DIFFERENTIAL MODULATION AND DEMODULATION OF MULTI-FREQUENCY
`DIGITAL COMMUNICATIONS SIGNALS
`
`Paul H. Moose
`Naval Postgraduate School
`Monterey, CA.
`
`Mercury Digital Communications, Inc.
`243 Eldorado St., Suite 201,
`Monteray, CA, 93940
`
`Abstract
`Multi-Frequency Modulation, is a
`bandwidth efficient digital
`communication signalling technique that
`may be employed effectively in mobile
`satellite communications links. A
`feature of MFM is that it requires good
`synchronization because it is coherent
`and signal symbols have high time-
`bandwidth products. In this paper we
`describe algorithims for generating and
`demodulating differentially encoded
`Multi-Frequency Quadrature Phased Shift
`Keyed signals using Discrete Fourier
`Transform techniques. Experimental
`results are presented showing good
`agreement with the theory.
`
`INTRODUCTION
`In order to increase the data rate
`of digital communications through
`bandwidth restricted channels, some
`MODEMS use multiple carrier frequencies
`spaced throughout the available
`bandwidth, each frequency independently
`modulated with digital information. The
`frequencies are transmitted
`simultaneously during each baud
`interval. This type of modulation is
`called Multiple Frequency Modulation
`(MFM). One advantage of MFM is that
`inter-baud interference can be reduced
`by introducing a small guard time
`between successive baud. In order to
`prevent inter-frequency interference,
`the frequencies must not be spaced too
`closely. Ideally, the frequencies are
`made orthogonal over one baud interval.
`The frequencies in the set will be
`mutually orthogonal so long as they are
`spaced at exact multiples of the
`reciprocal of the baud length, that is
`at the baud rate.
`
`One problem with the MFM systems
`described above is that although they
`can effectively eliminate inter-baud
`interference and inter-frequency
`interference within the bauds, they must
`be demodulated using fully coherent
`receivers for each frequency. Since the
`frequencies are subject to different and
`unknown amplitude and phase changes
`introduced by the transmission channel,
`coherent reception by inclusion of a
`pilot tone will not be effective as was
`discovered by Alard et a1 [l] in a
`prototype model of a UHF satellite sound
`broadcasting utilizing MFM signalling
`techniques. Cimini has suggested a
`system that would correct for the phase
`of the different carriers by sending a
`training or pilot signal set through the
`mobile channel [ 2 ] .
`Differential encoding of the
`carriers provides a more practical
`solution to this problem with of course
`an attendent loss of performance against
`additive noise of approximately 3 db.
`The conventional way to differentially
`encode data is from baud to baud as is
`found in ordinary DPSK and in the MFM
`system of [l]. However, this is not
`desireable when the bauds are long, due
`to channel instability such as that
`introduced by fading or if asynchronous
`or packet transmissions are envisoned
`because of the potential loss of data
`rate when only two or three bauds are
`sent per packet. (Differential encoding
`in time requires using one baud as a
`reference). The solution to this problem
`is to differentially encode the symbols
`in frequency. Differential encoding
`between frequencies only requires that
`the channel be constant in phase over
`very narrow bands corresponding to the
`baud rate. In MFM the baud rate can be
`chosen independentlv of the data rate
`and so, in principle it can be chosen
`to make the differential coding
`effective for any given channel.
`
`12.4.1.
`CH2831-619010000-0273 $1 .OO 0 1990 IEEE
`
`0273
`
`Petitioner Sirius XM Radio Inc. - Ex. 1006, p. 1
`
`

