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Page 1: Hybrid Amplitude-and-Phase Modulation for Analog Data Transmission

634 IEEE TRANSACTIONS ON COMhIUNICATIONS, VOL. conf-23, NO. 6, JUNE 1975

Hybrid Amplitude-and-Phase Modulation

for Analog Data Transmission C. R!IELVIL THOMAS, MEMBER, IESE, CUltTIS L. AIAY, MEMBER, IEEE,

AND GEORGE R. WEI,TI, MEMBER, I I ~ E

1 Abslract-Pulse-amplitude modulation (PAM) is extended to two

and four dimensions by means of a nonlinear, memoryless mapping to obtain pulse-amplitude-and-phase modulation (PAPM). This modulation requires either the same bandwidth as double-sideband PAM or twice as much but yields a signal-to-noise ratio (SNR) im- provement factor similar to that of FM with considerably less sacri- fice of spectral occupancy.

Several signal-space mappings are investigated for Gaussian data. Expressions for output SNR are derived in terms of a linear error above threshold and an anomalous threshold-producing error. PAPM is compared with various modulation techniques in terms of power and bandwidth requirements.

The degradations in output SNR caused by carrier phase, carrier amplitude, and sample timing reference errors are evaluated so that requirements for demodulator design can be specified.

I. INTRODUCTION

T H E INCRC 4 ASED emphasis on spectrum conservation coupled with increasing conlmunications traffic de-

mands has led to a renewed interest in modulation tech- niques which efficiently utilize bandwidth. Several recent papers discuss the use of hybrid amplitude-and-phase keying with large symbol alphabets for transmitting digital information [i]-[3]. This paper considers the use of hybrid amplitude-and-phase modulation (APM) for transmission of sampled analog data such as frequency- division-muitiplexed (FDM) telephony.

The most common modulation currently applied to FDA1 telephony is frequency modulation (FM) . FM, how- ever, uses a significant amount of bandwidth, although it does provide a signal-to-noise ratio (SNR) improvement factor, or process gain, in exchange for this bandwidth. Single-sideband amplitude modulation, a contrasting possibility, attains a bandwidth ratio of unity but pro- vides no SNR improvement.

A concept for achieving both SNR improvement and low bandwidth by mapping a sampled analog signal into multiple dimensions has recently been introduced by McRae [lo] as “mixed base modulation.” In this concept

Systems of the IEEE Communications Society for publication with- Paper approved by the Associate Editor for Data Communication

out oral presentation. Manuscript received May 10, 1974; revised

national Telecommunications Satellite Organization (INTELSAT). November 22, 1974. This work was supported in part by the Inter-

Views expressed in this paper are not necessarily those of INTELSAT.

C. M. Thomas is with the Advanced Analysis Staff, TRW Sys- tems, Redondo Beach, Calif.

C. L. May was with TRW Systems, Redondo Beach, Calif. He is now with the Advanced Programs Department, Magnavox Re- search Laboratories, Torrance, Calif., and the Department of Electrical Engineering, California State University, Long Beach, Calif.

G. R. Welti is with COMSAT Laboratories, Clarksburg, Md.

nonlinear, or “twisted,” modulation [4], [SI is employed to obtain a SNR improvcmcnt a t the expense of introduc- ing a demodulator threshold. Each dimension is trans- mitted as a separate orthogonal pulse by single-sideband or quadrature modulation (such as that used for the color subcarrier in. telcvision) so that the bandspread factor equals the mapping dimensionality.

For lorn dimensionality ( 5 5 ) McRae showed that ideal mixed-base modulation performs significantly closer to the rate distortion bound than other commonly employed digital and analog modulation techniques.

This paper reports the results of studies [SI, [7] which investigated the performance of certain representative one-pulse and two-pulse pulse-amplitude-and-phase modu- lation (PAPM) mapping transformations with a non- idealized transmission channel approximating a com- munication satellite system with realistic modem equip- ment. Not only are the basic results of McRae extended to the more interesting case of Gaussian data, but the effects of mapping types and parameters are illustrated. More importantly, the effects of reference errors in the demodulator are evaluated so that modem design require- ments can be specified to yield a desired output per- formance.

The basic signal structure of PAPM, as well as four of the mappings which have been investigated,.are described in Section 11. Section I11 derives the performance of the PAPM modulation schemes in terms of output SNR versus average carrier-to-noise ratio (CNR) . Two types of, error are shown to affect the PAPI\/I demodulator’s estimate of the transmitted baseband signal. The first is an error that arises from interference parallel to the signal mapping locus; this error is inversely proportional to CNR and predominates a t high CNR. The second is caused by interference normal to the mapping locus and produces large output errors, or “clicks,” when these anomalous events occur. At low CNR anomalous errors predominate and produce a demodulator threshold. As in FM, there is a tradeoff between SNR improvement and threshold level. The section concludes with a comparison o f PAPM and a number of conventional modulation systems.

