IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-33, NO. 5 , MAY 1985
`256-QAM Modem Performance in Distorted Channels
`
`fo
`is the carrier frequency
`is the symbol rate; for 256-QAM, T, = 8Tb
`1/T,
`l/Tb is the bit rate
`is a pulse defined by
`g ( t )
`
`487
`
`KUANG-TSAN WU 'AND KAMILO FEHER
`, I ,
`
`of 256-QAp .modems in dis-
`Absfracf-Performance degradatiqps
`torted channel5 are studied is this correspondepce. Illustrative 3 I' Inear,
`parabolic, and sinusoidal implitude and group delay cbannel distortions,
`caused by filter imptrfe$tidnq and/or
`by selective radio fades,
`are
`investigated. To e9able +an easy comparison with,,44-QAM systems, we
`present the degradation4 of these systems in the same figures as those of
`256-QAM- modem's., It is shown 'that linear' (slople) group delay and
`distortions cause the most significant
`sinusoidal (ripple) amplitude
`performance degradations.
`
`re-
`
`I. INTRODUCTIQN
`Digitial 64-QAM radio systems are described in many
`[ 11 -[4], [7] -[ 101. The theoretical RE
`ferences, including
`spectral eff.&i'ency of 64-QAM systems
`is 6 bits/s/Hz. With
`a! = 0.3-0.5
`rolloff raised-cosine Nyquist channel filters
`6 bits/s/Hz f
`[ 7 ] , [ 101 a practical spectral efficiency of
`( 1 -I- CY)-= 4.5 bits/s/Hz has beep achieved. Due to the
`in-
`creasing need for even'higher bandwidth efficiency digital radio
`(DUV), data-in-voice (DIV);
`and hybrid data-under-voice
`systems [ 71 , we consider
`and data-'above-voice/video (DAV)
`256-RAM 'as a next logical step. An
`(Y = 0.2 raised-cosine
`filtered 256-QAM modem could achieve a practical efficiency
`of 8 bits/s/Hz f 1.2 = 6.66' bits/s/Hz. For example, such a
`high spectral efficiency
`is required for the DIV transmission
`of a 1.544 M.bit/s rate signal in the standard analog super-
`group band of 240,kHz.
`of higher order
`Alt!lough
`it is known that the sensitivity
`modulatiop' techniques to different impairments .may in-
`crease significantly with the number
`of states, the perform-
`anc'e .of 256-QAM under vqrious effects
`such as channel
`distortions'is not yet .available in the.literatpre. The objective
`of channel fading
`o f this 'paper is, to investigate the effect
`and/or hardware imperfe'ctions on 256-QAM. The struc-
`ture 'of '256:QAM and the computer simulation model are
`briefly described. The effects
`of group delay and .amplitude
`distortions with linear, parabolic, or sinusoidal characteristics
`on 256-QAM are evaluated. Measured 256-QAM eye diqgrams
`which confirm' o'yr computer simulation results are also
`included.
`
`11. ANALYSIS AND DESCRIPTION OF COMPUTER SIMULATIONS
`Fig. 1 shows a block diagram'
`of a 256-QAM system. The
`modulated 256-QAM signal can be represented by (1)'
`
`r m
`
`1
`
`where
`Re [ ]
`
`denotes real part of
`
`In and Q, = +1, ir3, ..., +15 are the sampled values of the
`in-pha,se and quadrature symbols.
`The simulatioa is performed entirely in the complex baseband
`form as shown in Fig. 2.
`We report simulatjon results for
`systems having a symbol rate of 15 Mbaud (120 Mbits/s). The
`choice of this baud rate is for an easy comparison with the per-
`formance 90 Mbit/s rate 64-QAM systems. The results can be
`apRlied to another bit rate by appropriate scaling. In the simula-
`i s assumed. For fil-
`tion sa perfect carrier and symbol timing
`tering' we use FFT techniques to alternate between the fre-
`quency and time domains. The assumed PRBS sequence length
`is 16 384 bits. The number of samples per symbol is 32.
`as the ideal
`' The transmit and receive filters are assumed
`square root of raised-cosine filters with x/sin (x) equalization
`so that the whole system satisfies the
`in the transmitter,
`[ 71, [ log. The resulting eye 'diagrams
`Nyquist first criterion
`for CY = 0.1 and 0.4 are shown in Fig. 3(a) and
`(b) where a!
`denotes the rolloff factor, that
`is, the ratlo
`of the excess
`of an a =
`bandwidth to the Nyquist band. The eye diagram
`0.2 unequalized channel, having a sinusoidal group delay distor-
`i s illustrated in Fig. 3(c). Illustrative, measurement
`tion,
`results are shown in F g . 4 .
`Note that for the ideal filter there is no IS1 at the optimum
`sampling instant. However, data transition jitter
`is very signifi-
`(Y = 0.1 filtered case. This indicates
`Cant, particularly in the
`that a.high-precision sampling clock is required.
`For specified power of white Gaussian noise at the threshold
`detector input, the error probability
`of the ith symbol with
`respect to the in-phase channel is calculated as,follows;
`
`where Ii is the ith transmitted symbol of the in-phase channel,
`
`si is the magnitude of the ith received sample of the in-phase
`channel,
`T H R l i = I I i l - 1 ; T H R 2 i = I Z i ( + 4 ;
`erfc (x> LIZ e-r* d t
`
`2
`
`"
`
`
`
`Radio Communication of the IEEE
`Paper approved by the Editor for
`Communications Society for publication without oral presentation. Manu-
`script received May 11; 1984;
`revised October 18, 1984. This,work
`was
`supported by. the Natural Sciences and Engineering Research Council
`(NSERC) of Canada and by Karkar Electronics, Inc., San Francisco, CA.
`K. -T. Wu is with the Department of Electrical Engineering, University of
`Ottawa, Ottawa, Ont., Canada K1N 6N5, on leave from The National Taiwan
`University, Taipei, Taiwan, Republic of China.
`K. Feher is with the Department of Electrical Engineering, University of
`Ottawa, Ottawa, Ont., Canada
`KIN 6N5, and with Sianford University,
`Stanford,
`.
`.
`
`0090-6778/85/0500-0487$01.00 0 1985 IEEE
`
`N
`
`.
`
`
`
`and a2 is the received noise power at the threshold detector
`input. The error probability P,,Q with respect to the quadra-
`se-
`ture channel
`is obtained similarly from (2), and for a
`quence of N symbols in each channel, the average symbol
`error rate P, is calculated as follows:
`
`
`
`PMC Exhibit 2046
`Apple v. PMC
`IPR2016-00755
`Page 1
`
`

