Deterministic Identification Over Channels With Power Constraints

Abstract

The deterministic identification (DI) capacity is developed in multiple settings of channels with power constraints. A full characterization is established for the DI capacity of the discrete memoryless channel (DMC) with and without input constraints. Originally, Ahlswede and Dueck established the identification capacity with local randomness at the encoder, resulting in a double exponential number of messages in the block length
n
. In the deterministic setup, the number of messages scales exponentially, as in Shannon’s transmission paradigm, but the achievable identification rates are higher. An explicit proof was not provided for the deterministic setting. In this paper, a detailed proof is presented for the DMC. Furthermore, Gaussian channels with fast and slow fading are considered, when channel side information is available at the decoder. A new phenomenon is observed as we establish that the number of messages scales as
2nlog⁡(n)R2^{n\log (n)R}
by deriving lower and upper bounds on the DI capacity on this scale. Consequently, the DI capacity of the Gaussian channel is infinite in the exponential scale and zero in the double exponential scale, regardless of the channel noise.

Full text

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 68, NO. 1, JANUARY 2022 1
Deterministic Identification Over Channels
With Power Constraints
Mohammad J. Salariseddigh ,UziPereg ,Member, IEEE, Holger Boche ,Fellow, IEEE,
and Christian Deppe ,Member, IEEE
Abstract— The deterministic identification (DI) capacity is
developed in multiple settings of channels with power constraints.
A full characterization is established for the DI capacity of the
discrete memoryless channel (DMC) with and without input
constraints. Originally, Ahlswede and Dueck established the
identification capacity with local randomness at the encoder,
resulting in a double exponential number of messages in the block
length n. In the deterministic setup, the number of messages
scales exponentially, as in Shannon’s transmission paradigm,
but the achievable identification rates are higher. An explicit
proof was not provided for the deterministic setting. In this
paper, a detailed proof is presented for the DMC. Furthermore,
Gaussian channels with fast and slow fading are considered,
when channel side information is available at the decoder.
A new phenomenon is observed as we establish that the number
of messages scales as 2nlog(n)Rby deriving lower and upper
bounds on the DI capacity on this scale. Consequently, the DI
capacity of the Gaussian channel is infinite in the exponential
scale and zero in the double exponential scale, regardless of the
channel noise.
Index Terms—Channel capacity, identification, deterministic
codes, Gaussian fading channels, super exponential growth,
channel side information.
I. INTRODUCTION
IN THE fundamental communication paradigm considered
by Shannon [3], a sender wishes to convey a message
Manuscript received November 5, 2020; revised September 22, 2021;
accepted October 17, 2021. Date of publication October 26, 2021; date of
current version December 23, 2021. The work of Mohammad J. Salariseddigh,
Uzi Pereg, and Christian Deppe was supported by the Lehr- und Forschung-
seinheit für Nachrichtentechnik (LNT), New Communication Models-Post
Shannon Communication (NEWCOM) under Grant 16KIS1005. The work
of Holger Boche was supported in part by LTI, NEWCOM under Grant
16KIS1003K and in part by the Bundesministerium für Bildung und
Forschung (BMBF) within the National Initiative for Macroscopic Mole-
cular Communications (MAMOKO) under Grant 16KIS0914. An earlier
version of this paper was presented in part at the IEEE Information
Theory Workshop (ITW 2020) [DOI: 10.1109/ITW46852.2021.9457587],
in part at the IEEE International Conference on Communications (ICC
2021) [DOI: 10.1109/ICC42927.2021.9500406], in part at the Post-Shannon
Communication Session, London Symposium of Information Theory (LSIT
2021) (as a talk), and in part at the VDE-TUM Workshop on Molecu-
lar Communication and 6G in September 2021. (Corresponding author:
Mohammad J. Salariseddigh.)
Mohammad J. Salariseddigh, Uzi Pereg, and Christian Deppe are with the
Institute for Communications Engineering, Technische Universität München,
80333 Munich, Germany (e-mail: mjss@tum.de; uzi.pereg@tum.de;
christian.deppe@tum.de).