The effect of sync timing, carrier phase, and carrier amplitude references are evaluated in Section IV for modems employing raised-cosine filters. The great sensi- tivity of one-pulse mappings to all three types of error shows that PAPM modems must be carefully designed to achieve a high output SNR. Double-pulse modems can

Page 2: Hybrid Amplitude-and-Phase Modulation for Analog Data Transmission

THOMAS et al: HYBRID APM 635

SIN uol

SAMPLE AND HOLD

SAMPLE AND HOLD

4 REFERENCES

(b) Fig. 1. Block diagram of PAPM modem. (a) Modulator.

(b) Demodulator.

achieve the same output signal-to-interference ratio (SIR) as one-pulse modems with somewhat larger reference errors.

11. SIGNAL STRUCTURE

PAPM is a nonlinear multidimensional mapping of a sampled analog voltage into an RF signal. The nlodulator shown in the block diagram of Fig. 1 performs the map- ping.

The input voltage v ( t ) is a Gaussian process with an ideal flat spectrum bandlimited to B H a ; it has zero mean and a variance of u?. u ( t , is sampled a t a rate 1/T = 2B to obtain the message pulses

nzi = v ( i T ) , (i - l ) T 5 t < iT . (1)

The sampler overloads, or peak clips, a t a level of f l V, causing mi to be limited to the range ( - 1, + 1). The level of the input voltage is adjusted so that uU2 << 1 and thus the variance of the message pulse am2 is negligibly dif- ferent from u?. The overload factor is defined as

A simple message sample pulse is mapped by the modulation functions X,( .) and Y p ( .) into a P-pulse RF signal. The pulses may be either frequency or time orthogonal but in this paper time orthogonality is assumed. For each RF pulse, X, ( - ) and Y , ( - ) produce the desired carrier amplitude and phase by modulating orthogonal carrier components with the equivalent Cartesian reDre-

P

s(711i,t) = ( 3 / T ) 1 / 2 [ X p ( n l i ) cos tuut - Y P ( m i ) sin too t ] , p=l

(i - 1) T 5 t < iT. ( 3 )

A more compact vector representation is obtained with the orthonormal basis vectors

#, ( t ) = (2P/ T ) sin .wOt, 1 @,(t) = ( 2 P / T ) 113 cos toot ,

which vanish outside thc givcn tinw slot]. These allow ~ ( r n . ~ , t ) to be represented as

I’

s(’l1ti) = [sp(712;) + . jYp(???i ) ] , (i - 1)T 5 t < il’ p = l

(4) where :j = - 1.

The modulation mappings are describPd in terms of the locus of S(m) as nt. takes on all values within the range ( - 1,l). The nonlinear functions X p ( m ) and Y,(m.) deter- mine this locus.

The four representative signal mappings under con- sideration in this paper are shown in Figs. 2 and 3; the three in Fig. 2 consist of a single pulse (Le., P = 1) while that of Fig. 3 has two pulses ( P = 2) .

These mappings belong to a family of nonlinear modula- tion schemes known as “twisted modulation” which was first introduced by Kotelnikov [4], but has been employed and extended by several authors [SI, [7]-[ll]. The fundamental concept of these schemes is that nonlinear modulation “stretches” the baseband signal with a locus of length 2 into an RF signal X ( m ) with a locus of length 2L. A stretching func,tion g(m) specifies the nature and degree of stretching :

The stretch factor or length of the modulated locus is therefore

In this study the st,retc,h is uniform so that

A. Mapping for One-Pulse APM

For one-pulse APR4, each input sample is quantized in a N-level quantizer which produces two outputs: a logical output mc, which specifies one of N levels, and an analog output WLR, which is proportional to the re-

sentation. The resulting R F signal can be represented as mainder or quantizat,ion error. Output nlc selects the

Page 3: Hybrid Amplitude-and-Phase Modulation for Analog Data Transmission

636 IEEE TRANSACTIONS ON COMMUNICATIONS, JUNE 1975

m = -1 I i

m = t l ii

T - i m = -If

kYi m = t 1

( 4 Fig. 2. Single-pulse modulation mappings. (a) Linear ( N = 6).

(b) Skewed linear ( N = 6). (c) Spiral ( N = 6).

Y l

mC x, Y l I 1

-3

0.46 -0.5 ’ - 1 -0.41 -1.0 -2 -1.27 -0.5

0 0.0 -0.41

1 0.5

-1.27 0.5 3 -0.41 1.0 2 0.46

X2 = 0.71 mF

Y2 = 5.76 m d o ,

Fig. 3. Double-pulse modulation mapping. (a) First pulse. (b) Second pulse.

locus segment or semicircle while m R determines the signal location on this segment.

Linear Mappings: The mappings for the linear and skewed linear loci map the point - 1 to the bottom of the left-hand segment and +I to the top of the right-hand segment. The mapping on each segment is in an upward direction joining the top of a segment with the bottom of the next segment to the right.. The segments for the linear locus extend from - L / N to + L / N , but for the

skewed linear locus the end points of each segment in- crease in stairstep fashion as shown.

The peak carrier signal for all loci is given by the mapping of the f 1 points,

C p k = I S(1) 12. (8 )

The mean-square carrier power is found by integrating I S(,m) I2 over the message space

1

C m s = I ~ ( v l ) 12 p(m> am (9) -1

where

when N is even and

n - 1 n

N / 2 N / 2 < m < -

or

when -1; is odd and

2n - 1 2n + 1 < m < -

1V N ’ I, is the stretch factor.

