ISSN O280—53 16
`
`ISRN LUTFD2/TFRT—-5612--SE
`
`Synchronization in ADSL
`Modems
`
`Mikael Cordes
`
`Andreas Johansson
`
`Department of Automatic Control
`
`Lund Institute of Technology
`December 1998
`
`Dish
`Exhibit 1040, Page 1
`
`

`
`Department of Automatic Control
`Lund Institute of Technology
`BOX 118
`_
`d
`S 221 00 Lund SW6 en
`Aut12ar(s)
`Mikael Cordes
`Andreas Johansson
`
`i Document name
`l MASTER THESIS
`Dare ofissue
`December 1998
`Document Nunzber
`ISRN LUTFD2/TFRT--5612--SE
`Supervisor
`Bjorn Wittenmark
`Rob Evans, Iain Collings, Melbourne Univ.
`
`Spansazing organise tion
`
`Title and subtitle
`
`Synchronization in ADSL modems (Synkroniseiingi ADSL modem)
`
`Abstract
`
`We present two methods for timing recovery in ADSL-modems. Both techniques, the phase-lock loop and
`the digital phase correction technique, use the pilot tone in the ADSL-signal. The first has a variable
`receiver sample clock that is tuned to the transmitter sample clock. The latter, also known as the rotor
`technique, has a non-adjustable receiver clock, but compensates the arising phase errors by rotations of
`the received QAM-constellation points. We also present a method to achieve frame synchronization. The
`synchronization frame that is sent every 69th frame is of crucial importance for this method. Simulations
`show that these synchronization techniques work well in the presence of noise on twisted pair copper
`wires.
`
`Key words
`timing recovery, synchronization, ADSL
`
`Classzlfica tion system and/or index terms (ifany)
`
`
`Supplemen tazy bibliographical i1n"orma tion
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`ISSN and key title
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`Secuzity classification
`
`Number ofpages
`54
`
`Recipien :2 notes
`
`The report may be ordered from the Department ofAutomatic Control or borrowed through:
`University Library 2, Box 3, S-221 00 Lund, Sweden
`Fax +46‘ 46' 222 44 22
`E-mail ub2@ub2.1u.5e
`
`Dish
`Exhibit 1040, Page 2
`
`

`
`Preface
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`First of all we would like to thank our supervisor prof. Bjorn Wittenmark who initiated our
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`contact with the University of Melbourne, Australia. We would also like to thank our supervisors
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`prof. Rob Evans and Dr. Iain Collings at the Dept. of Elec. 85 Electr. Engineering, University of
`
`Melbourne. The teleconnnunication service provider Telstra made our work possible by funding
`
`the work on ADSL at the Dept. of Elec. & Electr. Engineering.
`
`Finally we would like to thank all our friends from Melbourne which made our stay abroad really
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`enjoyable.
`
`Dish
`Exhibit 1040, Page 3
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`

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`Contents
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`1
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`Introduction
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`2 ADSL Modem Principles
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`2.1 DMT for ADSL Modems .
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`2.2 Frequency Allocation in ADSL Modems
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`2.3 ADSL Transniitter
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`2 .4 ADSL Receiver
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`3 Timing Recovery
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`3.1 Pilot Tone .
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`3.2 Timing Error
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`3.3 Phase Lock Loop .
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`3.3.1 Controller .
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`3.3.2 Voltage Controlled Crystal Oscillator .
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`3.3.3
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`Phase Mea.sure1nent
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`3.3.4
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`l\/latheiiiatical Model
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`3.4 Rotor Technique
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`3.4.1
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`System Model .
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`4 Timing Recovery Simulations
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`4.1 PLLSimula.tions
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`4.1.1
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`Pl-Controller Design .
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`4.1.2
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`RST—Co11troller Design .
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`4.1.3
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`PLL Perforincuice .
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`Dish
`Exhibit 1040, Page 4
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`

