Chernoff bound of trellis-coded modulation over ... - Semantic Scholar
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 8, AUGUST 1994
2506
Chemoff Bound of Trellis-Coded Modulation Over Correlated Fading Channels K. Leeuwin, J. C. Belfiore. and G. Kawas Kaleh
Abstract-We derive a Chernoff upper bound for the pairwise error probability in the presence of an additive white Gaussian noise and a Rayleigh or Rice Correlated fading. The bound is useful for situations where perfect interleaving cannot be achieved We use it to determine some indications in the design of optimum trellis coded modulation for correlated fading channels.
I
I. INTRODUCTION
0
N radio and satellite channels, system performance is degraded by Rician or Rayleigh fading resulting from multipath propagation. Performance can be improved by channel coding combined with interleaving; the deinterleaver at the receiver allows decorrelation of the fading. A criterion, deduced from a Chemoff error probability bound, for the design of trellis-coded modulation for uncorrelated fading channel was given in [ 2 ] .It consists of maximizing the Hamming distance in symbols between the coded transmitted sequences, and the Euclidean distance between trellis branches. When deriving the error probability bound, it was assumed that the received samples are affected by uncorrelated fades. This implies that perfect interleaving has been accomplished. In some cases, like the mobile radio channel whose fading autocorrelation function is a slowly decreasing Bessel function, such interleaving cannot be achieved since it requires unacceptable large time delay. In this paper, we derive a Chernoff painvise error probability bound for the case of correlated fades. Obviously, uncorrelated fading represents a special case of our result. The bound is valid for any type of convolutional encoding and linear modulation. We use trellis-coded modulation as an application. We apply it to design the optimum codes for systems with correlated fading. The paper is organized as follows. Section I1 describes the system model under study. The calculation of the painvise error probability bound is presented in Section 111. In Section IV, we give indications on the design of codes for correlated fading channel. In order to illustrate our results, we apply them, in Section V, to the case of mobile radio channel. The conclusion is summarized in Section VI.
mapped into modulation symbols. A partial, or imperfect, interleaver is added to reduce the fading correlation. In the channel, the ith transmitted symbol, denoted z;, is multiplied by a complex Gaussian variable a; and a white Gaussian noise is added. Assuming coherent detection, the effect of the fading is limited to the multiplication of the symbol by the real random variable Iail. Therefore, the ith received baseband signal has the form y; = l a ; l a z ;
The general transmission model is depicted in Fig. 1. The input data bits are encoded by a convolutional encoder and Paper approved by S. G . Wilson, the Editor for Coding Theory and Applications of the IEEE Communications Society. Manuscript received October 26, 1990; revised September 27, 1991. This paper was presented in part at the EURO CODE '90 Conference, Udine, Italy, November 1990. K. Leeuwin is with CNETIPABISHM, 92131 Issy les Moulineaux, France. J. C. Belfiore and G. K. Kaleh are with Telecom Paris, Department of Communications, 75634 Paris 13, France. IEEE Log Number 9401939.
i = 1, 2 , . . . N
W;
(1)
where E, is the energy per symbol, 1z;I = 1 and w; is an additive complex white Gaussian noise with zero mean and variance 2No. The complex Gaussian variables a; have real mean p (equal to zero for Rayleigh distributed fading) and variance y2. Let k = p 2 / y 2 denote the ratio of the average power in the specular component to the variance of the fading. The signal-to-noise ratio (SNR) is equal to
r = E(laiVmkil2)- (r2+ P2)& -
(2) 2NO NO Consider the general case of time correlated fading. The probability density function (pdf) of the fading vector = ( a l , . . . ,isa ~ ) 1 1 p ( d ) = ~( ~ 7det~ K ) ~
Y2
and K is the N x N normalized covariance channel matrix with elements defined by
K , , - E { ( a ;- P I * ( % 23
11. SYSTEM MODEL
+
-
Y2
-
P)}
(4)
At the receiver, where channel state information (CSI) (or fading envelope estimate) is assumed available, the detection is performed by a soft Viterbi decoder. Assuming perfect CSI and a Gaussian decoding metric, the decision rule is to choose in the set of all possible symbol sequences, denoted C, the sequence (xl, . . . X N ) at minimum distance from the observation, i.e.,
0090-6778/94$04,00 0 1994 IEEE
~
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 8, AUGUST 1994
111. CHERNOFF BOUND OF THE PAIRWISE ERROR PROBABLITY IN A CORRELATED FADINGCHANNEL
Let Z denote the transmitted sequence ( 2 1 , . . . ZN). An upper bound on the bit error probability, based on the union bound, is given by
2507
and, for the Rayleigh channel, to the following:
which are the same expressions obtained by Divsalar and Simon.