Holger Boche is with the Institute of Theoretical Information Technology,
Technische Universität München, 80290 Munich, Germany, also with the
Munich Center for Quantum Science and Technology (MCQST), 80799
Munich, Germany, and also with the Cyber Security in the Age of Large-
Scale Adversaries—Exzellenzcluster (CASA), 44780 Bochum, Germany
(e-mail: boche@tum.de).
Communicated by F. Alajaji, Associate Editor for Shannon Theory.
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TIT.2021.3122811.
Digital Object Identifier 10.1109/TIT.2021.3122811
through a noisy channel in such a manner that the receiver
will be able to retrieve the original message. In other words,
the decoder’s task is to determine which message was sent.
Ahlswede and Dueck [4] introduced a scenario of a different
nature, where the decoder only performs identification and
determines whether a particular message was sent or not
[4]–[6]. Applications include the tactile internet [7], vehicle-
to-X communications [8], [9], digital watermarking [10]–[12],
online sales [13], [14], industry 4.0 [15], health care [16], and
other event-triggered systems.
We give two motivating examples for applications of
the identification paradigm. Molecular communication is a
promising contender for future applications such as the sixth
generation of cellular communication (6G) [17], [18], in which
a number of applications require alerts to be identified [18].
Furthermore, in other systems of molecular communication,
a nano-device needs to determine the occurrence of a specific
event. For example, in the course of targeted drug deliv-
ery [19], [20] or cancer treatment [21]–[23], a nano-device
will seek to know whether the blood pH exceeds a critical
threshold or not, whether a specific drug is released or not,
whether another nano-device has replicated itself, whether a
certain molecule was detected, whether a target location in
the vessels is identified, or whether the molecular storage is
empty, etc., [24]. A second application for identification is
vehicle-to-X communications, where a vehicle that collects
sensor data may ask whether a certain alert message concern-
ing the future movement of an adjacent vehicle was transmitted
or not [25, Sec. VII].
The identification problem [4] can be regarded as a
Post-Shannon [26] model where the decoder does not perform
an estimation, but rather a binary hypothesis test to decide
between the hypotheses ‘sent’ or ‘not sent’, based on the
observation of the channel output. As the sender has no knowl-
edge of the desired message that the receiver is interested in,
the identification problem can be regarded as a test of many
hypotheses occurring simultaneously. The scenario where the
receiver misses and does not identify his message is called
a type I error, or ‘missed identification’, whereas the event
where the receiver accepts a false message is called a type II
error, or ‘false identification’.
Ahlswede and Dueck [4] required randomized coding for
their identification-coding scheme. This means that a random-
ized source is available to the sender. The sender can make his
encoding dependent on the output of this source. It is known
that this resource cannot be used to increase the transmission
capacity of discrete memoryless channels [27]. A remarkable
result of identification theory is that given local randomness
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2IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 68, NO. 1, JANUARY 2022
at the encoder, reliable identification can be attained such
that the code size, i.e., the number of messages, grows
double exponentially in the block length n, i.e., ∼22nR [4].
This differs sharply from the traditional transmission setting
where the code size scales only exponentially, i.e., ∼2nR.
Beyond the exponential gain in identification, the extension
of the problem to more complex scenarios reveals that the
identification capacity has a very different behavior compared
to the transmission capacity [28]–[33]. For instance, feedback
can increase the identification capacity [28] of a memory-
less channel, as opposed to the transmission capacity [34].
Nevertheless, it is difficult to implement randomized-encoder
identification (RI) codes that will achieve such performance,
because it requires the encoder to process a bit string of
exponential length. The construction of identification codes
is considered in [6], [35]–[38]. Identification for Gaussian
channels is considered in [30], [39]–[42].
In the deterministic setup for a DMC, the number of mes-
sages scales exponentially in the blocklength [4], [43]–[45],
as in the traditional setting of transmission. Nevertheless,
the achievable identification rates are significantly higher
than those of transmission. In addition, deterministic codes
often have the advantage of simpler implementation and
simulation [46], explicit construction [47], and single-block
reliable performance. In particular, JáJá [44] showed that
the deterministic identification (DI) capacity1of a binary
symmetric channel is 1 bit per channel use, based on com-
binatorial methods. The meaning of this result is that one
can exhaust the entire input space and assign (almost) all
sequences in the n-dimensional space {0,1}nas codewords.