`

`‘igure 1 An MFDQPSK Transmitter/Receiver
`Even though phase coherent
`reception of a differentially encoded
`MFM tone set is not required, baud
`(symbol) sync is nessecary to support
`asynchronous modes. We provide this
`synchronization by sending a known sync
`baud at the beginning of each packet of
`bauds. The sync baud is an ordinary MFM
`baud with randomly selected phases for
`the tones. The sync baud is detected
`using a polarity only matched filter.
`The matched filter peak output occurs
`exactly at the end of the baud. This
`peak is detected and used to initiate
`signal acquisition of the data bauds. A
`block diagram showing the principal
`components of a MFDQPSX MODEM is shown
`in Figure 1.
`It is useful to note how generation
`and demodulation of an MFM baud using
`Inverse Discrete Fourier Transform
`(IDFT) and Discrete Fourier Transform
`techniques as suggested by Weinstein and
`Ebert [ 3 ] and later by Hirosaki [ 4 ]
`permits encoding and decoding directly
`in the frequency domain.
`
`THEORY OF MFDQPSK
`The following definitions are used
`in MFM :
`T: Packet length in seconds
`T: Baud length in seconds
`k,: Baud length in number of
`samples
`L: Number of bauds per packet
`t: Time between samples in seconds
`f,=l/ t: Sampling or clock
`frequency for D/A and A/D conversion in
`Hz.
`f=l/ T: Frequency spacing
`(minimum) between MFM tones.
`K: Number of MFM tones.
`
`Let the lth baud of the transmit signal
`be described by;
`
`x(u) =
`(1)
`xk(u)
`where ,
`x,(u) = Akcos(2rk fu+@,) ;OSuS T. (2)
`Here, U is time referenced to the
`beginning of the baud. Now the discrete
`time signal corresponding to the lth
`baud is found by sampling (1) and (2) at
`the sampling intervals
`t=l/f, and is
`given by ;
`x(n) = C x,(n)
`( 3 )
`where,
`x,(n) = A,~os(Zrkn/k,+@~) ;O<nlk,-1. ( 4 )
`Here, n is discrete time referenced
`to the beginning of the baud. Note that,
`in general, k may take on all integer
`values between 0 and k,/2-1. We will
`refer to k as the llharmonic numberll of
`the MFM tone of frequency k f.
`Consider the k, point Discrete
`Fourier Transform (DFT) of ( 3 ) . It is
`given by;
`
`Thus, it is apparent that the discrete
`time signal ( 3 ) is given by the k, point
`Inverse Discrete Fourier Transform
`(IDFT) of ( 5 ) and ( 6 ) , namely;
`x(n) = IDFT[X(k) 3 ; Olk,nlk,-1. ( 8 )
`
`0274
`
`12.4.2.
`
`Petitioner Sirius XM Radio Inc. - Ex. 1006, p. 2
`
`