x, t E C
where P ( 2 ) is the a priori probability of transmitting the sequence 2,P ( Z + fl is the pairwise error probability, Le., the probability of decoding the sequence Tin place of the transmitted sequence 2, and u(Z, i) is the number of resulting erroneous bits. The Chernoff bound of the pairwise error probability conditioned on the fading vector d was calculated in [2, eq (2.a)l. It is given by
IV. DESIGNOF TCM FOR THE CORRELATED FADINGCHANNEL
The union upper bound of the bit error probability given in (6) may be evaluated using a general transfer function [6]
approach. However, the result is very loose when there is fading correlation, for error rate ranging from 10-1 to The problem is that the union bound does not account for N the intersection of the pairwise decision regions [7]. Since (7) the pairwise error probability decays slowly with SNR in the i=l correlated fading channel, we expect that the tightness of the The painvise error probability is now obtained by averaging bound is obtained only for values of SNR much larger than that obtained for the Gaussian channel. (7) over 6, using its pdf (3). Expressions (1 1) and (12) were used in [2] to derive an P(2 z / Z ) p ( d )dd. appropriate code for uncorrelated fading channels; we use our bound to deduce a useful result on coding design for correlated The calculation is give! in the Appendix, and leads to (9) fading channels. Without loss of generality, we study the case and (10) below where 1 is the vector t ( l , l , . . .l), and 4i, of Rayleigh fading. (i = 1, 2 , . . . ,N ) , are the eigenvalues of the product matrix Since the bit error probability (6) is dominated by the D K , with K the covariance channel matrix defined above, pairwise error term with the smallest power of l/r, the aim and D an N x N diagonal matrix formed by the squared of the code designer is to maximize the degree of this term, Euclidean distances between the correct codeword 2 and the and, thus, increase the slope of the error probability curve with incorrect codeword t' l/r.To analyze (lo), we first rewrite it in the following form --f
P ( 2 + i)5 Notice that the case of uncorrelated fading (perfect interleaving), treated by Divsalar and Simon [2], represents a special case of our study. The corresponding Chernoff bound is obtained by replacing the normalized covariance channel matrix K in (4) by the identity matrix. This leads, for the Rician channel, to the following expression:
1
Det ( I
+ $DK)
where I is the identity matrix, and Det (.) denotes ,the determinant. As an example, for coded sequences Z and t of length N = 3, the determinant to evaluate is given by the equation at the bottom of the next page.After expansion, Det ( I
+ :DK)
Rician Channel
(9) Rayleigh Channel (k = 0):
2508
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 8, AUGUST 1994
Since Ki; = 1 (i = 1,. . . N ) , we shall omit this term in the following equations. By generalizing, the expansion of the determinant (13) leads to a polynomial in r Det
(I+
D K ) = PO
(i) (i) +
+ pl
+. . . + p ,
Suppose now that the fading correlation is reduced by a partial interleaver. Let f i denote the largest order of the nonzero determinants in the expansion (14). The value of 77Z depends on the interleaver efficiency (see Section V-B). Using (14), we can deduce the following rule for the appropriate design of codes for such partially correlated channels: 1) construct a code such that the length of the shortest error event, i.e, the smallest number of distinct symbols between the coded sequences, is equal to m; 2) maximize the product of the Euclidean distances along the error events of length f i . As an illustration, in the following section we study the case of a mobile radio channel.
v.
