Optimal Decoder for Channels with Estimation Errors - IEEE Xplore
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 6, JUNE 2009
2795
Optimal Decoder for Channels with Estimation Errors S. H. Song, Member, IEEE, and Q. T. Zhang, Fellow, IEEE
Abstract—Gaussian inputs and nearest neighbor decoder are optimal choices for the transceiver when perfect channel information is available. However, channel uncertainty is inevitable in practical systems due to additive noise and channel variation. Under such circumstances, Gaussian inputs are no longer optimal whereas the nearest neighbor decoder even loses part of the mutual information provided by Gaussian codebooks. In this letter, we tackle the issue of decoder optimization for channels with Gaussian inputs and channel estimation errors. The result is a normalized nearest neighbor decoder which is proved, by a semi-analytical method, capable of fully obtaining the mutual information with Gaussian inputs. We further show that the proposed decoder is capable of achieving the performance improvement provided by a better non-Gaussian codebook. Index Terms—Mutual information, channel uncertainty, wireless communications.
I. I NTRODUCTION
C
APACITY represents the maximum achievable rate of a wireless system and thus, its analysis is of practical importance. It is well known that the nearest neighbor (NN) decoder can achieve the capacity of flat fading channels with Gaussian codebooks at the transmitter and perfect side information available at the receiver. However, in many applications only an imperfect channel estimate is available to receiver [1]. Such channel uncertainty will cause an additional crossproduct term in the received signal, which is the product of the useful signal and detrimental channel estimation errors. Since channel estimation errors are unknown, this cross-product term will introduce additional uncertainty to the receiver. It is, therefore, important to fully understand the impacts of the cross-product term on system mutual information (MI), so as to design an optimal or suboptimal transceiver to fully achieve the possible mutual information. This question is partly answered in [2] by showing that the achievable rate of a nearest neighbor decoder is just the lower bound of the mutual information with Gaussian codebooks [3]. The cross-product term between signals and estimation errors is usually treated as “effective noise” in the derivation of achievable rate [2] and mutual information lower bound [3]. However, the cross-product term should definitely contribute to both useful signal and noise. This nature is more Manuscript received April 16, 2008; revised October 17, 2008; accepted February 20, 2009. The associate editor coordinating the review of this letter and approving it for publication was M. Valenti. S. H. Song was with the Department of Electronic Engineering, City University of Hong Kong, and is now with the Electronic and Computer Engineering Department, the Hong Kong University of Science and Technology (e-mail: [email protected]). Q. T. Zhang is with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong (e-mail: [email protected]). This work was fully supported by a strategic research grant from the City University of Hong Kong, Hong Kong, China (Project No.7002074). Digital Object Identifier 10.1109/TWC.2009.080529
accurately uncovered in [4] where an accurate result for the MI with Gaussian input is obtained and shown to exceed the achievable rate of NN decoders. A clear picture is that the NN decoder is only able to achieve a lower bound of the MI with Gaussian codebooks but fail to fully exploit it. In this letter, we show how to design a novel decoder to exploit the useful information carried in the cross-product term to achieve the gap between the achievable rate of NN decoders and the mutual information. Besides the efforts at the receiver side, we also note that with channel uncertainty, Gaussian codebooks are no longer optimal. In fact, as shown in previous studies [5], using codebooks with the Pearson’s type VII (PT-7) distribution can provide more mutual information than using Gaussian ones. The performance of the NN and our proposed decoders in channels with PT-7 codebooks is another focus of this letter. II. S YSTEM MODEL Without loss of generality, this letter just focuses on the realchannel case, since the methodology used here can be easily extended to its complex counterpart. Suppose that a random real symbol x with zero-mean and unknown distribution is transmitted over a single-input single-output (SISO) wireless fading channel with channel gain h ∼ N (0, σh2 ), and is corrupted by zero-mean additive white Gaussian noise (AWGN) n ∼ N (0, σn2 ), so that the received signal, in its baseband form, can be expressed as r = hx + n.
(1)
We further assume that the receiver knows only a channel es¯ of h with a Gaussian estimation error h ˜ ∼ N (0, σ 2 ). timate h e Namely, ¯+h ˜ h=h (2) which, when inserted into (1), gives r
=
¯ + h)x ˜ + n. (h
(3)
˜ is the cross product between signal and channel The term hx estimation error, which we have more to say in Section III.B. The Gaussian assumption on channel estimation errors is widely adopted by many researchers [6], and can be justified by the fact that channel estimation errors mainly come from AWGN and the time-selectivity of Gaussian distributed channel h. It can be shown that the variance of estimation errors, σe2 , is inversely proportional to the signal to noise power ratio (SNR), thus enabling us to write [2] σe2 = α(σn2 /σx2 )
(4)
where α > 0 is a constant depending on a given channel environment and σx2 = E[x2 ] denotes the variance of the
c 2009 IEEE 1536-1276/09$25.00
2796
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 6, JUNE 2009
transmit symbol. The case of α = 1 corresponds to using a maximum likelihood (ML) estimator in slow fading channels with one pilot per frame [4]. Clearly, the side information can be regarded as “perfect” if σe2 is negligibly small in comparison with the reciprocal of the SNR. We now define channel capacity and Gaussian mutual information for the system in (3).
