Analysis of the Effects of Phase-Noise in Orthogonal
`Frequency Division Multiplex (OFDM) Systems
`
`Patrick Robertson and Stefan Kaiser *
`German Aerospace Research Establishment (DLR), Institute for Communications Technology
`D-82230 Oberpfaffenhofen, Germany, Tel.: ++49 8153 28 2808 (2805); FAX: ++49 8153 28 1442
`
`Abstract
`In OFDM transmission schemes, phase-noise from oscillator insta-
`bilities in the receiver is a potentially serious problem, especially
`when bandwidth efficient, high order signal constellations are em-
`ployed. The paper analyses the two effects of phase-noise: inter-
`carrier interference (ICI) and a phase error common to all OFDM
`sub-carriers. Through numerical integration, the IC1 power can
`be evaluated and is shown as a function of the number of OFDM
`sub-carriers and various parameters of the phase-noise model. In-
`creasing the number of sub-carriers causes an increase in the IC1
`power, which our analysis indeed shows to become a potential
`problem, since it can lead to a BER floor. The analysis allows the
`design of low-cost tuners through specifying the required phase-
`noise characteristics. A similar technique is applied to calculate
`the variance of the common phase error. After showing that the
`common phase error is essentially uncorrelated from symbol to
`symbol, we propose a simple feed-forward correction technique
`based on pilot cells, that dramatically reduces the degradation
`due to phase-noise. This is confirmed by BER simulations of a
`coded OFDM scheme with 64 QAM.
`1 Introduction
`In this paper, we will discuss the effects of phase-noise on dig-
`ital transmission systems using OFDM with a high number
`of sub-carriers. Typical applications can be audio, T V and
`HDTV transmission over terrestrial channels. OFDM has
`been proposed for terrestrial broadcasting because of its high
`spectral efficiency and robustness in the case of long echoes,
`but has a number of disadvantages too, such as behaviour
`in the case of non-linearities and quite sever carrier synchro-
`nization requirements [1]. Phase-noise is also a potentially
`serious problem because of the common necessity to employ
`relatively low cost tuners in the receivers. Low cost tuners
`are associated with less good phase-noise characteristics, i.e.
`their output spectrum cannot be modelled by a Dirac delta
`at the centre frequency, but rather as a delta surrounded by
`noise with certain spectral characteristics.
`The effects of this phenomenon are heightened due to the
`fact that, for example, TV and HDTV transmissions often
`use high order modulation formats to transmit a signal with
`a high data rate over the available channel (e.g. 8 MHz).
`For transmission of T V and HDTV signals, 64 QAM modu-
`lation with OFDM has been suggested [a], and this constella-
`tion is particularly sensitive to phase-noise: on the one hand
`phase-noise causes inter-carrier interference (due to the non-
`
`*The authors are within the European dTTb-Race and German
`HDTV-T projects
`
`{
`
`L,,(f) = 10-c +
`
`orthogonality after mixing with a 'noisy' local oscillator); on
`the other hand a further degradation is the result of the so
`called common phase error.
`The paper is organized as follows: we shall begin by in-
`troducing a simple phase-noise model that is the basis for
`subsequent simulations and analysis. We will then focus on
`the two effects that phase-noise will have on the OFDM trans-
`mission: IC1 and the common phase error. This is followed
`by numerical evaluation of the effects, and we shall show the
`importance of the number of OFDM sub-carriers and param-
`eters of the phase-noise model. Motivated by the theoretical
`analysis, we will present a feed-forward method of combating
`phase-noise (as far as possible) and shall conclude the con-
`tribution by showing simulation results of the feed-forward
`correction technique applied to a coded OFDM scheme pro-
`posed for terrestrial transmission of digital television.
`2 The Phase-Noise Model
`To model the phase-noise in the receiver's local oscillator(s)
`we assume instability of the phase only, and no deviation of
`the centre frequency or amplitude. The model is taken from
`[3] which is based on [4]. It has been accepted as a reference
`in the European dTTb project.
