Qam Transmission through a Commanding ... - Semantic Scholar
IEEE TRANSACTIONS ON COMMUNICATIONS,VOL. 42, NO. 21314, FEBRUARYMAKCHIAPKIL 1994
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QAM Transmission Through a Companding Channel - Signal Constellations and Detection Irving Kalet and Burton R. Saltzberg,Fellow, IEEE Abstract-The effect of the companding process on QAM signals has been under investigation for the past several years. The compander, included in the PCM telephone network to improve voice performance, has an unusual affect on digital QAM data signals which are transmitted over the same channel. The quantization noise, generated by the companding process which is multiplicative (and asymmetric), degrades the detectability performance of the outermost points of the QAM constellation more than that of the inner points. The combined effect of the companding noise and the inherent white gaussian noise of the system, leads us to a re-examinationof signal constellation design. In this paper we investigate the detectability performance of a number of candidates for signal constellations including, a typical rectangular QAM constellation, the same constellation with the addition of a smear-desmear operation, and two new improved QAM constellation designs with two-dimensionalwarping.
I. INTRODUCTION Recently, there has been an interest [l, 2, 3,4] in the effect of the pulse code modulation (PCM) companding [5] mechanism (introduced to improve voice transmission), inherent in communication networks, on the detectability performance of digital Quadrature Amplitude Modulation (QAM) signals uansmitted through these networks. As has been previously shown [l, 2, 3,4], the effect of the companding process is to introduce quantization noise which is multiplicative in nature. This noise is amplitude dependent, i.e., large amplitude signal points in the QAM constellation will be surrounded by more noise than signals closer to the origin. The effect of this noise is such that the outermost points in a simple rectangular constellation suffer more performance degradation than the smaller internal signals. Saltzberg and Wang in [4]made the observation that not only is the noise around each point dependent on its amplitude, but that this noise is elliptical and not circular, Le., the noise components in the two dimensional QAM space are not equal. In [4],it was also shown that the two dimensional noise components are not only unequal, but also correlated for most choices of vector bases. These properties make the detection problem and the signal constellation design problem very interesting, and represent a new challenge in signal design. Paper approved by Nikolaos A. Zervos, the Editor for Transmission Systems of the IEEE Communications Society. Manuscript received October 3, 1991; revised March 16,1992. I. Kalet lives in Haifa, Israel and is currently with AT&T Bell Laboratories, Middletown, NJ 07748. B. R. Saltzberg is with AT&T Bell Laboratories, Middletown, NJ 07748. IEEE Log Number 9400925.
In this paper we consider the problem of digital signal constellation design for the companding channel. We first consider the performance of the typical rectangular QAM constellation, and then the same constellation with smear-desmear [6, 7,8] filtering applied to average the noise over the entire constellation, i.e., to spread out the multiplicative noise over all the constellation points. We then consider two versions of two-dimensional warping of the original constellation, which preserve a constant normalized distance between constellation points and their neighboring decision boundaries. It should be noted that Pahlavan has recently also considered the design of constellations so as to maintain normalized distance between constellation points [9], where normalization is with respect to the total noise power surrounding each constellation point. In the next section we review previous work on companding channels and describe the system model. In Section 111, we describe the combined effect, of additive white gaussian noise (AWGN) and companding, on performance. Sections IV and V describe transmitter filter optimization and a smear - desmear technique for reducing the effect of the companding noise. Sections VI and VI1 discuss the detectability properties of rectangular constellations and the proposed warped constellations, and finally the last two sections contain a discussion of results and conclusion. 11. SYSTEM MODEL The system model is shown in Fig. 1 [41. The QAM signal, (the input to the coder), is generated by a stream of digital data which modulates the carrier at the frequency,fo. This signal is digitally converted by the coder and transmitted through a digital companding channel to the receiver in which it is converted back to an analog QAM signal. The coder is in general of the p-law [5] or A-law type. In this work we will assume a p-law device, and approximate it as being logarithmic. The multiplicative noise is generated by the quantization and companding process. s(t),
As described in [4], the transmitted signal s ( t ) before companding is given by
where c, = a , + jb,, b , are independent)
0090-6778/94$04.000 1994 IEEE
a n r b n= t-1, t-3,- - - (a, and
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO 21314, FEBRUARYMARCHIAPRIL 1994
Analog End-Link Transmitter
:
AWGN
I
Coder
Receiver
r
Overall Baseband h(t) Equivalent Filter,
where * denotes convolution. Given that constellation point c, is transmitted, the radial and tangential noise variances abou? the received point are
Overall Baseband f(t) Equivalent Filter,
System model-h(t) includes the transmit filter, the transmitter endlink channel, and the codec input filter, f(t) includes the codec output filter, receiver end-link channel, receiver input filter, and equalizer.
Fig. 1.
and
- (t) = hr(t) h
+ jh,(t).
(2)
The signal h( t ) is the equivalent baseband impulse response of the transmit filter, the transmit end-link channel and the codec input filter. It is assumed that h ( t ) is bandlimited to frequencies in the
[
range - f m a x f m m whereto > f m a x . The mean square quantization error $, at the kth sample, caused by the companding process is given by [4], [5]
where k , and ke are multiplicative noise coefficients given by
[
k , = (1/2) AKT2 2z+ (0)
i
ke = (1/2) AKT2 22, (0)
9
a$ = K s2
[ kA + a ]
[Ln (1 + p)I2 / 3 N 2
-
(0) -22, (O)]
z - (0)
(9)
+ 2 z , (O)],
k, is the additive noise coefficient k;
=
AKf2x a + ( n T ) ,
(10)
n#O
(3)
where l / A is the sampling frequency, and a is an arbitrary random offset, assumed to be uniformly distributed between k A / 2 . The quantization errors at different samples are assumed uncorrelated. For the p- law codec with N levels, K is given by [ 5 , s ]
K
+ z-
and 0: is due to additional noise (e.g., AWGN) that is not signal dependent. (Similar definitions of the constants k , can be made if we use the x and y axes ) The cross - correlation of the radial and tangential noise components is
(4)
where N equals the number of codec levels. For typical values of p and N , p = 255 and N = 28(256), K equals 1.564 x 10-4, or -38.0 dB. At the receiver, the quantized signal is received and expanded back to s( l ) , now including the effect of the multiplicative quantization noise. It is assumed that the receiver contains a filterf(t) used to ideally equalize the QAM signal (i.e., no intersymbol interference) so that the output points equal c,. In [4] a general solution is found for the effect of the miltiplicative noise on the signal outputs, for a general h(t). The results may be summarized in a restated format, by first defining
This quantity is clearly zero whenf(t) and h(r) are real, and can be shown to be small when F ( f ) and H ( f ) are close to flat in amplitude. Assuming the noise components in (8) to be uncorrelated, the total noise, a$,is
-
and its average, cr2 ,over all constellation points is
0'
w
= 2AKT2
z+(nT) ?a?=--
+ 2 0 , '.
(13)
KALET AND SALTZBERG: QAM TRANSMISSION THROUGH A COMPANDING CHANNEL
111. THE EFFECT OF AWGN
.,"= 4 =!
In this section we investigate the quantitative effect of AWGN on the second-order output statistics. We will consider h( t ) andf (I ) which are real.
(J;=
[(3a:
2
+ b:)
z(0)
+
*IcT2 [(a; + 3 b 3 z(0) + 2
n+o
J
and the correlation, oV ,is given by
oxy= A K T 2 a o b oz(0)
.
If AWGN now appears at the input to the equivalent receiver channel whose impulse response is given byf(t) cos 27c fo t, ( f ( t ) is real), then the output noise n, ( t ) due to the AWGN, will be given by n , ( t ) = n,,(t)cos
2n f o t - nWq(t)sin2n f o t (19)
where
Fig. 2. Signal point 2. and unmrrelated quantization noise components, np and ne.
If we assume that h ( t ) andf(t) are real, then it follows from (11) that the output quantization noise surrounding each point is such that its radial and tangential components, as shown in Fig. 2, are uncorrelated, i.e., E i n p ne J = 0. This is not necessarily true for other axes, e.g., x and y . It follows from (5)-(lo), that c$ and 02 are given by.
-
02 p -
AKT2 2
13
(co12 z ( 0 ) + 2 7
z(nT4
Since the detector recovers n,,,,( t ) in the in-phase direction and n,, ( t ) in the quadrature direction, 0;will be equal to either E{n,:(t)] ,orE{n$ ( r ) ] ,or equivalentlyE{n;(t)J. We find G: by first determining the power spectral density, Pnw (f). If F(f) is the Fourier transform off(t) ,then
(14)
n#O
and
"
I
n
J
Therefore,
+ 0; If we consider the x and y components of the noise then the variances are given as [4]
2
df
IEEE TRANSAC:TIONS ON COMMUNICATIONS, VOL. 42, NO. 21314, FEBRUARYMARCWAPRL 1994
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=
-
NO I_
4
J
Denoting the summation by A 2 ( f ) , the quantity to be minimized is of the form
I F ( f ) I 2df.