Vertically skewing the segments of a linear locus reduces C,, for a Gaussian message probability density function (pdf) , C,, is minimized when the ‘(center of gravity” of the pdf on each segment is centered on the X axis. This results in a small (0.2- to 0.3-dB) increase in the SNR process gain. The offset given by

Ay = yi+l - yi = 0.08~m (12)

very closely approaches the optimum skew when d / a , = 4.5.

Spiral Mapping: The spiral mapping is composed of semicircular segments of radius (2k - 1) d / 2 , k = 1,2, - - *,

X / 2 . The locus is constructed so that each circular seg- ment begins and ends on the Y axis. For N segments, the stretch factor L is

If the message level 7n lies in the range

(72 - 1 ) 2 - < Lnz 5 ?a2 - dm dn 2 - 2 ’

t.hen S ( m ) lies on the nth segment from the origin and

Page 4: Hybrid Amplitude-and-Phase Modulation for Analog Data Transmission

I

THOMAS et at: HYBRID APM 637

The quantizer levels are therefore nonuniform for the spiral mapping.

The peak and mean-squhre carrier powers are again given by (8) and (9).

B. Mapping for Two-Pulse A P M For the two-pulse APM mapping each sample is

quantized in an NM-level quantizer, which produces three outputs: a logical output me, which specifies one of M coarse levels; a logical output m F , which specifies one of N fine levels within each coarse level; and an analog out- put mR, which is proportional to the remainder or quantiza- tion error.

Output mc is defined as the number in the set

{‘iM,’iM1 - - ... , - 2

which minimizes I m - 2m,/M I and mF is similarly the number in the set

\ 2 ’ 2 ” N - 2 l } 1 - - N 3 - N - - ... -

minimizing I m - 2mc/ M - 2 m ~ j M N 1 . Output mR is then

C . Noise

The received noise of spectral density No can represented with the basis vectors % ( t ) and + k ( t )

P

n ( t ) = c Cnz,Wt) + nu,+,(t)l p=1

or as a complex variable, P

N = C (nzp + jnup). p=l

The power in each component is

E(nzp2J = E{nu,2) = (N0/2) A an2

where E{ J denotes expected value. The total noise power within the transmitted signal

bandwidth of P I T Hz is 2Pun2. The carrier-to-noise power ratio (CNR) in the channel bandwidth is therefore

111. THERMAL NOISE PERFORMANCE

The demodulator in the receiver is assumed to perform an ideal maximum likelihood estimation (MLE) of X (m) in the presence of Gaussian noise. Such a demodulator is

the demodulator input be thc sum of the carrier signal plus noise,

2 = S ( m ) + N . (17)

The estimate of S(m) is chosen to minimize the distance 1 2 - 8 ( m ) 1 , and the MLE estimate of the baseband signal is the inverse mapping

G = S-l(m). (18)

Estimation errors for single- and double-pulse APM will be considered separately.

A . Si,ngle-Pulse APM Two types of errors arise in the estimation of X(m)

with single-pulse APM. The first occurs when N is small and results in a small displacement of S along its locus, as illustrated in Fig. 4. The second, which occurs when N is larger than half the locus separation dl results in a larger error because the wrong locus segment is chosen. This estimate bears little relation to the true S and thus is termed anomalous. It is the cause of the threshold be- havior in the output SNR.

The performance of the MLE can be expressed in terms of output SNR as a function of input CNR. The contribu- tion of both the linear and anomalous errors to the output SNR will now be evaluated.

Linear Error: When the noise is small X ( m ) can be approximated in the neighborhood of the point S(mo) by the straight line

where

dX dX(m) dY(m) dm dm. dm.

- + j - .

The MLE estimate S(m) is the projection of 2 onto the signal locus. Thus,

where (a lp ) denotes the inner product and the derivatives are evaluated at m = mo. The output signal error is then

with a mean square value

or

shown in Fig. 1 and opernt,es in the following manner. Let since g(m) is uniform.

Page 5: Hybrid Amplitude-and-Phase Modulation for Analog Data Transmission

638 IEEE TRANSACTIONS ON COMMUNICATIONS, JUNE 1975

(a) (b) Fig. 4. Illustration of errors caused by small and large noise vectors

(a) Small noise (linear error). (b) Large noise (anomalous error).

Anomalous Error: An anomalous error occurs when the noise vector is so large that Z lies closer to a folded section of the locus than to the true locus region. The probability of occurrence of such an event depends upon the distance between folded locus lines. Its effect depends on the length of the locus comprising the fold; thus it is different for each modulation scheme.

Consider first the linear raster of Fig. 2. With N line segments separated by cl, adjacent points on neighboring line segments are separated by a locus distance of 2L/N. Let & ( m N ) be the probability that the signal lies on an outer segment,

& ' ( m , N ) = J' p(m.) dm. ( 2 5 )

The anomalous error, which .is two sided except for the outer segments, can be found by using the expression

1-l /N

+ & ( m ~ ) 1 - erf - [ ( 2 A J l l

where erf ( - is the error function

2 " erf (x) = - exp ( - t 2 ) dt.