`

`488
`
`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-33, NO. 5, MAY 1985
`
`d a t a
`
`I
`
`2 5 6 QAM
`Modulator
`
`M =
`
`8 f o r
`
`6 4 QAM
`
`Fig. 1. 256 and/or 64 QAM system block diagram.
`
`Symbol e r r o r
`
`Calcula
`
`Sampler &
`M u l t i - l e v e l
`Threshold
`
`Fig. 2. Computer
`
`simulation model of 256-QAM and 64-QAM systems.
`
`(a)
`with sinusoidal group delay distortion, e(f)=
`Fig. 3. Eye diagrams of 256-QAM. (a) 01 = 0.1, (b) 01 = 0.4, (c) 01 = 0.2
`S, sin (2?rKf/2fBw), SD=
`12 ns, K = 4. Bit ratefb = 120 Mbits/s.
`
`PMC Exhibit 2046
`Apple v. PMC
`IPR2016-00755
`Page 2
`
`

`

`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-33, NO. 5, MAY 1985
`
`489
`
`a = 0 . 4
`
`Fig. 4. Measured 256-QAM and 64QAM demodulated I-channel eye
`diagrams. I-channel symbol rate f, = 200 kbaud corresponds to modem
`bit rate of 1.6 Mbits/s (256-QAM) and 1.2 Mbits/s (64-QAM). Raised
`cosine channel Nyquist filters (fN = 100 kHz) having a rolloff parameter
`attenuation beyond 120 kHz, designed by Karkar
`(Y = 0.2 and a 55 dB
`Electronics, Inc., are used in this experiment. Unequalized filters have a
`sinusoidal group delay distortion. (a), (b) 256-QAM, (c) 64-QAM.
`
`PMC Exhibit 2046
`Apple v. PMC
`IPR2016-00755
`Page 3
`
`