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`4.2 Rotor Simulations
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`4.2.1 Rotor Technique Performance .
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`4.3 Comparison .
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`25
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`26
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`Frame Synchronization
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`5.1 Statistical Properties of the Signals .
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`5.2 Maximum Likelihood Detector
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`5.3 A Criterion for Selecting the Beginning of a Frame .
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`5.3.1 Example of a Threshold .
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`5.4 Frame Position Averaging .
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`Frame Synchronization Simulations
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`6.1 Channel Modeled by a Kronecker Delta Function .
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`6.1.1 Cyclic Prefix for a Kronecker Delta Function .
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`6.1.2
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`VVhole Reverb Signal for a Kronecker Delta Function .
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`6.2 Realistic Channel Model
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`6.2.1 Cyclic Prefix for a Realistic Channel
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`6.2.2 Whole Signal for a Realistic Channel .
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`6.3 A Realistic Scenario
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`7 Conclusions
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`Dish
`Exhibit 1040, Page 5
`
`

`
`1
`
`Introduction
`
`Recently engineers have been attempting to extend the capabilities of existing telephone lines
`
`to achieve faster data communication. A technology called asymmetric digital subscriber line
`
`(ADSL) has been suggested for high speed data communication over the installed twisted pair
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`copper network, which was originally designed for the ”plain old telephone service” (POTS).
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`One of the main a.dvantages of ADSL compared to other techniques with similar performance
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`and functionality, such as coaxial cable, is that the twisted pair copper connections already exist.
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`Although the same wire is used, the ADSL bandwidth will be much higher than the bandwidth
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`used by current voiceband modems.
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`An ADSL connection consists of two information channels 4 a high speed downstream channel
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`with rates up to 8 Mbps and a duplex channel with rates up to 640 kbps. The high data
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`rates combined with cross talk, noise and the distorting effects of the channel makes the frame
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`synchronization and the timing recovery two challenging problems to solve.
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`In this paper we
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`describe solutions to these problems. It is shown that to achieve the high data rates, reliable
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`and robust timing recovery and frame synchronization must be carried out.
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`A brief overview covering the basic ideas and principles used in ADSL modems is given below.
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`This is followed by a thorough discussion on timing recovery and frame synchronization.
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`Section 2 discusses the basic principles and ideas behind ADSL modern design with particular
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`emphasis on multitone modulation and frequency allocation. This section also covers the physical
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`properties of the twisted pair copper subscriber loops.
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`Two different methods for timing recovery are discussed and evaluated in Section 3. The first
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`method, based on a phase—locl<ed loop, uses a receiver sample clock that is adjustable in fre-
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`quency. The second method, known as the rotor technique, corrects the timing error digitally.
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`The two techniques are simulated and compared in Section 4.
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`In Section 5, Frame Synchronization, a maximum likelihood detector is derived for detecting
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`the beginning of the frames. The statistical properties of this detector are derived theoretically.
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`VVith the help of the statistical properties a suitable criterion in the shape of a threshold is used
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`to determine the beginning of the frames. This frame synchronization algorithm is simulated in
`Section 6.
`
`Dish
`Exhibit 1040, Page 6
`
`

`
`2 ADSL Modem Principles
`
`An ADSL network consists basically of an ADSL Transmission Unit — Central ollice (ATU-C)
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`and on the customer side an ADSL Transmission Unit - Remote (ATU—R). These two units are
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`physically connected via the twisted pair copper cables. Since ADSL and POTS share the same
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`physical connection they need to be separated using a device called splitter. A low—pass filter is
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`used to pass signals to POTS and a high—pass filter passes on ADSL signals at roughly 25 kHz
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`and above. VVhen the POTS signal has been split from the ADSL signal it goes to the telephone
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`at the customer side and to the public switched telephone network (PSTN) at the network end.
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`The premises distribution network (PDN) on the customer‘ side is used to separate the incoming
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`ADSL signal. One of the benefits with ADSL modems is that POTS is maintained even when
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`the modem is used. The basic blocks are shown in Figure 1.
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`Some typical ADSL applications are Internet access and video on demand.
`
`2.1 Dl\/IT for ADSL lvlodems
`
`A channel can be thought of as a set of independent, parallel subchannels. This is the supposition
`
`that is made for the modulation technique used for ADSL modems which use Discrete Multitone
`
`(J_)l\/IT) modulation. There are ADSL modems that use difi’erent modulation schemes, such as
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`Carrierless Amplitude Modulation (CAP) and Quadrature Amplitude Modulation (QAM), but
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`these modulation techniques are not discussed here since DMT was selected by the American
`
`National Standards Institute (ANSI) to be the ADSL standard modulation technique. The
`
`division of the channel into a set of subchannels is called channel partitioning. To achieve
`
`the desired channel partitioning, a set of N orthogonal basis functions is used. These basis
`
`functions are regarded as orthogonal after the transmission of data through a channel with
`
`transfer function |H( and with additive White Gaussian noise (AWGN). A simplified block
`
`Dish
`Exhibit 1040, Page 7
`
`