-m
For the ideal-bandlimited Nyquist channel, with F ( f) = 1, for No
If1
1
< 1 / 2 T , E{n: ( t ) ] equals - - . These results remain 4 T the same for square-root raised-cosine, Nyquist filtering with non-zero roll off. Using the results above we can now find the total noise variance about any given received constellation point c,. In the sections that follow we will analyze a ncmber of different receivers in the combined AWGN, companding noise environment. However we first look at the general problem of minimizing the total quantization noise power at the output.
From the Schwartz inequality,
with equality if and only if A ( f ) = a / A ( f ),or
IV. TRANSMIT FILTER OPTIMIZATION We will first find the transmit filter which minimizes the total noise given by (13). The quantization noise is uncorrelated at the codec output, and we assume that any signalindependent noise ( e.g., AWGNj is also uncorrelated at that point. For simplicity, we ignore the latter noise component, but the end result is not affected. We assume that the receiver includes a linear equalizer, either zero forcing or minimummean-square with high signal-to-noise ratio. Under that condition, the output noise-to-signal ratio, E, is given by [5]
where ?I is the mean-square value of the constellation points, where H ( f ) is the Fourier transform of h ( t ) and N ( f) is the power spectral density of the quantization noise at the input to the receive leg. It is given by
-
= 2 c2
I
C
IH(f+klT)P = a .
(29)
k
This looks very much like the Nyquist criterion for no inter-
symbol interference, but note that the argument is the square magnitude of the transmit leg transfer function, rather than the transfer function itself. If the signal is bandlimited to the Nyquist interval [-1/2T,1/2T] , then only one term appears in (29). The optimization condition is then just a flat amplitude over the band. This is a good approximation for systems with sharp rolloff. Since the transmit leg includes the transmitter filter and the access channel, for sharp rolloff systems the amplitude of the optimum transmitter filter is the reciprocal of the transmit channel amplitude response. It is important to note that any added phase function will have no effect on the optimization, nor will any change in the transmit power. We will use the results of this section in Section VI where we describe the " Smear-DesmearTechnique." V. RECTANGULAR CONSTELLATION - OPTIMUM AWGN RECEIVER
112T
K'
E therefore achieves its minimum value of K'IT when
CIH(f+k/T)12df.
(24)
-1l2T k
where several constants have been combined into K'. Since the noise is uncorrelated, N ( f ) is constant, and can be removed from the integral in (23), which then becomes
In this section we describe the optimum detection technique normally used with a typical rectangular constellation, and that technique's detectability performance in a companding environment. It should be emphasized that a rectangular constellation is especially undesirable in this environment because of its high peak to average power ratio. However it is simple to analyze, and provides valid comparisons of different approaches. Since the quantization noise is not gaussian (although it is thought that it closely approximates a gaussian distribution [lo]), we will use as a measure of comparison for the different modulation and detection techniques, the distance squared, d 2 ,
42 1
KALET AND SALTZBERG: QAM TRANSMISSION THROUGH A COMPANDING CHANNEL
from the proposed decision boundaries normalized by the noise variance, oz in the same direction, i.e., d2/o:. If the noise were gaussian then the probability of symbol error, P r , { e ) , for large constellations could be closely approximated by
error probability greater than I I
I I I
I I
I I I I I
As a reference point we will consider that all of the modulation systems must have a normalized d2/ozratio which would be equivalent to a P r s { e } equal to lo4 for an AWGN channel. Using the inverse of (30) this reduces to d l o , equal to 5.09. We will look at constellation sizes of 64, 256 and 1024 points, and normalize so that d is equal to one for the normal rectangular constellations. The total noise variance in the direction of a decision boundary is therefore given by
02
= 1/(5.09)2 = .0386
1 EJN,
I
I
I
.
-1
.
0
1
I
0
I
-1
I I w
Fig. 3.
I I I I I I
.
1
I
3
I
tâ
I I I
.
-3
*
I I I I
I
I
I I
I I
I I
16-QAM rectangular constellation with optimum decision boundaries for AWGN.
VI. SMEAR - DESMEAR TECHNIQUE
where L is the number of signal levels in one dimension. Using (32) and L = 8, 16,32 respectively for constellations of size M = 64,256, and 1024 we find that for a P r , { e } equal to IO6, the E,IN, , required is given by the AWGN -ONLY :olumn in Table 1. TABLE 1 Performance Comparison (12.5%roll off) RequiredEJ N o , GB (Pr{e} = 10-5
27.4 dB 37.1 1024
* I 3 * , * I
I
-3
I
3
I I
I
I
+-L2-1
I
I
(31)
In the case of optimum rectangular constellation the detection region is the rectangular grid which is optimal for AWGN. (See Fig. 3.) If we consider AWGN - without companding, then the variance a$ (and o,â) from equations 16 (and 17) reduces to
I
39.5
Nhen companding is added to the system, the degradation for i 64 - points constellation is about one-half a dB, however for !56 - points there may be as much as a 3 dB difference, and for 024 the companding noise is greater than the value of .0386 if (31). In this case even if the total noise is not gaussian the ompanding will in all probability limit the performance to an
Now that we have treated the problem of minimizing the total quantization noise at the receiver sampler input, we are ready once again to consider the effect of the uneven distribution of that noise among the constellation points. In this section we describe one approach to overcome the multiplicative nature of the quantization noise, by spreading the total quantization noise over the entire constellation, thereby producing the same average noise for each point. Without this spreading, assuming low distorting end-links, the high amplitude constellation points will be subjected to more noise than the low amplitude points. The effect of the desmearing filter is to distribute the same total noise power more evenly among the constellation points, thereby substantially improving overall performance, as will be demonstrated here. The smear - desmear technique is essentially identical to that proposed many years ago to combat impulsive noise [6]. More recently, it has been proposed for use in source quantization, as a means of providing a more Gaussian probability distribution to the quantizer [111 when the source itself has arbitrary distribution. One measure of the uneven distribution of quantization noise is the ratio of the worst case noise variance about any constellation point in any direction, to the average one-dimensional noise variance. The maximum noise variance occurs in the
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 21314, FEBRUARYMARCHIAPRIL 1994
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radial direction, and about constellation points of maximum magnitude. It is determined from equation (8). The average one-dimensional noise is half that of (13). Treating only quantization noise, the performance measure can then be written as
(33)
1.125 2T We now multiply H ( f ) by the unity magnitude smearing function
where
P,, =
I 7 .
(34)
The peak-to-average constellation power P,, , is a function of the constellation shape and the position of the points within that boundary. For a dense square constellationof equally spaced points, P , = 3 . A dense, qui-spaced circular constellation has P,, = 2. An interpretation of the performance measure, R, is the factor by which the quantization noise would have to be reduced, so that the error probability given an outer point was transmitted, would be the same as would result if the quantization noise were the same average value, but independent of the transmitted point. Implicit in this interpretation are two assumptions: the probability distribution of the quantization noise has the same shape in both cases: and errors result in the direction of maximum noise. In the asymptotic case of very high signalto-noise ratio, almost all errors will result when outer points are transmitted, so the performance measure applies to overall error probabilities in the two cases. A specific realistic example was investigated numerically in order to test the effectiveness of smearing and desnearing in the presence of quantization noise. The base system was one in which both the transmit and receive legs have baseband equivalents that are real with 12.5 percent rolloff square-root raised-cosine shaping. This provides zero intersymbol interference and the transmit leg meets the total noise optimization requirement of (29).
This quadratic phase function provides a linear group delay versus frequency. The parameter d is a measure of the amount of smearing. The phase shift at the Nyquist frequencies are K d / 2 radians relative to the center frequency phase. The difference in group delay between the lower and upper Nyquist frequencies is 2dT. The desmear filter provides the negative of the phase function in (36). Intersymbol interference is still eliminated and transmitter optimization remains unchanged. From the definition (13), it is easily seen that the average total noise and therefore the denominator of (33) are unaffected. Fig. 4 shows the results of this evaluation. The performance measure R, is plotted in dB, as a function of the smearing parameter for a dense square constellation with P,, = 3. The performance loss would be less for a circular constellation. Without smearing, the performance loss is 5.6 dB. This can be reduced to 1.2 dB by using a smearing factor of d = 4, which will be shown to be reasonable. Greater smearing requires higher delay and cost, with only small improvement in performance. The use of a linear delay versus frequency smearing function is a reasonable choice, although it may not necessarily be optimum in terms of implementation complexity and delay. In [8] it is shown that this function is not optimum for combating impulse noise, and that considerable improvement can be obtained in terms of maximizing smearing efficiency and minimizing residual intersymbol interference with a given length filter. However these results are not directly applicable to the problem being dealt with here. There are many choices available to the modem designer in implementing a given smear/desmear function. Current technology typically involves the use of tapped delay lines @IR) for such linear filtering operations, but IIR implementation may be advantageous.
~
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KALET AND SALTZBERG: QAM TRANSMISSION THROUGH A COMPANDING CHANNEL
6 P
e !
5
0
r m a
n
4
C
e M e
3
a S
ur e i
n
2
1
d
B
0
0
1
2
3
4
5
6
7
Smear Parameter d
Performance loss for a circular dense constellation with 12.5% rolloff. Linear delay vs. frequency smearing, with delay difference of 2dT between Nyquist frequencies.
Fig. 4.