Ahlswede et al. [4], [43] stated that the DI capacity of a
discrete memoryless channel (DMC) with a stochastic matrix
Wis given by the logarithm of the number of distinct row
vectors of W(see Section IV. in [4] and abstract of [43]).
Nonetheless, an explicit proof for this result was not provided
in [4], [43]. Instead, Ahlswede and Cai [43] referred the reader
to a paper [48] which does not include identification and
addresses a completely different model of an arbitrarily vary-
ing channel [48]. Since then, the problem of proving this result
has remained unsolved, since a straightforward extension of
the methods in [48], using decoding territories, does not seem
to yield the desired result on the DI capacity [49].
Modern communications require the transfer of enormous
amounts of data in wireless systems, for cellular commu-
nication [37], sensor networks [50], smart appliances [51],
and the internet of things [52], etc. Wireless communication
is often modelled by fading channels with additive white
Gaussian noise [53]–[61]. In the fast fading regime, the
transmission spans over a large number of coherence time
intervals [62], hence the signal attenuation is characterized
by a stochastic process or a sequence of random parameters
[63]–[66]. In some applications, the receiver may acquire
channel side information (CSI) by instantaneous estimation of
the channel parameters [67]–[69]. On the other hand, in the
slow fading regime, the latency is short compared to the
1The DI capacity in the literature is also referred to as the non-randomized
identification (NRI) capacity [43] or the dID capacity [30].
coherence time [62], and the behaviour is that of a compound
channel [70]–[74].
In this paper, we establish the DI capacity of channels
subject to an input constraint. Such a constraint is often
associated with a limited power supply or regulation, as in
the case of the Gaussian channel. We consider the settings of
a DMC and of Gaussian channels with fast fading and slow
fading, with CSI available at the decoder. For a DMC, one may
assume without loss of generality that the rows of the channel
matrix are distinct (see Remarks 3 and 5). Our first result
is that the DI capacity of a DMC Wunder this assumption,
subject to the input constraint 1
nn
t=1 φ(xt)≤A,isgivenby
CDI(W)= max
pX:E{φ(X)}≤AH(X).(1)
We note that the DI capacity does not depend on the
specific values of the transition probabilities, as long as the
rows of the channel matrix are distinct. This result has the
following geometric interpretation. At first glance, it may
seem reasonable that for the purpose of identification, one
codeword could represent two messages. While identification
allows overlap between decoding regions [75], [76], it turns
out that overlap at the encoder is not allowed for deterministic
codes. However, if two messages are represented by the same
codeword, then the low probability of a type I error comes
at the expense of the high probability of a type II error, and
vice versa. Thus, DI coding imposes the restriction that every
message must have a distinct codeword. The converse proof
follows from this property in a straightforward manner, since
the volume of the input subset of sequences that satisfy the
input constraint is ≈2nCDI(W). A similar principle guides the
direct part as well. The input space is covered such that each
codeword is surrounded by a sphere of radius nε to separate
the codewords.
Gaussian channels have an entirely different and unusual
behaviour. We consider deterministic identification for
Gaussian channels with fast fading and slow fading, and
with channel side information (CSI) available at the decoder.
Applying discretization to our capacity result above (see (1)),
we obtain that the DI capacity of the standard Gaussian
channel is infinite in the exponential scale (as we have recently
observed in [2], [77]). However, for a finite blocklength n,the
number of codewords must be finite. Thereby, the meaning
of the infinite capacity result is that the number of messages
scales super-exponentially. This raises the question: What is
thetrueorderofthecodesize.In mathematical terms, what
is the scale L(n, R)for which the DI capacity is positive yet
finite. We show that for Gaussian channels, the number of
messages scales as 2nlog(n)R, and develop lower and upper
bounds on the DI capacity in this scale. As a consequence,
we deduce that the DI capacity of a Gaussian channel with
fast fading is infinite in the exponential scale and zero in the
double exponential scale, regardless of the channel noise. For
slow fading, the DI capacity in the exponential scale is infinite,
unless the fading gain can be zero or arbitrarily close to zero
(with positive probability), in which case the DI capacity is
zero. Note, however, that this scale is comparably lower than
the double exponential scale of RI coding.