,112 1 (27 )

The anomalous error of the skewed locus differs from that of the linear locus in two respects. First, the point on the adjacent locus segment on which the anomalous event is most likely to place the estimate S(m) is no longer separated from S(m) by a segment length 2 L / N . For the skew of (12) it is separated by

2 ( L - 0.08u,) / N . (28)

The second difference is caused by the change in error probability at the ends of the skewed segments. For a small skew, the end condition applies to only a small fraction of the segment length and its effect can be neglected.

For the spiral mapping, the anomalous error is more difficult to evaluate and cannot be given a convient form;

it is calculated in the Appendix. The boundary for the error is formed by the locus of a spiral located midway between the segments of the circular signal spiral. Anoma- lous errors near the origin must be treated separately. Likewise, on the outermost segment, no anomalous error boundary exists in the direction away from the origin.

Output SNR: The output SNR will now be evaluated and from it the SNR improvement factor, or process gain, which is the ratio of output SNR to input CNR in the data bandwidth. Since the output signal power is u,2, the output SNR is

SNR, = urn2/ ( ~ 1 ~ + ea2)

Threshold occurs when ea2 begins to contribute signifi- cantly to the denominator.

The input carrier power depends upon the modulation mapping as well as p ( n z ) and can be expressed in terms of either peak or average power, as discussed in Section 11.' The process gain (PG) , or SNR improvement factor, is

SNR, CNR '

PG = -

B. Double-Pulse APM

The second pulse of two-pulse APRI experiences linear and anonlalous errors in the same manner as one-pulse AI". In addition there is a second anomalous error caused by selection of the incorrect signal point for the first pulse.

Li,near Error: As in one-pulse APM, the linear, above- threshold error is given by

The stretch factor g for the second pulse is given by

1IMN J g(nz> dm = L~~ ( 32) -1INM

where Lzi is the segment length of 'the second pulse and g = g(m) is independent of m. For the mapping shown in Fig. 3,

L2i = 5.76(21113 m a x / ~ m )

and

so that

g = 5.76/am. (33)

Anonzalous Errors: Anomalous errors occurring on the first and second pulses will be denoted by and E=;,

respectively. to? is the same as that for the one-pulse APRI in ( 2 5 ) , except that neighboring points on adjacent loci are separated by 2 / M N instead of 2/AT. Hence

Page 6: Hybrid Amplitude-and-Phase Modulation for Analog Data Transmission

THOMAS et al: HYBRID APM

TO eva l~a te e,: for a first pulse consisting of M points, note that if the demodulator errs by choosing the jth signal point instead of the ith, the baseband signal is mapped to the incorrect quantization level. The resulting mean-square error can be expressed as

( 2 / M ) I i - .j 12. ( 3 5 )

The total mean-square anomalous error for the first pulse is then the average

(M-1)/2 ( M - 1 ) / 2 2

~

ea: = c C - ~ i ~ j i ( i - . j I 2 (36) i = - ( M - l ) D j=-(M--1)/2;j#i M

1 where p i is the probability of the message signal lying within the ith quantization range and pii is the probability of the demodulator selecting the jth RF signal point when the ith is transmitted. (For simplicity, M is assumed to be odd.) The probability pji depends upon the distance dii between the ith and j t h signal points in S ( m ) ahd is given by

(37)

For a Gaussian message pdf, p i is

Output SNR: For double-pulse APRl the output SNR is given by

(39)

C. Performance Curves of SNR, versus CNR have been computed for

the four mappings for various mapping parameters. Fig. 5 shows the linear single-pulse APM mapping with a 4a overload factor and unity stretch factor.' Curves 2 and 5 are square rasters for N = 8 and 4, while the other three illustrate varying segment spacing. Comparing curves 1 and 3 it is evident that reducing the segment separation, which reduces C,,, increases the process gain and the threshoid level. Curve 3 has the same segment length as curve 1 but half the segment spacing. It may also be viewed as a mapping with the same segment spacing as curve i (u,/d = 4.5) but with a stretch factor of L = 2 . The process gain varies from 6.4 to 13.3 dB with corre- sponding CNR thresholds between 12 and 22 dB. The linear skewed mapping provides an increase in the process gain of 0.2 dB in comparison to the linear mapping.

Performance characteristics of the spiral mapping are illustrated in Fig. 6. Curves 1, 3, and 5 have a unity stretch factor ( L = 8u,), while 2 , 4 , and 6 have L = 16u,. Comparing the even and odd curves, it is evident that the

Note that urn is the rms baseband voltage and (Cm8)1'z is the rms RF voltage. In general, urn2 # Cma.

to

M

40

VI u g30 $ a

w

L 3 0

IO 1 N = 8 , uJd = 4.5, Li =urn

2 N = 8 , u d d = 7 , Li =urn

0 4 N - 4 , uJd/cl=3, Li =2um

5 N = 4, uJd = 1.5, Li = 20,,

-10 I I 0 IO 20 30 40

INPUT CNR, DECIBELS

639

3

Fig. 5. Linear mapping noise performance (overload factor of 4).

to

40 VI Y m - M

v)

I-

5 30 3

3 t 0

20

10

0

-10 0 10

NO. CONDITION

1 N=8, o J d = n

-

~ 2 N = 8 , u J d = r / 2

3 N = 10, u d d =4.91

4 N = IO, a d d = 2.45

- 5 N = 4 , u d d =0.785

6 N = 4 , o d d =0.393 ' 30 40 3

INPUT CNR, DECIEELS

Fig. 6. Spiral mapping noise performance (overload factor of 4).

process gain is independent of cl/a,. However, doubling W increases the process gain by Fj.73 dB so that it is also very nearly proportional to N 2 with the Gaussian pdf.

