713
`Adaptive OFDM for wideband radio channels
`Andreas Czylwik
`
`Deutsche Telekom AG, Research Center
`Am Kavalleriesand 3, 64295 Darmstadt, Germany
`Tel. : -+49-6 15 1-83-3537, Fax: $49-6 15 1-83-4638, E-Mail: czylwikQfz. telekom.de
`
`Abstract
`An OFDM (orthogonal frequency division multiplexing)
`transmission system is simulated with time-variant transfer
`functions measured with a wideband channel sounder. The
`individual subcarriers are modulated with fixed and adaptive
`signal alphabets. Furthermore, a frequency-independent as
`well as the optimum power distribution are used.
`The simulations show that with adaptive OFDM, the required
`can be reduced
`signal power for an error probability of
`by 5 ... 15 dB compared with fixed OFDM. The fraction of
`channel capacity which can be achieved with adaptive OFDM
`depends on the average signal-to-noise ratio and the propa-
`gation scenario.
`1 Introduction
`Shannon has shown that, theoretically, it is possible to trans-
`mit information over a given channel with an arbitrary small
`error probability if the data rate is not greater than the chan-
`nel capacity (information rate). Therefore, the channel ca-
`pacity is the ultimate limit of the data rate for reliable com-
`munication. In the present contribution the channel capacity
`of a wideband radio channel with fixed antennas and with
`frequency-selective fading is compared with the achieveable
`data rate using adaptive ortthogonal frequency division multi-
`plexing (OFDM) transmission schemes. Radio channels with
`frequency-selective fading originate from multipath propaga-
`tion and can be found in broadband wireless local area net-
`works (LANs) or broadband wireless access networks.
`In an OFDM transmission system which uses the same fixed
`modulation scheme for all OFDM subcarriers, the error prob-
`ability is dominated by the OFDM subcarriers with highest
`attenuation. Therefore, in case of frequency-selective fad-
`ing the error probability decreases very slowly with increas-
`ing average signal-to-noise ratio (SNR). This problem can be
`mitigated if different modulation schemes are employed for
`the individual OFDM subcarriers. The modulation schemes
`have to be adapted to the SNRs of the individual subcarriers.
`Furthermore, the power distribution for the individual sub-
`carriers can be optimized. In the present contribution only
`uncoded modulation schemes are considered.
`In order to combat the temporal fluctuations of a radio chan-
`nel, adaptive single carrier imodulation schemes are proposed
`in literature [l, 21. Also in the field of OFDM, adaptive chan-
`nel usage is recommended [:3]. Adaptive OFDM based on the
`optimum water-pouring method is well-known in the field of
`transmission over twisted-pair lines [4, 5, 61. But in the field
`of multipath radio channels, the water-pouring method with
`variable modulation schemes is considered only sparcely [7].
`Therefore, in the present paper the performance of adaptive
`OFDM in a radio system is investigated in detail.
`The paper is organized as follows: In section 2 the channel
`
`0-7803-3336-5/96 $5.00 0 1996 II?EE
`
`capacity of a measured broadband radio channel is calculated
`and in section 3 transmission systems with adaptive OFDM
`are presented. Finally, section 4 contains simulation results.
`2 Channel capacity
`The channel capacity of an ideal transmission channel with
`additive white Gaussian noise (AWGN) can be calculated us-
`ing Shannons formula [8]:
`
`where B denotes the (single-sided) channel bandwidth, P, the
`average signal power and P, the average noise power in the
`bandwidth B . Equation (1) shows that the channel capacity
`only depends on the bandwidth B and on the signal-to-noise
`ratio (SNR = Px/Pn). In case of a frequency-selective channel
`the capacity is determined by the ratio of the signal’s power
`spectral density (PSD) Sx(w) to the noise PSD Sn(w). The
`transmission channel has to be divided into a large number
`of narrowband subchannels where the transfer function can
`be considered as constant. The channel capacity results from
`integration over all subchannels [9, 101:
`
`The integrand in Eq. (2)
`
`C’ = l d ( l + #)
`
`(3)
`
`will be called the density of the channel capacity. C’ deter-
`mines the maximum data rate per bandwidth. Therefore, it
`is the ultimate limitation of bandwidth efficiency. Its unit is
`
`It is assumed that the transmitted signal is limited by its
`mean power P,.
