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`2908
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`IEEE TRANSACTIONS ON COMMUNICWIONS, V a . 42, NO. 10, OCIDBER 1994
`
`Technique for Orthogonal Frequency Division
`Multiplexing Frequency Offset Correcti*on
`Paul H. MQose, Member, IEEE
`
`is being suggested as an efficient modulation for applications
`ranging from modems [4], to digital audio broadcast [5].
`One of the principal advantages of OFDM is its utility
`for transmission at very nearly optimum performance in un-
`equalized channels and in multipath channels. As described in
`[31451, intersymbol interference (ISI) and intercarrim interfer-
`ence (ICI) can be enthly eliminated by the
`of inserting between symbols a small time in
`a guard interval. The length of the guard
`
`symbols do not overlap because of the guard interval, IS1 is
`eliminated, too.
`One of the principal disadvantages of O M is sensitivity to
`
`frequency offset in the channel. For example, the coded OFDM
`system developed by CCETT (Centre Commun. d'Etudes de
`Telediffusion et Telecommuications) for digital sound broad-
`casting to mobile receivers incorporates an AFC (automatic
`frequency control) loop in the receiver to reduce frequency
`offset caused by tuning oscillator inaccuracies and doppler
`shift [6].
`There are two deleterious effects caused by frequency offset;
`one is the reduction of signal amplitude in the output of
`the filters matched to each of the carriers and the second is
`introduction of IC1 from the other carriers which are now no
`longer orthogonal to the filter. Because, in OFDM, the carriers
`are inherently closely spaced in frequency compared to the
`channel bandwidth, the tolerable frequency offset becomes
`a very small fraction of the channel bandwidth. Maintaining
`sufficient open loop frequency accuracy can become difficult
`in links, such as satellite links with multiple frequency transla-
`tions or, as mentioned previously, in mobile digital radio links
`that may also introduce significant Doppler shift. The effects
`of frequency offset are presented in Section II.
`In Section 111, we present an algorithm to estimate fresuency
`offset from the demodulated data signals in the receiver. The
`algorithm extends to OFDM, with important differences, a
`method described by Simon and Divsalar [7] for single carrier
`MPSK. The technique involves repetition of a data symbol
`and comparison of the phases of each of the carriers between
`the successive symbols. Since the modulation phase values are
`not changed, the phase shift of each of the carriers between
`successive repeated symbols is due to the frequency offset.
`The frequency offset is estimated using a maximum likelihood
`estimate (MLE) algorithm. Performance of the algorithm as
`
`I. INTRODUCTION
`HE TECHNIQUE described in this paper has been de-
`
`T veloped to correct frequency offset errors in digital com-
`
`munications systems employing orthogonal frequency division
`multiplexing (OFDM) as the method of modulation. The aim
`of the paper is twofold; to show the effect offset errors have on
`the signal-to-noise ratio of the OFDM carriers and to present
`an algorithm to estimate offset so that it may be removed
`prior to demodulhhn.
`OFDM is a bandwidth efficient signalling scheme for dig-
`ital communications that was first proposed by Chang [l].
`The main difference bepeen frequency division multiplexing
`(FDM) and OFDM, is that in OFDM the spectrum of the
`individual carriers mutually overlap, giving therefore an opti-
`mun spectrum efficiency (asymptotically Q b/Hz for 2Q-ary
`modulation of each carrier). Nevertheless, the OFDM carriers
`exhibit orthogonality on a symbol interval if synthesized such
`that they are spaced in frequency exactly at the reciprocal
`of the symbol interval. Fortunately, this synthesis can be
`accomplished perfectly, in principle, utilizing the discrete
`Fourier ttansform (dft) as first described by Darlington [2] and
`later, for data modems, by Weinstein and Ebert [3]. With the
`recent evolution of integrated circuit digital signal processing
`(dsp) chips, OFDM has become practical to implement and
`Paper approved by M. J. Joindot, the Editor for Radio Communications of
`the IEFE Comtnunications Society. Manuscript received November 27,1991;
`revised June 29,1992.
`The author is with the Electrical and Computer Engineering Department,
`Naval FWgraduate School. Monteaey, CA 93943. USA.
`EEE Log Number 9401947.
