SIGNAL PROCESSING V
`THEORIES AND APPLICATIONS
`
`Proceedings of EUSIPC0-90
`Rfth European Signal Processing Conference
`Barcelona, Spain, September 18-21, 1990
`
`Edited by
`Luis TORRES
`Enrique MASGRAU
`Miguel A. LAGUNAS
`Department of Signal Theory and Communications
`ETSIT-UPC
`Barcelona, Spain
`
`VOLUME Ill
`
`1990
`
`ELSEVIER
`AMSTERDAM•NEWYORK •OXFORD•TOKYO
`
`Petitioner Sirius XM Radio Inc. - Ex. 1015, p. 1
`
`

`

`ELSEVIER SCIENCE PUBLISHERS B.V.
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`
`© Elsevier Science Publishers B.V., 1990
`©British Crown Copyright, 1990: pp. 433-436
`
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`
`Petitioner Sirius XM Radio Inc. - Ex. 1015, p. 2
`
`

`

`SIGNAL PROCESSING V: Theori9s and Applications
`L. To"9s, E. Masgrau, and M.A. Lagunas (9ds.)
`Els9Vi9r SciMC9 Publish9rs B. V., 1990
`
`1807
`
`DIFFERENTIALLY CODED MULTI-FREQUENCY MODULATION FOR DIGITAL COMMUNICATIONS
`
`Paul H. Moose
`
`Department of Electrical and computer Enqineerinq
`Naval Postqraduate School
`Monterey, CA 93943
`and
`Mercury Diqital Communications, Inc.
`243 Eldorado st., suite 201,
`Monterey, CA, 93940
`
`Multi-frequency Modulation (MFMN), utilizing a multiplicity of orthogonal
`carrier tones simultaneously, produces a robust, bandwidth efficient signal
`for digital communications. Siqnals are generated at baseband or bandpass
`with minimal hardware requirements. Differential coding between adjacent
`carrier tones, providing the tones are closely spaced, eliminates the need
`for coherent carrier reference signals and for channel equalization.
`Encoding/decoding take place utilizing fast Fourier transforms.
`
`1.
`
`INTRODUCTION
`
`Multi-frequency Modulation (MFMN) is a
`new method for data communications that
`relys on digital signal processing
`capabilities resident in the host
`sending and receiving microcomputers to
`qenerate and demodulate the actual
`physical analog signals sent over the
`link. Interfacing to the link is via
`digital-to-analog (D/A), and analog-to(cid:173)
`digital (A/D) converters. The frequency
`spectrum of the signal, either bandpass
`or baseband, is controlled by an
`externally supplied clock to the D/A and
`A/D. MFM is a packet oriented signalling
`format that sends K tones per baud for L
`bauds. These KL signals form an
`orthogonal signal set. Data are encoded
`in the amplitude and phase of each of
`the KL signals. In differentially
`encoded MFM, data are encoded as the
`change in amplitude and/or of phase
`between two adjacent tones within the
`same baud. Differential encoding of MFM
`signals is extremely effective when
`successive bauds or adjacent frequencies
`are subject to identical but unknown
`amplitude andjor phase changes between
`the transmit and receive computers. In
`this paper we describe the encoding,
`generation, demodulation, decoding and
`performance of Multi-Frequency
`Differential Quadrature Phase Shift
`Keyed (MFDQPSK), and of Multi-Frequency
`Differential 16-QAM (MFD16-QAM).
`
`2. THEORY
`
`MFM signals are generated inside the
`host transmit microcomputer using an
`
`Inverse Discrete Fourier Transform
`(IDFT). The OFT technique was first
`suggested by Weinstein and Ebert [1] and
`has subsequently been further described
`by others {see, for example, [2],[3],
`and [4]}. Each baud consists of a
`digital signal of kx real values. When
`clocked out upon command through an I/O
`port to a D/A converter at fx samples
`per second, a baud of length ~T = kxlfx
`seconds is sent over the channel. The
`signal consists of tones spaced at
`·intervals ~f = 1/~T Hz. A bandpass
`signal is generated in the b~nd f 1 =
`k1 ~f to f 2 = k,~f by assigning non-zero
`amplituded only to those digital
`frequencies between k 1 and k2 = k 1+K. A
`baseband signal is generated by
`assigning all digital frequencies
`between one and kx/2-1 non-zero
`amplitudes. In both cases, the actual
`frequency spectrum occupied by the
`signal is controlled by the clock
`frequency fx. Concatenation of L signal
`bauds produced by an L-fold repetition
`of this process creates a signal packet
`of length L~T.
`
`From the discussion above, we see that
`in MFM the data to be transmitted with
`each baud are encoded directly in the
`frequency domain as complex numbers. In
`MFQPSK,
`two bits (a di-bit) are sent
`with each digital frequency using the
`state diagram of Figure 1. Since the
`signal occupies a bandwidth of K/~T, the
`throughput rate is 2 bits per Hz of
`occupied channel bandwidth. In MF16-QAM,
`four bits are sent with each digital
`frequency using the state diagram shown
`in Figure 1. Three bits are encoded into
`
`Petitioner Sirius XM Radio Inc. - Ex. 1015, p. 3
`
`

