hence, acorrect capture decision based on the SIR during preamble and PLCP header does notnecessarily mean that the frame will also be received correctly.

19

(a) SF (Sender First) case (b) SLC (Sender Last, Clear interference)case

(c) SLG (Sender Last, Garbled interference)case

Figure 11: Required SIR for various FC scenarios. (Source: Figure 7 of [23])

4.3 802.11p scenario predictions

An important problem we need to take into account is that the above illustrated sce-narios are all based on literature experimenting with 802.11a wireless network setups,and as described in Table 2 and Chapter 3, some physical layer parameters in 802.11pare different from 802.11a. A very important question that rises thus is whether theframe capture scenarios in 802.11a also apply to 802.11p. The changes have influence ofcourse on the reliability of the signal under harsh channel environments, but does it alsoinfluence the behavior of a chipset when two frames collide, or when it makes a capturedecision?

To answer this, we need to closely look at the OFDM symbols, how they react to inter-ference and what happens if two OFDM symbols arrive simultaneously at the receiver.The 802.11p channel bandwidth is halved, but so is the data rate so the amount of trans-mitted information per Hz (or information rate) of the channel is the same. The way thesymbols are modulated is also exactly the same, as described in Section 3.3. The samesubcarrier modulations are used as well as the same combinations of modulation andcoding rates. Depending on the signal strength of the interfering signal, the reception ofthe OFDM symbol is disturbed because energy not belonging to the original symbol isbeing received. Thinking again of the modulation constellations in Figure 5, interferencewill add extra energy to the original signal, changing the amplitude (either with con-

20

structive or destructive interference) and changing the perceived phase of the signal. InFigure 12 one can see various types of noise superimposed on a 16-QAM constellation.Phase noise (a slight distortion of the phase of the original signal) causes the point inthe constellation to shift either clockwise or counterclockwise on a circle. White noise(also called AWGN, or Additive White Gaussian Noise) is usually everywhere on thespectrum, can be any phase and is normally of a very low amplitude, so it shows as a’rotating pointer’ superimposed on the original constellation point. If no other interfer-ers are present, the received constellation point can be anywhere inside the white noisecircle.

I

Q

(a) An ideal constellation

I

Q

(b) Phase noise

I

Q

(c) White noise

Figure 12: Various noise types superimposed on a 16-QAM constellation.

In the case of interference from other (non-802.11) transmitters or white noise, the onlyrelevant variable is the noise power. The source of the energy can of course also benoise or other transmitters (not necessarily 802.11) in the same frequency band, so thepossible modulation and data rate of an interfering signal is not relevant to the receptionprobability of the signal itself.

Another very important factor in the Frame Capture behavior is the chipset design.Ultimately it is the chipset that, when detecting an energy increase above a certainlevel, decides that it drops the current frame and starts locking on the newly arrivedframe, taking eventual bit errors for granted. A chipset can decide to keep demodulatinga weaker frame, but depending on the RSS of the interference this frame will without adoubt contain so many bit errors that the frame can be discarded altogether. We havealso seen the variance in capture behavior among different chipsets in Section 4.1. Ifwe assume the same chipset design for 802.11p, it is fair to assume that it will exhibitthe same behavior as it did in 802.11a, given that the circumstances are equal (such astiming and RSS). It will therefore also make the same capture decisions.

There are a few small differences between 802.11a and p frames however. For examplethe preamble of an 802.11p frame is a lot longer. This is shown in Figure 13, where ’P’is the preamble and ’H’ is the PLCP header.

The preamble is an important phase of the reception process. A receiver does thefollowing during the preamble:

1. It detects and measures the signal power; this signal power has to be greater than

21

µs 0 10 20 30 40 50 60 70 80 90 100 110

P H 802.11p frame

P H 802.11a frame

Figure 13: Comparison between the first parts of typical 802.11a and p frames

or equal to RXSens, which is the receiver’s minimum coding and demodulationsensitivity. Any signals with a power level below this threshold will be consideredas noise. RXSens for Atheros chipsets is around -88 dBm. [27, 26]

2. It performs automatic gain control (AGC) during which it sets the LNA (low-noiseamplifier, see Figure 10) to a level appropriate for the received signal power. Thishappens because a signal when it arrives at the receiver is very weak; it needsto amplified before further processing (while preferably the noise is not furtheramplified).