`

`490
`
`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-33, NO. 5 , MAY 1985
`
`(4 1
`
`. i r i GROUP DELAY DISTORTIONS
`.To investigate the effects
`of selective fading and/or radio
`system hardware imperfections, we insert between the trans-
`mit and
`tfie, receive filters a linear filter with the equivalent
`baseband transfer function C ( f ) . as shown in Fig. 2. We may
`express the frequency response C ( f ) as
`~ ( f ) = I cV) Jeie%f)
`and
`'4 (f = C ( f )
`-1 W f )
`D ( f ) = - - .
`27t
`df
`To simulate the group delay distortion we assume there
`i.e., A ( f ) = 1. The group delay
`no amplitude distortion,
`distortions are defined as follows:
`D ( f 1
`
`( 5 )
`
`( 6 )
`
`is
`
`I NO DISTORTION
`2 L D = 0.095 n s / M H z
`3 L D = 0.190 n s / M H z
`4 P,= 0 . 2 7 0 n s / M H 2 * * 2
`
`,
`
`6 S,=
`
`1 s,=
`
`6 . 0 0
`n s
`12.0 n s
`
`GROUP DELAY DiSTORTlON
`
`\
`
`Dlfl = SD sin(2nKf/2fgWl
`K = 4
`
`(sinusoidal1
`
`1b.00
`
`2'4.00
`
`'0
`e'.oo
`A.oo
`15.00
`2b.00
`2b.00
`55.00
`C/N I N d B
`Fig. 5. P, versus C / N for linear, parabolic, and, sinusoihl group delay
`distoitions for 256-QAM with a bit rate of 120 Mbitsls; Le., 15 Mbaud
`and CY = 0.4. Noise is definM in the double-sided Nyquist bandwidth,
`i.e., the equivalent noise bandwidth is 15 MHz.
`
`3 k . w
`
`4 b . w
`
`4
`
`16
`
`8
`
`
`
`12
`
`I
`0
`
`12
`
`24
`
`Parabolic ( n s )
`36
`
`4 8
`
`6 0
`
`7,
`in ns
`Fig: 6. Degradation of C / N versus 7, for linea;, parabolic, and sinusoidal
`group delay distortions for 256-QAM and 64-QAM with symbol rate of 15
`Mbaud and cy = 0.4 raised-cosine filters, where
`(linear) .,
`ns
`7, = LD(2fBW)
`= PO ( f ~ w ) '
`ns
`(parabolic)
`= s,
`ns
`(sinusoidal)
`few = (1 + CY) = 1.4f~ = 10.5 MHz.
`
`SD sin (2nKf/2fBW)
`
`for linear group delay
`,
`.
`
`for parabolic group delay
`for sinusoidal group delay
`
`(7)
`
`where
`f B W 2 (1 + b)fN
`(1 +
`and fN is the Nyquist bandwidth
`in baseband. This
`a)fN bandwidth definition (for group delay distortion) is more
`of the
`appropriate than fN, as it includes the critical effect
`grbup delay at the edge
`of the filter attenuation band. In
`the case of sinusoidal group delay, we present only results.for
`K = 4. The symbol error rate P, versus average C/N is com-
`5. We define a maxi-
`puted. Typical results are shown in Fig.
`mum group delay T , in the filter bandwidth ( 2 f B w) as follows:
`for linear group delay
`for parabolic group delay
`for sinusoidal.group delay. (8)
`
`The degradation of C/N as a function of T?, relative to the
`P, of loL4, is shown in Fig. 6.
`case with no distortion, for
`Note that here we simulate a 120 Mbit/s system with.
`a! =
`0.4. We also include the performance of a 90 Mbit/s 64-QAM
`01 = 0.4 in Fig. 6, which is confifmed to be the
`system with
`same as in [ 11. Again we notice that for a given value of maxi-
`in the filter bandw'idth, linear group
`mum group delay
`T,,
`delay ,causes the most severe degradation to the system's
`performance as compared to parabolic or sinusoidal group
`delay distortions [ i ] .
`
`IV. AMPLITUDE DISTORTIONS
`For the simulation of the amplitude distortion, we assume
`is equalued, i.e., D ( f ) is equal to
`the group delay distortion
`a constant. Three different characteristics for the amplitude
`distortion are defined as follows:
`A (f)
`L A * f
`P A 'f2
`
` for linear amplitude distortion
`
`
`
`
`
`
`
`
`
` for parabolic amplitude
`
`distortion
`- -
`SA sin ( 2 7 r K f / 2 f ~ W ) for sinusoidal amplitude
`distortion.
`
`(9)
`
`PMC Exhibit 2046
`Apple v. PMC
`IPR2016-00755
`Page 4
`
`