`The complex oscillator output (in baseband notation) can
`be written as
`~ ( t ) e j P N ( t ) 1 + j . Prv(t),
`(1)
`the approximation holding when c p ~ ( t ) << 1, which is a valid
`assumption (no frequency deviation). The power density
`spectrum (PDS) of q(t) is given through the PDS of c p ~ ( t ) ,
`which can be obtained through measurements of a real tuner
`using a PLL. For the PDS of p~ the model specifies [3]:
`: If1 5 fl
`: f l < f
`f < -fl
`Typical parameters are: a = 6.5, b = 4, c = 10.5, fi = 1
`kHz and fi = 10 kHz. A plot of L P N ( f ) for these values
`is shown in Fig. 1. The parameter c determines the noise
`floor, here at -105 dB, parameter a and fi the characteristics
`of the PLL. The steepness of the linear slope of the curve
`is given by b, here we assume a reduction in the noise level
`by 40 dB/decade. The frequency f 2 is where the noise-floor
`becomes dominant. Later we will vary the important param-
`eters a and cl since they are particularly dependent on the
`tuner technology used.
`
`10-(f-f1).Z%-a
`lo-"
`l O ( f f f h 5 T - a
`
`.
`
`(2)
`
`0-7803-2486-2195 $4.00 0 1995 IEEE
`
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`
`Petitioner Sirius XM Radio Inc. - Ex. 1010, p. 1
`
`

`

`-70
`
`-80
`
`-90
`
`Phase
`N o i s e
`PDS
`[dBl
`
`-100
`
`.
`
`I
`
`-20000
`
`-10000
`
`0
`f [Hzl
`
`10000
`
`201
`
`Figure 1: PDS L V N ( f ) of the modelled phase-noise process
`
`Figure 2: OFDM transmission scheme including phase-noise
`
`Transmission
`OFDM
`3 The
`Scheme Including Phase-Noise
`The principle of OFDM is to transmit information over a
`large number of orthogonal sub-carriers, thus making the
`symbol time very long compared to single carrier systems.
`By including a guard interval longer than the longest echoes
`of the channel in each OFDM symbol, no traditional channel
`equalizer to resolve IS1 is needed [5], but a channel estima-
`tion operating in the frequency domain instead. For correct
`demodulation, orthogonality of the sub-carriers is essential,
`and this is threatened in the receiver either by phase-noise or
`incorrect carrier frequency synchronization.
`Fig. 2 shows the OFDM transmission scheme with N sub-
`carriers which we have modelled here with a perfect channel.
`The transmitted signal only suffers from phase-noise due to
`oscillator instabilities in the tuner. d k , , are the transmitted
`data symbols where k indicates the OFDM sub-carrier and
`d the OFDM symbol. The demodulated data symbols in the
`receiver are written as r k , l and c p ~ ( t ) represents the phase-
`noise. The sub-carrier frequencies are W k = 2 n f k , spaced
`by 1/T where T = N I B equals the symbol duration. The
`signal bandwidth B is assumed to be constant at 8 MHz in
`the examples. The OFDM signal can be formed by a simple
`inverse FFT as shown in [5, 61. The received OFDM signal
`has to be sampled with the frequency 1/T. For simplicity we
`do not consider the guard interval in the following, without
`loss of generality.
`
`3.1 Simplified model assuming one trans-
`mitted sub-carrier
`
`In the sequel we shall assume that only one OFDM sub-
`carrier with index R is active and all other sub-carriers m # R
`have dm,l = 0. The reason this is allowed for the analysis of
`the IC1 is that under the assumptions of linearity and that
`normally all dk,l are independent, the IC1 noise components
`from all sub-carriers can be superimposed. Furthermore, we
`shall be interested only in one received sub-carrier (index k).
`The common phase error can also be analysed for this one
`sub-carrier (see section 5). The transmission model is shown
`in Fig. 3 a).
`
`Figure 3: Simplified transmission models: a> assuming only
`one active sub-carrier and observing received sub-carrier k;
`b) setting dn,j = 1 VI allows one to ignore the sampling when
`determining the variance of the samples rk,j
`
`4 Inter-Carrier Interference due to
`Phase-Noise
`
`We shall now assume that sub-carrier n will disturb sub-
`carrier IC, since we can later superimpose disturbances from
`all sub-carriers # IC, by simply summing over n. In order to
`continue, the modiel of Fig. 3 a) must be further simplified.