3. It performs frequency synchronization using a Phase-Locked Loop (PLL). For de-tails on how a PLL works, review Chapter 3-7 of [15].

4. It performs time synchronization [27].

The normally observed arrival time differences in a contended medium are between 0 and20 µs at the receiver due to RX/TX turnaround time delay and inherent uncertaintiesin the 802.11 firmware clock synchronization. However as said before, in the hiddenterminal situation a new frame might arrive at any time during the ongoing framereception. The larger preamble in 802.11p will make thus a bit more probable that twoframes are interfering in each other’s preamble (Scenario 2 and 3 of Table 3). Also notethat this does not say anything about the behavior of the chipset under collisions, justthat these scenarios will occur more often [27, 30].

Given the fact that the shapes of the signals of 802.11a and p are equal with just differenttiming parameters, it is fair to conclude that given the same chipset design, the capturebehavior will not change: the receiver will use the same thresholds to decide if theadditional energy detected during frame reception is belonging to another, stronger signaland is worth capturing or not. The bit-error rate of the received frames could changeof course, but the modulation, multiplexing scheme and the amount of information perHz of bandwidth are the same, so this will depend more on other factors related tothe channel conditions than on the used modulation. Of course the channel conditionswill be different in 802.11p compared to a, so these channel conditions also have to beconsidered in our simulations.

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5 Potential impact of FC on vehicular networks

The simulations we will perform later will be mainly focused on the potential impactof FC. VANETs are networks with an extremely high degree of mobility. The type oftraffic in a VANET will largely depend on the implementations of safety applicationson a higher layer, but we expect that a large portion of the traffic in a VANET will bebroadcast traffic, because independent of the application it is reasonable to assume thatsafety information should be disseminated as much as possible over the network andbroadcast traffic achieves this information dissemination a lot faster than unicast traffic([31] for example proposes a smart network flooding mechanism to distribute trafficinformation).

It is this broadcast traffic scenario that we are interested in. Because of the design of the802.11a and p MAC layers, broadcast frames are not transmitted using the RTS/CTSmechanism and no acknowledgment is sent after reception. Also, depending on the typeof information transmitted and the safety-application at a higher layer, these broad-cast messages may be retransmitted for dissemination beyond the original transmitter’srange. If the real-world congestion increases (more cars), the same might also happenwith the network congestion (more frames are transmitted so the load on the networkincreases), up to a point where collisions occur on a regular basis. For this high networkload scenario it is very interesting to know if Frame Capture has a significant effect onnetwork throughput. Simulations demonstrating Frame Capture have been done, butmostly with infrastructure networks and not all network simulators currently supportFrame Capture or it is implemented in a way which doesn’t reflect all possible scenariosidentified. One paper shows that ns-2 only captures a stronger frame if it arrives first atthe receiver, and they show with experiments that Prism chipsets also capture a frameif it arrives during the weaker frames’ preamble [30]. Another paper expands this byshowing that with different chipsets (Atheros) the capture behavior is again different,also capturing stronger frames which arrive at any time during a weaker frames’ recep-tion [23]. Both only experiment with infrastructure networks however. There have beenefforts to implement Frame Capture in ns-2 and QualNet, the results of this are shownin [30, 32]. However for the simulator we prefer to use, OMNeT++, nothing has beendone that we’re aware of yet. Some performed simulations indicate a substantial TCPthroughput increase (up to 430%, based on various physical layer models [25]). In [32]the ns-2 simulator is updated and also put to the test in a VANET context, but only as asupport to show an implementation of FC in ns-2. The authors demonstrate that thereis a lot to gain when correctly modeling the physical layer, so for VANET simulationsis OMNeT++ implementing this behavior is an important task. We thus suspect thatFrame Capture throughput gain might be equally or even more beneficial to vehicularnetworks, due to the increased amount of broadcast traffic.