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`
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`
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`diagram of a continuos transmitter and receiver is shown in Figure 2.
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`n(t)
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`Figure 2: Simplified block diagram of a multitone transceiver system.
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`DMT is a multicarrier modulation technique that uses several different orthogonal carriers. The
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`orthogonality between the different carriers is expressed as
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`
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`1Xl f ( i,
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`t) f (j, t)
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`xo
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`
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`functions that are used, x0 and x1 are the modulation are the different where f (i, t) and j(j, t)
`when i = j and O otherwise.
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`which is equal to 1 and the end of one period and Oij is a Kronecker delta function
`beginning
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`
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`This relationship is true for the different carriers that are used in the
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`(1)
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`DMT modulation for ADSL. The carriers used in DMT are cyclic functions which use different
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`center frequencies to obtain the orthogonality that is needed for maximum performance. Figure
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`
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`3 shows an example of two carriers that are orthogonal to each other.
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`Figure 3: Example of two carriers that are orthogonal
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`There are N different subchannels centered on frequencies
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`is able to carry Each subchannel that the transmitted signal is a sum of these N frequencies.
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`fk, k = 1, ... , N which means
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`5
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`Dish
`Exhibit 1040, Page 8
`
`

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`data symbols. Subchannel number 64 is used to send a pilot tone and it is only used for
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`this. The subchannels use basically an JVI-point Quadrature Amplitude Modulation (l\/l—QAl\/I)
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`constellation to code tl1e different symbols. A typical transmitted power spectrum is shown in
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`Figure 4.
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`Transmitted Power
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`' FTGQUSHCV
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`Figure 4: The transmitted power versus the different center frequencies
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`The modulation technique that uses cyclic functions as carriers is also used in Wireless communi-
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`cation, Where it is called Orthogonal Frequency Division l\/lultiplexing (OFDM). The difference
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`between OFDM and l)l\/IT is that the latter uses loading to assign the numbers of bits per sub-
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`channel while OFDM uses a fixed number of data bits per subchannel. The reason for this is that
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`the channel normally changes more rapidly in time for wireless communication. This means that
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`the transceiver cannot estimate the channel characteristics as well as a communication system
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`designed for communication over a twisted pair copper cable. The twisted pair channel can be
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`regarded as very slowly varying in time. A source that influences the channel in a slow way is
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`temperature. Other, more sudden, disturbances are impulse noise, AM interference, near—end
`and far—end crosstalk.
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`The loading of information data bits onto diflerent carriers uses a variety of algorithms to
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`maximize the performance, see
`
`One of the features of the DMT multicarrier modulation
`
`technique is that it is possible to assign a different number of information bits to each of the
`
`subchannels. The number of bits that are assigned to the subchannels is chosen to maximize
`
`the performance of the multicarrier communication system. The basic idea is to assign a larger
`
`number of bits to those subchannels that have higher SNR. In the standard [2] it is specified
`
`that as many as 15 bits can be assigned to each subchannel. This results in a 32768—QAl\/I
`
`constellation. If there are subchannels that are affected by severe distortion, for example AM
`
`interference, it is possible to exclude these subchannels. Figure 5 shows some examples of how
`
`the bit allocation can be carried out for the different frequencies when there are disturbances
`
`Dish
`Exhibit 1040, Page 9
`
`