The nearest neighbor decoder chooses message m, for which the metric DN N (m) =
N 1 ˆ k xk (m))2 (rk − h N
is minimized. Alternatively, we can write DN N (m) ≤ DN N (m ), ∀ m = m.
A. Channel capacity By definition, the capacity of the channel (3) is given by ˆ h = h) C = sup I(r, x | ¯
(5)
¯ = h) ˆ = H(r | h ¯ = h) ˆ − H(r | h ¯ = h, ˆ x) I(r, x | h
(6)
f (x)
where
denotes the conditional MI between r and x for a given ˆ The first term on the right represents channel estimate ¯ h = h. ¯ = h, ˆ and the second term can the entropy of r for given h be defined in a similar manner. We need to optimize the input probability density function f (x), so that the conditional MI is maximized. B. Gaussian mutual information In this letter, the term Gaussian mutual information is used for mutual information with a Gaussian codebook. Thus, Gaussian MI is defined as the mutual information ¯ = h) ˆ IMI = I(r, x | h
(7)
with Gaussian input x ∼ N (0, σx2 ). It should be noted that IMI does not represent the channel capacity, since a Gaussian codebook is no longer optimal for the case with imperfect channel information. In fact, Gaussian mutual information represents the largest achievable rate when Gaussian codebooks are used. III. O PTIMAL DECODER FOR G AUSSIAN CODEBOOKS We first tackle the issue of finding the optimal decoder for Gaussian codebooks. A. Nearest neighbor decoder Given that Gaussian codebooks are used, let us begin with the achievable rate of a nearest neighbor decoder. Consider block coding in which we utilize the N-tuple X(m) = (x1 (m), ..., xN (m)) ∈ RN to represent message m ∈ M = {1, ..., eN R } [2] where N and R are the block length and rate of the code, respectively. The codewords should satisfy N 1 2 xk (m) ≤ σx2 . N
(8)
k=1
Thus, {xk (m)} can be chosen from independent and identical Gaussian population with xk (m) ∼ N (0, σx2 )
(9)
so that Eq. (8) is typically satisfied as N → ∞. The received ˆ is then expressible as signal conditioned on ¯ h=h ˜ k xk (m) + nk . rk = ˆ hk xk (m) + h
(10)
(11)
k=1
(12)
It has been shown [8] that the achievable rate of the NN decoder with Gaussian codebooks is upper bounded by ˆ 2 σ2 h 1 IN N ≤ log 1 + 2 2k x 2 . (13) 2 σe σx + σn The right hand side of (13), as shown by M´ edard [3], is just the lower bound of IMI implying that IN N < IMI . The challenge is how to design a decoder to fill the mutual information gap IMI − IN N for a better performance. B. Normalized nearest neighbor decoder The NN decoder described in (11) is the ML decoder derived by assuming perfect side information and AWGN independent of the codeword. The presence of channel estimation errors breaks this assumption producing a cross-product term between signals and channel estimation errors. This crossproduct term behaves somewhat like a ‘noise’ component but with power dependent on the codeword xk (m). Consequently, a larger amplitude xk (m) will introduce more ‘noise’ making the minimum distance metric no longer optimum and thus, calling for a new performance metric. To this end, we rewrite the received signal for message m in vector form: ˆ ˜ r = diag(h)x(m) + diag(h)x(m) +n
(14)
ˆ h ˜ where x(m) = [x1 (m), ..., xN (m)]T and we have defined h, ˆ and n in a similar manner. Given h, x(m) and the Gaussianity ˜ the received signal r follows the multivariate Gaussian of h, distribution with its pdf given by ˆ x(m)) = f (r | h, (15) 1 T −1 ˆ ˆ exp − 2 (r − diag(h)x(m)) Q (r − diag(h)x(m)) (2π)N/2 (det Q)1/2 where
T
˜ ˜ Q = E diag(h)x(m) + n diag(h)x(m) + n .
