低精度ADC通信极限：挑战与优化策略

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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。

and the input-output mutual information I(X; Y ), expressed

explicitly as a function of F is

I(F ) =

∞

−∞

i=1

(x) log

(x)

R(y

; F )

dF (x) .

(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 =

F : E[X

] =

∞

−∞

dF (x) ≤ P

, 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.

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

i=1

(x) log

(x)

R(y

; F

∗

)

+ γ(P − x

) ≤ C (7)

for all x, with equality if x is in the support

of F

∗

, where

the transition probability function W

(x), and the output prob-

ability R(y

; 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

(x), i = 1, . . . , K} and the output

PMF {R(y

; F ), i = 1, . . . , K}. For convenience, let us denote

this divergence function by d(x; F ), that is,

d(x; F ) =

i=1

(x) log

(x)

R(y

; 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

; F ).

(b) There exists a ﬁnite constant A

such that

for x > A

, d(x; F ) < −log R(y

; F ).

The logarithm is base 2 throughout the paper, so the mutual information

is measured in bits.

That there exists a capacity achieving distribution follows by standard

function analytic arguments [23]. For details, see our technical report [24].

The support of a distribution F (or the set of increase points of F ) is the

set S

(F ) = {x : F (x + ²) − F (x − ²) > 0, ∀² > 0}.

The constant A

depends on the choice of the input F . For notational

simplicity, we do not explicitly show this dependence.

Proof: We have

d(x; F ) =

i=1

(x) log

(x)

R(y

; F )

i=1

(x) log W

(x) −

i=1

(x) log R(y

; F ) .

As x → ∞, the PMF {W

(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

; F ) = −log R(y

; F ).

Next we prove part (b). For x > q

K−1

, W

(x) is a strictly

decreasing function of x for i ≤ K −1 and strictly increasing

function of x for i = K. Since {W

(x)} → 1(i = K) as

x → ∞, it follows that there is a constant A

such that

) < R( y

; F ) for i ≤ K −1 and W

) > R( y

; F ).

Therefore, it follows that for x > A

d(x; F ) =

i=1

(x) log

(x)

R(y

; F )

< W

(x) log

(x)

R(y

; F )

< −log R(y

; 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

∗

such that it is

not possible to attain equality in (7) for any x > A

∗

From Lemma 1, we get lim

x→∞

d(x; F

∗

)=−log R(y

; F

∗

)=:L.

Also, there exists a ﬁnite constant A

such that for

x > A

, d(x; F

∗

) < L.

We consider two possible cases.

• Case 1: γ > 0.

For x > A

, we have d(x; F

∗

) < L.

For x >

max{0, (L − C + γP )/γ} =:

A, we have

γ(P − x

) < C −L.

Deﬁning A

∗

= max{A

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

∗

= A

, for x > A

∗

we get,

d(x; F

∗

) < L ≤ C, that is, d(x; F

∗

) + γ(P − x

) < C.

Combining the two cases, we have shown that the support

of the distribution F

∗

has a ﬁnite upper bound A

∗

. Using

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