m
. . .+ P N
THE
{ k p T y
where the coefficients are given by Po = 1
MOBILERADIOCHANNEL
We limit our investigation to the case of Rayleigh fading. The baseband energy spectral density of the multiplicative fading variable /ail in (1) is given by [3]
If1 5 f d
S(f)=
(17)
If I > f d i=I A'
N
N
where f d is the maximum Doppler frequency, and f o is the carrier frequency. Taking the inverse Fourier transform of S ( f ) , we get the autocorrelation function in the form of a Bessel function. The covariance matrix K is then
Kij =
and K ; ~ ~ . .represents . ~ ~ a minor, of order m x m, of the matrix K . The choice of appropriate codes for correlated fading channel depends on the determinants of K and of its minors. In highly correlated fading channels, the determinants ~ ~close ) to zero for m > 1 (see Section V-A). Det ( K ; ~ ~ . . .are Therefore, with a good approximation and for any code, the expansion (14) is limited to the first and second terms. In this case, the pairwise error probability bound (13) is given by
This is equivalent to the performance of an uncoded system. We can, therefore, conclude that coding is not useful when the fading is highly correlated.
{
1
&(2Tfd(i - j ) T )
ifi=j elsewhere
(18)
where J o ( . )is the zero-order Bessel function of the first kind, and T is the symbol interval duration. In the following, f o is equal to 900 MHz, the bit rate is 8 kbit/s, and the vehicle speed ranges from 20 to 150 kmh, i.e., the Doppler frequency spread ranges from 16 to 125 Hz. We examine now the value of m defined by (15) for different transmission systems. A. Without Interleaving
Table I gives the values of the determinants of K of order N and the corresponding value of 77Z for different vehicle speeds V. For any vehicle speed ranging from 20 km/h to 150 km/h, the determinant of the covariance channel matrix K is close to zero with a matrix order m 2 2. This is due to the slowly decreasing autocorrelation function of this channel. Then 77Z = 1. The performance of any code is asymptotically proportional to l/r as was indicated in Section IV. B. With Partial Interleaving The size of the interleaver is limited because of the time delay constraint. Assuming a convolutional interleaver [5] with
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 8, AUGUST 1994
TABLE I DETERMINANT OF THE MOBILE CHANNEL MATRIX K OF SIZEi\r x COVARIANCE
Pb
100
I' OtmfhYN
E
4 m 2 0,342e-03 E 1 20 50 0,214e-02 6 1 100 0,853e-02 F 1 150 0,191e-01 E 1 is close to zero of a precision inferior to the computer finite precision
I
(
2506
Chemoff Bound of Trellis-Coded Modulation Over Correlated Fading Channels K. Leeuwin, J. C. Belfiore. and G. Kawas Kaleh
Abstract-We derive a Chernoff upper bound for the pairwise error probability in the presence of an additive white Gaussian noise and a Rayleigh or Rice Correlated fading. The bound is useful for situations where perfect interleaving cannot be achieved We use it to determine some indications in the design of optimum trellis coded modulation for correlated fading channels.
I
I. INTRODUCTION
0
N radio and satellite channels, system performance is degraded by Rician or Rayleigh fading resulting from multipath propagation. Performance can be improved by channel coding combined with interleaving; the deinterleaver at the receiver allows decorrelation of the fading. A criterion, deduced from a Chemoff error probability bound, for the design of trellis-coded modulation for uncorrelated fading channel was given in [ 2 ] .It consists of maximizing the Hamming distance in symbols between the coded transmitted sequences, and the Euclidean distance between trellis branches. When deriving the error probability bound, it was assumed that the received samples are affected by uncorrelated fades. This implies that perfect interleaving has been accomplished. In some cases, like the mobile radio channel whose fading autocorrelation function is a slowly decreasing Bessel function, such interleaving cannot be achieved since it requires unacceptable large time delay. In this paper, we derive a Chernoff painvise error probability bound for the case of correlated fades. Obviously, uncorrelated fading represents a special case of our result. The bound is valid for any type of convolutional encoding and linear modulation. We use trellis-coded modulation as an application. We apply it to design the optimum codes for systems with correlated fading. The paper is organized as follows. Section I1 describes the system model under study. The calculation of the painvise error probability bound is presented in Section 111. In Section IV, we give indications on the design of codes for correlated fading channel. In order to illustrate our results, we apply them, in Section V, to the case of mobile radio channel. The conclusion is summarized in Section VI.