The process gain for the illustrated spiral parameters varies from 8.2 to 16 dB with corresponding CNR thresh- olds occurring from 11 to 21 dB.

The performance of the two-pulse APM of Fig. 3 with 3 . 5 ~ overload factor is shown in Fig. 7. Its processing gain

Page 7: Hybrid Amplitude-and-Phase Modulation for Analog Data Transmission

640 IEEE TRANSACTIONS ON COMMUNICATIONS, JUNE 1975

0 10 m 30 . 40 50 INPUT CNR IN P/T Hz, dB

Fig. 7. Comparison of one-pulse and two-pulse APM noise per- formance.

above threshold is 20.9 dB. Threshold occurs at CNR = 13.5 dB and SNR, = 33.4 dB.

A linear single-pulse APM mapping with N = S and 3.5a overload is shown for conlparison of single and double pulse mappings. Clearly the double~pulse configuration offers a very significant advantage in both process gain and threshold.

Fig. 7 also shows the rate distortion bounds for cine- and two-pulse APM (i.e., for bandwidth expansion factors of 2 and 4) given by SNR, = (1 + CNR jzp . At threshold the CNR is about 4 dB from the one-pulse bound, while the two-pulse CNR is about 6 dB from the two-pulse bound.

D. Comparison with Other Systems

Fig. 8 compares the ideal performance of various, modu- lation systems, including one- and two-pulse APM. Peak and average power and bandwidth requirements are shown relative to the requirements for single-sideband. Also shown are contours of constant RF CNR, and Shannon’s rate distortion bound for a Gaussian baseband noise- power ratio (NPR) of 32 dB, which is appropriate for large FDM telephone multiplex assemblies. All realizable systems lie above this bound.

Fig. S does not include practical system degradations, which vary between modulation systems. Nevertheless, it may be used as a fairly accurate guide for comparing well-engineered systems. It may be noted that two-pulse APM uses less power and less bandwidth than any of the angle-modulation systems.

Iv. EFFECT OF REFERENCE ERRORS The performance of PAPM is affected by three types of

reference error: pulse sample t h i n g , carrier phase, and

0 I I l l I I I I t I . I I I I , I I l l , 0.01 0.1 1 10

RELATIVE POWER

IAVE- PEAK)

Fig. 8. Comparison of modulation techniques.

carrier gain. Errors in the carrier phase and gain references cause a. distortion of the locus of the received signal so that the inverse mapping at the receiver introduces an error in the demodulated output message. An error in pulse sample timing has two effects. First, it reduces the signai energy in the main pulse. Second, it increases the interpulse interference (IPI) due to adjacent~pulses.

To evaluate the effect of these errors, consider again the modem illustrated in Fig. 1 with its transmitter and receiver filters and demodulator reference circuits. The filters are ideai raised-cosine filters with a low-pass equiva- lent frequency response of

H ( w ) = (T/P)’”, 0 5 I w I 5 - (1 - a!) TP T

= [ $ [ 1 - sin (&(a - $))]}’”,

where a! has been chosen to de 1/3 for this study. ( P is the number of transmitted pulses per sample period T.) The impulse response of this ideal filter passes through zero at all multiples of T I P , so that the IPI is zero with a perfect sampling sync reference .i.

The gain reference 0 controls an automatic gain control (AGC) amplifier at the filter output. The local oscillator reference 8 for the I and Q channels is supplied from a variable phase oscillator. Both I and Q channels are samples a t time i. The maximum likelihood inverse mapping converts these sampled values to a PAM signal from which the continuous message waveform is recon- structed by appropriate filtering.

The’ linear and anomalous errors caused by sync timing, phase, and gain reference errors are derived in this section. Curves of output signal-to-noise ratio are then presented to illustrate the performance degradation due to each type of error.

Page 8: Hybrid Amplitude-and-Phase Modulation for Analog Data Transmission

THOMAS et al: HYBRID APM

A . Sync Timing The demodulator input consists of an infinite sequence

of APM impulses 00

p ( t ) = S ( 7 7 7 k ) G ( t - IcT/P) (41) k=- m

which have passed through the transmitter and receiver filters. The effect of these two raised-cosine filter com- ponents can be represented by the channel function

C(7,k) = h ( k T / P - 7 0 - T) (42)

where

h ( t ) inlpulse response of conlposite filter TO optimum sync delay T sync delay error.

C(0,O) is the desired pulse output with no sync error and C(7 ,k ) is the response a t 70 + 7 to a pulse transmitted k intervals after the desired pulse. The effect of IPI with sync timing error can be determined by evaluating the mean-square interference components parallel and nornlal to the signal locus. The parallel component increases the linear error above threshold while the normal component increases the probability of an anomaly.