`
`0090-6778/94$04.00 0 1994 IEEE
`
`Petitioner Sirius XM Radio Inc. - Ex. 1011, p. 1
`
`

`

`MOOSE: OFDM FREQUENCY OFFSET CORRECTION
`
`2909
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`a function EJN, (individual OFDM carrier energy to one-
`sided. spectral density of additive white Gaussian noise) and
`in Section III.
`frequency offset is @lu&d
`In the event that the frequency offset exceeds f1/2 the
`i n t e d e r spacing, the maximum limits of the algorithm, a
`strategy is requkd for initial acquisition. One such strategy
`is described in Section IV.
`
`H. OFDM ”SLATION IN A
`CHANNEL WITH FREQUENCY OFFSET
`An OFDM transmission symbol is given by the N point
`complex modulation sequence
`
`k=-K
`n = 0 , 1 , 2 ,... N - 1 ; N L 2 K + 1 . (1)
`It consists of 2K + 1 complex sinusoids which have been
`modulated with 2K + 1 complex modulation values { X k } .
`We note that the iadividual sinusoids are orthogonal on the
`symbol interval, that is
`
`N - 1
`
`= ( 1 / N ) ) X k ) 2 a k l
`
`(2)
`
`n=O
`where Z n k = ( 1 / N ) X k e 2 * j n k / ” ’ .
`We also note that the N point discrete Fourier transform
`(dft) of (1) is ,the N point sequence
`D m N (xn}
`
`= {XO, X I , . - X K , 0, 0,. -. 0, 0, x-K . . . x--2, X - l }
`(3)
`of modulation values. Equation (1) is the inverse discrete
`Fourier transform (IDFT) of (3) and defines a practical
`modulation-carrier synthesis technique for generating OFDM
`with perfect orthogonality.
`After passing through a bandpass channel, the complex
`envelope of the received q e n c e can be expressed as
`
`n = 0 , 1 , 2 ,..., N - 1
`(4)
`where H k is the transfer function of the channel at the
`fresueacy of the kth carrier, E is the relative frequency offset
`of the cbilmel (the ratio of the actual frequency offset to
`t h e i n k ” ‘er sjmcing), and ‘UIn is the complex envelope of
`additive white Gaussian noise (AWGN). Let the actual symbol
`transmitted be the N + Ng point sequence
`
`(5)
`{ X N - - N g , . . - r x N - 2 , Z N - 1 , 2 0 , Z l , . . - , ~ N - l }
`with Ng greater than or equal to the time spread of the
`channel. The Ng point p m o r signal allows the received
`symbol sequence to reach steady state by n = 0 (we assume
`
`synchronization at this stage of the receiver) leading to a
`received sequence as given by (4). It is assumed that the
`impulse response of the channel does not change (much)
`during the symbol plus guaTd interval (this corresponds to
`“slow-fading” in a radio frequency channel).
`The insertion of guard intervals renders the received carriers
`orthogonal on the N point symbol interval. However, the
`demodulation process, which is implemented with a dft (the
`dft is equivalent to matched filter reception in the absence of
`frequency offset) is affected by frequency offset. That is,
`
`N-I
`
`n=O
`the kth element of the dft sequence, consists of three com-
`ponents;
`
`The first component is the modulation value Xk modified
`by the channel transfer function. This component experiences
`an amplitude reduction and phase shift due to the frequency
`offset. Since N is always much greater than TE, N sin ( m / N )
`may be replaced by m.
`The second term is the IC1 caused by the frequency offset
`and is given by
`(XIH~){(sin7FE)/(Nsin(n(l- k + E ) / N ) ) )
`. , j m ( N - 1 ) / N e - j * ( 1 - k ) / N
`
`K
`
`I k =
`
`I = - K
`I f k
`
`(8)
`
`In order to evaluate the statistical properties of the ICI, some
`further assumptions are necessary. Specifically, it will be
`aSSUmed that E[X&] = 0 and E[X&X;] = IxI2bl&, that is,
`the modulation values have zero mean and are uncorrelated.
`With this provision E[&] = 0, and
`K
`
`I = - K
`l # k
`
`-{sin
`
`/ { N sin (.(a - IC -I- C ) / N ) } ~ . (9)
`
`The average channel gain, E { l H l I 2 } = [HI2, is constant so it
`can be separated from the sum and (9) becomes
`
`E [ I I k 1 2 ] = IX121H12{sinnE}2
`K - k
`
`.