(16)
Obviously, the codewords will influence both the mean value and covariance of the distribution for received signals. However, the NN decoder only takes the mean value into consideration. As in [2], we assume that a genie provides the decoder with a noisy measurement where the estimation error is independent of the codebook, transmitted message and AWGN noise. With this assumption, (16) can be simplified to give Q = diag{x21 (m)σe2 + σn2 , ..., x2N (m)σe2 + σn2 }
(17)
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 6, JUNE 2009
which, along with (15), enables representing the log likelihood of r as
ˆ x(m)) log f (r | h,
N ˆ k xk (m))2 1 (rk − h = − (18) 2 x2k (m)σe2 + σn2
− −
1 2
k=1 N
2797
where N →∞
The function ψ(s) is the scaled cumulant moment-generating function for D(m ) defined as
log(x2k (m)σe2 + σn2 )
ψ(s)
k=1
N log(2π). 2
Since xk (m) is generated from i.i.d. (independent and identically distributed) Gaussian codebook, the second term of (18) is asymptotically independent of the codeword index m, as shown by N 2 2 2 k=1 log(xk (m)σe + σn ) lim N →∞ N N 2 2 2 k=1 log(xk (m )σe + σn ) . (19) = lim N →∞ N Hence, the decision metric can be simply built on the first term taking the form N ˆ k xk (m))2 1 (rk − h D(m) = . 2 N xk (m)σe2 + σn2
(20)
C. Achievable rate for normalized nearest neighbor decoder We now calculate the maximum rate that can be obtained by the NNN decoder. Similar to the NN decoder, the achievable rate for the NNN decoder IN N N is defined as the rate at which the probability of decoding error Pe approaches zero as the size, N , of the Gaussian vector x(m) tends to infinity. ˆ Namely, for given received signal r and channel estimate h Pe ≤ exp(−N (IN N N − δ)), ∀ δ > 0,
(21)
and the detailed discussion can be found in [8]. Assuming the transmission of message m, the probability of error is expressible as Pe = P r(D(m ) < D(m)),
m = m.
(22)
To calculate Pe , we need to first determine the pdf of D(m ) which, however, is difficult as we are considering the case with N → ∞. Fortunately, since we are concerned with the small probability exp(−N (IN N N − δ)) as N → ∞, the problem of finding IN N N can be solved by using the large deviation principle (LDP) [7]. Specifically, by virtue of the Gartner-Ellis theorem [7], it has been shown [8] that IN N N = sup(sT − ψ(s)) s
2795
Optimal Decoder for Channels with Estimation Errors S. H. Song, Member, IEEE, and Q. T. Zhang, Fellow, IEEE
Abstract—Gaussian inputs and nearest neighbor decoder are optimal choices for the transceiver when perfect channel information is available. However, channel uncertainty is inevitable in practical systems due to additive noise and channel variation. Under such circumstances, Gaussian inputs are no longer optimal whereas the nearest neighbor decoder even loses part of the mutual information provided by Gaussian codebooks. In this letter, we tackle the issue of decoder optimization for channels with Gaussian inputs and channel estimation errors. The result is a normalized nearest neighbor decoder which is proved, by a semi-analytical method, capable of fully obtaining the mutual information with Gaussian inputs. We further show that the proposed decoder is capable of achieving the performance improvement provided by a better non-Gaussian codebook. Index Terms—Mutual information, channel uncertainty, wireless communications.