Either the smear or the desmear operation can take place at baseband or at passband. The smear filter can be combined with the transmit filter, but the resultant will require approximately the same number of taps as the sum of separate filters. Similarly, the desmear filter can be eliminated and the receive equalizer allowed to automatically provide this function. This however will increase the required number of adjustable taps. A few test cases indicate that a compromise receiver design can be cost effective, in which a relatively short desmear filter relieves the required increase in equalizer taps. To give some idea of the complexity and delay involved in smearing, filters were designed for the d = 4 case described above. A combined transmitter shaping and smearing filter was designed using 88 complex taps, spaced T/4, whereas a comparable shaping filter alone would require approximately 44 taps. At the receiver, a 43-tap equalizer was required to reduce residual mean-square intersymbol interference to 40 dB below the signal. Another measure of performance, which will also be used in subsequent sections, is the additional noise that can be tolerated so that the total noise variance, for the worst received constellation point and in the worst direction, is equal to a given value. Here we will use that variance value, which if the noise ere gaussian, would produce a symbol error probability of 10' . Again we note that quantization noise is not gaussian, but may be well approximated as such [lo], and is more so if smearing is employed than if not. We have evaluated this measure for square constellations with the filtering described above, and a symbol rate l/r of
27
3000 Hz and a codec sampling rate, l / A , of 8000 Hz , The noise variance used is that in the horizontal or vertical direction because of the mechanism which produces errors. The results shown in Table 2 (in Section VIII) for various size constellations indicate the permissible added noise in terms of E J N , in dB. First we show the permitted E , / N , if there were no quantization. Next is shown that value if there is quantization but if no smearing is employed. Finally we give that measure if perfect smearing were obtained, that is the quantization noise would correspond to R = 1 and d+w in the above analysis. We note fro the table that for a desired error rate in the -g . . . vicinity of 10 , quantizauon noise has little effect on performance of a 64-point constellation, degrading the added noise measure by only 0.5 dB even without smearing. On the other hand, for a 1024- point constellation, the quantization noise dominates performance, exceeding the given noise variance with or without smearing. The most interesting case that illustrates the value of smearing is that of the 256-point constellation. Here the added noise performance is degraded by 3.7 dB if the quantization noise is not smeared, but only 1.3 dB if infinite smearing could be accomplished. VII, CONSTELLATION DESIGN WITH WARPING (TWO-DIMENSIONAL
AND RADIAL) As was mentioned in previous sections one of the important characteristics of the logarithmic quantizing noise is its multiplicative nature. In the constellations described in this section we attempt to design new QAM - constellations taking into account the effects of both the additive white gaussian noise of the channel and the quantizing noise as well. This design differs from that of [l-31 in that not only is the multiplicative nature of the noise taken into account, but also the elliptic shape of this noise as well.
Two-Dimensional Warping The design involves warping the signal constellation so that the normalized variance of the noise components in the x and y directions are the same for all points in the constellation. A normalized noise component is defined (as previously) as the distance from the nearest decision boundary divided by the noise component in that direction. This seems like a logical design, and would result in equal error probability in each of the x and y coordinates for all points, if the quantization variables were gaussian. The proposed signal design may not be optimum since the exact probability density function of the output companding noise is not known (in which an optimum constellation may be hard to define). The actual design is based on the following steps to be shown in first quadrant only (See Fig. 5). The calculation of a number of the constants to be used in design is shown in Appendix A.
1. Assume a QPSK (4-point) constellation as an initial
IEEE TRANSACTlONS ON COMMUNICATIONS, VOL. 42, NO. 21314, FEBRUARYMARCWAPRIL 1994
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normalized distance required by (A.4). The resulting constellations for 64,256 and 1024-point constellations are shown in Fig. 6. Notice the warping effect on the constellations in this design. Table 2 (in the next section) contains the calculation of E , I N , to achieve the normalized distance for an error probability equal to lom6,for this constellation. We notice that in this design, even transmission using a 1024-pointconstellation is possible. c
*
*
*
*
*
*
*
*
*
*
*
* 6 -
4 -
I
I
1
I
2
I
3
+
I
4
I
5
2 Fig. 5.
Design of warped constellation for 16-QAM. ( " X represents origlnal constellation point," "represents new point).
modulation condition and use symmetrical decision regions (which would be optimal for AWGN). Each coordinate has a length of one. We also assume that the noise components in the x and y directions have variance ot,which is determined by some performance criterion. If we were to expand this constellation symmetrically to a rectangular 16-QAM, the next point of height one would be at (3,l). However because the noise is multiplicative we know that the noise variance in the x direction, CY^,^, of this point will be greater than q l . Therefore to keep the normalized distance to the decision boundary equal we temporarily place this point at ((2 + CY,,,CY^) ,1) where CT,,, will be determined so as to keep the normalized distance property intact. We repeat the same process in the y direction using the noise variance in the y direction, oY,,and finally place this point at [(z
+ cXu/ol),
/vl I], For larger constellations, we continue in a similar manner for all the
points with an original height of one.
2. We then repeat this process for y equal to three and so on until we fill in that part of the constellation between the horizontal axis and a 45" line. By symmetry we can fill in the remainder of the constellation. The calculations of the constants ol, CT,,, and C T ~ , ,are shown in the Appendix A. After we complete the design we then calculate the E,IN, , as shown in the Appendix, to achieve the
*
0
Fig, 6a.
* I
I
I
2
4
6
8
Warped constellation design -64 points (first quadrant).
Radial Warping
We have also tried a simpler, sub-optimum approach to constellation warping, in which points in an initial rectangular grid constellation are moved in a radial direction only, as a function only of their distance from the origin. This approach shows promise of being automated. A single warping parameter is determined at the receiver based on a measurement of noise spread differences during a training period, and that parameter is sent to the transmitter for constellation construction. To motivate the choice of warping function, consider a very dense constellation so that the noises associated with adjacent signal points are approximately equal. Furthermore, assume that we are concerned with adjacent points on the same radius. Original adjacent points with magnitude p and p+Ap are mapped onto new points c and c + Ac by the warping function, in a manner such that Ac is proportional to the standard deviation of the radial noise at c (and c + Ac) A p = k- AC OP
(37)
KALET AND SALTZBERG: QAM TRANSMISSION THROUGH A COMPANDING CHANNEL
425
where 0: includes both additive quantization noise and signal-independent noise. Approximating increments by differentials, (37) becomes
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
$
*
*
*
- *
*
*
*
*
*
*
*
*
*
*
*
*
*
* *
*
*
*
*
*
*
* *
*
*
*
*
*
*
*
*
* O t
15
lo
-
5 - *
* *
*.
which has the solution
where
Solving (40) for c gives the warping function 1
Fig. 6b.
c = -sinh(gp). l?
-
We may further scale the points to achieve the same average power as in the original constellation. What we propose is to use the warping function (41) even though the assumptions leading to its derivation do not precisely apply to the problem at hand. In particular, we are concerned with the horizontal and vertical spacing of adjacent points, rather than radial spacing. However the following example shows that we come very close to the desired objective of equal noise performances for all constellation points.
Warped constellation design 256 points (first quadrant).
j0 -
*
*
* *
The procedure used is: Choose an arbitrary value of g ,and radially move each point in a rectangular grid according to the warping function (41). I
Calculate the new average power, and multiply all points by a constant so as to achieve the same average power as the original grid. Assume an allowable 0: and 0; ,normalized by the distance between adjacent points, as in the previous design. Calculate (7: from (16) and (17) for all point pairs.
0
Fig. 6c.
10
20
30
40
50
Warped constellation design - 1024 points (first quadrant).
From (8), bpis of the form
60
If the spread of 6, 's is large, repeat the procedure using a different warping factor g , We have performed this procedure for the same 0.125 rolloff system treated previously. After several iterations using different values of g we were able to find added noise tolerances values for all point pairs such that maximum and minimum of such values differed by only 0.12 dB, for the 256-point constellation. The resultin performance, again in terms of added E , / N , to achieve 10-dsymbol error probability, is given in Table 2.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 42, NO. 21314, FEBRUARYIMARCWAPRIL 1994
For the case of most interest, the 256-point constellation, this warping approach achieves a performance loss of 1.1 dB compared to the case of no quantization effect. We have repeated this evaluation using an X - X 3 I6 approximation to sinhX, with almost identical results to those given above. In this case it is interesting to note again that even for a 1024-point constellation it is possible to achieve the normalized m'nimum distance required for a symbol error probability of 10-b (under the gaussian assumption). However there is an increase of almost 6 dB in required power. VIII. DISCUSSION OF RESULTS We have considered four possible constellation designs and transmission methods. These are the rectangular QAM constellation and optimum AWGN receiver, a smear-desmear technique with the same constellation, two-dimensional warping and radial warping with appropriate detection. It is expected that the comparative results would be very similar if a non-rectangular constellation were used as the starting point, although the absolute performance could be substantially
IX. CONCLUSIONS We have examined the problems of performance of QAM constellations over a companding channel, including the design of a constellation to overcome the multiplicative nature of the combined quantization noise and AWGN. The companding process designed to improve analog voice transmission, has a detrimental effect on high-speed data transmission. For example, as shown, if a 256-point ( 8 bits/symbol) rectangular QAM constellation is used, which corresponds to a data rate of 24 kbps (3000 bauds) the performance degrades by about 4 dB, as compared to that of a channel without companding. For a 1024-point signal the companding n ise is so great that transmission at a symbol error rate of 10-% is not possible. Three methods which improve performance were investigated. The "Smear-Desmear" method does not change the shape of the constellation but spreads the total quantization noise over all the points. In this case, it is possible to return the performance to within 1.1 dB of that of a system without companding. But 1024-point signaling is not possible. The two other methods warp the constellation to try to maintain different normalized distance criteria. Both of these techniques are a little bit better than the SMEAR-DESMEAR method for 64 QAM. For a 1024-constellation it is possible wi h the Two-Dimensional Warping technique to signal at a 10- error rate but with a degradation of 6 dB as compared to the noncompanding channel. All of the results above indicate that with proper design of signal constellation and receiver structure, high data rate transmission is possible over companding channels.