`
`1/{Nsin(a(p+e)/N))2. (10)
`
`p = - K - k
`P#O
`The sum in (10) can be bounded for E = 0. It consists of 2 K
`positive terms. The interval of the sum is contained within the
`longer interval -2K 5 p 5 2K, its looation dependent on k.
`Recall that 2K 5 N- 1. Also note the following; the argument
`of the sum is periodic with period N, it is an even function of
`p , and it is even about p = N/2. Thus the 2K terms of the sum
`
`Petitioner Sirius XM Radio Inc. - Ex. 1011, p. 2
`
`

`

`2910
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`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 10, OCIOBER 1994
`
`m. m Q U E N C Y OFFSET ESTIMATION
`If an OFDM transmission symbol is repeated, one receives,
`ih the absence of noise, the 2N point sequence
`
`I K
`
`k=-K
`
`rn = (1/N)
`
`I
`
`Expcrimnt: ++++++
`Thwry (Lowrr BoMd): sdidlinca
`
`am 0.1
`ais
`a s
`a3 0.35
`a2
`5~
`Fig. 1. SNR versus relative frequency offset.
`
`0.4
`
`0.4s
`
`0.5
`
`are a subset of the N terms in the intervals -N/2 5 p 5 -1
`and 1 5 p 5 N/2 for every k. Consequently,
`
`Observe that (sin?rp/N)” 2 (2p/N)2 for lpl 5 N/2. There-
`fore,
`
`N f 2
`2 C 1 / ( N sin (rP/”2
`p=l
`
`XkHkeznjn‘k+e)lN i
`= 0, 1 ,..., 2 N - 1. (16)
`The kth element of the N point dft of the first N points of
`(16) is
`Rlk = xrne-zzjnk/N; k = 0, 1, 2,. . . , N - 1, (17)
`
`N-1
`
`rne-2xjnklN
`
`n=O
`and the kth element of the dft of the second half of the
`sequence is
`2N-1
`R~~ =
`n=N
`N -1
`rn+Ne-2”jnkIN; k = 0, 1, I .
`=
`n=O
`But from (16),
`rn+N = rneznje --f R2k = R1keznj‘.
`Including the AWGN one obtains
`
`. , N - 1. (18)
`
`(19)
`
`N / 2
`
`(12)
`
`Observe that between the first and second DWs, both the
`IC1 and the signal are altered in exactly the same way, by
`a phase shift proportional to frequency offset. Therefore, if
`offset E is estimated using observations (20) it is possible to
`obtain accurate estimates even when the offset is too large for
`satisfactory data demodulation.
`It is shown in the Appendix that the maximum likelihood
`estimate (MLE) of E is given by
`; = (1/27r)tan-l
`
`l o o
`< 2c1/(2p)’ < - c l / p 2 = 2 / 1 2 = 0.882
`p=l
`Zp=1
`upper bounds the sum for small E. Numerically, we have
`determined that the sum in (10) is bounded by 0.5947 for
`E < 0.5 so that
`E[11k12] 5 0.59471X121H12(~in~~)2; 161 5 0.5
`(13)
`upper bounds the variance of the IC1 for values of frequency
`offset up to plus or minus one half the carrier spacing.
`Equation (13) may be used to give a lower bound for the
`S N R at the output of the dft for the OFDM carriers in a channel
`with AWGN and frequency offset. Thus,
`This is an intuitively satisfying result since, in the absence of
`SNR 2 1x121H12{sin?rE/(TE)}2/
`{0.5947)X121H12(sinx€)2 + E[Iwk12]). (14)
`for each k. Fig. 2 shows
`noise, the angle of Y2kYik is ~ A E
`simulation results for the estimate of E obtained using (21)
`versus E for values of E,/No corresponding to 17 and 5 dB.
`= &/No
`It is easily established that ~ x ~ 2 ~ ~ ~ 2 / ~ [ ~ ~ k ~ 2 ]
`
`where E, is the averaged received energy of the individual
`carriers and N0/2 is the power spectral density of the AWGN
`in the bandpass transmission channel. Therefore, (14) may be
`more conveniently expressed as
`L { ~ c / ~ o ) { s ~ ~ E / ( ~ E ) } 2 /
`(1 + 0.5947(E,/N,)(~inm)~} (15)
`with equality at E = 0. Quation (15) is plotted in Fig. 1 versus
`fresuency offset E for values of E,/N, equal to 11, 17, 23,
`and 29 dB. Simulation results for N = 256, K = 96, and 8
`PSK modulation in a flat channel are included for validation.