mapped into modulation symbols. A partial, or imperfect, interleaver is added to reduce the fading correlation. In the channel, the ith transmitted symbol, denoted z;, is multiplied by a complex Gaussian variable a; and a white Gaussian noise is added. Assuming coherent detection, the effect of the fading is limited to the multiplication of the symbol by the real random variable Iail. Therefore, the ith received baseband signal has the form y; = l a ; l a z ;
The general transmission model is depicted in Fig. 1. The input data bits are encoded by a convolutional encoder and Paper approved by S. G . Wilson, the Editor for Coding Theory and Applications of the IEEE Communications Society. Manuscript received October 26, 1990; revised September 27, 1991. This paper was presented in part at the EURO CODE '90 Conference, Udine, Italy, November 1990. K. Leeuwin is with CNETIPABISHM, 92131 Issy les Moulineaux, France. J. C. Belfiore and G. K. Kaleh are with Telecom Paris, Department of Communications, 75634 Paris 13, France. IEEE Log Number 9401939.
i = 1, 2 , . . . N
W;
(1)
where E, is the energy per symbol, 1z;I = 1 and w; is an additive complex white Gaussian noise with zero mean and variance 2No. The complex Gaussian variables a; have real mean p (equal to zero for Rayleigh distributed fading) and variance y2. Let k = p 2 / y 2 denote the ratio of the average power in the specular component to the variance of the fading. The signal-to-noise ratio (SNR) is equal to
r = E(laiVmkil2)- (r2+ P2)& -
(2) 2NO NO Consider the general case of time correlated fading. The probability density function (pdf) of the fading vector = ( a l , . . . ,isa ~ ) 1 1 p ( d ) = ~( ~ 7det~ K ) ~
Y2
and K is the N x N normalized covariance channel matrix with elements defined by
K , , - E { ( a ;- P I * ( % 23
11. SYSTEM MODEL
+
-
Y2
-
P)}
(4)
At the receiver, where channel state information (CSI) (or fading envelope estimate) is assumed available, the detection is performed by a soft Viterbi decoder. Assuming perfect CSI and a Gaussian decoding metric, the decision rule is to choose in the set of all possible symbol sequences, denoted C, the sequence (xl, . . . X N ) at minimum distance from the observation, i.e.,
0090-6778/94$04,00 0 1994 IEEE
~
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 8, AUGUST 1994
111. CHERNOFF BOUND OF THE PAIRWISE ERROR PROBABLITY IN A CORRELATED FADINGCHANNEL
Let Z denote the transmitted sequence ( 2 1 , . . . ZN). An upper bound on the bit error probability, based on the union bound, is given by
2507
and, for the Rayleigh channel, to the following:
which are the same expressions obtained by Divsalar and Simon.
x, t E C
where P ( 2 ) is the a priori probability of transmitting the sequence 2,P ( Z + fl is the pairwise error probability, Le., the probability of decoding the sequence Tin place of the transmitted sequence 2, and u(Z, i) is the number of resulting erroneous bits. The Chernoff bound of the pairwise error probability conditioned on the fading vector d was calculated in [2, eq (2.a)l. It is given by
IV. DESIGNOF TCM FOR THE CORRELATED FADINGCHANNEL
The union upper bound of the bit error probability given in (6) may be evaluated using a general transfer function [6]
approach. However, the result is very loose when there is fading correlation, for error rate ranging from 10-1 to The problem is that the union bound does not account for N the intersection of the pairwise decision regions [7]. Since (7) the pairwise error probability decays slowly with SNR in the i=l correlated fading channel, we expect that the tightness of the The painvise error probability is now obtained by averaging bound is obtained only for values of SNR much larger than that obtained for the Gaussian channel. (7) over 6, using its pdf (3). Expressions (1 1) and (12) were used in [2] to derive an P(2 z / Z ) p ( d )dd. appropriate code for uncorrelated fading channels; we use our bound to deduce a useful result on coding design for correlated The calculation is give! in the Appendix, and leads to (9) fading channels. Without loss of generality, we study the case and (10) below where 1 is the vector t ( l , l , . . .l), and 4i, of Rayleigh fading. (i = 1, 2 , . . . ,N ) , are the eigenvalues of the product matrix Since the bit error probability (6) is dominated by the D K , with K the covariance channel matrix defined above, pairwise error term with the smallest power of l/r, the aim and D an N x N diagonal matrix formed by the squared of the code designer is to maximize the degree of this term, Euclidean distances between the correct codeword 2 and the and, thus, increase the slope of the error probability curve with incorrect codeword t' l/r.To analyze (lo), we first rewrite it in the following form --f