`

`a. Compute
`X,(k)=Y(k)Y*(k-1) (l+j) ;k,+llklk,+K. (lo)
`b.If Re[X (k)]IO, then the
`least significant bft of the kth di-bit
`of the baud is 1; otherwise it is 0.
`c.If Im[x,(k)]rO, the the most
`significant bit of the kth di-bit of the
`baud is 1; otherwise it is 0.
`Sisnal-to-Noise Ratios /SNR‘s) and the
`- DFT
`Let the receiver input signal-to-
`noise ratio, SNR,, be defined as the
`signal power in bandwidth W divided by
`the noise power in bandwidth W. Also
`define the average tone signal power as;
`
`then ,
`
`P,/K
`
`(11)
`
`Po =
`(12)
`[SNR], = KPJWN, = PJ fN,
`Note that the narrowband input SNR;
`[SNR],, = Pk/ fN, = mE,,/N,
`(13)
`is the same as the wideband SNR of (11)
`if the power in tone k is the same as
`the average tone power. E,, is the
`average received energy per bit carried
`by the tone k and m is the number of
`bits per tone.
`Now consider the k,-point DFT
`coefficients, Y(k,)of the input sequence
`Y (n) ;
`
`Differential Encodins OPSK in the
`Frequency Domain
`
`In QPSK, the real and imaginary
`parts of X(k) each carry one bit of
`information for each tone in the tone
`set within the transmission bandwidth.
`In satellite transmissions the original
`information band will be bandshifted to
`the appropriate channel location for
`combination and transmission with the
`other channels. Using MFM, the data band
`can be selected in such a way to
`simplify the band shifting operation.
`For example, a 45KHz wide MFM signal may
`be generated directly between 60KHz and
`105KHz, leaving a 120KHz guard band
`between the alias spectrum to be
`filtered out in the bandshifting
`operation. In this example, we could
`select a sampling frequency f, of
`256KHz, a tone spacing and baud rate of
`lKHz and select harmonic numbers 60 thru
`105 to carry information. The amplitudes
`of all other harmonics between 0 and 127
`would be set to zero.
`Differential encoding in the
`frequency domain is accomplished by
`setting
`
`X (k) =X (k-1) D (k)
`where D(k) are modulation values
`generated by input message di-bits in
`accordance with Table 1 for k,+llklk,+K.
`
`(9)
`
`Input Di-bit
`1+3 o
`00
`01
`O+ j
`0- j
`10
`-I+ j 0
`11
`TABLE 1
`Differential Encoding Algorithim
`
`The phase of the initial tone, k, is
`arbitrary.
`
`Demodulation and Differential Decodinq
`Demodulation of MFM is accomplished
`by a process inverse to its generation.
`Given the time domain signal at the
`receiver y(u) on the interval O<ul T:
`The signal is sampled at f samples per
`second and converted to digital format
`with an A/D converter. The k, real
`values thus obtained are loaded into a
`k, point complex valued array (the
`imaginary parts are set to zero). This
`becomes the array for the lth baud. Its
`k, point DFT yields the complex valued
`array Y(k) containing, in its first
`half, the amplitude and phase modulation
`information, of the K harmonics
`employed in the generation of the
`transmit signal.
`To differentially decode MFDQPSK,
`we proceed as follows:
`
`Y(k) = S ( k ) + W(k)
`(14)
`where ,
`S (k) = 4 ( 2P,) ‘k,exp ( j@,) ; k,<klk, (15)
`and the W(k) are the DFT coefficients of
`the white noise sequence. It is easily
`shown that
`
`E[W(k)] = 0
`(16)
`and ,
`E[Re(W(k) ) 2 ]=E[Im(W(k) ) 2 ]= fN,kX2/4 (17)
`E[Re{W(k) )Im{W(k) )] = 0.
`(18)
`Furthermore, the Real and Imaginary
`parts of W(k) and W(i) are uncorrelated
`with one another for k z i.
`Let us define the output signal-to-
`noise ratio for the kth tone, SNR,, of
`the MFDQPSK receiver as the ratio of the
`square of the mean of the Real (or
`Imaginary) part of X,(k) to its
`variance. It can be shown [5], that
`these are given by
`l+. 5/ [ SNR] ,,,
`[SNR], = rSNRl,&
`
`= &&,
`(19)
`l+. 25/ ( Ebk/No)
`
`12.4.3.
`
`0275
`
`Petitioner Sirius XM Radio Inc. - Ex. 1006, p. 3
`
`