`It can be seen that the bound of (15) is quite tight for small
`values of E but is about 3 dB too low at E = 0.5.
`
`A. Statistical Properties of the Estimate
`The conditional mean and variance of i given E and {&}
`can be approximated as follows. Consider the complex prod-
`ucts Y2kYck from which we estimate E. For a given E, subtract
`the corresponding phase, ~ K E , from each product to obtain the
`tangent of the phase error
`K
`
`tan [2?r(i - E)] =
`
`Petitioner Sirius XM Radio Inc. - Ex. 1011, p. 3
`
`

`

`MOOSE O€?DM FREQUENCY OFFSET CORRECTION
`
`2911
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`0.5
`
`ai 0.15
`o.os
`a4 0.6 a5
`0.3
`'0
`a2
`0.25
`a35
`Fig. 2. MLE relative frequency offset estimate versus relative frequency
`Offset
`
`For 12 - E ( << 1/2n, ?he tangent can be approximated by its
`argument so that
`i - E "N (1/27r)
`
`(23)
`At high signal-to-noise ratios, a condition compatible with suc-
`cessful communikations signaling, (23) may be approximated
`bY
`I
`2 - E M (1/27$ I
`
`/ K
`
`\ \
`
`from which we find that
`E[; - € I €, {Rk}] = 0.
`(25)
`Therefore, for small errors, the estimate is conditionally un-
`biased.
`Tk conditional variance of the estimate is easily determined
`for (24).
`
`vu[; 1 % {Rk}] = 1/{(2r)2(Es/No)}
`
`(26)
`
`where
`
`N-I
`
`n=O
`is the total symbol energy. Since the total energy is the sum
`of the energies of the 2K + 1 carriers, the error variance
`of the offset estimate will in practice be very low. Fig. 3
`shows the sample standard deviation of the error in the relative
`offset estimate for 100 simulation trials of (21)
`f " c y
`versus E,/N, for 2K + 1 =193 carriers and for offsets of
`
`Fig. 3. MLE relative frequency offset estimate error standard deviation
`versus EsINo.
`
`E = 0 and 0.45. The theoretical standard deviation from (25) is
`plotted for comparison. We conclude that (21) will give very
`accurate estimates of the relative frequency offset E. Under
`normal conditions for communications signalling, the accuracy
`is sufficient to correct E to well within tolerances (see (14)
`and Fig. 1) for negligible losses due any residual error in the
`offset estimate.
`The limits of accurate estimation by (21) are 161 10.5, that
`is, f112 the intercarrier spacing. As E 40.5, 2 may, due to
`noise and the discontinuity of the arctangent, jump to -0.5.
`When this happens the estimate is no longer unbiased and,
`in practice, it becomes useless. Thus, for frequency offsets
`exceeding one half the carrier spacing, an initial acquisition
`strategy must be prescribed. One such strategy is discussed in
`Section IV.
`
`B. Frequency W s e t Estimution in a Multipath Channel
`It is evident from (25) and (26) that the mean and variance
`of the offset estimate do not depend on the actual received
`frequency coefficients {Rk}. Furthermore, if the symbol pair
`has been received through an unknown multipath channel, and
`as described in Section I it has been preceded by a periodic
`precursor of length Ng 2 N, the time spread of the channel
`then the carriers remain at their steady state values throughout
`the duration of both symbols because the modulation values
`are repeated. Thus, as no guard interval is required between
`the symbol pair, the algorithm of (21) can be used without
`modification.
`Fig. 4 shows six amplitude responses of a channel with five
`multipaths whose arrival times have been uniformly distributed
`over an interval T, = T/16. The paths have equal weight and
`random phases so that the overall channel exhibits frequency
`selective Rayleigh fading as is evident b m the figure. Fig. 5
`shows estimates of E from (21) with Ng = N/16 for the
`same conditions as Fig. 2. It can be seen that the estimate is
`unaffected by the multipath.
`
`IV. ACQUISITION
`In the event that the frequency offset is greater than f1/2
`the carrier spacing, a strategy for initial acquisition to bring
`the offset within the limits of the algorithm must be developed.