`In this case the maximum of channel ca-
`pacity is obtained if the PSD of the transmitted signal S, (w)
`is adapted to the transfer function of the channel. The op-
`timum method for distribution of power is the well-known
`“water-pouring” method [lo]:
`
`(4)
`The parameter Sm, has to be chosen such that the boundary
`condition
`
`P, = 7T
`
`is fulfilled.
`
`00
`
`Sz(w)dw
`
`(5)
`
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`714
`
`A radio channel is described by a time-variant transfer
`function H ( t , w ) [ll] and an additive white Gaussian noise
`(AWGN) source (thermal noise nth(t)). The corresponding
`block diagram is shown in Fig. 1. Because of the time-
`variant behaviour, the channel capacity changes with time,
`too. Propagation measurements of radio channels with fixed
`antennas show that the transfer function varies very slowly
`with time. Because of this reason it is assumed that the
`instantaneous transfer function of the radio channel can be
`estimated at the receiver and can be communicated back to
`the transmitter via signalling channels with noticable delay.
`Therefore, only in a bidirectional transmission system, the
`transmitted power spectrum can be optimized.
`
`Figure 1: Block diagram of a time-variant radio channel.
`
`The power spectral density of the equivalent input noise of a
`time-variant radio channel is given by:
`
`where F denotes the noise figure, IC Boltzmann’s constant,
`and To the reference temperature. Since the equivalent input
`noise is time-variant, also the optimum PSD SsoPt(w) and the
`channel capacity become time-variant.
`From a qualitative point of view, with the water-pouring
`method, most of the power is concentrated in frequency
`ranges where the channel attenuation is small. Numerical
`evaluations of channel capacity have shown that there is only
`a very small loss of channel capacity if a white power spec-
`trum is used instead of the optimum power spectrum [12].
`This result holds for the case that the average SNR is high.
`Only in case of a low average SNR, the channel capacity can
`be increased significantly with an optimized power spectrum.
`Therefore, only the case of a frequency-independent signal
`power spectrum is considered for the calculation of channel
`capacity.
`For the numerical evaluation, data from wideband outdoor
`propagation measurements in an industrial area in Darm-
`stadt, Germany, were used. The measurements were car-
`ried out with fixed antennas. The transfer function of a first
`propagation measurement with two omni-directional anten-
`nas and under line-of-sight (LOS) conditions is displayed in
`Fig. 2a. In a second measurement an omni-directional and a
`sectional antenna (angle of aperture: 110’) were used. The
`omni-directional antenna was located inside a building so that
`there were non-line-of-sight (NLOS) conditions. The most im-
`portant parameters of both measurement scenarios are sum-
`marized in Table 1.
`The channel capacity was evaluated within a bandwidth of
`B = 5 MHz. The resulting temporal fluctuations of chan-
`nel capacity are shown in Fig. 3 (The channel capacity is
`normalized with the bandwidth B: fl = C / B ) . The level
`of transmitted power was adjusted so that for both measure-
`ments a similar average SNR results. The curves show that
`the fluctuations in the LOS case are much stronger than in
`
`distance of antennas
`carrier frequency
`bandwidth of
`measurement
`average attenuation
`-lOlg( i H ( t , W ) 1 2 )
`delay-spread
`
`Figure 2: Time-variant transfer function of measurement 1 (a) and mea-
`surement 2 (b).
`I
`
`I measurement 1
`95 m
`
`I measurement 2 I
`230 m
`
`1.8 GHz
`6 MHz
`
`77.4 dB
`
`0.31 ps
`
`112.0 dB
`
`1.15 ps
`
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`715
`
`individual subcarriers is adapted to the shape of the trans-
`fer function of the radio channel. Modulator B optimizes
`simultaneously the distribution of bits and the distribution
`of signal power with respect to frequency. The algorithms for
`the distribution of bits and power are described later in this
`section.
`Different QAM modulation formats can be selected: no
`modulation, 2-PSK, 4-PSK, 8-QAM, 16-QAM1 32-QAM, 64-
`QAM, 128-QAM1 and 256-QAM (see Fig. 5). This means
`that 0, 1, 2, 3, ... 8 bit per subcarrier and FFT block can be
`transmitted. In order to get a minimum overall error prob-
`ability, the error probabilities for all used carriers should be
`approximately equal.
`
`rlmmn
`
`16-QAM
`
`;E;ii?:iigiii?