2
Const. Rectangular Constellation TWO-D Sinh Size Optimum Detection SmearAWGN AWGN Desmear Warping Warping ONLY Companding 27.9 21.6 21.6 64 27.4dB 21.1 33.4 37.1 34.1 34.4 256 34.5 1024 39.5 45.2 48.3 I
-
As we see from the table above the performance of 64-QAM is slightly degraded by the companding (about 0.5 dB), and all of the techniques designed to improve performance give some small improvement. For a 256-point constellation, all three improvement techniques result in about a 2.5 dB improvement. However for the large 1024-point constellation only the warping techniques are able to achieve the normalized distance criterion (under th gaussian assumption) for a symbol error rate of less than 10- , The warping methods accomplish the above with a degradation of more than 6 dB, but at least in this case companding does not dominate performance to an extent that desired communication is not possible. All of the results above do not include coding, which should lower the required E / N o , by more than 3 dB, depending on the coding complexity. In addition, it appears that the warping technique should be modified for use with trellis coded signals, in order to account for sequence dependencies. This remains a subject for further investigation.
8
X. ACKNOWLEDGMENTS
The authors thank W. Beth, G. Golden, and J. Mazo for many stimulating discussions, and 6. D. Forney for his valuable comments. APPENDIX A WARPED-CONSTELLATION
We will describe the calculation of the variables required for the signal constellation design of Section VII. At every iteration of the process described in Section VI1 for finding the coordinates of the warped constellation, determine these values by using the equations below
~
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KALET AND SALTZBERG: QAM TRANSMISSION THROUGH A COMPANDING CHANNEL
-
(where o2 = c 2 )
where the initial values of x i , j and yi,j with i or j equal to and minus one, are q u a l to zero, e.g., x - l j = 0. Also cry,, are equal to zero for i or j equal to minus one, e.g., =x_,J = 0. The values i a n d j refer to the original location of the points in the rectangular grid constellation, i.e., ~ 3 , refers l to the new x location of the point that was originally at (3,l). Also, i and j take on odd values, i.e., 1,3,5,.... At each point we determine the values of ox,,,and by,,by solving the equations below which are derived from the requirement that the normalized distance be the same in the x direction and y direction for a given constellation point, and therefore be the same for all points.
k2+4 2
= 3
A M = --K T 4 3
= -M
+A
We start the process by choosing a b?. If we assume that the combined noise is gaussian then we can choose offor a given Pr {a} by using the equation below
We then find A from equation (A.6), After we calculate all the new constellation points, we calculate 0 2 ,and then from equation A.4, we find the appropriate value of E, IN,. APPENDIX B OPTIMUM RECEPTION
and
The variables k l , k 2 , b,, by,cl, A and M are defined below.
and
1
A=u2 --M+-j 1 E,IN,
The actual probability distribution of the companding noise is not known, even though its second order statistics are well defined [4]. It may even be true that the distribution is a function of the constellation size, and rolloff factor, r. There is however some reason to assume that it is approximately gaussian in nature [lo]. If the total noise is assumed gaussian then an optimum receiver can be found for any given constellation using the radial representation of the signals (e.g., the two signals in Fig. 2) and applying the likelihood rates as shown below. Consider the two constellation points A l and A2, shown in Fig. B.l, where A (or A 2 ) is the amplifude of the vector A l (or A ) and 6 (or 6 2 ) is the phase angle of the vector A ('or A 2 ) r The received vector r, has amplitude equal to R i n d a bhase equal to 0 R. We can-write the vector z, as a function of either a two dimensional set of axes around A 1, or a similar set of axes around A2. If the axes are chosen as shown then the noise components are uncorrelated [4],and if we assume that they are gaussian, then they are also independent. Therefore we can write the likelihood ratio, A ( r ) in the following manner.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO U314,FEBRUARYMARCWAPRIL 1994
I
Decide A
-2
We see that the terms within the brackets resemble a distance, and so we define a normalized distance dN,,between c and 4 as
We can then rewrite equation (B.3) as
I
I
We can normalize one step further by defining another normalized distance dhz,where
Therefore we may rewrite (B .5) as Fig. B.l. Decision regions - general binary case.
where
oi; = 05
=
Pi
=
ei
=
Except for a bias factor this has the same form as the optimum decision rule for signals in additive white gaussian noise. An optimum receiver, may now be built by "simply" finding the normalized distances in the receiver, and comparing them, taking into account the bias factor b ~ p L n ( b ~ p ~ ~ e l b l p ' 3. 1 e ) 2 The actual decision boundary is elliptical in nature.
variance of multiplicative noise in direction of A variance of muItiplicative noise tangential to 4 component of -r in direction of Ai component of c tangential to 4 i .
I
REFERENCES
p and 8 are given by pi = m s ( e R - e i ) 8 , = Rsin(OR-Oi)
where R is the amplitude of c, and O R its phase angle.
[l] K. Pahlavan and J. L. Holsinger, "A Model for the Effects of PCM Compandors on the Performance of High Speed Modems," GLOBECOM '85, New Orleans, December 1985,pp. 24.8.1-24.8.5.
(B3
[2] K. Pahlavan and J. L. Holsinger, "A Method to Counteract rhe Effects of PCM Systems on fne Perfonnance of Ultra High Speed Voice Band Modems", ICC '86, Toronto, Canada, June 1986, pp. 50.2.1-50.2.5. [3] K. Pahlavan and J. L. Holsinger, "Expanded TCM for Channels with MultiplicativeNoise," ICC '87, June 1987,pp. 12.3.1-12.3.5. [4]
B. R. Saltzberg and 3.-D. Wang, "Second-Order Statistics of Logarithmic Quantization Noise in QAM Data Communication.'' IEEE
Trans. on'commun., Vol. 39, October 1991.
After some manipulation equation (B.l) reduces to
[SI K. W. Cattermole,"Principles of Pulse Code Modulation",Iliffe Books Ltd., London, 1969 - ChaptetIII
[6] R. A. Wainwright, "On the Potential Advantage of a SmeanngDesmearing Filter Technique in Overcoming Impulse Noise Problems in Data Systems," IRE Trans. on Comm. Systems, Vol. CS-9, December 1961, pp. 362-366.
[7] B. R. Saltzberg, "Smear-Desmear Technique for Combating Quantization Noise," COMSPHERE '91, Herzliya, Israel, December 1991.
[8] G . F. M. Beaker, T. A. C. M. Claason, and P. J. VanGerwen," Design of Smearing Filters for Data Transmission Systems",IEEE Trans. Commun., Vol. COM - 33, Sept. 1985, pp. 955 - 963.
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KALET AND SALTZBEUG: QAM TRANSMISSION THROUGH A COMPANDING CHANNEL
[9] K. Pahlavan, "Nonlinear Quantization and the Design of Coded and Uncoded Signal Constellations", IEEE Trans. Commun., Vol. 39, August 1991, pp. 1207 - 1215. [ 101 J. E. Mam, "Quantization Noise and Data Transmission", BSTJ, Oct.
1968, pp. 1737 - 1753.
[ l l ] A. C. Popat and K. Zeger, "Robust Quantization of Memoryless Sources", 1990 Int. Symp. on Info. Theory and Appl., Hawaii, Nov. 1990, Pawr No. 33-2.
Irving Kalet was born in The Bronx, New York. He received his BEE in 1962 from CCNY, and his MS(EE) and Dr. Eng. Sc. from Columbia University in 1964and 1969 respectively. He taught in CCNY, and for the past twenty years has been teaching in Israel, where he lives. He is currently teaching seminars in topics mceming modulation techniques, and also acts as a consultant in this area in Israel and in the United States. He is presently spending a year at AT&T Bell Laboratories, Middletown, New Jersey working in the area of wireless data communication. He has previously worked in the areas of satellite communication, at MIT Lincoln Laboratories, and high-speed data transmission over the twisted pair channel. He is a member of Eta Kappa Nu Tau Beta Pi and Sigma Xi.
Burton R. Saltzberg (S152-M'55-F'76) received the B.E.E. degree from New York University in 1954, the M.S.from the University of Wismnsin in 1955, and the Eng.Sc.D. from New York University in 1964. Dr. Saltzberg joined AT&T Bell Laboratories in 1957. Since that time, he has been primarily engaged in the development, analysis, and initiation of data communication systems. He has published many articles in this field, and has been issued 23 patents. He is currently supervisor of the Data Theory Group in the Data CommunicationResearch Department. He served as chairman and in other offices of the Data Communications technical committee of the IEEE Communications Society. Dr. Saltzberg was general chairman of the 1989 IEEE Communication Theory Workshop, and Guest Editor of the August 1991 issue of JSAC on High Speed Digital Subscriber Lines. He is a member of Tau Beta Pi, Eta Kappa Nu, and Sigma Xi. He received the IEEE Communication Society Armstrong Achievement Award in 1991, "for sustained major contributions to the theory and practice of data communications".