`
`Petitioner Sirius XM Radio Inc. - Ex. 1011, p. 4
`
`

`

`I
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`2912
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`10
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`Fig.4. Multipathf
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`'r/
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`0 5 ,
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`035
`a3
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`au
`ai
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`-50
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`I
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`0
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`. ion channel.
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`50
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`100
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`IEEE TRANSACTIONS ON C O M M U " S , VOL. 42, NO. 10, OCTOBEX 1994
`
`1
`
`I
`150
`
`I
`
`repeated symbols by a repetition of the fist full length data
`symboi or by the use of an AFC loop as shown in [6].
`To illustrate, consider the following example for a digital
`audio broadcasting service. Assume an intercarrier spacing for
`the data stream of 1 kHz, and a frequency offset uncertainty
`in the system dominated by the long term accuracy of the
`oscillators in the receiver that heterodyne the received signal to
`IF and quadrature demodulate to obtain the complex envelope.
`Assume VHF radio transmission at 150 MHz and overall
`oscillator uncertainty (long term stability) of 1 part in lo5.
`Thus, S,,
`= 1500 Hz and Afinitial must be greater than 3000
`Hz. At regular intervals in the data stream of 1 ms (plus guard
`interval) symbols insert a short symbol of length 250 ps (4
`kHz carrier spacing) and repeat it once. From this repeated
`shortened symbol estimate E. Assume Ec/No is 11 dB for the
`data symbols and that there are 200 carriers, so that Es/No is
`34 d33. The shortened symbols have 114th energy so Es/No is
`only 28 dB for the initial estimate of E . From (25) we find that
`Uainitial = 0.0063 SO that Uedata = 0.025. From (15) we See
`that residual offset errors of this magnitude cause IC1 resulting
`in a signal-to-interference ratio of 24.3 dB in the data symbols
`for only 0.2 dB loss in overall S N R at &/No of 11 dB.
`This example illustrates a situation for which the initial
`offset acquisition estimate with shortened data symbols is of
`sufficient accuracy that refinement of the estimate with longer
`symbols is not necessary.
`
`Fi 5. MLE relative frequency offset estimate versus relative frequency
`in a multipatb channel.
`o&
`
`We envision that, if continuous, the OFDM symbol stream will
`be punctuated at appropriate intervals with repeated symbols.
`A continuous symbol stream occurs in applications such as
`digital audio broadcast [53. A second possibility is that OFDM
`modulation is used in session oriented digital data or voice
`communications such as digital radio [8]. Here, we envision
`that the session initiation interval will include one or more
`lepemzd symbols.
`The basic strategy for initial fresuency offset acquisition,
`in &her case, is to shorten the dft's and use larger carrier
`spacings such that the phase shift does not exceed f ? r . The
`fkqtwncy offset in Hz is S = c/T = EA f where A f is the
`intercarrier spacing and T is the symbol interval. Let us assume
`that the initial frequency offset is no greater than *Smm. Then
`
`determines the minimum initial carrier spacing, and corre-
`sponding dft lengths. If the average power of the shortened
`symbols is kept the same, the variance of the estimate of €initial
`will'be larger than for the longer data symbols since there
`is less symbol energy. Also, the offset estimate error for the
`shortened symbols, since it estimates the fraction of carrier
`spacing, corresponds to a proportionately larger fractional
`offwt for the longer data symbols. However, the MLE estimate
`is sa accurate that in practice the initial estimate still may
`be adequate. If not, it is refined by following the shortened
`
`T:
`
`V. DISCUSSION AND CONCLUSIONS
`We have seen that, as expected, frequency offset in OFDM
`causes serious loss of S N R of the dft outputs due primarily to
`ICI. A lower bound for S N R has been derived and simulation
`results show that the bound is quite accurate for small offsets,
`but about 3 dB too pessimistic as the offset approaches 1/2
`the carrier spacing.
`An algorithm for maximum likelihd estimate
`of frequency offset using the dft values of a repeated
`symbol has been presented. It has been shown that for small
`error in the estimate, the estimate is conditionally unbiased
`and is consistent in the sense that the variance is inversely
`proportional to the number of carriers in the OFDM signal.
