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首页低精度ADC通信极限:挑战与优化策略
低精度ADC通信极限:挑战与优化策略
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更新于2024-09-10
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随着现代通信系统在速度和带宽上的不断提升,传统的高精度(如8-12位)模拟到数字转换器(ADC)的成本和功耗问题逐渐成为数字信号处理架构下无线收发器设计的关键瓶颈。本文主要关注"低精度量化"这一主题,深入探讨了降低ADC精度对通信链路性能的影响。 作者Jaspreet Singh、Onkar Dabeer和Upamanyu Madhow作为IEEE的成员,针对接收端采用低精度ADC(如1-3位)的无线接收机性能进行了深入研究。他们研究的核心内容是,在给定输入信号平均功率约束的条件下,如何通过减少ADC的位数来优化通信性能,尤其是在面对实值离散时间加性白高斯噪声(AWGN)通道时。 他们的研究发现,当ADC具有K个量化级(即,K比特的精度)时,输入分布实际上只需要有K+1个质心点就能达到信道容量。这是一个重要的理论贡献,因为它表明,即使在低精度量化的情况下,通过合理的信号处理策略,依然有可能接近或实现最优的通信效果。 对于二进制对称量化(1比特)的情况,这项研究进一步细化,证明了二进制反极性信号(如正负两个符号)在任何信号噪声比(SNR)下都是最优化的编码方式。这意味着,即使是在如此简化的量化方案下,通过精心设计的编码策略,依然可以保证通信的稳定性和有效性。 这篇文章揭示了低精度量化在现代通信系统中的潜在价值,尤其是在追求低成本和低功耗的前提下,它为设计者提供了一种可能的解决方案,即通过牺牲部分精度来换取系统整体性能的平衡。这不仅对于无线通信技术的发展具有重要意义,也对其他领域,如物联网(IoT)和无线传感器网络(WSN),有着广泛的应用前景。
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3
and the input-output mutual information I(X; Y ), expressed
explicitly as a function of F is
I(F ) =
Z
∞
−∞
K
X
i=1
W
i
(x) log
W
i
(x)
R(y
i
; F )
dF (x) .
1
(5)
Under an average power constraint P , we wish to find the
capacity of the channel (1), given by
C = sup
F ∈F
I(F ), (6)
where F =
n
F : E[X
2
] =
R
∞
−∞
x
2
dF (x) ≤ P
o
, i.e., the set
of all average power constrained distributions on R.
Structural Properties of Optimal Inputs: We begin by em-
ploying the Karush-Kuhn-Tucker (KKT) optimality condition
to show that, even though we have not imposed a peak power
constraint on the input, it is automatically induced by the
average power constraint. Specifically, a capacity achieving
distribution for the AWGN-QO channel (1) must have bounded
support.
2
A. An Implicit Peak Power Constraint
Using convex optimization principles, the following
necessary and sufficient KKT optimality condition can
be derived for our problem, in a manner similar to the
development in [25], [26]. An input distribution F
∗
achieves
the capacity C in (6) if and only if there exists γ ≥ 0 such that
K
X
i=1
W
i
(x) log
W
i
(x)
R(y
i
; F
∗
)
+ γ(P − x
2
) ≤ C (7)
for all x, with equality if x is in the support
3
of F
∗
, where
the transition probability function W
i
(x), and the output prob-
ability R(y
i
; F
∗
) are as specified in (2) and (4), respectively.
The summation on the left-hand side (LHS) of (7) is the
Kullback-Leibler divergence (or the relative entropy) between
the transition PMF {W
i
(x), i = 1, . . . , K} and the output
PMF {R(y
i
; F ), i = 1, . . . , K}. For convenience, let us denote
this divergence function by d(x; F ), that is,
d(x; F ) =
K
X
i=1
W
i
(x) log
W
i
(x)
R(y
i
; F )
. (8)
We begin by studying the behavior of this function in the
limit as x → ∞.