`

`1808
`
`the 8 phases and one bit is encoded as
`amplitude of the digital frequencies.
`Using this constellation, data is
`transmitted at a throughput rate of 4
`bits per Hz of channel bandwidth. An
`MFM packet is illustrated in Figure 2.
`
`I.
`
`• !)~
`
`• Sa
`
`'S, s,
`•
`
`• Sq
`
`• s,'l..
`
`,s,
`s
`4•
`
`.s3
`
`.s~
`
`• s.
`
`•St.
`
`•So
`
`R
`.s,
`•S\4- • s,s-
`•S,o •s.'l.
`Su l • 5,3
`
`1
`
`Figure 1
`MFM constellations
`
`Tone~~
`k,
`
`ltL
`
`[ ERII 1 c7J~tl\ .J (l!A
`±
`' " .~"
`T
`AT- 1
`"''..'/, llll
`'' ~--.....
`-4 AT~ I
`I
`~ L~T ----4
`
`t
`(~+
`~
`
`Figure 2
`An MFM Signal Packet
`
`The demodulation and decoding of MFM is
`accomplished as the inverse of the
`encoding and modulation process. That
`is, at the receiver L real valued
`sequences of kx points are obtaine~
`from the packet of L bauds by sampl1ng
`the received analog signal at fl samples
`per second and storing the samp es in
`the host receiver microcomputer's RAM.
`The kx point DFTs are obtained of the L
`sequences, but only those complex
`coefficients that correspond to
`transmitted digital frequencies are
`retained for decoding. In an additive
`white gaussian noise memoryless channel,
`the 2KL values obtained in this manner
`are statistically independent gaussian
`random variables with identical standard
`deviations and with means that depend on
`the transmitted data values. Dividing
`each of these values by the standard
`deviation yields a set of 2KL
`statistically independent, unit
`
`variance, gaussian random variables
`that have mean values given by
`
`E[Rl (k)]
`
`E[Il(k)]
`
`{2Ekl/N0 }~'~ cos¢kl
`{ 2Ekl/N0 } y, sin¢kl
`where EIU is the energy and <Pl~ is the
`phase or the kth tone during the lth
`signal baud and the white noise has
`power spectral density Nof2 (See the
`Appendix).
`
`(1)
`
`The bit error rate for MFQPSK is
`identical to the bit error rate of
`ordinary QPSK and the symbol error rate
`of the 3 phase bits of MF16-QAM is the
`same as the symbol error rate of
`ordinary 8 PSK given the same Ekl/N0
`3.
`DIFFERENTIAL CODING IN THE
`FREQUENCY DOMAIN
`
`•
`
`Demodulation of MFM is strictly coherent
`and requires that phase synchronization
`between the transmitter and receivers be
`maintained for each of the multiplicity
`of carrier frequencies in the MFM
`signal. For links involving radio
`frequency ( or acoustic) propagation
`between the transmitting and receiving
`microcomputers these requirements may be
`difficult or impossible to meet. In such
`cases, differential encoding should be
`employed.
`
`In Multi-frequency Differential
`Modulation (MFDM) , symbols are
`differentially encoded within each baud
`between adjacent tones. The differential
`encoding algorithims that we employ are
`given in Table I. The first digital
`frequency, k 1 , is always assigned state
`Sq. K+1 digital frequencies are sent
`w1th each signal baud. At the receiver,
`following the OFT, the complex product
`between the OFT coefficient of digital
`frequency k and the complex conjugate of
`the OFT coefficient of digital frequency
`k-1 is formed. In the case of MFDQPSK,
`the result is multiplied by exp(j~/4);
`in the case of MFD16-QAM, the result is
`multiplied by exp(j~/8). Consideration
`of Table I shows that this realigns the
`differentially encoded phase-bits to the
`constellations of Figure 1. It is shown
`in the Appendix
`that the 2KL values
`thus obtained are approximately gaussian
`random variables and that after
`normalization by dividing by their
`standard deviations they have mean
`values given by
`
`E[Rl(k)]
`
`Akcos¢kl
`
`E[Il(k)]
`with,
`
`Aksin¢kl
`
`(2)
`
`Petitioner Sirius XM Radio Inc. - Ex. 1015, p. 4
`
`