`
`present .
`“*"“”'E' 9' '1“
`
`Ch-nnr-3‘ raivmsa
`
`i
`" "ff If i
`I,
`I
`L
`IA
`t,
`
`1,
`' *7"
`t,
`I,
`
`1.
`
`Number av sis
`
`1,
`
`1.
`
`I,
`
`iii
`r,
`1.
`
`i,
`
`r,
`
`r,
`
`Figure 5: Description of how the bit allocation can be performed. The upper diagram is when
`there are no disturbances and the lower is when there is AM interference.
`
`Other disturbances are near— and far—end crosstalk, which are disturbances that are generated
`
`through a coupling path in the twisted pair cable.
`
`The ordinary twisted pair copper cable channel that is used in subscriber loops has large varia-
`
`tions both in gain and phase for different frequencies. There are many different channels since
`
`the various subscriber loops can vary in length, wire gauge and might also contain bridged
`
`taps. A bridged tap is a connection to a subscriber loop that is not connected to anything. The
`
`bridged taps are usually not terminated which may lead to reflections and distortion. For further
`
`information about channels and disturbances, see
`
`Each of the frames that are transmitted
`
`is preceded by a cyclic prefix. The cyclic prefix is defined as :E_k = :17gN_k,
`
`is = 1. .
`
`.
`
`. 32.
`
`2.2 Frequency Allocation in ADSL Modems
`
`The frequencies used in ADSL modems range between DC and the Nyquist frequency which is
`
`1.104 lVlHZ. This means that the sampling rate is equal to 2.208 MHZ if no oversampling is used.
`
`Ordinary telephone calls use frequencies below 4 kHz, which means that this frequency band is
`
`not allowed be used for data Communication in ADSL modems since this service is required to be
`
`available all the time. The channel is divided into 256 different subchannels, these subchannels
`
`can be used to carry information. The carrier spacing between the subchannels is equal to 4.3125
`kHz.
`
`There are two possibilites to divide the bitstrearns in two separate directions. The first one is to
`
`use frequency division and the second is to use echo cancellation. The former divides the channel
`
`Dish
`Exhibit 1040, Page 10
`
`

`
`into two frequency bands, one for downstream and one for upstream data. In the latter, echo
`
`cancellation, the two bitstreams are overlapping each other and an echo cancellation algorithm
`1'
`She frequency use of the two techniques are described in Figure
`
`needs to used to separate them.
`6.
`
`POTS
`
`ii
`
`Downstream
`
`25kHz
`
`|2‘C]Oi<Hz
`
`.
`
`7
`
`zuokai
`
`Wit/ii-ii
`
`Frequency division
`
`Echo cancellation
`
`Figure 6: Description of the frequency use for the dividing of the upstream and downstream
`data.
`
`2.3 ADSL Transmitter
`
`The use of orthogonal carriers to send information over a band limited fading channel is known
`
`to be optimum. see
`
`To obtain the desired orthogonality between the different Carriers the
`
`inverse discrete Fourier transform (IDFT) is used. The IDFT is easily implemented in most
`
`DSP’s and if 2N is a power of two it only requires 2N loge; (2N) operations. This is the case in
`
`AD SL modems which uses D3/IT, the IDFT is then called inverse fast Fourier transform (lFl~"l‘).
`
`The coded data from the M — QAJVI constellation is called Xi where 2' = O,
`
`,N -— 1. The
`
`encoder generates only N complex Values X7; but to make ask real the vector X must be augmented
`L
`-forz'=N—|—l, ..., 2N—l.
`
`according to Xi = ’§N_
`
`Figure 7 shows a simplified diagram of the transmitter.
`
`Dish
`Exhibit 1040, Page 11
`
`