Lemma 1: For the AWGN-QO channel (1), the divergence
function d(x; F ) satisfies the following properties
(a) lim
x→∞
d(x; F ) = −log R(y
K
; F ).
(b) There exists a finite constant A
0
such that
for x > A
0
, d(x; F ) < −log R(y
K
; F ).
4
1
The logarithm is base 2 throughout the paper, so the mutual information
is measured in bits.
2
That there exists a capacity achieving distribution follows by standard
function analytic arguments [23]. For details, see our technical report [24].
3
The support of a distribution F (or the set of increase points of F ) is the
set S
X
(F ) = {x : F (x + ²) − F (x − ²) > 0, ∀² > 0}.
4
The constant A
0
depends on the choice of the input F . For notational
simplicity, we do not explicitly show this dependence.
Proof: We have
d(x; F ) =
K
X
i=1
W
i
(x) log
W
i
(x)
R(y
i
; F )
=
K
X
i=1
W
i
(x) log W
i
(x) −
K
X
i=1
W
i
(x) log R(y
i
; F ) .
As x → ∞, the PMF {W
i
(x), i = 1, . . . , K} → 1(i = K),
where 1(·) is the indicator function. This observation,
combined with the fact that the entropy of a finite alphabet
random variable is a continuous function of its probability law,
gives lim
x→∞
d(x; F ) = 0 − log R(y
K
; F ) = −log R(y
K
; F ).
Next we prove part (b). For x > q
K−1
, W
i
(x) is a strictly
decreasing function of x for i ≤ K −1 and strictly increasing
function of x for i = K. Since {W
i
(x)} → 1(i = K) as
x → ∞, it follows that there is a constant A
0
such that
W
i
(A
0
) < R( y
i
; F ) for i ≤ K −1 and W
K
(A
0
) > R( y
K
; F ).
Therefore, it follows that for x > A
0
,
d(x; F ) =
K
X
i=1
W
i
(x) log
W
i
(x)
R(y
i
; F )
< W
K
(x) log
W
K
(x)
R(y
K
; F )
< −log R(y
K
; F ).
The saturating nature of the divergence function for the
AWGN-QO channel, as stated above, coupled with the KKT
condition, is now used to prove that a capacity achieving
distribution must have bounded support.
Proposition 1: For the average power constrained AWGN-
QO channel (1), an optimal input distribution must have
bounded support.
Proof: Let F
∗
be an optimal input, so that there exists
γ ≥ 0 such that (7) is satisfied with equality at every point
in the support of F
∗
. We exploit this necessary condition to
show that the support of F
∗
is upper bounded. Specifically,
we prove that there exists a finite constant A
2
∗
such that it is
not possible to attain equality in (7) for any x > A
2
∗
.
From Lemma 1, we get lim
x→∞
d(x; F
∗
)=−log R(y
K
; F
∗
)=:L.
Also, there exists a finite constant A
0
such that for
x > A
0
, d(x; F
∗
) < L.
We consider two possible cases.
• Case 1: γ > 0.
For x > A
0
, we have d(x; F
∗
) < L.
For x >
p
max{0, (L − C + γP )/γ} =:
˜
A, we have
γ(P − x
2
) < C −L.
Defining A
∗
2
= max{A
0
,
˜
A}, we get the desired result.
• Case 2: γ = 0.
Since γ = 0, the KKT condition (7) reduces to
d(x; F
∗
) ≤ C , ∀x.
Taking limit x → ∞ on both sides, we get
L = lim
x→∞
d(x; F
∗
) ≤ C.
Hence, choosing A
∗
2
= A
0
, for x > A
∗
2
we get,
d(x; F
∗
) < L ≤ C, that is, d(x; F
∗
) + γ(P − x
2
) < C.
Combining the two cases, we have shown that the support
of the distribution F
∗
has a finite upper bound A
2
∗
. Using
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