23

6 Conclusion

In this report we have discussed the basics of the 802.11 MAC and PHY layers. Wespecifically had a look at 802.11a and 802.11p because of their similarity and because802.11p is the standard currently under development for vehicular networks. We per-formed a literature study to gain as much knowledge as currently available on FrameCapture, a phenomenon where under some circumstances 802.11 chipsets can receive aframe even during a collision. We looked in detail into OFDM and the mathematicsbehind it, in order to make a reasonable prediction on when capture decisions are madein a chipset and what the effect will be on the reception of a frame. We concluded thatbecause 802.11a and p have almost equally designed PHY layers and the informationrate per Hz of bandwidth is the same, the changes at the PHY layer (halved bandwidth,doubled symbol- and guard time) and the changes to the shape of the signal will notaffect the behavior of an 802.11 chipset. Required SIRs found in literature describing FCfor 802.11a are thus still valid, although the differences in channel conditions will in reallife change the BER of the received and captured frames. The capture decision behavioritself will remain the same. We note however that the chipset design is of great influenceto this capture behavior, and for our future simulations we will confine ourselves to thebehavior of the Atheros chipset as it is described in literature. It is reasonable to assumethat not much will change when implementing 802.11p based on a current (802.11a)Atheros chipset.

6.1 Master’ thesis lookahead

We have clearly laid out the different parameters we have to incorporate in our model,have compared 802.11a to 802.11p and demonstrated how Frame Capture will mostprobably behave in an 802.11p environment. To conclude this report, we can look aheadto outline our future research. After all, the work in this article only contains the basicgroundwork we can now use for further research.

Our research questions, including possible directions to take, are the following:

• How do we correctly model and implement FC behavior in our simulator of choice,OMNeT++?

• Does FC increase throughput in an ad-hoc 802.11p vehicular environment? Inwhat way does the road traffic density influence this?

• Which types of network traffic and traffic flows can we expect? Does FC positivelyinfluence the desired behavior given these network traffic flows?

• How does the type of traffic (unicast / broadcast / perhaps various types of flows)influence the throughput?

24

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A Appendix

A.1 Orthogonality of multiple sinusoidals and their frequency spac-ing

For two sinusoidals to be orthogonal to each other, the following must be true:

∫ T

0cos(2π(f1t+ φ) · cos(2πf2t)dt = 0 (2)

where T is the symbol period, f1 and f2 are the frequencies of the two sinusoidals andφ is the phase difference of the first cosine. Integrating and applying the limits, thissimplifies to

cos(φ)

[sin(2π(f1 + f2)T )

2π(f1 + f2)+sin(2π(f1 − f2)T )

2π(f1 − f2)

]+

sin(φ)

[cos(2π(f1 + f2)T )

2π(f1 + f2)+cos(2π(f1 − f2)T )

2π(f1 − f2)

]= 0

(3)

Also note that for any integer value of n, sin(nπ) = 0 and cos(2nπ) = 1. Then if weassume that (f1 + f2)T is an integer, we see that a few terms in the equation abovedisappear because sin(2π(f1 + f2)T ) = 0 and cos(2π(f1 + f2)T ) = 1. Substitutingsimplifies Equation 3 to

cos(φ)sin(

2π(f1 − f2)T)

2π(f1 − f2)+ sin(φ)

cos(

2π(f1 − f2)T)− 1

2π(f1 − f2)= 0 (4)

So, for a random phase (φ) between 0 and 2π, the numerators of both fractions needto be 0 in order for the whole equation to be zero, and thus have orthogonality. Sosin(2π(f1 − f2)T ) has to be 0 and cos(2π(f1 − f1)T ) has to be 1, this is the case when2π(f1 − f2)T = 2nπ, n being an integer. This condition, 2π(f1 − f2)T = 2nπ, can besimplified to f1 − f2 = n

T . The minimum value of n is 1, so the minimum frequencydifference at arbitrary phase difference between f1 and f2 is 1

T , and the two sinusoidalsare orthogonal at frequency differences n

T [34].

28