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SALARISEDDIGH et al.: DETERMINISTIC IDENTIFICATION OVER CHANNELS WITH POWER CONSTRAINTS 3
By providing a detailed proof for the DMC, we thus fill the
gap in the previous analysis [4], [43] as well. In the proof,
we use the method of types, while the derivation is based
on ideas that are analogous to the combinatoric analysis of
Hamming distances by JáJá [44]. Although the codebook con-
struction is similar to that of Ahlswede’s coding scheme [48],
the decoder is significantly different. In particular, we do not
use decoding territories as in [48], but rather perform a typical-
ity check. Nonetheless, the type-class intersection lemma and
the message-set analysis in [48] turn out to be useful in our
analysis as well. Hence, our proof combines techniques and
ideas from both works, by JáJá [44] and by Ahlswede [48],
to derive the DI capacity both with and without an input
constraint. The analysis for Gaussian channels also relies
on geometric considerations, using sphere packing. Based
on fundamental properties of packing arrangements [78], the
optimal packing of non-overlapping spheres of radius √nε
contains an exponential number of spheres, and by decreasing
the radius of the codeword spheres, the exponential rate can be
made arbitrarily large. However, in the derivation of our lower
bound in the 2nlog(n)R-scale, we pack spheres of a sub-linear
radius √nεn∼n1/4, which results in ∼21
4nlog(n)codewords.
The paper is organized as follows. In Section II we give the
definitions and a brief review of related work. In Section III
we address deterministic identification for the DMC with and
without an input constraint. In Subsection III-A, a channel
reduction procedure is described such that high-cost identical
rows are removed from the channel matrix. The capacity
theorem is stated in Subsection III-B. The direct part is
proved in Subsection III-C, and the converse proof is in
Subsection III-D. We consider deterministic identification over
fading channels in Section IV. Definitions for fading channels
are given in Subsection IV-A and IV-B.
Lower and upper bounds in the 2nlog(n)R-scale are pre-
sented in Subsection IV-C, and the respective proofs in IV-D
and IV-E. Section V is dedicated to summary and discussion.
II. DEFINITIONS AND RELATED WORKS
In this section we introduce the channel model and coding
definitions. Here we only consider the discrete memoryless
channel (DMC). The channel description and coding definition
for the Gaussian channel will be presented in Section IV.
A. Notation
We use the following notation conventions throughout. Cal-
ligraphic letters X,Y,Z,... are used for finite sets. Lowercase
letters x,y,z,... stand for constants and values of random
variables, and uppercase letters X,Y,Z,... stand for random
variables. The distribution of a random variable Xis specified
by a probability mass function (pmf) pX(x)over a finite
set X. The set of all pmfs over Xis denoted by P(X).
H(X)and I(X;Y)are the entropy and mutual information,
respectively; H2(p)=−(1 −p) log(1 −p)−plog(p)is the
binary entropy function for a given 0<p<1; all logarithms
and information quantities are taken to the base 2.Weuse
xj=(x1,x
2,...,x
j)to denote a sequence of letters from
X. A random sequence Xnand its distribution pXn(xn)are
defined accordingly. The set of consecutive natural numbers
from 1through Mis denoted by [[ M]] . The Hamming distance
between two sequences anand bnis defined as the number
of positions for which the sequences have different symbols,
i.e., dH(an,b
n)=|{t∈[[ n]] ; at=bt}|. The notation x=
(x1,x
2,...,x
n)is used instead of xnwhen it is understood
from the context that the length of the sequence is n,andthe
2-norm of xis denoted by x.Then-dimensional Hamming
sphere of radius nε that is centered at anis defined as
Sε(an)={xn∈Xn:dH(xn,a
n)<nε}.(2)
We denote the hyper-sphere of radius raround x0by
Sx0(n, r)=x∈Rn:x−x0≤r,(3)
and its volume by Vol(S). The closure of a set Ais denoted
by cl(A). In the continuous case, we use the cumulative
distribution function FX(x)=Pr(X≤x)for x∈R,
or alternatively, the probability density function (pdf) fX(x),
when it exists. The element-wise product of vectors is denoted
by x◦y=(xtyt)n
t=1. A random sequence Xand its
distribution FX(x)are defined accordingly.