I. I NTRODUCTION
C
APACITY represents the maximum achievable rate of a wireless system and thus, its analysis is of practical importance. It is well known that the nearest neighbor (NN) decoder can achieve the capacity of flat fading channels with Gaussian codebooks at the transmitter and perfect side information available at the receiver. However, in many applications only an imperfect channel estimate is available to receiver [1]. Such channel uncertainty will cause an additional crossproduct term in the received signal, which is the product of the useful signal and detrimental channel estimation errors. Since channel estimation errors are unknown, this cross-product term will introduce additional uncertainty to the receiver. It is, therefore, important to fully understand the impacts of the cross-product term on system mutual information (MI), so as to design an optimal or suboptimal transceiver to fully achieve the possible mutual information. This question is partly answered in [2] by showing that the achievable rate of a nearest neighbor decoder is just the lower bound of the mutual information with Gaussian codebooks [3]. The cross-product term between signals and estimation errors is usually treated as “effective noise” in the derivation of achievable rate [2] and mutual information lower bound [3]. However, the cross-product term should definitely contribute to both useful signal and noise. This nature is more Manuscript received April 16, 2008; revised October 17, 2008; accepted February 20, 2009. The associate editor coordinating the review of this letter and approving it for publication was M. Valenti. S. H. Song was with the Department of Electronic Engineering, City University of Hong Kong, and is now with the Electronic and Computer Engineering Department, the Hong Kong University of Science and Technology (e-mail: [email protected]). Q. T. Zhang is with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong (e-mail: [email protected]). This work was fully supported by a strategic research grant from the City University of Hong Kong, Hong Kong, China (Project No.7002074). Digital Object Identifier 10.1109/TWC.2009.080529
accurately uncovered in [4] where an accurate result for the MI with Gaussian input is obtained and shown to exceed the achievable rate of NN decoders. A clear picture is that the NN decoder is only able to achieve a lower bound of the MI with Gaussian codebooks but fail to fully exploit it. In this letter, we show how to design a novel decoder to exploit the useful information carried in the cross-product term to achieve the gap between the achievable rate of NN decoders and the mutual information. Besides the efforts at the receiver side, we also note that with channel uncertainty, Gaussian codebooks are no longer optimal. In fact, as shown in previous studies [5], using codebooks with the Pearson’s type VII (PT-7) distribution can provide more mutual information than using Gaussian ones. The performance of the NN and our proposed decoders in channels with PT-7 codebooks is another focus of this letter. II. S YSTEM MODEL Without loss of generality, this letter just focuses on the realchannel case, since the methodology used here can be easily extended to its complex counterpart. Suppose that a random real symbol x with zero-mean and unknown distribution is transmitted over a single-input single-output (SISO) wireless fading channel with channel gain h ∼ N (0, σh2 ), and is corrupted by zero-mean additive white Gaussian noise (AWGN) n ∼ N (0, σn2 ), so that the received signal, in its baseband form, can be expressed as r = hx + n.
(1)
We further assume that the receiver knows only a channel es¯ of h with a Gaussian estimation error h ˜ ∼ N (0, σ 2 ). timate h e Namely, ¯+h ˜ h=h (2) which, when inserted into (1), gives r
=
¯ + h)x ˜ + n. (h
(3)
˜ is the cross product between signal and channel The term hx estimation error, which we have more to say in Section III.B. The Gaussian assumption on channel estimation errors is widely adopted by many researchers [6], and can be justified by the fact that channel estimation errors mainly come from AWGN and the time-selectivity of Gaussian distributed channel h. It can be shown that the variance of estimation errors, σe2 , is inversely proportional to the signal to noise power ratio (SNR), thus enabling us to write [2] σe2 = α(σn2 /σx2 )
(4)
where α > 0 is a constant depending on a given channel environment and σx2 = E[x2 ] denotes the variance of the
c 2009 IEEE 1536-1276/09$25.00
2796
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 6, JUNE 2009
transmit symbol. The case of α = 1 corresponds to using a maximum likelihood (ML) estimator in slow fading channels with one pilot per frame [4]. Clearly, the side information can be regarded as “perfect” if σe2 is negligibly small in comparison with the reciprocal of the SNR. We now define channel capacity and Gaussian mutual information for the system in (3).
The nearest neighbor decoder chooses message m, for which the metric DN N (m) =
N 1 ˆ k xk (m))2 (rk − h N
is minimized. Alternatively, we can write DN N (m) ≤ DN N (m ), ∀ m = m.
A. Channel capacity By definition, the capacity of the channel (3) is given by ˆ h = h) C = sup I(r, x | ¯
(5)
¯ = h) ˆ = H(r | h ¯ = h) ˆ − H(r | h ¯ = h, ˆ x) I(r, x | h
(6)
f (x)
where