417
QAM Transmission Through a Companding Channel - Signal Constellations and Detection Irving Kalet and Burton R. Saltzberg,Fellow, IEEE Abstract-The effect of the companding process on QAM signals has been under investigation for the past several years. The compander, included in the PCM telephone network to improve voice performance, has an unusual affect on digital QAM data signals which are transmitted over the same channel. The quantization noise, generated by the companding process which is multiplicative (and asymmetric), degrades the detectability performance of the outermost points of the QAM constellation more than that of the inner points. The combined effect of the companding noise and the inherent white gaussian noise of the system, leads us to a re-examinationof signal constellation design. In this paper we investigate the detectability performance of a number of candidates for signal constellations including, a typical rectangular QAM constellation, the same constellation with the addition of a smear-desmear operation, and two new improved QAM constellation designs with two-dimensionalwarping.
I. INTRODUCTION Recently, there has been an interest [l, 2, 3,4] in the effect of the pulse code modulation (PCM) companding [5] mechanism (introduced to improve voice transmission), inherent in communication networks, on the detectability performance of digital Quadrature Amplitude Modulation (QAM) signals uansmitted through these networks. As has been previously shown [l, 2, 3,4], the effect of the companding process is to introduce quantization noise which is multiplicative in nature. This noise is amplitude dependent, i.e., large amplitude signal points in the QAM constellation will be surrounded by more noise than signals closer to the origin. The effect of this noise is such that the outermost points in a simple rectangular constellation suffer more performance degradation than the smaller internal signals. Saltzberg and Wang in [4]made the observation that not only is the noise around each point dependent on its amplitude, but that this noise is elliptical and not circular, Le., the noise components in the two dimensional QAM space are not equal. In [4],it was also shown that the two dimensional noise components are not only unequal, but also correlated for most choices of vector bases. These properties make the detection problem and the signal constellation design problem very interesting, and represent a new challenge in signal design. Paper approved by Nikolaos A. Zervos, the Editor for Transmission Systems of the IEEE Communications Society. Manuscript received October 3, 1991; revised March 16,1992. I. Kalet lives in Haifa, Israel and is currently with AT&T Bell Laboratories, Middletown, NJ 07748. B. R. Saltzberg is with AT&T Bell Laboratories, Middletown, NJ 07748. IEEE Log Number 9400925.
In this paper we consider the problem of digital signal constellation design for the companding channel. We first consider the performance of the typical rectangular QAM constellation, and then the same constellation with smear-desmear [6, 7,8] filtering applied to average the noise over the entire constellation, i.e., to spread out the multiplicative noise over all the constellation points. We then consider two versions of two-dimensional warping of the original constellation, which preserve a constant normalized distance between constellation points and their neighboring decision boundaries. It should be noted that Pahlavan has recently also considered the design of constellations so as to maintain normalized distance between constellation points [9], where normalization is with respect to the total noise power surrounding each constellation point. In the next section we review previous work on companding channels and describe the system model. In Section 111, we describe the combined effect, of additive white gaussian noise (AWGN) and companding, on performance. Sections IV and V describe transmitter filter optimization and a smear - desmear technique for reducing the effect of the companding noise. Sections VI and VI1 discuss the detectability properties of rectangular constellations and the proposed warped constellations, and finally the last two sections contain a discussion of results and conclusion. 11. SYSTEM MODEL The system model is shown in Fig. 1 [41. The QAM signal, (the input to the coder), is generated by a stream of digital data which modulates the carrier at the frequency,fo. This signal is digitally converted by the coder and transmitted through a digital companding channel to the receiver in which it is converted back to an analog QAM signal. The coder is in general of the p-law [5] or A-law type. In this work we will assume a p-law device, and approximate it as being logarithmic. The multiplicative noise is generated by the quantization and companding process. s(t),
As described in [4], the transmitted signal s ( t ) before companding is given by
where c, = a , + jb,, b , are independent)
0090-6778/94$04.000 1994 IEEE
a n r b n= t-1, t-3,- - - (a, and
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Analog End-Link Transmitter
:
AWGN
I
Coder
Receiver
r
Overall Baseband h(t) Equivalent Filter,
where * denotes convolution. Given that constellation point c, is transmitted, the radial and tangential noise variances abou? the received point are
Overall Baseband f(t) Equivalent Filter,
System model-h(t) includes the transmit filter, the transmitter endlink channel, and the codec input filter, f(t) includes the codec output filter, receiver end-link channel, receiver input filter, and equalizer.
Fig. 1.
and
- (t) = hr(t) h
+ jh,(t).
(2)
The signal h( t ) is the equivalent baseband impulse response of the transmit filter, the transmit end-link channel and the codec input filter. It is assumed that h ( t ) is bandlimited to frequencies in the
[
range - f m a x f m m whereto > f m a x . The mean square quantization error $, at the kth sample, caused by the companding process is given by [4], [5]
where k , and ke are multiplicative noise coefficients given by
[
k , = (1/2) AKT2 2z+ (0)
i
ke = (1/2) AKT2 22, (0)
9
a$ = K s2
[ kA + a ]
[Ln (1 + p)I2 / 3 N 2
-
(0) -22, (O)]
z - (0)
(9)
+ 2 z , (O)],
k, is the additive noise coefficient k;
=
AKf2x a + ( n T ) ,
(10)
n#O
(3)
where l / A is the sampling frequency, and a is an arbitrary random offset, assumed to be uniformly distributed between k A / 2 . The quantization errors at different samples are assumed uncorrelated. For the p- law codec with N levels, K is given by [ 5 , s ]
K
+ z-
and 0: is due to additional noise (e.g., AWGN) that is not signal dependent. (Similar definitions of the constants k , can be made if we use the x and y axes ) The cross - correlation of the radial and tangential noise components is
(4)
where N equals the number of codec levels. For typical values of p and N , p = 255 and N = 28(256), K equals 1.564 x 10-4, or -38.0 dB. At the receiver, the quantized signal is received and expanded back to s( l ) , now including the effect of the multiplicative quantization noise. It is assumed that the receiver contains a filterf(t) used to ideally equalize the QAM signal (i.e., no intersymbol interference) so that the output points equal c,. In [4] a general solution is found for the effect of the miltiplicative noise on the signal outputs, for a general h(t). The results may be summarized in a restated format, by first defining
This quantity is clearly zero whenf(t) and h(r) are real, and can be shown to be small when F ( f ) and H ( f ) are close to flat in amplitude. Assuming the noise components in (8) to be uncorrelated, the total noise, a$,is
-
and its average, cr2 ,over all constellation points is
0'
w
= 2AKT2
z+(nT) ?a?=--
+ 2 0 , '.
(13)
KALET AND SALTZBERG: QAM TRANSMISSION THROUGH A COMPANDING CHANNEL
111. THE EFFECT OF AWGN
.,"= 4 =!
In this section we investigate the quantitative effect of AWGN on the second-order output statistics. We will consider h( t ) andf (I ) which are real.
(J;=
[(3a:
2
+ b:)
z(0)
+
*IcT2 [(a; + 3 b 3 z(0) + 2
n+o
J
and the correlation, oV ,is given by
oxy= A K T 2 a o b oz(0)
.
If AWGN now appears at the input to the equivalent receiver channel whose impulse response is given byf(t) cos 27c fo t, ( f ( t ) is real), then the output noise n, ( t ) due to the AWGN, will be given by n , ( t ) = n,,(t)cos
2n f o t - nWq(t)sin2n f o t (19)
where
Fig. 2. Signal point 2. and unmrrelated quantization noise components, np and ne.
If we assume that h ( t ) andf(t) are real, then it follows from (11) that the output quantization noise surrounding each point is such that its radial and tangential components, as shown in Fig. 2, are uncorrelated, i.e., E i n p ne J = 0. This is not necessarily true for other axes, e.g., x and y . It follows from (5)-(lo), that c$ and 02 are given by.
-
02 p -
AKT2 2
13
(co12 z ( 0 ) + 2 7
z(nT4
Since the detector recovers n,,,,( t ) in the in-phase direction and n,, ( t ) in the quadrature direction, 0;will be equal to either E{n,:(t)] ,orE{n$ ( r ) ] ,or equivalentlyE{n;(t)J. We find G: by first determining the power spectral density, Pnw (f). If F(f) is the Fourier transform off(t) ,then
(14)
n#O
and
"
I
n
J
Therefore,
+ 0; If we consider the x and y components of the noise then the variances are given as [4]
2
df
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420
=
-
NO I_
4
J
Denoting the summation by A 2 ( f ) , the quantity to be minimized is of the form
I F ( f ) I 2df.
-m
For the ideal-bandlimited Nyquist channel, with F ( f) = 1, for No
If1
1
< 1 / 2 T , E{n: ( t ) ] equals - - . These results remain 4 T the same for square-root raised-cosine, Nyquist filtering with non-zero roll off. Using the results above we can now find the total noise variance about any given received constellation point c,. In the sections that follow we will analyze a ncmber of different receivers in the combined AWGN, companding noise environment. However we first look at the general problem of minimizing the total quantization noise power at the output.