B. Channel Description
ADMC(X,Y,W)consists of finite input and output
alphabets Xand Y, respectively, and a conditional pmf
W(y|x). The channel is memoryless without feedback, and
therefore Wn(yn|xn)=n
t=1 W(yt|xt). We denote a DMC
by W=(X,Y,W). Next, we consider an input constraint.
Let φ:X→[0,∞)be some given bounded cost function,
and define
φn(xn)=1
n
n

t=1
φ(xt).(4)
Given an input constraint A>0corresponding to the cost
function φn(xn), the channel input xnmust satisfy
φn(xn)≤A. (5)
We may assume without loss of generality that 0≤A≤
φmax,whereφmax =max
x∈X φ(x).Itisalsoassumedthatfor
some x0∈X,φ(x0)=0.
C. Coding
The definitions for DI codes, achievable rates, and capacity
are given below. In this paper we consider codes with different
size orders. For instance when we discuss the exponential
scale, we refer to a code size that scales as L(n, R)=2
nR.
On the other hand, in the double exponential scale, the code
size is L(n, R)=2
2nR . Later, in Section IV where we con-
sider Gaussian channels, we will see that the appropriate scale
turns out to be neither exponential nor double exponential, but
in between. Throughout the paper we use the mathematical
convention that L:N×R+→Ndenotes a map, and
L(n, R)∈Nis its value for a given blocklength nand
rate R.
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4IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 68, NO. 1, JANUARY 2022
Definition 1: Let L1(n, R)and L2(n, R)be two coding
scales. We say that L1dominates L2if
lim
n→∞
L2(n, b)
L1(n, a)=0,(6)
for all a, b > 0. We will denote this relation by L2≺L1.
In complexity theory of computer science, the relation
above is denoted by the ‘small o-notation’, L2(n, 1) =
o(L1(n, 1)) [79]. Beyond exponential, other orders that com-
monly appear in complexity theory are the linear, logarithmic,
and polynomial scales, nR,log(nR),and(nR)k. The corre-
sponding ordering is
log(nR)≺nR ≺(nR)k≺2nR ≺2nlog(n)R≺22nR .(7)
Definition 2: An (L(n, R),n)DI code for a DMC W
under input constraint A, assuming L(n, R)is an integer,
is defined as a system (U,D)that consists of a codebook
U={ui}i∈[[ L(n,R)]],U⊂X
n, such that
φn(ui)≤A, for all i∈[[ L(n, R)]],(8)
and a collection of decoding regions D={Di}i∈[[ L(n,R)]]
with L(n,R)
i=1 Di⊂Y
n. Given a message i∈[[ L(n, R)]],
the encoder transmits ui. The decoder’s aim is to answer
the following question: Was a desired message jsent or
not? Two types of errors may occur: Rejecting of the true
message, or accepting a false message. Those error events are
often referred to as type I and type II errors, respectively.
Specifically, P(n)
e,1(i)is the type I error probability for rejecting
thetruemessagei, while P(n)
e,2(i, j)is the type II error
probability for accepting the false message j, given that the
message iwas sent.
The error probabilities of the identification code (U,D)are
given by
Pe,1(i)=Wn(Dc
i|ui)(missed-identification error),(9)
Pe,2(i, j)=Wn(Dj|ui)(false identification error).(10)
An (L(n, R),n,λ
1,λ
2)DI code further satisfies
Pe,1(i)≤λ1,(11)
Pe,2(i, j)≤λ2,(12)
for all i, j ∈[[ L(n, R)]] such that i=j.
ArateR>0is called achievable if for every λ1,λ
2>0
and sufficiently large nthere exists an (L(n, R),n,λ
1,λ
2)DI
code. The operational DI capacity is defined as the supremum
of achievable rates and will be denoted by CDI (W,L).
As mentioned earlier, Ahlswede and Dueck [4] needed
randomized encoding for their identification-coding scheme.