`To determine the variance of the samples r k , l (for a given
`power of dn,l), the sampling can be ignored if we are able to
`set dn,l = l , V 1 and if the phase-noise process is stationary.
`The former restriction can be justified since the system is
`linear and the data of one OFDM symbol 1 will not affect
`the output of the receiver for other OFDM symbols # 1. In
`this case the variance of
`will equal that of r k , ~ as long
`as E{d;,,} = 1. This leads to the simple model of Fig. 3
`b) where we need to calculate the variance of y ~ , since the
`IC1 from sub-carrier n on k is the signal yI: the real valued
`(useful) component of r](t) in (1) results in no component in
`yr as the sub-carriers n and k # n are spaced by multiples of
`1/T. It is easy to show using (1) that the PDS of the IC1 is
`approximately
`
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`
`Petitioner Sirius XM Radio Inc. - Ex. 1010, p. 2
`
`

`

`Now the IC1 variance is the integral
`
`03
`
`to one, the symbols d k , l will be disturbed by a common phase
`error, which for OFDM symbol 1 we define as:
`
`as the IC1 has zero mean. Unfortunately, the integral did not
`yield to analytical solution as far as the lower two terms in the
`bracket in (2) are concerned. The frequency constant term
`and the term for If1 5 fi are both integrateable since there
`the noise PDS is flat. Therefore, in subsequent evaluation
`the two remaining integrals in
`
`-IT+
`
`where we assume arg { z}
`
`is small.
`
`5.1 Variance
`It is useful to evaluate the variance of @ E ~ , since this is an im-
`portant measure used to determine the degradation of trans-
`mission quality due to the common phase error. We obtain:
`
`- f l t f n - f k
`
`& = E {@&} - ( E { @ ’ E t } ) 2 z E
`
`J
`-CO
`
`(5)
`
`cos(:! x z ) + 2 x Si(2 x 2) . z - 1
`N - 1
`~-
`+
`10CT
`2 x 2 l o a (fn - fk - f l )
`cos(:! A z’) + 2 x Si(2 x 2’) z’ - 1
`2 x2 10” (fn - f k + f i ) T2 ’
`had to be solved numerically. For brevity, we have defined
`z = n - k - fiT and z’ = n - k + fiT.
`We can assume the worst case when the affected sub-carrier
`k is located in the center of the frequency band, which cor-
`responds to k = N / 2 , and sum over n to finally obtain the
`following bound (equality for the worst case IC = N / 2 ) for the
`IC1 power:
`
`N-1
`
`n=0
`
`n+ $
`One can model this noise as Gaussian distributed, because of
`the large number of contributing sub-carriers (central limit
`theorem). Note that the useful power in the case of no phase-
`noise is equal to 1.
`5 The Common Phase Error
`So far, we have analysed the IC1 due to phase-noise and
`we will now look at the additional perturbations caused by
`phase-noise besides the ICI. We will prove that each sub-
`carrier is affected by a common phase error Q E ~ that is only
`a function of the OFDM symbol index d . Let us consider the
`special case in Fig. 3 a) where n = k. The samples r k , l are
`given by
`
`-IT-
`
`-IT- T
`
`Hence we have shown that the output samples are multiplied
`by a disturbance that is identical for all sub-carriers IC and
`that only varies from one OFDM symbol tjo the next. Since
`the magnitude of the integral in the right half of (7) is close
`
`(9)
`since the phase-noise has zero mean. Because the phase-noise
`process is stationary, we can write the PDS of the common
`phase error as
`L@,(f) = T2 .
`1
`L,,(f)
`
`. sinc2(afT) . T 2 ,
`
`(10)
`
`and
`
`7
`
`2 x
`trQE
`
`L V N ( f ) . sinc2(rfT)df,
`
`(11)
`
`-03
`
`(which is equivalent to setting k = n in (4)). Further evalu-
`ation is possible by inserting IC = n into (5).
`
`5.2 Auto-correlation function
`It is interesting to look at the correlation of the common
`phase error between adjacent OFDM symbols because if the
`correlation is high, feeding back the common phase error to
`correct before the F F T (frequency de-multiplexing) in the
`receiver can help demodulation of future OFDM symbols.