`

`( 3)
`
`Comparing (1) and (2), we see that when
`adjacent frequencies have equal E~ there
`will be a theoretical loss of 3 db in
`output signal-to-noise ratio compared to
`coherent demodulation of each tone
`separately. However, actual measurements
`on a prototype system show that MFDQPSK
`performs within one to two db of MFQPSK
`over an Ek/N0 range of 5 to 20 db [Ref
`6]. The reasons for this are as of now
`not completely clear, but are believed
`to be due to positive correlation of the
`noise of adjacent tones.
`
`As can be seen from (2), the signal-to(cid:173)
`noise ratio for decoding the phase bits
`in MFD16-QAM depends on the amplitude
`bit. If the amplitude bit is a zero,
`then one amplitude is high and one is
`low; if the amplitude bit is one, then
`adjacent tones have equal amplitudes
`which are either both high or both low.
`Phase bit decoding errors are dominated
`by the case when both amplitudes are
`low. The probability of a differential
`phase symbol decoding error, given both
`amplitudes low, is [See, for example,
`Hakin [5], pg. 317]
`
`(4)
`
`For moderately high signal-to-noise
`ratios, the probability of an amplitude
`bit error is closely approximated by
`
`(5)
`
`where we have used high energy symbols
`with 25/4 the energy of the low ones.
`Overall, the symbol error probability
`for MFD16-QAM is bounded by
`
`(6)
`
`Figure 3 shows theoretical upper bounds
`for 4-bit symbol error probabilities for
`MFDQPSK and MFD16-QAM versus average
`• MFQPSK is shown for comparison.
`EtJN0
`4.
`
`CONCLUSIONS
`
`Multi-frequency modulation is an
`extremely robust, bandwidth efficient
`technique for digital data
`communications. It relys on OFT
`algorithims to modulate and demodulate
`the data. The practical application of
`MFM is greatly enhanced by
`differentially encoding the information
`to be transmitted between adjacent
`digital frequencies. Differential
`coding\decoding algorithims have been
`described and theoretical performance
`results have been given herein for
`MFDQPSK and MFD16-QAM. The theoretical
`
`1809
`
`-+
`41.10
`
`-"
`
`4'1..10
`
`:z.o
`
`10 E~fNo IS (~ \,)
`Figure 3
`MFDM 4 Bit Symbol
`Error Probabilities
`
`INPUT DATA
`FOR
`MFD16-QAM/MFDQPSK
`
`STATE FOR TONE k
`GIVEN TONE k-1 IS
`IN STATE Sz,/S2n+1
`
`0000
`0001/00
`0010
`0011
`0110
`0111/01
`0100
`0101
`1100
`1101/11
`1110
`1111
`1010
`1011/10
`1000
`1001
`
`TABLE
`I
`(Index addition modulo 16)
`Differential Encoding
`
`performance is approximately 3db lower
`than for non-differential MFM. Actual
`data suggest these results may in fact
`be too pessimistic; however, even with
`the 3 db reduction, differential coding
`should be employed because it reduces or
`eliminates the need for channel
`equalization.
`
`APPENDIX
`
`Let the lth baud of an MFM signal at the
`receiver be given by
`
`Petitioner Sirius XM Radio Inc. - Ex. 1015, p. 5
`
`