This means that a randomized source is available to the
sender. The sender can make his encoding dependent on
the output of this source. Therefore, a randomized-encoder
identification (RI) code is defined in a similar manner where
the encoder is allowed to select a codeword Uiat random,
according to some conditional input distribution Q(xn|i).The
RI capacity is then denoted by CRI (W,L). Given local ran-
domness at the encoder, reliable identification can be attained
such that the number of messages grows double exponentially
Fig. 1. Geometric illustration of identification errors in the deterministic
setting. The arrows indicate three scenarios for the channel output, given that
the encoder transmitted the codeword u1corresponding to i=1.Ifthe
channel output is outside D1, then a type I error has occurred, as indicated
by the bottom red arrow. This kind of error is also considered in traditional
transmission. In identification, the decoding sets can overlap. If the channel
output belongs to D1but also belongs to D2, then a type II error has occurred,
as indicated by the middle brown arrow. Correct identification occurs when
the channel output belongs only in D1, which is marked in blue.
in the block length n,i.e.,L(n, R)=Ldouble(n, R)
22nR [4]. This differs sharply from the traditional transmis-
sion setting where the code size scales only exponentially,
i.e., L(n, R)=Lexp(n, R)2nR . Remarkably, in [4] it
was shown that CRI (W,L
double)=CT(W,L
exp),where
CT(W,L
exp)denotes the transmission capacity of the channel
in the exponential scale.
Remark 1: The code scale can also be thought of as a
sequence of monotonically increasing functions Ln(R)of the
rate. Hence, given a code of size M=Ln(R), the coding
rate can be obtained from the inverse relation R=L−1
n(M).
In particular, for the transmission setting [3], or DI coding for
a DMC [44], the coding rate is defined as
R=1
nlog(M).(13)
Whereas for RI coding [4], the rate was defined as
R=1
nlog log(M).(14)
On the other hand, using the scale L(n, R)=2
nlog(n)Ras
for Gaussian channels stated in Theorem 10, the coding rate
is
R=log M
nlog n.(15)
Remark 2: It can be readily shown that in general, if the
capacity in an exponential scale is finite, then it is zero in
the double exponential scale. Conversely, if the capacity in a
double exponential scale is positive, then the capacity in the
exponential scale is +∞. This principle can be generalized to
any pair of scales L1and L2,whereL2is dominated by L1.
We come back to this in Subsection IV-C.
A geometric illustration for the type I and II error prob-
abilities is given in Figure 1. When the encoder sends the
message ibut the channel output is outside Di, then type I
error occurs. This kind of error is also considered in traditional
transmission. In identification, the decoding sets can overlap.
A type II error covers the case where the output sequence
belongs to the intersection of Diand Djfor j=i.
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SALARISEDDIGH et al.: DETERMINISTIC IDENTIFICATION OVER CHANNELS WITH POWER CONSTRAINTS 5
D. Related Work
We briefly review Ahlswede and Dueck’s result [4] on
the RI capacity, i.e., when the encoder uses a stochastic
mapping. As mentioned above, using RI codes, it is possible
to identify a double exponential number of messages in the
block length n. That is, given a rate R<CRI (W,L),there
exists a sequence of (L(n, R)=2
2nR ,n)RI codes with
vanishing error probabilities. Despite the significant difference
between the definitions in the identification setting and in the
transmission setting, it was shown that the value of the RI
capacity in the double exponential scale equals the Shannon
capacity of transmission.
Theorem 1 (See [4], [80]): The RI capacity in the double
exponential scale of a DMC Wis given by
CRI (W,L)= max
pX:E{φ(X)}≤AI(X;Y),for L(n, R)=2
2nR .
(16)
Hence, the RI capacity in the exponential scale is infinite,
i.e.,
CRI (W,L)=∞,for L(n, R)=2
nR.(17)
In the next sections, we will consider the identification set-
ting when the encoder does not have access to randomization.
Theorem 2 (See [4], [40]): Let Gdenote the standard
Gaussian channel, where the input-ouptut relation is given by
Y=gX +Z, with Z∼N(0,σ
2
Z)and a fixed known gain
g>0. Then, the RI capacity in the double exponential scale
is given by
CRI (G,L)=1
2log 1+g2A
σ2
Z,for L(n, R)=2
2nR .