`

`49 I
`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-33, NO. 5, MAY 1985
`The symbol error rate P, versus C/N is computed. A typical
`of C/N for 256-QAM
`result is shown in Fig. 7. Degradations
`15 Mbaud and a = 0.4, versus maximum
`and 64-QAM with
`amplitude distortion A , in the filter bandwidth, are plotted
`in Fig. 8, where
`
`I NO DISTORTION
`2 L A = 0.038 dB/MHz
`
`5 P A = 0.008 d B / M H r * * 2
`6 S A = 0 . 2
` d B
`7 S A = 0 . 4
`
`d B
`
`AMPLITUDE DISTORTION
`
`A ( f ) = LA f
`
`(linear)
`
`(parabolic)
`A ( f ) = PA f2
`A I € ) = SA sin(2nKf/2fgW)
`K = 4
`
`(smusoidal)
`
`i
`
`and sinusoidal amplitude
`P, versus C / N for linear, parabolic,
`Fig. 7.
`distortions for 256-QAM with a bit rate of 120 Mbitsls, i.e.. 15 Mbaud
`and CY = 0.4. Noise is defined in the double-sided Nyquist bandwidth,
`Le., the equivalent noise bandwidth is 15 MHz.
`
`’ T -
`2 5 6 QRM (120 Mb/s)
`I
`6 4 OHM
`( 90 E?b/s)
`-------
`
`- Parabolic
`
`0.0
`
`I
`0.0
`
`0.5
`
`0 . 2
`
`1.0
`
`Sinusoidal
`0 . 4
`
`115
`
`016
`
`2 .o
`
`ON. 8
`
`A,
`
`in dB
`Fig. 8. Degradation of C/N versus A , for linear, parabolic, and sinusoidal
`amplitude distortions for 256-QAM and 64-QAM with symbol rate of 15
`Mbaud and a = 0.4 raised-cosine filters, where
`oinear)
`A m = LA(2fBW)
`(ParabOliC)
`= P A f S W Y
`(sinusoidal)
`= S A
`
`f s w = (1 + CY^, = 1.4fN = 10.5 MHz.
`
`for Iinear amplitude distortion
`for parabolic distortion
`for sinusoidal distortion.
`
`(1 0)
`
`We note that for a given value of maximum amplitude distor-
`tion ( A m z ) , linear amplitude distortion causes the least degrada-
`tion, followed in order of increasing degradation by parabolic
`and sinusoidal amplitude distortions.
`
`V. CONCLUSION
`The effects of amplitude and group delay distortions on the
`error performance on 256-QAM have been studied. Computer
`simulation and measurement results indicate that linear group
`delay distortion is the most critical group delay parameter.
`REFERENCES
`[l] T. Hill and K. Feher, “A performance study of NLA @-state QAM,”
`IEEE Trans. Commun., vol. COM-31, pp. 821-826, June 1983.
`of 16, 32, 64 and 128 QAM modulation
`[2] M. Borgne, “Comparison
`schemes for digital radio systems,” presented at IEEE GLOBECOM,
`San Diego, CA, 1983.
`[3] B. T. Bynum and E. W. Allen, “135 Mb/s-6 GHz transmission
`system design considerations.” presented at IEEE lnt. Conf. Com-
`mun.. Boston, MA, 1983.
`[4] T. Noguchi, T. Ryu, and Y . Koizumi, “6 GHz 135 MBPS digital radio
`system with 64 QAM modulation,”
`presented at IEEE
`Int. Conf.
`Commun., Boston, MA, 1983.
`[5] M. Subramanian, K. C. O’Brien, and P. J. Puglis, “Phase dispersion
`characteristics during fade in a microwave line-of-sight radio channel,”
`Bell Syst. Tech. .I., vol. 52, Dec. 1973.
`[6] G. M. Babler,
`“Selectively faded nondiversity and space diversity
`narrowband microwave radio channels,” Bell Syst. Tech. J . , vol. 52,
`Feb. 1973.
`Digital Communications: Microwave Applications.
`[7] K. Feher,
`Englewood Cliffs, NJ: Prentice-Hall, 1981.
`[8] Y. Saito, S . Komaki, and M. Murotani, “Feasibility considerations of
`high level QAM multi-carrier system,” presented at IEEE Int. Conf.
`Commun., Amsterdam, The Netherlands, 1984.
`[9] J. D. McNicol, S . G. Barber, and F. Rivest, “Design and application
`of the R D 4 A and RD-6A 64-QAM digital radio systems,” presented at
`IEEE Int. Conf. Commun., Amsterdam, The Netherlands, 1984.
`[ 101 K. Feher, Digital Communications: Satellite/Earth Station Engi-
`neering. Englewood Cliffs, NJ: Prentice-Hall, 1983.
`
`On Variable Length Codes Under Hardware Constraints
`
`HAIM GARTEN
`
`under the
`Abstract-A method for finding a variable length code set
`constraint of maximal clock rate is presented. Given a probability vector
`and requiring that the length of codewords would be an integer multiple
`of some K ( K > 2) reduces the maximal clock rate needed in implementing
`the compression system. On the other hand, given a maximal clock rate,
`the method enables us to use a quantizer with more levels than the usual
`
`Paper approved by the Editor for Signal Processing and Communication
`Electronics of the IEEE Communications Society for publication without oral
`pre.sentation. Manuscrip received February 17, 1984; revised August
`15, 1984.
`The author is with Rafael, State of Israel, Haifa.Israel.
`1$01.00 0 1985 IEEE
`
`PMC Exhibit 2046
`Apple v. PMC
`IPR2016-00755
`Page 5
`
`