P ( 2 + i)5 Notice that the case of uncorrelated fading (perfect interleaving), treated by Divsalar and Simon [2], represents a special case of our study. The corresponding Chernoff bound is obtained by replacing the normalized covariance channel matrix K in (4) by the identity matrix. This leads, for the Rician channel, to the following expression:
1
Det ( I
+ $DK)
where I is the identity matrix, and Det (.) denotes ,the determinant. As an example, for coded sequences Z and t of length N = 3, the determinant to evaluate is given by the equation at the bottom of the next page.After expansion, Det ( I
+ :DK)
Rician Channel
(9) Rayleigh Channel (k = 0):
2508
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 8, AUGUST 1994
Since Ki; = 1 (i = 1,. . . N ) , we shall omit this term in the following equations. By generalizing, the expansion of the determinant (13) leads to a polynomial in r Det
(I+
D K ) = PO
(i) (i) +
+ pl
+. . . + p ,
Suppose now that the fading correlation is reduced by a partial interleaver. Let f i denote the largest order of the nonzero determinants in the expansion (14). The value of 77Z depends on the interleaver efficiency (see Section V-B). Using (14), we can deduce the following rule for the appropriate design of codes for such partially correlated channels: 1) construct a code such that the length of the shortest error event, i.e, the smallest number of distinct symbols between the coded sequences, is equal to m; 2) maximize the product of the Euclidean distances along the error events of length f i . As an illustration, in the following section we study the case of a mobile radio channel.
v.
m
. . .+ P N
THE
{ k p T y
where the coefficients are given by Po = 1
MOBILERADIOCHANNEL
We limit our investigation to the case of Rayleigh fading. The baseband energy spectral density of the multiplicative fading variable /ail in (1) is given by [3]
If1 5 f d
S(f)=
(17)
If I > f d i=I A'
N
N
where f d is the maximum Doppler frequency, and f o is the carrier frequency. Taking the inverse Fourier transform of S ( f ) , we get the autocorrelation function in the form of a Bessel function. The covariance matrix K is then
Kij =
and K ; ~ ~ . .represents . ~ ~ a minor, of order m x m, of the matrix K . The choice of appropriate codes for correlated fading channel depends on the determinants of K and of its minors. In highly correlated fading channels, the determinants ~ ~close ) to zero for m > 1 (see Section V-A). Det ( K ; ~ ~ . . .are Therefore, with a good approximation and for any code, the expansion (14) is limited to the first and second terms. In this case, the pairwise error probability bound (13) is given by
This is equivalent to the performance of an uncoded system. We can, therefore, conclude that coding is not useful when the fading is highly correlated.
{
1
&(2Tfd(i - j ) T )
ifi=j elsewhere
(18)
where J o ( . )is the zero-order Bessel function of the first kind, and T is the symbol interval duration. In the following, f o is equal to 900 MHz, the bit rate is 8 kbit/s, and the vehicle speed ranges from 20 to 150 kmh, i.e., the Doppler frequency spread ranges from 16 to 125 Hz. We examine now the value of m defined by (15) for different transmission systems. A. Without Interleaving
Table I gives the values of the determinants of K of order N and the corresponding value of 77Z for different vehicle speeds V. For any vehicle speed ranging from 20 km/h to 150 km/h, the determinant of the covariance channel matrix K is close to zero with a matrix order m 2 2. This is due to the slowly decreasing autocorrelation function of this channel. Then 77Z = 1. The performance of any code is asymptotically proportional to l/r as was indicated in Section IV. B. With Partial Interleaving The size of the interleaver is limited because of the time delay constraint. Assuming a convolutional interleaver [5] with
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 8, AUGUST 1994
TABLE I DETERMINANT OF THE MOBILE CHANNEL MATRIX K OF SIZEi\r x COVARIANCE
Pb
100
I' OtmfhYN
E
4 m 2 0,342e-03 E 1 20 50 0,214e-02 6 1 100 0,853e-02 F 1 150 0,191e-01 E 1 is close to zero of a precision inferior to the computer finite precision
I
(