`
`Input bf: stream
`
`Bi buffer
`and
`encoder
`
`P/S
`
`DAC ,,,*,“,L
`
`QAM~symbols
`
`Tme domain
`
`Figure 7: Simplified block chart of an ADSL transmitter.
`
`The sent data is divided into blocks (frames). A set of 69 frames is called a superframe. The
`
`superframe consists of 68 data—carrying frames and one known synch frame. This 69’th frame is
`
`a synchronization frame which is used to recover the frames when there has been an interruption
`in the received bitstream.
`
`2.4 ADSL Receiver
`
`The ADSL receiver block diagram appears in Figure 8. The important blocks are described in
`
`the following sections.
`
`channel output
`
`i
`l‘_::eail“?S_
`flhar and E
`equalizer
`
`.
`
`AGC
`.
`
`V
`
`9
`
`.
`
`W
`
`i
`H TEQ I
`i
`
`‘V
`.
`‘
`
`’_
`:
`’..,.
`
`bitstieam
`Decoder ~¥>
`
`7'j
`:
`Z,“
`
`Figure 8: ADSL receiver block diagram.
`
`Analog Pre-Equalizer
`
`The incoming analog signal is low—pass filtered to avoid aliasing. The signal is also pre-equalized
`
`with an analog filter. This is done because a twisted pair copper channel usually attenuates
`
`higher frequencies more than lower frequencies.
`
`If this equalization is not done before the
`
`A/D—convcrtcr, the resolution will be less in the higher frequency range. The antialiasing and
`
`the pre-equalizer filter can of course be combined in one analog filter. This analog filter is
`
`Dish
`Exhibit 1040, Page 12
`
`

`
`implemented in hardware and will not equalize the channel completely. It is possible to use a
`
`set of analog filters and choose the most appropriate one for the actual channel. The number of
`
`filters must be limited however, since analog filters are expensive to implement.
`
`Automatic Gain Controller
`
`The Automatic Gain Controller (AGC) is used to adjust the amplitude of the incoming signal
`
`so the precision of the A/D-converter is used in an optimal way. To ensure that large values
`
`of the signal are not clipped, the average output level of the AGC is set to PAR dR below the
`
`highest value tolerated by the A /D-converter, where PAR is the Peak—to-Average Ratio of the
`
`ADSL signal. The AGC is trained during the first initialization sequence, known as C-REVERB
`
`in the ADSL standard [
`
`A/D-Converter
`
`According to the ADSL standard
`
`a normal linear A/D—converter should be used. The number
`
`of bits required is a. function of the actual SNR. Basically the quantization noise introduced by
`
`the A/D-converter should be much smaller than the received noise. In an ADSL envirornnent
`
`with up to 60 dB SNR this implies that 12 bits or more should be used [
`
`Time Domain Equalizer
`
`The Time Equalizer
`
`is used to shorten the channel impulse response to a delay spread of
`
`less than the prefix of 32 samples. This is essential to avoid lntersymbol Interference (ISI), but
`
`is not part of the ADSL standard
`
`The TEQ is also trained with a known pseudo—random
`
`sequence. C-REVER3, during the initialization. After this training the combined channel —
`
`TEQ impulse response should be less than or equal to 32 bits long. lt is also possible to update
`
`the TFIQ parameters continuously during data exchange by using the known synchronization
`
`frame that is sent every 69th frame, although this is probably not necessary since the twisted
`
`pair copper channel response will not change much in time.
`
`Frequency Domain Equalizer
`
`The Frequency Equalizer (FEQ) is the final and most accurate equalizer. The FEQ consists
`
`of 256 one—tap complex filters, one for each frequency. This means that every frequency com-
`
`10
`
`Dish
`Exhibit 1040, Page 13
`
`

`
`ponent is adjusted in amplitude and phase. The FEQ is also trained with the pseuddrandom
`
`iliitialization sequence. Au LMS algorithm can be used to update the filter taps. Similar to the
`
`TEQ, the filter taps can be updated during data. exchange by using the synchronization frame.
`
`The errors from the FEQ when the synchronization frame is sent can also be used to estiiiiate
`
`the SNR in each frequency channel.
`
`Dish
`Exhibit 1040, Page 14
`
`