For the single-pulse linear mapping the mean-square linear error caused by pulse interference in the direction of the locus is given in terms of the channel function C ( T , k ) :

I;

cIP12 = ~ 2 ( 7 , k ) / (Im [ ~ ( m ) ] ) z r ) ( m ) c ~ v l (43)

when 2K adjacent pulses are taken into account. This value of qp12 is added to the noise variance un2 in the calculation of output SNR. The effective noise variance causing an anomalous error is increased by

UIPI' = C I C ( ? , k ) l2 / (Re [S(m)])2p(n l ) dm.

k--K ; k f O

K

k=-K; k#O

(44)

For the spiral mapping we make the approxinmtion that the locus is uniformly distributed in amplitude and phase. In this case, the composite IPI is uniformly dis- tributed in angle and its effect can be treated in the same manner as thermal noise. The mean-square contribution to the linear and anomaly-producing variances for the spiral locus is therefore

K

(TIP12 = €IPI2 = +ems CZ(7,Ic). (45) k-K; k#O

With double-pulse APM, the interference power along the signal locus of the second pulse is caused by both the first pulse of that particular message sample and double pulses from adjacent samples. Letting Sl(m) and &(m) represent the first and second pulses, respectively, we can write the mean-square error as

641

K

tIP12 = J p ( m ) [ (Im [~,(nz) 3 2 ~ 2 ( T , k ) k=-K; k odd

K + (Im [ & ( w z ) ] ) ~ C 2 ( 7 , k ) dm (46)

for 2 K significant adjacent pulses. Note that ~ I P I ' is minimized by rotating X1 relative t o X2 so as to minimize its mean-square power in the Y direction. The increase in anonlalous error for the gcneral double-pulse APM map- ping must account for both pulses. However, for the mapping of Fig. 3, the anomalous error of the second pulse predominates at threshold. Since the second pulse is a linear mapping, the effective noisc variance producing anomalous error e,: is found by analogy between (+3), (44), and (46) to bc

k=-K; k even; h,#O I

K

uIpI2 = J p(717) { ( ~ c cs1(m)1)2 c CY+) k-K; R odd

K

+ (Re [&(n2)] )2 C2(7,Ic)} clm. (47)

In both the single- and double-pulse APM the energy in the main pulse is reduced by a sync error in addition to the IPI being increased. This loss in energy appears to the receiver AGC control as a loss in gain of (C (0,O) /C ( 7 , O ) ) 2.

B. Phase Reference Error

k=-K; k evon; k#O

When the phase reference is in error by an angle 0, the demodulator's signal mapping is rotated by 0. The received signal-plus-noise is projected onto this rotated mapping and an error is introduced into the demodulated output.

Consider first the single-pulse linear mapping. If the transmitted signal is S ( m ) , the received signal is X(m) exp (j0) and t,he linear error is the component in the Y direction

e ~ ( m ) = Im [S(m) - S ( m ) exp ( j e ) ]

= (1 - case) Im [ X ( m ) ] - sinORe [ S ( m ) ]

M sin e Re [ S ( ? ~ I . ) ] (48)

for small e. The mean-square linear error is therefore

= sin2B/ (Re [X(71z)])~p(m) d m . (49)

The anomalous error probability is increased due to the warped boundaries of the receiver mapping. The effective boundary displacement is

e~,(nz) = Re [S (m) - X(m) exp ( $9) 3 % sin 0 Im [X(m)] (50)

and an anomalous event occurs whenever the noise exceeds ( d j 2 ) f sin 0 Im [X(m) 3. The mean-square anomalous error of (26 ) therefore becomes

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642 IEEE TRANSACTIONS ON COMMUNICATIONS, JUNE 1975

cg2 = Ay2 / (Im [S(m)])2p(m) dm,. (56)

- erf ( d / 2 + sin8 Im [ S ( m ) ] ) ] dm

The decision boundary for anomalous errors shifts from d / 2 to d / 2 f Ag Re [S(rn)].

For the spiral mapping the linear error is the inner product

e , (m) = Ag(v t (?n.> ,S(m)) (57)

with mean-square value of d / 2 - sin 0 Im [S(m)]

ea2 = (Ag)2/p(m)(v,(m),S(m))2dm. (58)

- erf ( d / 2 + sin e Im [ S ( m ) ] a a ) ] d m ) * (51)

With the spiral mapping, the linear error is caused by the tangential error component resulting from the rotated mapping. Let the unit vector tanget to the spiral locus at S(m) be v,(m). The linear error is then. the inner product

ee (m) = (vt(m) ,S(m) [1 - exp ( 1) yielding a mean-square error of

eo2 = J p(m> (v t (m) , ~ ( m ) [I - exp ( j e ) 3 1 2 dm. ( 5 2 )

The boundary displacement contributing to the anomalous error is the inner product with the unit normal vector vn. (m)

ee.(m) = (v,(?n) ,S(m) 11 - exp ( j e ) 3). (53)

As for the linear mapping, an anomalous error occurs whenever the noise exceeds d / 2 f eea(mj.