`The auto-correlation function (ACF) of the common phase
`error can be evaluated from the inverse Fourier transform of
`LaE (f), corresponding to
`
`6 Numerical Results
`With equation (5) and (6) the IC1 power P ~ c r could be eval-
`uated numerically. For different numbers of sub-carriers N
`the influence of the phase-noise model parameters a and c on
`the IC1 power is shown in Figs. 4 and 5. The standard devi-
`ation of the common phase error
`for different numbers
`of sub-carriers can be seen in Fig. 6. The calculation of the
`standard deviation of the common phase error was confirmed
`by simulations. With an increasing number of sub-carriers
`
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`Petitioner Sirius XM Radio Inc. - Ex. 1010, p. 3
`
`

`

`corresponding to an increasing OFDM symbol time, uaE de-
`creases. For optimal tuner design, the maximal allowable
`IC1 power should be set several dB below the channel noise
`level, otherwise a BER floor will be the result; the technique
`proposed here can be used in such a trade-off between tuner
`complexity and performance.
`
`Ecruivalent Noise due to IC1 -influence of Darameter a
`
`- 2 6
`
`0.06
`
`0.05
`
`Standard o . O4
`deviation
`of common
`phase
`0 . 0 3
`error
`
`0 . 0 2
`
`0 . 0 1
`
`-28
`
`IC1
`power
`IdBl
`-30
`from
`useful
`signal
`- 3 2
`
`-34 ,’ I
`
`/ I
`
`I
`
`I
`
`I
`
`I
`
`2 0 0 0
`
`4000
`
`10000
`6000
`8000
`Number of carriers N
`
`1 2 0 0 0
`
`Figure 4: Influence of parameter a on the IC1 power, PICI
`for c = 10.5.
`
`c=9.5
`
`-30
`
`IC1
`power
`[dBl
`from
`- 3 2
`useful
`signal
`- 3 4
`
`- 3 6
`
`- 3 8
`
`2 0 0 0
`
`4 0 0 0
`6000
`Number of carriers N
`
`8000
`
`10000
`
`Figure 5: Influence of parameter c on the IC1 power, PICI for
`a = 6.5. If the tuner noise-floor is too high, any high order
`modulation OFDM system will suffer badly.
`
`The auto-correlation function laE (1T) for different FFT
`sizes is shown in Fig. 7. As expected, only for a decreasing
`FFT size, which corresponds to a decreasing OFDM symbol
`duration, the correlation of the common phase error between
`adjacent OFDM symbols increases. That irnplies that only
`for a small number of sub-carriers a feed-back correction of
`the common phase error before the FFT (i.e. a correction of
`the next several OFDM symbols see Fig. 9) could be effective.
`
`7 Eliminating the Common Phase
`Error
`As we have seen above, the auto correlation function of the
`common phase error is small even for shifts of one OFDM
`
`Figure 6: Standard deviation u+E of common phase error
`[radians].
`
`0.0012
`
`0.001
`
`o.oooa
`
`0.0006
`
`ACF
`
`0.0004
`
`0.0002
`
`0
`
`10
`
`ACF of Common Phase Error
`
`,
`
`I
`
`-5
`
`0
`symbol index 1
`
`5
`
`10
`
`Figure 7: ACF laE of the common phase error where the
`parameter is the number of OFDM sub-carriers.
`
`symbol time T when N is large. This means that knowl-
`edge of previous values of
`will not help in estimating
`future values. Fortunately, there exists a simple method of
`estimating @ E and using this estimate to directly correct all
`the sub-channel in OFDM symbol I in a feed-forward struc-
`ture, before demodulation and even further processing such
`as channel estimation.