`::::::&.:::I:
`x::n:n::::::::
`?;$;?;$$;ii;z::
`:::=:::I:::::::
`$;;;E;$?;;$;g
`
`.
`
`.
`
`.
`
`,
`100
`time t in s
`
`.
`
` I 1 ;I
`.
`150
`
`200
`
`.
`
`a ) ; 4
`
`;
`
`0
`
`50
`
`1
`
`h
`
`b ) j 4 ; .
`0
`
`, 50 ,
`
`,
`
`,
`
`,
`
`,
`
`,
`
`,
`, 100
`
`,
`
`,
`
`,
`
`,
`, 150
`
`,
`
`,
`
`,
`
`, 200 I
`
`time t in s
`-
`Figure 3: Temporal fluctuation of the normalized channel capacity =
`C/B for a) measurement 1 and b) measurement 2. C’ is the average
`density of channel capacity with respect to frequency for the bandwidth
`B = 5 MHz. Transmitted power: a) 0 dBm, 10 dBm, 20 dBm; b) 30
`dBm, 40 dBm, 50 dBm.
`
`to demodulation, the signal is equalized in frequency domain
`with the inverse of the transfer function. Finally, an adaptive
`demodulator detects the taansmitted symbols and generates
`the output bit stream.
`
`H Hh
`
`Inverse
`
`Cyclic
`Ex1 ension
`
`Binary
`Source
`
`Adaptive
`Modulation
`
`Bit Error
`Detection
`
`I
`
`Linear
`Time-Variant
`Channel
`
`emodulatio
`
`Extension
`Figure 4: Block diagram of an OFDM transrnission system.
`
`The temporal variation of the transfer function of a radio
`channel makes it necessary to adapt the modulation schemes
`of the transmitted subcarriers continuous1.y. The adaptive
`modulator and demodulator have to be synchronized via a
`signalling channel which is disregarded in the present con-
`tribution. Furthermore, ideal carrier and clock recovery are
`assumed.
`The input signal of the adaptive modulator is processed block-
`wise due to the blockwise signal processing of the FFT blocks.
`For most applications of broadband radio systems a constant
`data rate is required. Therefore, also the adaptive modula-
`tor has to transmit with a constant data rate. This means
`that with each FFT block the same number of bits M is
`transmitted. Two different ~modulator/demodulator pairs are
`considered. In modulator AI the distribution of bits to the
`
`2-PSK
`
`4-PSK
`
`8-QAM
`
`........ ............
`..........................
`..........................
`............
`.............. ::::::::::::
`........
`............
`...... ....................
`........
`
`256-QAM
`128-QAM
`64-QAM
`32-QAM
`Figure 5: Constellation diagrams of the used QAM schemes.
`
`The required SNR for the above mentioned modulation
`schemes is displayed in Fig. 6 for given symbol error prob-
`abilities. Using Gray coding, the bit error probability is ap-
`proximately equal to the symbol error probability. The fig-
`ure shows the bandwidth efficiency of different modulation
`schemes as a function of SNR. It is obvious that the band-
`width efficiency follows approximately a straight line as a
`function of SNR (measured in dB). In the following, this linear
`approximation (for a symbol error probability Per, =
`is used to distribute bits to individual subcarriers:
`S
`C b A M M 0.31 (101g - - 6.7).
`N
`Especially for low SNRs (2-PSK) Eq. (7) is a significantly
`better approximation than the “gap approximation” [13, 61:
`
`(7)
`
`where 10 lg I’ denotes the SNR gap between channel capac-
`ity and the bandwidth efficiency of real modulation schemes
`(approximately 8 dB for Per, =
`In the following, the principle of the bit distribution algo-
`rithm for modulator A is described: In a first step, from the
`given SNRs of the individual subcarriers and the required
`the capacities CbAM, are cal-
`error probability Per, =
`culated with Eq. (7). These values are rounded to the max-
`imum integers m; which are smaller than CLAM,. If in all
`subchannels i a number of m; bits are mapped to complex
`symbols of corresponding QAM schemes, the error probabil-
`
`ity is smaller than required. A sum of mc = xi mi bits can
`
`be transmitted per OFDM symbol with less than the required
`error probability.