`
`3 Timing Recovery
`
`It is very important that an ADSL—syste1n is synchronized to make the correct symbol decisions.
`
`Since the downstream bit rate (from ATU-C to ATU—R) is much higher than the upstream bit
`
`rate, tl1e most diflicult synchronization problem will appear in the ATU-R. Hence this problem
`
`is discussed here. Basically the bit synclironization can be carried out in two different ways. The
`
`first is to adjust the phase of the receiver sample clock relative to the phase of the transmitter
`
`sample clock. This is commonly known as a Phase—Lock Loop (PLL). A practical solution based
`
`on this approach is discussed in Section 3.3. When the loop is locked the frequencies of the
`
`transmitting and receiving sample clocks are equal. The pilot phase can be locked to its original
`
`Value which is 7r/4 rad. However, because of the phase shift in the channel, it is necessary to
`
`adjust the phase reference so that the correct frame position is maintained, see Section 5.4. If
`
`the ATU—R uses this extracted received clock for transmission it is called loop timing. Thus
`
`only one PLL is needed because the ATU-C may use its transmitter sample clock to derive its
`
`receiver sample clock. The phase might still be adjusted because of the delay in the channel.
`
`The second approach is known as rotors.
`
`It uses two very accurate crystal oscillators for the
`
`transmitter and receiver sample clocks. If the frequency difference between the crystals is very
`
`small the drifting phase can be tracked. A consequence of the drifting phase is that samples
`
`may need to be added or deleted. Note that the crystal oscillators must be very stable and close
`
`in frequency for the basic DMT relationship to hold. The phase is not locked with this solution.
`
`Consequently the phase needs to be adjusted for all channels in proportion to the drift in the
`
`pilot phase. Adjusting the phase is equivalent to a rotation of the constellation point. This
`
`synchronization technique is further discussed in Section 3.4.
`
`3.1 Pilot Tone
`
`In the ADSL standard [2] tone # 64 at 276 kliz in the downstream direction is assigned to a pilot
`
`with constant phase (same phase in all frames). The pilot tone is locked to the transmitted data
`
`rate. Any measured phase offset from the known nominal can be used to adjust the sampling
`
`rate. Thus this tone cannot be used for data transmission. How much this degrades the total
`
`data rate will depend on how many bits are assigned to the other subchannels and on how many
`
`bits it would have been possible to assign to the pilot channel. However, under normal conditions
`
`it will probably reduce the total data rate by less than one percent. Although it is not necessary
`
`to use a pilot tone to extract the incoming bit rate it will make the synchronization task easier.
`
`Dish
`Exhibit 1040, Page 15
`
`

`
`In OFDM one suggestion is to extract the data rate without a pilot tone
`
`Even if no bits are
`
`assigned to the pilot channel it is possible that the total bit rate can be greater by using a pilot
`
`tone. This is because a more accurate and stable estimate of the receiver bit rate may make
`
`it possible to assign more bits to the other channels. However, if the bit synchronization o11ly
`
`depends on the pilot tone a small SNR in the pilot channel can degrade the whole transmission.
`
`3.2 Timing Error
`
`It is essential to keep the timing error iii an ADSL transmission very small. VVhen many bits
`
`are assigned to each frequency bin the received constellation points become very close. A timing
`
`error is equal to a rotation of the constellation points. The FEQ is used to adjust the amplitude
`
`and phase of all the sine waves. By rotating the received constellation points in the complex
`
`domain the phase error can be compensated. The total misadjustment in a constellation point
`
`is a combination of the timing error and the noise from the channel. A good design goal of
`
`the receiver is to make the noise added by the timing error much smaller than the noise from
`
`the channel.
`
`If the timing error is small and considered constant during one frame it can be
`
`expressed as
`
`-'II(t+€)¢=>X(f)-e"2"f‘~X(f)~(1-j27Tf6)
`
`If the noise added by the channel is N(f) then
`
`X(f)2(27rf)20§ << NW Vf
`
`Thus the maximal tolerable timing error is given by
`
`03 << mfin
`
`1
`(2wf)2S1\’R(f)
`
`(2)
`
`(3)
`
`(4)
`
`Thus the maximal tolerable timing error is determined by the maximum of the product between
`
`the frequency f and the SNR(f In ADSL the maximum will probably occur at approximately
`
`100 kHz with a maximal SNR of 60 dB
`
`With the values above the standard deviation of
`
`the timing error should be much less than 1.6 ns. This is approximately equal to 3.5 X 10*?’
`
`samples. Of course a constant offset in the timing, or equivalent frame misadjustment, can be
`
`tolerated because it can be adjusted with the FEQ.
`
`The number of bits assigned to each subchannel is determined by the SNR in the subchannels.
`
`However the number of bits per channel is limited to 15. The maximum number of bits will be
`
`assigned if the SNR is approxim:.tely 60 dB. Thus SNR( f ) is limited to approximately 60 dB in
`
`(4).
`
`Dish
`Exhibit 1040, Page 16
`
`