For two-pulse APM, the linear error is caused by rota- tion of the second pulse. Thus by comparison with (49) we have

€ 8 = sin2 8 I (Re [S2(m)] ) 2 p (m) dm. (54)

The boundary displacement for anomalous error is

ea,(m) = &(vn(m),S(m> ) . (59) For two-pulse APRII, the mean-square error is

,ea2 = Ag2 (Im [S2(m)])2p(m) dm J (60)

and the boundary for anomalous errors in the second pulse is d / 2 f Ag Re [ S ( m ) ] .

D. Performance with Reference Errors Reference errors impose a limit, or ceiling, on the output

signal quality that can be obtained under very high CNR conditions. This ceiling SIR is plotted in Fig.' 9 for the three mappings and three references, using the linear mean-square error components. Note that the ceiling SIR is inversely proportional to 'the square of the reference error.

Representative cases of output SNR have been cal- culated as a fynction of CNR for steady reference errors of T = 0.02T, 0 = 0.05 radians, and Ag = 0.05. Results for the three assumed mappings, shown.in Figs. 10 to 12, also illustrate how each type of reference errors imposes a ceiling on output SNR.2 For sufficiently accurate refer- ences, the ceiling is high enough so that threshold per- formance is .not significantly affected. Based on ' the

An anomalous event occurs whenever the noise in the

events in the first pulse can be neglected near threshold for the mapping of Fig. 3.)

amount of threshold shift produced by reference errors, both single-pulse mappings have about the same tolerance

is slightly lower' than that of the two-pulse mapping. The tolerance to gain errors is low for the linear single-pulse

second Pulse exceeds d /2 * sin e Im (m) 1' to sync timing and phase reference this tolerance

C . Gain Reference Error If the gain reference in the demodulator is in error, the

locus maps of the transmitter and receiver will not match. Consider first the linear mapping and let the relative gain error be Ag so that the received s'ignal is (1 + Ag) S(m). The linear error component is then

e,(m) = Im [S(m) - (1 + Ag)S(m)]

= Ag Im [ S ( m ) ] (551

yielding a mean-square error of

mapping and high for the spiral single-pulse and the double-pulse mappings.

V. CONCLUSIONS

Pulse-amplitude modulation (PAM) has been extended to two dimensions to form one-pulse APM, and to four dimensions to form two-pulse APM. The bandwidth occupancy of one-pulse PAPM is the same as that of

The two-pulse performance of Fig. 12 was actually obtained by computer simulation instead of using the analysis above.

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THOMAS et al: HYBRID APM 643

RELATIVE SYNC ERROR, PT/T PHASE ERROR, e , RADIANS RELATIVE G A I N ERROR, Ag

Fig. 9. Output signal-to-interference ratio versus reference errors in absence of thermal noise (conditions: linear- P = 3.5,N = 8, L;/d = 4.5; spiral-p = 4, N = 8, a,/d = 4.5; two-pulse-p = 3.5).

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644 IEEE TRANSACTIONS ON COMMUNICATIONS, JUNE 1975

INPUT CNR, dB

Fig. 12. Two-pulse mapping performance with reference errors (P = 3.5).

double sideband PAM, but its twisted modulation is shown to lead to typical process gains of about 10 dB in contrast to the 3-dB gain of PAM. The bandwidth of two- pulse APM is twice as great, but process gains of about 20 dB can be achieved. The bandwidth and power require- ments for PAPM have been shown to be smaller than those of exponential modulation systems.

PAPM is shown to be quite sensitive to sample timing, carrier phase, and carrier gain reference errors. For linear one-pulse APM near threshold, approximately 1-dB degradation in required CNR is produced by each of the following reference errors: 4 percent sync timing error, 3' phase error, and 5 percent voltage gain error. Two- pulse PAPM can tolerate references of somewhat lower accuracy.

The choice between one- and two-pulse APM in a com- munication system depends upon the relative importance of bandwidth, carrier power, and modem complexity. Results presented in this paper form the basis for this tradeoff.

APPENDIX

ANOMALOUS ERROR FOR SPIRAL MAPPING

The mean-square anomalous error for the spiral map- ping is evaluated in this Appendix. The boundary for this error is formed by the locus of a spiral located midway between the segments of t.he circular signal spiral. Anoma- lous errors near the origin are treated separately and will

be discussed later. Likewise, on the outermost segment, no anomalous error boundary exists in the direction away from the origin.

If n is defined by the requirement t:hat the message voltage lie in the range

( *d /2 ) (n - 1 ) Z 5 Lm 5 (7rd/2)n2

then S(m) can be expressed in polar form as

S(m) = r(n) exp Cje(m,n)l+ j(d/2j.

An anomalous error transforms the point S(mj into

S(m> = r ( n =t 1 ) exp c~~(wL,+J + j ( d / 2 ) ,

2 5 n 5 N / 2 - 1.

The resulting demodulated message point is &l(m) for an error s-l(m) - ,W(m) .