`

`noise is highest, some 23.5 db, for the
`smallest tone spacing of 15 Hz. Since
`data rate in MFM is constant at m bits
`per Hz of channel bandwidth ( 2 bits per
`Kz for MFDQPSK), system performance in
`badly unequalized channels is enhanced
`significantly by using low baud rates,
`that is small tone spacings, so that
`adjacent tones incur nearly identical
`phase and amplitude changes through the
`channel.
`White noise was added to the analog
`signal x(t) and output signal-to-noise
`ratios were measured versus EdNo. The
`results are shown in Figure 4 for the
`same baud rates used in Figure 3. Note
`that as EdN, increases, the output S N R s
`tend toward the system noise as
`expected. We also note that performance
`seems two to three db better than theory
`for most cases. The source of this
`unexpected benefit has not yet been
`determined but it may be due to the
`noise not being exactly whitelso that
`there is some correlation in the phase
`noise from harmonic to harmonic in the
`receive DFT.
`
`a
`
`0 ..o
`
`66.0
`
`120.0
`AF
`
`leil.0
`
`I
`
`‘igure 3 System Noise Signal-to-Nois
`Ratio vs. Baud Rate
`
`where we have used the fact that for
`QPSK, m=2 ;that is two bits are sent
`with each tone. For relatively high
`values of E d N o , the demodulated
`outputs, Re[X,(k) ] and Im[X,(k) ] are
`very nearly gaussian random variables
`with output signal-to-noise ratios just
`1/2 those of ordinary QPsK (or MFQPSK).
`Under the gaussian approximation, the
`probability of a bit error using the
`decision rules of (12) and (13) is given
`by
`
`and Q(x) is the error probability Q-
`function. Pb is shown versus E d N , in
`
`Figure 2 Probability of Bit Error foi
`MFDQPSK
`
`Figure 2.
`Emerimental Results
`An MFM system has been configured
`to transmit MFDQPSK over a 4 KHz
`bandpass channel. Tone spacings, that is
`baud rates, of 15, 30, 60, 120, and 240
`Hz were tested. Output signal-to-noise
`ratios were estimated by computing
`sample means and variances of the real
`and imaginary parts of Xa. Data were
`averaged across all tones in the MFM
`baud and over several bauds in order to
`reduce the variance of the estimates.
`System noise is shown in Figure 3.
`sources of system noise are numerical
`noise in the FFT‘s, assummed minimal,
`quantization noise induced by the 8 bit
`representation of the discrete time
`signal values x(n) at the transmitter
`and 12 bit representation of the y(n) at
`the receiver, and phase error in the
`differential decoding due to slightly
`different phase shifts of adjacent tones
`[ 6 ] . As expected, since the differential
`phase is less for tones closer together
`than for tones far apart, the maximum
`signal-to-noise as set by the system
`
`0276
`
`12.4.4.
`
`Petitioner Sirius XM Radio Inc. - Ex. 1006, p. 4
`
`

`

`0
`0
`
`- ? g!! -
`*
`
`3
`0
`(L Z ?
`' " 2
`
`?
`
`I
`
`X - AF OF 30
`
`. 0 - AF OF 15
`
`l1.b
`
`2.0
`
`7!0
`lJ.0
`(DB)
`Et/No
`C Figure 4 Output SNR vs. E d N o
`Conclusions
`MFM has been shown to be an
`attractive approach for digital
`satellite communications to mobile
`receivers [l]. In this work, we have
`developed the theory and a prototype
`system for differentially encoding and
`decoding MFQPSK in the frequency domain,
`a practical way to solve problems of
`coherent reception with minimal
`performance loss. Furthermore, by using
`long baud intervals and corresponding
`small spacing of the carrier tones,
`problems associated with channel fading
`should be greatly relieved with respect
`to the previous method of differentially
`encoding the multiple carrier tones from
`baud to baud. A frequency differential
`encoded 16-QAM system capable of 4
`bits/Hz transmission rate has been
`developed and is presently being
`evaluated.
`
`REFERENCES
`[l] M. Alard and R. Halbert, "Principles
`of Modulation and Channel Coding for
`Digital Broadcasting for Mobile
`Receivers", EBU Review, No. 224, August
`1987.
`[2] L. J. Cimini, Jr. IIAnalysis and
`Simulation of a Digital Mobile Channel
`Using Orthogonal Frequency Division
`Multiplexing18, IEEE Trans. on Comm.,VOl.
`Con-33, No. 7, July 1985.
`[3] S. E. Weinstein and P. M. Ebert, I'
`Data Transmission by Frequency-Division
`Multiplexing Using the Discrete Fourier
`Transform", IEEE Trans. on CO".
`Tech.,
`Vol. Com-19, No. 5, Oct. 1971.
`
`[ 4 3 B. Hirosaki , "An Orthogonaly
`Multiplexed QAM System Using the
`Discrete Fourier TransformIt, IEEE Trans.
`Vol Com-29, NO. 7, July 1981.
`on CO".,
`[5] P. H. Moose, "Performance of
`Decoding Algorithims for Differentially
`Encoded Multi-Frequency Modulation",
`Naval Postqraduate School Tech Rpt 62-
`90-012, July, 1990.
`[6] T. K. Gantenbein,Implementation of
`Multi-Frequency Modulation on an
`Industry Standard Computer, Naval
`Postgraduate School, Monterey, CA., MSEE
`(Sept 1989), March 1990.
`
`12.4.5.
`
`0277
`
`Petitioner Sirius XM Radio Inc. - Ex. 1006, p. 5
`
`