`
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`

`/y#
`
`716
`
`channel capacity
`
`In the following, the power and bit distribution algorithm
`of modulator B is described: In a first step, an initial bit
`distribution is calculated. A possible initial bit distribution
`is e.g. the result of modulator A for a frequency-independent
`Pi for
`power distribution. Next, the total required power
`the desired error probability is calculated from Eq. (7):
`
`(9)
`
`Successively, at each of the subcarriers the number of bits
`is reduced by one bit (if possible). The reduction of the total
`required power is calculated for each subcarrier. At the sub-
`carrier with the highest power reduction, the number of bits
`is reduced by a single bit. Next, successively at each of the
`subcarriers the number of bits is increased by one bit. The
`increase of the total required power is calculated for each sub-
`carrier. At the subcarrier with the smallest power increase,
`the number of bits is increased by a single bit. This procedure
`of decreasing and increasing the number of bits per subcar-
`rier is repeated until the bit distribution does not change any
`more. At this point, the optimum bit distribution is found.
`Simultaneously, the total required power for the desired error
`probability is minimized.
`Finally the (average) power of all subcarriers is adjusted by
`the same factor P x / P, so that the total power equals the
`available power P,. The result of this procedure is that the
`same SNR margin for all subcarriers is achieved. The ob-
`tained SNR margin is the maximum possible so that the er-
`ror probability becomes minimum. Therefore, modulator B
`calculates the optimum distribution of power and bits.
`The results of the optimization processes of both, modulator
`A and modulator B, are shown in Fig. 7. For comparison,
`the upper diagram shows the absolute value of the transfer
`function in dB. The lower diagram shows that for this specific
`example, both modulators yield the same distribution of bits.
`Furthermore, the power distribution and SNR is shown for
`both modulators. The transmit power for modulator B varies
`within a range of approx. -1.5 dB to 1.5 dB corresponding
`to the difference of approx. 3 dB between SNRs of adjacent
`QAM schemes.
`
`4 Simulation results
`
`length of FFT interval
`length of guard interval
`1 1 1 bandwidth
`sampling rate in complex baseband
`noise figure of the receiver
`number of transmitted QAM symbols
`
`256 samples
`25 or 50 samples
`5 MHz
`5 MHz
`6 dB
`5 x 106
`
`C ' / b i t
`1 0
`- -
`9
`8
`--
`--
`7
`6
`5
`4
`3
`
`~~
`
`-~
`
`--
`--
`
`,
`
`I
`
`
`
`,u::w~56-QAM
`,P',+'lZE-QAM
`32 - QAM $AM ""
`a*,,* ,++'&i-
`
`#
`
`,
`
`
`
`/d,iCl 6 -QAM
`,' ,'
`,'O*,h 0-QAM
`
`35 1 0 l g s / ~
`5
`-5
`30
`25
`10
`15
`20
`-10
`0
`Figure 6: Bandwidth efficiency of different modulation schemes versus
`SNR for a symbol error probability Per, = lo-'
`(circles) and Per, =
`(crosses).
`
`The required number of bits M per OFDM symbol is deter-
`mined by the requested service and usually does not depend
`on time (constant bit rate transmission is assumed). The
`number of bits m x that can actually be transmitted with
`the given quality will usually not coincide with the requested
`number M .
`In case the total number of bits m x is smaller than M ,
`the numbers of bits in individual subchannels have to be in-
`creased. Of course, with this measure the error probability
`will increase above the desired one. The additional bits have
`to be distributed on the subchannels such that the error prob-
`ability is increased as small as possible. The corresponding
`algorithm is described in the following: The additional bits
`are distributed with the target to minimize the maximum
`difference mi - CbAM,. One (computationally not efficient)
`method is to add successively bits to subchannels where the
`difference m, - CbAM, is minimal.
`In case the total number of bits m x is greater than M , the
`numbers of bits in individual subchannels have to be reduced.
`With this measure the error probability will decrease below
`the desired one. The reduction of bits in the subchannels is
`carried out such that the error probability is decreased as far
`as possible. This is done by reducing bits with the target to
`maximize the minimum difference CkAM - W . One (compu-
`tationally not efficient) method is to reduce successively bits
`at subchannels where the difference C&M, - is minimal.
`For a small SNR the values CbAM, can also become negative.
`This only means that the error probability is usually higher
`than desired. But nevertheless with the above described al-
`gorithm, the distribution of bits is carried out in an optimum
`way so that the overall error probability becomes minimum.