`We can estimate @ E if we know the transmitted phase
`and channel phase of a set of sub-channels with frequency
`index ci,l where 1 5 i 5 N / and N/ << N is the number
`of such sub-channels used in OFDM symbol 1 -i.e. a symbol
`variant pattern of known phases. The number and position in
`frequency (index c) of these sub-channels may be a function
`of index I . Averaging over the phase error of each such sub-
`channel yields the estimate
`
`N ; C yl,c,,l . 4 i , c , , l
`= i=l
`c
`,
`-
`N1”
`
`(13)
`
`Y l , C , , l
`%=I
`is a reliability estimate -for instance the estimated
`here, ~ l
`, ~ ,
`, ~
`sub-channel amplitude for sub-channel cz,l at symbol I- and
`
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`Petitioner Sirius XM Radio Inc. - Ex. 1010, p. 4
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`

`

`41,c,,l is the phase difference between the transmitted phase
`of sub-channel ca,l and the received phase. The transmit-
`ted phase can be made known to the receiver by designating
`sub-channels ci,l for pilot cells with known phase: a possible
`structure of pilot cells is shown in Fig. 8. Channel estimation
`can be performed using these pilot cells as well, for instance
`with Wiener filters as described in [7].
`
`cj Pilot cells
`
`N-1
`N-2
`N-3
`
`41
`
`21
`
`z i 1
`
`t
`
`data
`
`rei
`I
`
`I i i
`
`Frequency
`
`OFDM-symbol
`
`Figure 8: A possible arrangement of pilot cells
`
`The situation is complicated somewhat if the transmission
`channel introduces a frequency selective phase offset; in this
`case this phase difference must be estimated and subtracted
`from h , c , , l too.
`An illustration of such a scheme can be seen in Fig. 9. It
`should be ensured that the channel estimate used to correct
`c$l,c*,l is reasonably new and accurate, this enables the input
`to the channel estimator to be corrected by ( a ~ , also. Im-
`portant is that there are pilot cells in each OFDM symbol,
`otherwise
`cannot be estimated. Alternatively, one might
`envisage data directed techniques instead of pilot symbols,
`then using the estimate @ E ~ to enhance a second detection
`pro cess.
`
`7.1 Simulation results
`+ Q I C I of the
`=
`In Fig. 10 the total phase errors
`sub-carriers per OFDM symbol are illustrated for a OFDM
`system with a 2k FFT. The common phase error of each
`OFDM symbol can be seen in addition to the phase devia-
`tions in one OFDM symbol due to the IC1 ( @ I ~ I ) . The feed-
`forward method estimates and corrects this common phase
`error for each symbol, what remains are the effects of ICI. As
`expected from Fig. 7 the common phase errors of adjacent
`OFDM symbols are nearly uncorrelated.
`For a system with hierarchical multi-level coding [2] using
`64 &AM, suffering from Ricean fading and employing real
`
`tion
`t,
`
`recent channel phase estimate
`
`channel
`c -
`I
`L
`,/ estimation
`feed-fomard correction
`
`I
`
`Figure 9: A possible implementation of the feed-forward cor-
`rection (the doted lines indicate how a feed-back correction
`might be implemented). Thick lines denote complex signals.
`
`channel estimation, the power degradations due to phase-
`noise with and without feed-forward correction of the com-
`mon phase error for different parameters a are plotted in
`Fig. 11. The OFDM system again has a 2k FFT and the re-
`sults are valid for a BER of 4 .
`(after inner decoding -we
`assume an outer error correcting/detecting block code). The
`simulations include real channel estimation with sub-carrier
`PLL's [8]. It could be seen that if a feed-forward correction of
`the common phase error is performed] the degradation due to
`phase-noise is small. Furthermore] by choosing c 2 10.5 and
`a 2 6.3 there is no visible impact on the residual degradation
`from IC1 for this coding scheme where the channel SNR is
`about 22 dB; this is in accordance with our numerical results
`for the ICI.
`It must be added that without common phase error correc-
`tion the error structure (before channel decoding) is bursty,
`since only comparatively few symbols will suffer from a large
`common phase error. A coded system (assuming no interleav-
`ing across symbols) would suffer severely during the reception
`of such badly affected symbols.
`Finally, the complete BER curve of the above coded OFDM
`system is shown without phase-noise, with phase-noise and
`with phase-noise and feed-forward correction of the common
`phase error in Fig. 12. The good performance with phase-
`noise and feed-forward correction is evident. Comparing the
`operating SNR of roughly 22 dB with Figs. 4 and 5, we can
`conclude that the IC1 is insignificant here.