`
`3.3 Phase Lock Loop
`
`In a PLL a phase reference needs to be extracted from the received signal. This can be done in
`
`different ways. In ADSL the pilot tone is used as a phase reference. The received signal needs
`
`to be bandpass filtered to extract the pilot. In a DMT receiver this bandpass filtering is easily
`
`done with the FFT. The phase of the pilot is given directly from the FFT as the phase of the
`
`complex number. The proposed PLL is shown as a block diagram in Figure 9, and discussed in
`the remainder of this section.
`
`analog iupul
`
`.xl'C.\'O
`{-5
`
`F
`>
`
`.
`
`lei
`
`‘
`>%—>} HA2)
`
`’ ,
`
`l!
`
`Figure 9: ADSL PLL block diagram.
`
`3.3.1 Controller
`
`In most PLL literature the controller is considered to be a lowpass filter and the choice of cutoff
`
`frequency is a key design problem. A high cutoff frequency makes it possible to track fast
`
`wander in the transmitted bit rate. However, it may also track unwanted noise. It is easy to
`
`show that with a standard PI-controller the phase error will go to zero in the presence of an
`
`initial frequency offset between the transmitter sample clock and the receiver sample clock.
`
`Here we consider a more general controller structure. Note that the sample period of the control
`
`system is not constant. The control loop is sampled once each frame. The length in time of
`
`one sample period or frame depends on the current frequency from the sample clock. However
`
`the change in the sample rate is Very small and can be considered constant. An important
`
`distinction from an ordinary PLL is that the phase of the pilot tone is only measured once in
`
`each frame. This means that the pilot runs for 68 periods (64 when no prefix is sent) between
`
`each update of the local crystal oscillator.
`
`Dish
`Exhibit 1040, Page 17
`
`

`
`3.3.2 Voltage Controlled Crystal Oscillator
`
`Because of the high demands in ADSL an ordinary Voltage Controlled Oscillator (VCO) will not
`
`be accurate enough. Instead a Voltage Controlled Crystal Oscillator (VCXO) based on a quartz
`
`crystal must be used. There are very accurate VCXO’s available on the market today, but the
`
`most accurate ones are very expensive and would greatly increase the cost of an ADSL modem.
`
`The frequency sweeping range of the VCXO needs to be at least twice as big as the sum of the
`
`transmitter crystal frequency tolerance and the free running frequency tolerance of the VCXO.
`
`This typically means that a sweeping range of :: 50 ppm for the VCXO is necessary. The fact
`
`that the phase of the pilot is only measured once in each frame makes it Very important to use
`
`stable crystals.
`
`3.3.3 Phase 1\/Ieasurernent
`
`The FFT returns a complex number for every channel. The phase of the pilot tone is given
`
`by the phase of the complex number in channel # 64. Since the frequency of the transmitter
`
`crystal oscillator is (exactly) 8 times higher than the pilot frequency the phase measurement is
`
`multiplied with 8 to give the phase of the transinitter crystal oscillator.
`
`3.3.4 Mathematical Model
`
`The pilot phase computed by the FFT will be an average of the actual pilot phases at the
`
`beginning and end of the frame. All phases mentioned in this section are phase differences
`
`between the transmitter sample clock and the VOXO. The actual or true phase at the end of
`
`frame I: is denoted 6k. The number of bits in each frame, L, is equal to 544 when the prefix is
`
`sent and normal sampling rate is used. The VCXO is modeled as
`
`fVC'XO(k) = ffr + fc ' VVCXo(/Cl
`
`(5)
`
`where ff,« is the free running frequency and fc is the constant VCO gain with unit Hz /V. The
`
`phase difference at the end of frame is + 1 can be expressed in time as
`
`9n
`H1
`
`:9»
`k+
`
`ftr — fr/OX0 (/9)
`fr/CXOUC)
`
`- 277 L
`
`(6)
`
`wl1ere f,g,~ is the frequency of the transmitter sample clock. Since fa - Vvcxo
`
`<< ffr
`
`0k+1 W 0;;
`
`1
`
`ft?‘ ’ ff?‘ ’ fa’ l/VCXO(k) -27rL
`ffr
`
`15
`
`Dish
`Exhibit 1040, Page 18
`
`