`The above described algorithm for modulator A maximizes
`the minimum (with respect to all subcarriers) SNR margin
`(difference between actual and desired SNR for a given error
`probability).
`The modulator B optimizes the power spectrum (PSD) and
`distribution of bits simultaneously. The optimum Hughes-
`Hartog algorithm [5] which was designed for the twisted-pair
`channel, cannot be applied since it maximizes the data rate.
`Because of the temporal fluctuations of the radio channel, the
`data rate varies with time. Therefore, the Hughes-Hartog al-
`gorithm cannot be used for constant bit rate communications.
`
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`

`a 4-QAM (guard = 25)
`E 4-QAM (guard = 50)
`0 adaptive QAM (guard = 25)
`
`-
`
`100
`
`10-1
`
`10-2
`
`0 5 E
`
`io-'
`
`a> 10-8
`
`I
`
`100 7
`
`E I 64-QAM (AWGN)
`64-QAM (guard = 25)
`E 64-QAM (guard = 50)
`0 adaptive QAM (guard = 25)
`E adaptive QAM (guard = 50)
`
`transfer function IH(w)l in dB
`
`717
`
`-85.
`
`-90.
`
`0.
`
`100.
`
`zoo.
`
`number of
`subcarrier
`
`".OV td
`...c >-%-
`
`10.0
`
`\
`VR in dB for modulator B \
`SI
`
`bandwidth efficiency miin bit
`
`number of
`subcarrier
`
`\
`
`'transmit power in dBm for modulator A
`transmit power in dBm for modulator B
`Figure 7: Illustration of adaptive modulation: absolute value of transfer
`function IH(w)l, SNR, bandwidth efficiency m;, and normalized power
`per OFDM channel.
`(The average normalised power per subcarrier
`equals total average power.) M:odulator A optimizes only the modu-
`lation schemes, modulator B optimizes both, modulation schemes and
`power distribution.
`
`error ratio as a function of the transmitted power for mea-
`surement l and 2, respectively.
`In particular, for modulation schemes with high bandwidth
`efficiency an error floor can be noticed in case of insufficient
`length of the guard interval due to interbllock interference.
`The error floor is reduced or vanishes if the guard interval is
`extended. Due to the larger delay spread in measurement 2,
`a higher error floor can be observed. The figures show that
`fixed OFDM is more sensitive against interblock interference
`than adaptive OFDM. This can be explained by the fact that
`in the adaptive system, bad channels are not used or only
`used with small signal alphabets where a small amount of
`interblock interference is not so critical.
`In the following, the power required for a bit error ratio of
`is compared for fixed and adaptive modulation. For a
`sufficient length of the guard interval a gain due to adaptive
`modulation of 5 ... 15 dB ca:n be observed for the investigated
`channels. If a lower bit error ratio is considered, the gain is
`even higher. Especially for the NLOS channel, the bit error
`ratio can be reduced dramatically with adaptive OFDM. The
`loss for using modulator A .with frequency-independent PSD
`instead of modulator B with optimum PSI1 is less than 1.1
`dB. Because of this small loss, it is recommended to use a
`constant power spectrum in order to save computational or
`signalling effort.
`Finally, the channel capacity is compared with the data rates
`achievable with optimum adaptive OFDM (modulator B). It
`can be toler-
`is assumed that a bit error probability of
`ated. The results of the comiparison are summarized in Table
`3. Because of its superior performance, only adaptive modu-
`
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`

`718
`
`4-QAM (AWGN)
`4-QAM (guard = 25)
`4-QAM (guard = 50)
`adaptive QAM (guard = 25)
`adapt. QAM with opt. PSD (guard = 25)
`
`average
`bandwidth
`efficiency
`
`2 bit/symb.
`4 bit/symb.
`6 bit/symb.
`
`average
`bandwidth
`required
`density of
`efficiency
`transmit
`channel
`incl. guard
`power
`capacity
`interval
`level
`measurement 1 (LOS)
`1.82 bit
`6.02 bit
`-4.1 dBm
`3.64 bit
`8.25 bit
`2.7 dBm
`5.02 bit
`10.26 bit
`8.8 dBm
`
`%
`(AWGN)
`
`30.2 (45.9)
`44.1 (59.9)
`48.9 (68.5)
`
`2 bit/symb.
`4 bit/symb.
`6 bit/symb.