`Furthermore, both the signal values and the IC1 contribute
`coherently to the estimate so that it is possible to obtain very
`accurate estimates even when the offset is too great, that is
`there is too much ICI, to demodulate the data values. Since
`the estimation error depends only on total symbol energy, the
`algorithm works equally well in multipath spread channels.
`However, it is required that the frequency offset as well as
`the channel impulse response be constant for a period of two
`symbols.
`The accuracy required of frequency offset correction de-
`pends on how much residual offset can be tolerated. Offset
`induced IC1 can be treated quite satisfactorily as additional
`AWGN since its source is the multiplicity of other OFDM
`carriers that are zero mean and uncorrelated random processes.
`Note W the SNR defined in Section 11 [see (14) and (15)]
`is just &/No in the absence of offset. Thus we may interpret
`(15) as the effective &/No of the carriers, or if divided
`
`Petitioner Sirius XM Radio Inc. - Ex. 1011, p. 5
`
`

`

`by tbe number of bits encoded in each of the carriers, their
`effective Eb/No. Required &,/No of course depends qxm the
`maddation constellation, the fading statistics of the chamel
`the forward error control coding, if any, employed in the
`OPDM system and the desired BER (see, for example, [5,
`Figs. 11-13]).
`Tbe acquisition range of the algorithm presented here is
`f1/2 the intercarrier spacing of the repeated symbol. It is
`independeat of the modulation constellations chosen for the
`carriers and whether the symbols are coherently or differen-
`tially encoded. The AFC loop shown in [6] does not require a
`repeated symbol. However its acquisition range is only f1/2m
`of the intercarrier spacing for m-ary PSK. The initial frequency
`offset at the time of the initiation of the communication session
`may be greater than 1/2 the intercarrier spacing and thus
`even outside the range of the MLE algorithm. In this event,
`a strategy is required for initial acquisition. We propose to
`use a pair of shortened data symbols whose carrier spacing
`is sufticiently large to insure that the algorithm will operate
`within its range. Due to the low variance of the initial estimate,
`fiuther refinement will normally not be required. It may be
`advantageous to use shortened repeated symbols for tracking
`offset variations too, instead of an AFC loop, because this
`reduces the time during which the channel must be stable.
`
`2 2 1 * * . ZMI]
`
`where
`
`(A.1)
`
`Therefore,
`
`(A.2)
`(A.3)
`
`APPENDIX
`MAXIMUM LIKELIHOOD ESTIMATE OF DIFFERENTIAL PHASE
`Let M complex values {Zk} be represented by a length
`2M row vector
`Z = [ZIR Z ~ R
`* . ZMR Zir
`= [ Z R ZZ].
`Consider the random vectors
`Y1 =R1 +Wl
`Y2 = R l H ( 6 ) + W2
`W ) = [.
`
`2913
`
`so that
`
`To find the conditional density function in (A.8), note that
`Y2 = (Y1- W l ) H ( @ ) + W2
`(A.9)
`Y2 = Y i H ( 8 ) + W2 - W i H ( 8 ) .
`(A.10)
`If W1 and W2 are Gaussian, zero mean white random
`vectors with variance u2, then the conditional density function
`in (A.6) is multivariate Gaussian with mean value vector
`Y l H ( 8 ) and 2M x 2M covariance matrix
`K = E[(W2 - W l H ( Q ) ) t ( W 2 - w , H ( e ) ) ] = 2 2 1 .
`(A. 11)
`We note that K is independent of 8, therefore,
`6 = megx[f(y2 I 8, Y,)] = m $ [ ~ ( e ) J
`
`(A. 12)
`
`with
`
`J(@) = (Y2 - Y i H ( @ ) ) ( Y 2 - Y I H ( B ) ) ~ . (A.13)
`Using the fact that
`H ( 8 ) [ d H ( 8 ) / d 8 ] t + [ d H ( 8 ) / d 8 ] H t ( 8 ) = 0
`we can find that
`dJ(B)/dQ = -Y2[dH(Q)/d8ltYt - Yl[dH(O)/dQ]Yd.
`(A. 15)
`Using (A.4), it follows directly that (A.15) is identically zero
`when 8 = Q such that
`sin ( ~ ) F ~ R Y ~ R + Y ~ I Y ~ , , ] = COS ( 6 ) [ ~ 2 r y 4 R - Y~RY,~,I.