`
`8 Conclusions
`
`Based on a simple phase-noise model, we have presented
`numerical techniques for calculation of the power of inter-
`
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`

`

`64 QAM, Ricean Channel: a=6.5, b=4, e10.5
`
`10’‘ r ’
`
`-0.20
`
`0
`
`5
`
`10
`
`15
`
`20
`
`25
`
`30
`
`OFDM-Symbol number 1
`Figure 10: Mixed time/frequency representation of phase-
`noise: Within each OFDM symbol we have plotted the total
`phase error @ E of each sub-carrier.
`
`10 ,-
`
`Power efficiency degradation at BER of 4xi04
`
`I
`
`-
`
`6.5
`
`-
`
`=m
`6.6
`
`a
`
`ma
`6.7
`
`6.8
`
`6.3
`
`I
`
`6.4
`
`Figure 11: Degradation due to phase noise and the benefits
`of the feed-forward correction as a function of the phase-noise
`model parameters.
`
`carrier interference from phase-noise in OFDM schemes. The
`analysis will allow the design of low cost tuners that just
`meet the necessary phase-noise requirements and hence en-
`sure satisfactory overall system performance. The most im-
`portant factor -for a fixed phase-noise model and total sys-
`tem bandwidth- governing the IC1 power is the number of
`OFDM sub-carriers. The ICI level increases as the number
`of sub-carriers increases, and will pose a serious problem if,
`for instance, 8k sub-carriers are used in conjunction with high
`order modulation, such as multi-resolution modulation [9].
`Apart from ICI, a common phase error also results in
`degradation of transmission quality, that becomes less severe
`with an increasing number of sub-carriers. To combat this
`common phase error, we proposed a feed-forward correction
`scheme based on pilot cells, that is simple to implement and
`very efficient as simulations have confirmed.
`Further work could be focussed on defining more refined
`phase-noise models and their optimization with respect to
`IC1 and tuner complexity.
`
`20
`
`21
`
`22
`
`24
`23
`Average E, I No
`
`25
`
`26
`
`27
`
`Figure 12: Effects of phase noise on the BER of a typical
`OFDM system.
`
`The authors would like to thank those members within the
`dTTb and HDTV-T projects invclved with coding and mod-
`ulation for useful discussions.
`
`[l] H. Sari, “Channel equalization and carrier synchronization in
`OFDM systems,” in Proc. Sixths Tirrenia InternationaZ Work-
`shop on Communications ’93, September 1993.
`[2] K. Fazel and M. J. Ruf, “A hierarchical (digital HDTV trans-
`mission scheme for terrestrial broadcasting,” in Yroc. GLOBE-
`COM ’93, December 1993.
`[3] Members of Module 3 , “Reducing the effects of phase-noise.’’
`Deliverable, Race-dTTb/WP3.33, October 1994.
`[4] V. Kroupa, “Noise properties of PLL systems,” KEEE Trans.
`Commun., vol. 30, pp. 2244-2252, October 1982;.
`[5] M. Alard and R. Lassalle, “Principles of modulation and chan-
`nel coding for digital broadcasting for mobile receivers,” EB U
`Review, pp. 47-69, August 1987.
`[6] S. Weinstein and P. M. Ebert, “Data transmission by
`frequency-division multiplexing using
`the discrete fourier
`transform,” IEEE Trans. Commun., vol. 19, pp. 628-634, Qc-
`tober 1971.
`[7] P. Hoeher, “TCM on frequency-selective land-mobile fading
`channels,” in Proc. Fifth Tirrenia International Workshop on
`Communications ’92, pp. 317-328, September 1992.
`[8] IS. Fazel, S. Kaiser, P. Robertson, and M. J. Ruf, “A concept of
`a reconfignrable digital-TV/HDTV transmission scheme with
`flexible channel coding & modulation for terrestrial broadcast-
`ing,” in Proc. International Workshop on. H D T V ’94, October
`1994.
`[9] T. Cover, “Broadcast channels,” IEEE Trans. IT, vol. 18,
`pp. 2-14, January 1972.
`
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`Authorized licensed use limited to: Oxford University Libraries. Downloaded on March 31,2010 at 11:30:57 EDT from IEEE Xplore. Restrictions apply.
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`Petitioner Sirius XM Radio Inc. - Ex. 1010, p. 6
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