denotes the conditional MI between r and x for a given ˆ The first term on the right represents channel estimate ¯ h = h. ¯ = h, ˆ and the second term can the entropy of r for given h be defined in a similar manner. We need to optimize the input probability density function f (x), so that the conditional MI is maximized. B. Gaussian mutual information In this letter, the term Gaussian mutual information is used for mutual information with a Gaussian codebook. Thus, Gaussian MI is defined as the mutual information ¯ = h) ˆ IMI = I(r, x | h
(7)
with Gaussian input x ∼ N (0, σx2 ). It should be noted that IMI does not represent the channel capacity, since a Gaussian codebook is no longer optimal for the case with imperfect channel information. In fact, Gaussian mutual information represents the largest achievable rate when Gaussian codebooks are used. III. O PTIMAL DECODER FOR G AUSSIAN CODEBOOKS We first tackle the issue of finding the optimal decoder for Gaussian codebooks. A. Nearest neighbor decoder Given that Gaussian codebooks are used, let us begin with the achievable rate of a nearest neighbor decoder. Consider block coding in which we utilize the N-tuple X(m) = (x1 (m), ..., xN (m)) ∈ RN to represent message m ∈ M = {1, ..., eN R } [2] where N and R are the block length and rate of the code, respectively. The codewords should satisfy N 1 2 xk (m) ≤ σx2 . N
(8)
k=1
Thus, {xk (m)} can be chosen from independent and identical Gaussian population with xk (m) ∼ N (0, σx2 )
(9)
so that Eq. (8) is typically satisfied as N → ∞. The received ˆ is then expressible as signal conditioned on ¯ h=h ˜ k xk (m) + nk . rk = ˆ hk xk (m) + h
(10)
(11)
k=1
(12)
It has been shown [8] that the achievable rate of the NN decoder with Gaussian codebooks is upper bounded by ˆ 2 σ2 h 1 IN N ≤ log 1 + 2 2k x 2 . (13) 2 σe σx + σn The right hand side of (13), as shown by M´ edard [3], is just the lower bound of IMI implying that IN N < IMI . The challenge is how to design a decoder to fill the mutual information gap IMI − IN N for a better performance. B. Normalized nearest neighbor decoder The NN decoder described in (11) is the ML decoder derived by assuming perfect side information and AWGN independent of the codeword. The presence of channel estimation errors breaks this assumption producing a cross-product term between signals and channel estimation errors. This crossproduct term behaves somewhat like a ‘noise’ component but with power dependent on the codeword xk (m). Consequently, a larger amplitude xk (m) will introduce more ‘noise’ making the minimum distance metric no longer optimum and thus, calling for a new performance metric. To this end, we rewrite the received signal for message m in vector form: ˆ ˜ r = diag(h)x(m) + diag(h)x(m) +n
(14)
ˆ h ˜ where x(m) = [x1 (m), ..., xN (m)]T and we have defined h, ˆ and n in a similar manner. Given h, x(m) and the Gaussianity ˜ the received signal r follows the multivariate Gaussian of h, distribution with its pdf given by ˆ x(m)) = f (r | h, (15) 1 T −1 ˆ ˆ exp − 2 (r − diag(h)x(m)) Q (r − diag(h)x(m)) (2π)N/2 (det Q)1/2 where
T
˜ ˜ Q = E diag(h)x(m) + n diag(h)x(m) + n .
(16)
Obviously, the codewords will influence both the mean value and covariance of the distribution for received signals. However, the NN decoder only takes the mean value into consideration. As in [2], we assume that a genie provides the decoder with a noisy measurement where the estimation error is independent of the codebook, transmitted message and AWGN noise. With this assumption, (16) can be simplified to give Q = diag{x21 (m)σe2 + σn2 , ..., x2N (m)σe2 + σn2 }
(17)
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 6, JUNE 2009
which, along with (15), enables representing the log likelihood of r as
ˆ x(m)) log f (r | h,
N ˆ k xk (m))2 1 (rk − h = − (18) 2 x2k (m)σe2 + σn2
− −
1 2
k=1 N
2797
where N →∞
The function ψ(s) is the scaled cumulant moment-generating function for D(m ) defined as
log(x2k (m)σe2 + σn2 )
ψ(s)
k=1
N log(2π). 2
Since xk (m) is generated from i.i.d. (independent and identically distributed) Gaussian codebook, the second term of (18) is asymptotically independent of the codeword index m, as shown by N 2 2 2 k=1 log(xk (m)σe + σn ) lim N →∞ N N 2 2 2 k=1 log(xk (m )σe + σn ) . (19) = lim N →∞ N Hence, the decision metric can be simply built on the first term taking the form N ˆ k xk (m))2 1 (rk − h D(m) = . 2 N xk (m)σe2 + σn2
(20)
C. Achievable rate for normalized nearest neighbor decoder We now calculate the maximum rate that can be obtained by the NNN decoder. Similar to the NN decoder, the achievable rate for the NNN decoder IN N N is defined as the rate at which the probability of decoding error Pe approaches zero as the size, N , of the Gaussian vector x(m) tends to infinity. ˆ Namely, for given received signal r and channel estimate h Pe ≤ exp(−N (IN N N − δ)), ∀ δ > 0,
(21)
and the detailed discussion can be found in [8]. Assuming the transmission of message m, the probability of error is expressible as Pe = P r(D(m ) < D(m)),
m = m.
(22)
To calculate Pe , we need to first determine the pdf of D(m ) which, however, is difficult as we are considering the case with N → ∞. Fortunately, since we are concerned with the small probability exp(−N (IN N N − δ)) as N → ∞, the problem of finding IN N N can be solved by using the large deviation principle (LDP) [7]. Specifically, by virtue of the Gartner-Ellis theorem [7], it has been shown [8] that IN N N = sup(sT − ψ(s)) s