From the Schwartz inequality,
with equality if and only if A ( f ) = a / A ( f ),or
IV. TRANSMIT FILTER OPTIMIZATION We will first find the transmit filter which minimizes the total noise given by (13). The quantization noise is uncorrelated at the codec output, and we assume that any signalindependent noise ( e.g., AWGNj is also uncorrelated at that point. For simplicity, we ignore the latter noise component, but the end result is not affected. We assume that the receiver includes a linear equalizer, either zero forcing or minimummean-square with high signal-to-noise ratio. Under that condition, the output noise-to-signal ratio, E, is given by [5]
where ?I is the mean-square value of the constellation points, where H ( f ) is the Fourier transform of h ( t ) and N ( f) is the power spectral density of the quantization noise at the input to the receive leg. It is given by
-
= 2 c2
I
C
IH(f+klT)P = a .
(29)
k
This looks very much like the Nyquist criterion for no inter-
symbol interference, but note that the argument is the square magnitude of the transmit leg transfer function, rather than the transfer function itself. If the signal is bandlimited to the Nyquist interval [-1/2T,1/2T] , then only one term appears in (29). The optimization condition is then just a flat amplitude over the band. This is a good approximation for systems with sharp rolloff. Since the transmit leg includes the transmitter filter and the access channel, for sharp rolloff systems the amplitude of the optimum transmitter filter is the reciprocal of the transmit channel amplitude response. It is important to note that any added phase function will have no effect on the optimization, nor will any change in the transmit power. We will use the results of this section in Section VI where we describe the " Smear-DesmearTechnique." V. RECTANGULAR CONSTELLATION - OPTIMUM AWGN RECEIVER
112T
K'
E therefore achieves its minimum value of K'IT when
CIH(f+k/T)12df.
(24)
-1l2T k
where several constants have been combined into K'. Since the noise is uncorrelated, N ( f ) is constant, and can be removed from the integral in (23), which then becomes
In this section we describe the optimum detection technique normally used with a typical rectangular constellation, and that technique's detectability performance in a companding environment. It should be emphasized that a rectangular constellation is especially undesirable in this environment because of its high peak to average power ratio. However it is simple to analyze, and provides valid comparisons of different approaches. Since the quantization noise is not gaussian (although it is thought that it closely approximates a gaussian distribution [lo]), we will use as a measure of comparison for the different modulation and detection techniques, the distance squared, d 2 ,
42 1
KALET AND SALTZBERG: QAM TRANSMISSION THROUGH A COMPANDING CHANNEL
from the proposed decision boundaries normalized by the noise variance, oz in the same direction, i.e., d2/o:. If the noise were gaussian then the probability of symbol error, P r , { e ) , for large constellations could be closely approximated by
error probability greater than I I
I I I
I I
I I I I I
As a reference point we will consider that all of the modulation systems must have a normalized d2/ozratio which would be equivalent to a P r s { e } equal to lo4 for an AWGN channel. Using the inverse of (30) this reduces to d l o , equal to 5.09. We will look at constellation sizes of 64, 256 and 1024 points, and normalize so that d is equal to one for the normal rectangular constellations. The total noise variance in the direction of a decision boundary is therefore given by
02
= 1/(5.09)2 = .0386
1 EJN,
I
I
I
.
-1
.
0
1
I
0
I
-1
I I w
Fig. 3.
I I I I I I
.
1
I
3
I
tâ
I I I
.
-3
*
I I I I
I
I
I I
I I
I I
16-QAM rectangular constellation with optimum decision boundaries for AWGN.
VI. SMEAR - DESMEAR TECHNIQUE
where L is the number of signal levels in one dimension. Using (32) and L = 8, 16,32 respectively for constellations of size M = 64,256, and 1024 we find that for a P r , { e } equal to IO6, the E,IN, , required is given by the AWGN -ONLY :olumn in Table 1. TABLE 1 Performance Comparison (12.5%roll off) RequiredEJ N o , GB (Pr{e} = 10-5
27.4 dB 37.1 1024
* I 3 * , * I
I
-3
I
3
I I
I
I
+-L2-1
I
I
(31)
In the case of optimum rectangular constellation the detection region is the rectangular grid which is optimal for AWGN. (See Fig. 3.) If we consider AWGN - without companding, then the variance a$ (and o,â) from equations 16 (and 17) reduces to
I
39.5
Nhen companding is added to the system, the degradation for i 64 - points constellation is about one-half a dB, however for !56 - points there may be as much as a 3 dB difference, and for 024 the companding noise is greater than the value of .0386 if (31). In this case even if the total noise is not gaussian the ompanding will in all probability limit the performance to an
Now that we have treated the problem of minimizing the total quantization noise at the receiver sampler input, we are ready once again to consider the effect of the uneven distribution of that noise among the constellation points. In this section we describe one approach to overcome the multiplicative nature of the quantization noise, by spreading the total quantization noise over the entire constellation, thereby producing the same average noise for each point. Without this spreading, assuming low distorting end-links, the high amplitude constellation points will be subjected to more noise than the low amplitude points. The effect of the desmearing filter is to distribute the same total noise power more evenly among the constellation points, thereby substantially improving overall performance, as will be demonstrated here. The smear - desmear technique is essentially identical to that proposed many years ago to combat impulsive noise [6]. More recently, it has been proposed for use in source quantization, as a means of providing a more Gaussian probability distribution to the quantizer [111 when the source itself has arbitrary distribution. One measure of the uneven distribution of quantization noise is the ratio of the worst case noise variance about any constellation point in any direction, to the average one-dimensional noise variance. The maximum noise variance occurs in the
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radial direction, and about constellation points of maximum magnitude. It is determined from equation (8). The average one-dimensional noise is half that of (13). Treating only quantization noise, the performance measure can then be written as
(33)
1.125 2T We now multiply H ( f ) by the unity magnitude smearing function
where
P,, =
I 7 .
(34)
The peak-to-average constellation power P,, , is a function of the constellation shape and the position of the points within that boundary. For a dense square constellationof equally spaced points, P , = 3 . A dense, qui-spaced circular constellation has P,, = 2. An interpretation of the performance measure, R, is the factor by which the quantization noise would have to be reduced, so that the error probability given an outer point was transmitted, would be the same as would result if the quantization noise were the same average value, but independent of the transmitted point. Implicit in this interpretation are two assumptions: the probability distribution of the quantization noise has the same shape in both cases: and errors result in the direction of maximum noise. In the asymptotic case of very high signalto-noise ratio, almost all errors will result when outer points are transmitted, so the performance measure applies to overall error probabilities in the two cases. A specific realistic example was investigated numerically in order to test the effectiveness of smearing and desnearing in the presence of quantization noise. The base system was one in which both the transmit and receive legs have baseband equivalents that are real with 12.5 percent rolloff square-root raised-cosine shaping. This provides zero intersymbol interference and the transmit leg meets the total noise optimization requirement of (29).
This quadratic phase function provides a linear group delay versus frequency. The parameter d is a measure of the amount of smearing. The phase shift at the Nyquist frequencies are K d / 2 radians relative to the center frequency phase. The difference in group delay between the lower and upper Nyquist frequencies is 2dT. The desmear filter provides the negative of the phase function in (36). Intersymbol interference is still eliminated and transmitter optimization remains unchanged. From the definition (13), it is easily seen that the average total noise and therefore the denominator of (33) are unaffected. Fig. 4 shows the results of this evaluation. The performance measure R, is plotted in dB, as a function of the smearing parameter for a dense square constellation with P,, = 3. The performance loss would be less for a circular constellation. Without smearing, the performance loss is 5.6 dB. This can be reduced to 1.2 dB by using a smearing factor of d = 4, which will be shown to be reasonable. Greater smearing requires higher delay and cost, with only small improvement in performance. The use of a linear delay versus frequency smearing function is a reasonable choice, although it may not necessarily be optimum in terms of implementation complexity and delay. In [8] it is shown that this function is not optimum for combating impulse noise, and that considerable improvement can be obtained in terms of maximizing smearing efficiency and minimizing residual intersymbol interference with a given length filter. However these results are not directly applicable to the problem being dealt with here. There are many choices available to the modem designer in implementing a given smear/desmear function. Current technology typically involves the use of tapped delay lines @IR) for such linear filtering operations, but IIR implementation may be advantageous.
~
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KALET AND SALTZBERG: QAM TRANSMISSION THROUGH A COMPANDING CHANNEL
6 P
e !
5
0
r m a
n
4
C
e M e
3
a S
ur e i
n
2
1
d
B
0
0
1
2
3
4
5
6
7
Smear Parameter d
Performance loss for a circular dense constellation with 12.5% rolloff. Linear delay vs. frequency smearing, with delay difference of 2dT between Nyquist frequencies.
Fig. 4.