`
`26 dBm
`32.7 dBm
`39 dBm
`
`1.82 bit
`3.35 bit
`5.02 bit
`
`4.48 bit
`6.56 bit
`8.61 bit
`
`40.6 (45.9)
`51.1 (59.9)
`58.3 (68.5)
`
`30.
`40.
`20.
`transmitted power in dBm
`
`50.
`
`a 16-QAM (guard = 25)
`
`16-QAM (AWGN)
`
`16-QAM (guard = 50)
`adaptive QAM (guard = 25)
`adaptive QAM (guard = 50)
`
`100
`
`10-1
`
`10-2
`
`10-3
`
`IO-‘
`
`0 .+
`4 E
`L 2
`
`5
`P
`
`a> 10-8
`
`L
`
`100 7
`
`100 7
`
`64-QAM (guard = 25)
`64-QAM (guard = 50)
`e adaptive QAM (guard = 50)
`0 adaptive QAM (guard = 25)
`
`transmitted power in dBm
`Figure 9: Bit error ratio versus transmitted power with different modu-
`lation schemes and guard lengths (measured in samples), radio channel
`data from measurement 2. Average bandwidth efficiency for all modu-
`lation schemes: a) C’ = 2 bit, b) C’ = 4 bit, c) C’ = 6 bit.
`
`tion also depends on the average SNR: The higher the SNR,
`the higher is the fraction of channel capacity - this also holds
`for the AWGN channel.
`
`5 Conclusion
`By using adaptive modulation schemes for the individual sub-
`carriers in an OFDM transmission system, the required signal
`power can be reduced dramatically compared with fixed mod-
`ulation. Simulations show that for a bit error ratio of
`
`References
`[l] S. SAMPEI, S. KOMAKI, AND N. MORINAGA: Adaptive modula-
`tion/TDMA scheme for large capacity personal multi-media com-
`IEICE Trans. on Communtcattons E77-B
`munication systems.
`(1994), pp. 1096-1103.
`[2] W. T. WEBB AND R. STEELE: Variable rate QAM for mobile radio.
`IEEE Trans. on Communications 43 (1995), pp. 2223-2230.
`[3] Q. CHEN, E. s. SouSA, AND s. PASUPATHY: Multi-carrier DS-
`CDMA with adaptive sub-carrier hopping for fading channels. In
`Proceedings of the PIMRC ’95, Toronto, pp. 76-80 (1995).
`[4] B. HIROSAKI, A. YOSHIDA, 0. TANAKA, S. HASEGAWA, K. INOUE,
`AND K. WATANABE: A 19.2 kbps voiceband data modem based on
`orthogonally multiplexed QAM techniques. In IEEE Intemataonal
`Conference on Communrcattons, pp. 661-665 (1985).
`[5] D. HUGHES-HARTOGS: Ensemble modem structure for imperfect
`transmission media. U. S. Patent 4,679,227 (1987).
`[6] P. S. CHOW, J. M. CIOFFI, AND J. A. C. BINGHAM: A practicaldis-
`Crete multitone transceiver loading algorithm for data transmission
`over spectrally shaped channels. IEEE Trans. on Communications
`43 (1995), pp. 773-775.
`[7] J. LINDNER: Channel coding and modulation for transmission over
`multipath channels. Archiv f . efektrische Ubertragung 49 (1995),
`pp. 110-119.
`[8] C. E. SHANNON AND W. WEAVER: The mathematical theory of
`communication. The University of Illinois Press, Urbana 1949.
`[9] R. G. GALLAGER: Information theory and reliable communrcation.
`John Wiley & Sons, New York 1968.
`[lo] G. SODER AND K. TRONDLE: Digitale Ubertragungssysteme.
`Springer-Verlag, Berlin 1985.
`[ll] D. PARSONS: The mobrle radio propagation channel. London: Pen-
`tech Press 1992.
`[12] A. CZYLWIK: Kanalkapazitat von Breitband-Funkkanalen. Klein-
`heubacher Berichte 39 (1996), pp. 297-310.
`[13] J. M. CIOFFI: Asymmetric digital subscriber lines. Stanford Uni-
`versity, Stanford, U.S.A. 1994.
`
`HTC Corp., HTC America, Inc. - Ex. 1023, Page 6
`IPR2018-01555 and IPR2018-01581 (HTC and Apple v. INVT SPE)
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