`
`(A.14)
`
`(A. 16)
`
`(A. 17)
`
`is the maximum likelihood estimate (MLE) of 0.
`
`ACKNOWLEDGMENT
`The author would like to thank Dr. S. Pupolin of University
`of Padova for his helpful suggestions to the original manu-
`script. The author is also indebted to the anonymous reviewers
`whose constructive remarks have improved the quality of this
`Paper.
`
`REFERENCES
`[I] R. W. Chang, “Synthesis of band-limited orthogonal signals for multi-
`channel data transmission,” Bell Syst. Tech. J., vol. 45, pp. 1775-1796.
`Dec. 1%.
`[2] S. Darling, “On digital single-sideband modulators,’’ IEEE Truns. Circuit
`Theoty, V O ~ . CT-17, W. 409-414, Aug. 1970.
`[3] S. B. Weinstein and P. M. Ebert, “Data transmission by frequency-
`division multiplexing using the discrcte Fourier transform,” IEEE Trans.
`C o m n . Technol., vol. COM-19, pp. 62-34.
`Oct. 1971.
`[4] J. A. C. Bingham, ‘’Multicarrier modulation for data transmission: An
`idea whose time has come,” ZEEE Commun. Mug., vol. 28, pp. 17-25,
`Mar. 1990.
`
`4
`
`1
`
` c=COS(Q)I & S = sin(8)I
`04.4)
`is a 2M x 2M rotation matrix. The maximum likelihood
`estimate of the parameter 8, given the observations Y 1 and
`Y2 (see, for example, Sage and Melsa, [9, p. 1961) is the value
`of 8 that maximizes the conditional joint density function of
`the observations. That is
`6 = m a [ f ( Y i ,
`Y2 I Q)]
`8
`which can be written as
`e = m e [ f ( Y 2 I Q, YI)f(Yl I 811.
`But 9 gives no information about Y 1 , that is
`f W 1 I Q) = W l >
`
`(A.5)
`
`(A.6)
`
`(A.7)
`
`so that
`
`Q = m g W 2 I 8, Y l ) ] .
`
`(A.8)
`
`Petitioner Sirius XM Radio Inc. - Ex. 1011, p. 6
`
`

`

`pJr-----------
`
`--__
`
`~
`
`~ _ _
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`- - _ ”
`
`- __
`
`-
`
`__
`
`1
`
`j
`
`2914
`
`IEEE TRANSACTIONS ON COMMUNICATIONS. VOL. 42, NO. 10, OClDBER 1994
`
`[SI M. Nard and R Halbert, “Principles of modulation and channel coding
`for digital broadcasting for mobile receivers,” EBU Rev., no. 224, pp.
`3-25, Aug. 1989.
`[6] B. LeFloch, R. Halbert-Lassalle, and D. Castelain, “Digital sound
`bmdcashg to mbde receivers,” IEEE Trans. Conswn Elec.. vol. 35.
`no. 3, pp. 493-503, Aug. 1989.
`[fl M. K. Simon and D. Divsalar, “Doppleramted diffmntial detection
`of MPSK,” IEEE Tram. Commun.. vol. 37, no. 2, pp. 99-109, Feb. 1989.
`[8] D. C. Cox, W. S. M o d , and H. Sherry, “Low-power digital radigas a
`ubiphus subscriber loop,” IEEE Commun. Mag., vol. 29, pp. 92-95,
`Mar. 1991.
`[9] A. P. Sage and J. L. Melsa, Esrimarion Theory with Applications to
`Communications nnd C~nhOl. NCW YO& MCCraw-Hill, 1971.
`
`Paul M. Moose (M’79) was born in Omaha, NE,
`on July 22. 1938. He received the B.S., M.S., and
`Ph.D. d e w s in electrical engineering in 1960,
`1966. and 1970, respectively, from the University
`of Washington. Seattle.
`Since 1977 he has been with the Department of
`Electrical and Computer Engineering. Naval Post-
`graduate School, Monterey, CA, where he is an
`As&
`Professor of Ekctrical and Computer En-
`gineeaing. His research inters& are in digital com-
`munications. He is a -founder of and consultant
`Digital Co-~ni~ationS hc., Monterey, CA.
`for ~ a ~ r
`y
`
`
`Petitioner Sirius XM Radio Inc. - Ex. 1011, p. 7
`
`