From the way S(m) is formed, S-l(mj for n odd can be written as

s-'(m I n,en) = ( -1 )n+1(2 /N)2 [ (n - 1 ) ~

+ (2n - 1) (en/*) l where

e,(m) = - ?r (m.W - 4(n - 1)2) 4 n2 - (n - 1 1 2 '

Hence

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THOMAS et al: HYBRID APM 645

[X-l(m. I n.,e,> - X-’(m I ?& + 1,0,)1”

Near the origin, anomalous errors are mapped as follows. Poin.ts on the lowcr half of the first segment are mapped to the endpoint of that segment, 0 + j d . Points on the uppcr half of this scgmcnt are mapped to the origin. Therefore, for 0 5 m 5 4 4 , we have approxirnatcly

X-l(m, 1 1,e) - X-l(m. 1 o,e) = ( 2 / ~ ) 2 m

and, for sd/4 < m 5 nd/2 ,

~ - l ( m . I 1,e) - s-l(m I o,e) = ( 2 / ~ ) 2 ( 4 2 - m ) ,

The total valuc for the mcan-square anomalous error is found by summing these components

- S-l(m I 72 + 1,e) 1” + [X-l(m J n,e) - X-’(m 1 n - 1,e)l2) dm

- X-l(m I $N - i ,e) l2 dm. 1 where

m, = n2 ( n d / 2 ) .

ACKNOWLEDGMENT

The authors wish to thank Drs. 0 . Shimbo and R. J. Fang of COMSAT Laboratories for their assistance.

REFERENCES [l] C. M. Thomas, M. Y. Weidner, and S. H. Durrani, “Digital

amplitude-phase keying with M-ary alphabets,” ZEEE Trans. Commun., vol. COM-22, pp. 168-180, Eeb. 1974.

[2] J. Saltz, J. R. Sheehan, an!, D. J. Paris, “Data transmission by combined AM and PM, Bell Syst. Tech. J . , vol. 50, pp.

[3] G. J. Foschini, R. D. Gitlin, and S. B. Weinstein, ‘‘On the 2399-2419, Sept. 1971.

selection of a two-dimensional signal constellation in the presence of phase jitter and Gaussian noise,” Bell Syst. Tech. J. ,

[4] V. A. Kotel’nikov, The Theory of Optimum Noise Immunity .

[5] J. M. Wpzencraft and I. M. Jacobs, Principles of Communica-

V O ~ . 52, pp. 927-965, July 1973.

New York: McGraw-Hill, 1959.

tion Engzneering. New York: Wiley, 1967.

[6] C. M. Thomas, C. 1,. May, and W. n. Adams, “Final report on hybrid modulation,” prepared by TRW Systems for COM- SAT Laboratories as Manager of INTELSAT under Cont,ract

[7] G. R. Welti, “Application of hybrid modulation to TDMA telephony via satellite,” COMSAT Tech. Rev., vol. 3, pp. 419-

[8] J. Ziv, “The behavior of analog communication systems,” Z E E E 430, Fall 1973.

[9] U. Timor, “Design of signals for analog communication,” Trans. Inform. Theory, vol. IT-16, pp. 587-594, Sept. 1970.

IEEE Trans . In form. Theory , vol. IT-16, pp. 581-587, Sept. 1970. [lo] D. D. McRae, “Performance evaluation of a new modulation technique,” ZEEE Trans. Commun. Technol., vol. COM-19,

[Ill G. R. Welti, “Pulse amplitude-and-phase modulation,” in Proc., 2nd Znt. Conf. Digital Satellite Communications. Paris, France: Editions Chirons, 1972, pp. 208-217.

CSC-18-428, NOV. 1972.

pp. 431-445, A u ~ . 1971.

* C. Melvil Thomas (”61) was born in Glendale, Calif., on March 7, 1935. He re- ceived the B.S.E.E. degree from the Univer- sity of Texas, Austin, in 1957, the M.S.E.E. degree from Stanford University, Stanford, Calif., in 19-33, and the Ph.D. degree from the University o f Southern California, Los An- geles, in 1967.

He was employed by General Dynamics, Pomona, Calif. from 1958 to 1961 and by Lear Siegler Electronics, Anaheim, Calif.

from 1961 to 1963. In 1963 he joined Tl tW Systems, Redondo Beach, Calif., where he is now a member of the Advanced Analysis Staff. At TRW he has engaged in the anaIysis of all aspects of comunica- tion and telemetry systems including nonlinear distortions, spread- spectrum modulation, tracking loops, and convolutional coding. His current interest is t,he application of programmable processors to the reception of digital communication signals.

Curtis L. May (S’6.5-M’SS) was born in Hous- ton, Tex., on April 8, 1941. He received the B.S. degree from Itice Universit,y, Houston, Tex., in 1964, the MS. degree from Stanford University, Stanford, Calif., in 196.5, and the

.Ph.D. degree from Rice University in 1970. From 1965 through 1967, he worked in

Signal Processing and Display Group at Hughes Aircraft Company, Culver City, Calif. From 1970 to 1974, he worked in the Systems Engineering Laboratory of TRW

Systems, Itedondo Beach, Calif. 6 s research interests are in the fields of satellite communication systems, digital signal processing, and data compression of multispectral imagery. He is currently a Senior Staff Engineer in the Advanced Programs Department of Magnavox Research Laboratories, Torrance, Calif. He also teaches in the Electrical Engineering Department of the California State University a t Long Beach.

*