`

`1810
`
`y(t)
`
`x(t) + w(t)
`
`with,
`
`(7)
`
`(8)
`
`where ak is the phase of the
`differentially encoded phase bits
`realigned to the constellations of
`Figure 1. The real and imaginary parts
`of D(k) are uncorrelated and have equal
`variances
`
`and w(t) white gaussian lowpass noise in
`the band o to kx/2~T with power spectral
`density Nof2. Ek is the received signal
`energy and ¢
`is the phase of tone k
`during the l~h baud. Sampling (7) at
`~T/kx samples per second produces the
`discrete time signal
`
`y(n) = x(n) + w(n)
`
`; 0:5n:5kx-1
`
`(9)
`
`) •
`
`Var{Re[D(k)]}=Var{Im[D(k)]}=
`2 (Ek+Ek_ 1+N0
`Dividing the real and imaginary parts of
`(15) by the square root of (17) yields
`the unit variance differentially
`demodulated random variables with mean
`values given by (2).
`
`(17)
`
`with,
`
`N A trademark of Mercury Digital
`Communications, Inc.
`
`(10)
`
`REFERENCES
`[1] s. B. Weinstein and P. M. Ebert, "
`Data Transmission by Frequency-Division
`Multiplexing Using the Discrete Fourier
`Transform", IEEE Trans. on comm. Tech.,
`Vol. com-19, No. s, Oct. 1971.
`
`[2] M. Alard and R. Halbert, "Principles
`of Modulation and Channel Coding for
`Digital Broadcasting for Mobile
`Receivers", EBU Review, No. 224, August
`1987.
`
`[3] L. J. Cimini, Jr. "Analysis and
`Simulation of a Digital Mobile Channel
`Using Orthogonal Frequency Division
`Multiplexing", IEEE Trans. on Comm.,Vol.
`com-33, No. 7, July 1985.
`
`[4] B. Hirosaki, "An Orthogonaly
`Multiplexed QAM System Using the
`Discrete Fourier Transform", IEEE Trans.
`on comm., Vol Com-29, No. 7, July 1981.
`
`[5) Simon Hakin, Digital Communications,
`John Wiley and Sons, New York,1988.
`
`[6) T. K. Gantenbein,Implementation of
`Multi-Frequency Modulation on an
`Industry Standard Computer, MSEE Thesis,
`NPS, Monterey, CA., March 1990.
`
`and white noise sequence w(n) with zero
`mean and variance N kx/2~T. The k point
`OFT of (10) is the frequency dom~in
`random complex sequence
`
`with mean value,
`
`and white gaussian complex noise
`sequence with uncorrelated real and
`imaginary parts and with
`
`Var{Re[W(k))}=Var{Im[W(k)}=
`\k/Nc/ ~T.
`
`( 13)
`
`rhe properties of the noise components
`of the OFT coefficients follow directly
`from the properties of w(n) and the
`~efinition of the OFT. Dividing (12) by
`the square root of (13) yields a
`frequency domain random sequence
`
`F(k) = R(k) + ji(k)
`
`(14)
`
`with unit variance, gaussian
`uncorrelated real and imaginary parts
`and with data dependent mean values
`given by (1).
`Differential decoding the phase
`bits is accomplished by computing the
`random data sequence
`
`D(k) = F(k) F* (k-1) exp[j71j2m]
`k,+1:5k:5k2
`
`;
`
`(15)
`
`for the m bits that are differentially
`encoded as phase between adjacent
`digital frequencies in each MFDM baud.
`From the properites of W(k), it follows
`directly that the mean value of D(k) is
`given by
`E[D(k)] = 2 (EkEk_ 1)"'N0exp[j6k]
`
`(16)
`
`Petitioner Sirius XM Radio Inc. - Ex. 1015, p. 6
`
`