`
`Taking the Z-transform of (7) gives
`
`_ _f_C.
`f7‘
`
`271' L. V‘/,CXO(z) +
`
`ft7'”“.ff‘r'_ 2,” L‘
`ffr
`
`1
`1_Z_1
`
`1
`Z_
`
`‘
`
`1'l4/'0X0(Z)+L*fL'
`ff7’
`
`.,.—
`
`277 L" 2
`i ~~
`(Z _ D2
`
`(9)
`
`The phase difference in frame is computed by tl1e FFT, (pk, can be expressed approximately as
`
`an average between the phase at the beginning of the frame and the phase at the end of the
`Ir aine .
`
`9k+1 + 6k‘
`
`gbk+1 z
`
`: 9 A +
`A
`
`fir — fVoXo(’?) _
`ft/CXOUC)
`
`7r L
`
`(10)
`
`Again since fC- VVC«XO(lc) << ff?
`
`I
`®k+1rQjQk[
`
`ftr “ ffr — fa‘ VVCXOU9)
`ffr
`
`Taking the Z—transfor1n of (11) gives
`
`Zq)(Z) Z
`
`W
`
`ff?"
`
`- 7?‘ L - l/VC'XO(Z) +
`
`ftr _ ffr .
`ffr
`
`7‘.
`
`Substituting (9) for 9(2), we have
`
`<I>(z) =
`
`+ 1
`
`Z
`fa ‘W I«'
`
`_ffr z(z ~1>he
`
`'VvCXo{Z) +
`
`Figure 10: Simplified block diagram of the PLL.
`
`A simplified block diagram of the PLL is shown in Figure 10, Where the phase noise N is
`
`modeled as white Gaussian noise with zero average value, i.e.
`
`the phase noise in two different
`
`Dish
`Exhibit 1040, Page 19
`
`

`
`frames are independent. The transfer function H and the disturbance \lV(z) are given by (13)
`&S
`
`2+1
`, -1-
`TL z(z—1)
`
`NL-
`
`4
`(1)
`
`(15)
`
`The disturbance, \IJ(z), is modeled as a constant disturbance. The frequency variations in the
`
`transmitter crystal oscillator and the free running frequency of the VCXO are assumed to be
`
`very small. Basically it is only the surrounding temperature that will affect the frequencies of
`
`the crystals. Compared to the sample period of the PLL the temperature changes will be Very
`slow.
`
`3.4 Rotor Technique
`
`ln the rotor technique a receiver crystal with constant frequency is used. Thus the two crystals
`
`are not synchronized. The receiver crystal frequency needs to be Very close to the transmit-
`
`ter crystal frequency for the DMT modulation/demodulation principles to be Valid. All the
`
`transmitted tones are in theory orthogonal, but if there is an offset between the crystals each
`
`transmitted tone will give a contribution to all the received tones. This problem is similar to
`
`what happens in the presence of narrow band interference. As discussed in [7] the influence of
`
`the disturbance strongly depends of the location of the interference.
`
`Dish
`Exhibit 1040, Page 20
`
`

`
`Figure 11: DMT frequency basis functions.
`
`In Figure 11 some of the basis functions for the transmitted tones are shown. They are orthogonal
`
`at the discrete frequency Values, but if there is an offset in the frequencies between the two clocks
`
`the basis functions will be sampled with an