Either the smear or the desmear operation can take place at baseband or at passband. The smear filter can be combined with the transmit filter, but the resultant will require approximately the same number of taps as the sum of separate filters. Similarly, the desmear filter can be eliminated and the receive equalizer allowed to automatically provide this function. This however will increase the required number of adjustable taps. A few test cases indicate that a compromise receiver design can be cost effective, in which a relatively short desmear filter relieves the required increase in equalizer taps. To give some idea of the complexity and delay involved in smearing, filters were designed for the d = 4 case described above. A combined transmitter shaping and smearing filter was designed using 88 complex taps, spaced T/4, whereas a comparable shaping filter alone would require approximately 44 taps. At the receiver, a 43-tap equalizer was required to reduce residual mean-square intersymbol interference to 40 dB below the signal. Another measure of performance, which will also be used in subsequent sections, is the additional noise that can be tolerated so that the total noise variance, for the worst received constellation point and in the worst direction, is equal to a given value. Here we will use that variance value, which if the noise ere gaussian, would produce a symbol error probability of 10' . Again we note that quantization noise is not gaussian, but may be well approximated as such [lo], and is more so if smearing is employed than if not. We have evaluated this measure for square constellations with the filtering described above, and a symbol rate l/r of
27
3000 Hz and a codec sampling rate, l / A , of 8000 Hz , The noise variance used is that in the horizontal or vertical direction because of the mechanism which produces errors. The results shown in Table 2 (in Section VIII) for various size constellations indicate the permissible added noise in terms of E J N , in dB. First we show the permitted E , / N , if there were no quantization. Next is shown that value if there is quantization but if no smearing is employed. Finally we give that measure if perfect smearing were obtained, that is the quantization noise would correspond to R = 1 and d+w in the above analysis. We note fro the table that for a desired error rate in the -g . . . vicinity of 10 , quantizauon noise has little effect on performance of a 64-point constellation, degrading the added noise measure by only 0.5 dB even without smearing. On the other hand, for a 1024- point constellation, the quantization noise dominates performance, exceeding the given noise variance with or without smearing. The most interesting case that illustrates the value of smearing is that of the 256-point constellation. Here the added noise performance is degraded by 3.7 dB if the quantization noise is not smeared, but only 1.3 dB if infinite smearing could be accomplished. VII, CONSTELLATION DESIGN WITH WARPING (TWO-DIMENSIONAL
AND RADIAL) As was mentioned in previous sections one of the important characteristics of the logarithmic quantizing noise is its multiplicative nature. In the constellations described in this section we attempt to design new QAM - constellations taking into account the effects of both the additive white gaussian noise of the channel and the quantizing noise as well. This design differs from that of [l-31 in that not only is the multiplicative nature of the noise taken into account, but also the elliptic shape of this noise as well.
Two-Dimensional Warping The design involves warping the signal constellation so that the normalized variance of the noise components in the x and y directions are the same for all points in the constellation. A normalized noise component is defined (as previously) as the distance from the nearest decision boundary divided by the noise component in that direction. This seems like a logical design, and would result in equal error probability in each of the x and y coordinates for all points, if the quantization variables were gaussian. The proposed signal design may not be optimum since the exact probability density function of the output companding noise is not known (in which an optimum constellation may be hard to define). The actual design is based on the following steps to be shown in first quadrant only (See Fig. 5). The calculation of a number of the constants to be used in design is shown in Appendix A.
1. Assume a QPSK (4-point) constellation as an initial
IEEE TRANSACTlONS ON COMMUNICATIONS, VOL. 42, NO. 21314, FEBRUARYMARCWAPRIL 1994
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normalized distance required by (A.4). The resulting constellations for 64,256 and 1024-point constellations are shown in Fig. 6. Notice the warping effect on the constellations in this design. Table 2 (in the next section) contains the calculation of E , I N , to achieve the normalized distance for an error probability equal to lom6,for this constellation. We notice that in this design, even transmission using a 1024-pointconstellation is possible. c
*
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* 6 -
4 -
I
I
1
I
2
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3
+
I
4
I
5
2 Fig. 5.
Design of warped constellation for 16-QAM. ( " X represents origlnal constellation point," "represents new point).
modulation condition and use symmetrical decision regions (which would be optimal for AWGN). Each coordinate has a length of one. We also assume that the noise components in the x and y directions have variance ot,which is determined by some performance criterion. If we were to expand this constellation symmetrically to a rectangular 16-QAM, the next point of height one would be at (3,l). However because the noise is multiplicative we know that the noise variance in the x direction, CY^,^, of this point will be greater than q l . Therefore to keep the normalized distance to the decision boundary equal we temporarily place this point at ((2 + CY,,,CY^) ,1) where CT,,, will be determined so as to keep the normalized distance property intact. We repeat the same process in the y direction using the noise variance in the y direction, oY,,and finally place this point at [(z
+ cXu/ol),
/vl I], For larger constellations, we continue in a similar manner for all the
points with an original height of one.
2. We then repeat this process for y equal to three and so on until we fill in that part of the constellation between the horizontal axis and a 45" line. By symmetry we can fill in the remainder of the constellation. The calculations of the constants ol, CT,,, and C T ~ , ,are shown in the Appendix A. After we complete the design we then calculate the E,IN, , as shown in the Appendix, to achieve the
*
0
Fig, 6a.
* I
I
I
2
4
6
8
Warped constellation design -64 points (first quadrant).
Radial Warping
We have also tried a simpler, sub-optimum approach to constellation warping, in which points in an initial rectangular grid constellation are moved in a radial direction only, as a function only of their distance from the origin. This approach shows promise of being automated. A single warping parameter is determined at the receiver based on a measurement of noise spread differences during a training period, and that parameter is sent to the transmitter for constellation construction. To motivate the choice of warping function, consider a very dense constellation so that the noises associated with adjacent signal points are approximately equal. Furthermore, assume that we are concerned with adjacent points on the same radius. Original adjacent points with magnitude p and p+Ap are mapped onto new points c and c + Ac by the warping function, in a manner such that Ac is proportional to the standard deviation of the radial noise at c (and c + Ac) A p = k- AC OP
(37)
KALET AND SALTZBERG: QAM TRANSMISSION THROUGH A COMPANDING CHANNEL
425
where 0: includes both additive quantization noise and signal-independent noise. Approximating increments by differentials, (37) becomes
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which has the solution
where
Solving (40) for c gives the warping function 1
Fig. 6b.
c = -sinh(gp). l?
-
We may further scale the points to achieve the same average power as in the original constellation. What we propose is to use the warping function (41) even though the assumptions leading to its derivation do not precisely apply to the problem at hand. In particular, we are concerned with the horizontal and vertical spacing of adjacent points, rather than radial spacing. However the following example shows that we come very close to the desired objective of equal noise performances for all constellation points.
Warped constellation design 256 points (first quadrant).
j0 -
*
*
* *
The procedure used is: Choose an arbitrary value of g ,and radially move each point in a rectangular grid according to the warping function (41). I
Calculate the new average power, and multiply all points by a constant so as to achieve the same average power as the original grid. Assume an allowable 0: and 0; ,normalized by the distance between adjacent points, as in the previous design. Calculate (7: from (16) and (17) for all point pairs.
0
Fig. 6c.
10
20
30
40
50
Warped constellation design - 1024 points (first quadrant).
From (8), bpis of the form
60
If the spread of 6, 's is large, repeat the procedure using a different warping factor g , We have performed this procedure for the same 0.125 rolloff system treated previously. After several iterations using different values of g we were able to find added noise tolerances values for all point pairs such that maximum and minimum of such values differed by only 0.12 dB, for the 256-point constellation. The resultin performance, again in terms of added E , / N , to achieve 10-dsymbol error probability, is given in Table 2.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 42, NO. 21314, FEBRUARYIMARCWAPRIL 1994
For the case of most interest, the 256-point constellation, this warping approach achieves a performance loss of 1.1 dB compared to the case of no quantization effect. We have repeated this evaluation using an X - X 3 I6 approximation to sinhX, with almost identical results to those given above. In this case it is interesting to note again that even for a 1024-point constellation it is possible to achieve the normalized m'nimum distance required for a symbol error probability of 10-b (under the gaussian assumption). However there is an increase of almost 6 dB in required power. VIII. DISCUSSION OF RESULTS We have considered four possible constellation designs and transmission methods. These are the rectangular QAM constellation and optimum AWGN receiver, a smear-desmear technique with the same constellation, two-dimensional warping and radial warping with appropriate detection. It is expected that the comparative results would be very similar if a non-rectangular constellation were used as the starting point, although the absolute performance could be substantially
IX. CONCLUSIONS We have examined the problems of performance of QAM constellations over a companding channel, including the design of a constellation to overcome the multiplicative nature of the combined quantization noise and AWGN. The companding process designed to improve analog voice transmission, has a detrimental effect on high-speed data transmission. For example, as shown, if a 256-point ( 8 bits/symbol) rectangular QAM constellation is used, which corresponds to a data rate of 24 kbps (3000 bauds) the performance degrades by about 4 dB, as compared to that of a channel without companding. For a 1024-point signal the companding n ise is so great that transmission at a symbol error rate of 10-% is not possible. Three methods which improve performance were investigated. The "Smear-Desmear" method does not change the shape of the constellation but spreads the total quantization noise over all the points. In this case, it is possible to return the performance to within 1.1 dB of that of a system without companding. But 1024-point signaling is not possible. The two other methods warp the constellation to try to maintain different normalized distance criteria. Both of these techniques are a little bit better than the SMEAR-DESMEAR method for 64 QAM. For a 1024-constellation it is possible wi h the Two-Dimensional Warping technique to signal at a 10- error rate but with a degradation of 6 dB as compared to the noncompanding channel. All of the results above indicate that with proper design of signal constellation and receiver structure, high data rate transmission is possible over companding channels.
2
Const. Rectangular Constellation TWO-D Sinh Size Optimum Detection SmearAWGN AWGN Desmear Warping Warping ONLY Companding 27.9 21.6 21.6 64 27.4dB 21.1 33.4 37.1 34.1 34.4 256 34.5 1024 39.5 45.2 48.3 I
-
As we see from the table above the performance of 64-QAM is slightly degraded by the companding (about 0.5 dB), and all of the techniques designed to improve performance give some small improvement. For a 256-point constellation, all three improvement techniques result in about a 2.5 dB improvement. However for the large 1024-point constellation only the warping techniques are able to achieve the normalized distance criterion (under th gaussian assumption) for a symbol error rate of less than 10- , The warping methods accomplish the above with a degradation of more than 6 dB, but at least in this case companding does not dominate performance to an extent that desired communication is not possible. All of the results above do not include coding, which should lower the required E / N o , by more than 3 dB, depending on the coding complexity. In addition, it appears that the warping technique should be modified for use with trellis coded signals, in order to account for sequence dependencies. This remains a subject for further investigation.
8
X. ACKNOWLEDGMENTS
The authors thank W. Beth, G. Golden, and J. Mazo for many stimulating discussions, and 6. D. Forney for his valuable comments. APPENDIX A WARPED-CONSTELLATION
We will describe the calculation of the variables required for the signal constellation design of Section VII. At every iteration of the process described in Section VI1 for finding the coordinates of the warped constellation, determine these values by using the equations below
~
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KALET AND SALTZBERG: QAM TRANSMISSION THROUGH A COMPANDING CHANNEL
-
(where o2 = c 2 )
where the initial values of x i , j and yi,j with i or j equal to and minus one, are q u a l to zero, e.g., x - l j = 0. Also cry,, are equal to zero for i or j equal to minus one, e.g., =x_,J = 0. The values i a n d j refer to the original location of the points in the rectangular grid constellation, i.e., ~ 3 , refers l to the new x location of the point that was originally at (3,l). Also, i and j take on odd values, i.e., 1,3,5,.... At each point we determine the values of ox,,,and by,,by solving the equations below which are derived from the requirement that the normalized distance be the same in the x direction and y direction for a given constellation point, and therefore be the same for all points.
k2+4 2
= 3
A M = --K T 4 3
= -M
+A
We start the process by choosing a b?. If we assume that the combined noise is gaussian then we can choose offor a given Pr {a} by using the equation below
We then find A from equation (A.6), After we calculate all the new constellation points, we calculate 0 2 ,and then from equation A.4, we find the appropriate value of E, IN,. APPENDIX B OPTIMUM RECEPTION
and
The variables k l , k 2 , b,, by,cl, A and M are defined below.
and
1
A=u2 --M+-j 1 E,IN,
The actual probability distribution of the companding noise is not known, even though its second order statistics are well defined [4]. It may even be true that the distribution is a function of the constellation size, and rolloff factor, r. There is however some reason to assume that it is approximately gaussian in nature [lo]. If the total noise is assumed gaussian then an optimum receiver can be found for any given constellation using the radial representation of the signals (e.g., the two signals in Fig. 2) and applying the likelihood rates as shown below. Consider the two constellation points A l and A2, shown in Fig. B.l, where A (or A 2 ) is the amplifude of the vector A l (or A ) and 6 (or 6 2 ) is the phase angle of the vector A ('or A 2 ) r The received vector r, has amplitude equal to R i n d a bhase equal to 0 R. We can-write the vector z, as a function of either a two dimensional set of axes around A 1, or a similar set of axes around A2. If the axes are chosen as shown then the noise components are uncorrelated [4],and if we assume that they are gaussian, then they are also independent. Therefore we can write the likelihood ratio, A ( r ) in the following manner.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO U314,FEBRUARYMARCWAPRIL 1994
I
Decide A
-2
We see that the terms within the brackets resemble a distance, and so we define a normalized distance dN,,between c and 4 as
We can then rewrite equation (B.3) as
I
I
We can normalize one step further by defining another normalized distance dhz,where
Therefore we may rewrite (B .5) as Fig. B.l. Decision regions - general binary case.
where
oi; = 05
=
Pi
=
ei
=
Except for a bias factor this has the same form as the optimum decision rule for signals in additive white gaussian noise. An optimum receiver, may now be built by "simply" finding the normalized distances in the receiver, and comparing them, taking into account the bias factor b ~ p L n ( b ~ p ~ ~ e l b l p ' 3. 1 e ) 2 The actual decision boundary is elliptical in nature.
variance of multiplicative noise in direction of A variance of muItiplicative noise tangential to 4 component of -r in direction of Ai component of c tangential to 4 i .
I
REFERENCES
p and 8 are given by pi = m s ( e R - e i ) 8 , = Rsin(OR-Oi)
where R is the amplitude of c, and O R its phase angle.
[l] K. Pahlavan and J. L. Holsinger, "A Model for the Effects of PCM Compandors on the Performance of High Speed Modems," GLOBECOM '85, New Orleans, December 1985,pp. 24.8.1-24.8.5.
(B3
[2] K. Pahlavan and J. L. Holsinger, "A Method to Counteract rhe Effects of PCM Systems on fne Perfonnance of Ultra High Speed Voice Band Modems", ICC '86, Toronto, Canada, June 1986, pp. 50.2.1-50.2.5. [3] K. Pahlavan and J. L. Holsinger, "Expanded TCM for Channels with MultiplicativeNoise," ICC '87, June 1987,pp. 12.3.1-12.3.5. [4]
B. R. Saltzberg and 3.-D. Wang, "Second-Order Statistics of Logarithmic Quantization Noise in QAM Data Communication.'' IEEE
Trans. on'commun., Vol. 39, October 1991.
After some manipulation equation (B.l) reduces to
[SI K. W. Cattermole,"Principles of Pulse Code Modulation",Iliffe Books Ltd., London, 1969 - ChaptetIII
[6] R. A. Wainwright, "On the Potential Advantage of a SmeanngDesmearing Filter Technique in Overcoming Impulse Noise Problems in Data Systems," IRE Trans. on Comm. Systems, Vol. CS-9, December 1961, pp. 362-366.
[7] B. R. Saltzberg, "Smear-Desmear Technique for Combating Quantization Noise," COMSPHERE '91, Herzliya, Israel, December 1991.
[8] G . F. M. Beaker, T. A. C. M. Claason, and P. J. VanGerwen," Design of Smearing Filters for Data Transmission Systems",IEEE Trans. Commun., Vol. COM - 33, Sept. 1985, pp. 955 - 963.
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KALET AND SALTZBEUG: QAM TRANSMISSION THROUGH A COMPANDING CHANNEL
[9] K. Pahlavan, "Nonlinear Quantization and the Design of Coded and Uncoded Signal Constellations", IEEE Trans. Commun., Vol. 39, August 1991, pp. 1207 - 1215. [ 101 J. E. Mam, "Quantization Noise and Data Transmission", BSTJ, Oct.
1968, pp. 1737 - 1753.
[ l l ] A. C. Popat and K. Zeger, "Robust Quantization of Memoryless Sources", 1990 Int. Symp. on Info. Theory and Appl., Hawaii, Nov. 1990, Pawr No. 33-2.
Irving Kalet was born in The Bronx, New York. He received his BEE in 1962 from CCNY, and his MS(EE) and Dr. Eng. Sc. from Columbia University in 1964and 1969 respectively. He taught in CCNY, and for the past twenty years has been teaching in Israel, where he lives. He is currently teaching seminars in topics mceming modulation techniques, and also acts as a consultant in this area in Israel and in the United States. He is presently spending a year at AT&T Bell Laboratories, Middletown, New Jersey working in the area of wireless data communication. He has previously worked in the areas of satellite communication, at MIT Lincoln Laboratories, and high-speed data transmission over the twisted pair channel. He is a member of Eta Kappa Nu Tau Beta Pi and Sigma Xi.
Burton R. Saltzberg (S152-M'55-F'76) received the B.E.E. degree from New York University in 1954, the M.S.from the University of Wismnsin in 1955, and the Eng.Sc.D. from New York University in 1964. Dr. Saltzberg joined AT&T Bell Laboratories in 1957. Since that time, he has been primarily engaged in the development, analysis, and initiation of data communication systems. He has published many articles in this field, and has been issued 23 patents. He is currently supervisor of the Data Theory Group in the Data CommunicationResearch Department. He served as chairman and in other offices of the Data Communications technical committee of the IEEE Communications Society. Dr. Saltzberg was general chairman of the 1989 IEEE Communication Theory Workshop, and Guest Editor of the August 1991 issue of JSAC on High Speed Digital Subscriber Lines. He is a member of Tau Beta Pi, Eta Kappa Nu, and Sigma Xi. He received the IEEE Communication Society Armstrong Achievement Award in 1991, "for sustained major contributions to the theory and practice of data communications".