(18)
Hence, the RI capacity in the exponential scale is infinite,
i.e.,
CRI (G,L)=∞,for L(n, R)=2
nR.(19)
III. MAIN RESULTS—DMC
We give our main results on the DI capacity of the DMC.
For a DI code, as opposed to the randomized case, the number
of messages 2nR is only exponential in the blocklength. In this
sense, DI codes are similar to transmission codes. However,
the achievable rates for identification are significantly higher,
as the DI capacity is given in terms of the input entropy instead
of the mutual information.
A. Channel Reduction
We begin with a procedure of channel reduction where we
remove identical rows from the channel matrix, so that the
remaining input letters have a lower cost compared to the
deleted letters. As will be seen below, the DI capacity remains
the same following this reduction. The characterization of the
DI capacity will be given in the next section in terms of the
reduced input alphabet.
We begin with the definition of the reduced channel.
Definition 3 (Reduced Channel): Given a DMC Wwith
a stochastic matrix W:X→Y, we define the reduced
DMC Wras follows. Let {X()}be a partition of Xinto
equivalent classes, so that two letters xand xbelong to the
same equivalent class if and only if the corresponding rows
are identical, namely
x, x∈X()⇔W(y|x)=W(y|x)∀y∈Y.(20)
For every class X(), assign a representative element
z()=arg min
x∈X()φ(x),(21)
which is associated with the lowest input cost. If there is more
than one letter that is associated with the lowest input cost in
X(), then choose one of them arbitrarily. Then the reduced
input alphabet is defined as
Xr={z()},(22)
and the reduced DMC Wris defined by a channel matrix
Wr:Xr→Y, consisting of the rows in Xr,i.e.,
Wr(y|x)=W(y|x),(23)
for x∈X
rand y∈Y.
Lemma 3: The operational capacities of the reduced channel
Wrand the original channel Ware the same:
CDI (W,L)=CDI(Wr,L),for L(n, R)=2
nR.(24)
We give the proof of Lemma 3 in Appendix A. As we will
see shortly, the DI capacity of a DMC Wdepends on Wonly
through Xr. That is, the DI capacity does not depend on the
individual values of the channel matrix and depends solely on
the distinctness of its rows.
Remark 3: Based on Lemma 3, it is sufficient to consider
a channel with distinct rows. That is, if we establish the
DI capacity for channels with distinct rows, we can then
determine the DI capacity for a general channel. In other
words, in order to derive a capacity result, we may assume
without loss of generality that the channel rows are distinct.
B. Capacity Theorem
In this section, we give our main result on the DI capacity
of a channel subject to input constraint. The capacity result
is stated in terms of the reduced channel as defined in the
previous section. Let Wbe a DMC channel with input cost
function φ(x)and input constraint Aas specified in (5). Define
CDI (W)= max
pX:E{φ(X)}≤AH(X),(25)
for X∼pX.
Theorem 4: The DI capacity of a DMC Wunder input
constraint is given by
CDI(W,L)=CDI(Wr),for L(n, R)=2
nR,(26)
where Wrdenotes the reduced channel (see Definition 3).
Hence, the DI capacity in the double exponential scale is zero.
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6IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 68, NO. 1, JANUARY 2022
We prove the direct part in Subsection III-C and the
converse part in Subsection III-D. As can be seen in Sub-
section III-D, the strong converse property holds for the DI
capacity [5, see Def. 3.5.1]. Notice that we have characterized
the DI capacity of the DMC Win terms of its reduced version,
as specified in Lemma 3.
Corollary 5 (Also in [4], [43]): The DI capacity of a DMC
Wwithout constraints, i.e., with A=φmax,isgivenby
CDI (W,L)=lognrow(W),(27)
for L(n, R)=2
nR where nrow(W)is the number of distinct
rows of W.
The corollary above is an immediate consequence of The-
orem 4. Indeed, for A=φmax,wehave
CDI (Wr)= max
pX0∈P(Xr)H(X0)
=log|Xr|
=lognrow(W),(28)
where X0is a random variable, the support of which is in
the reduced input alphabet Xras defined in Definition 3. The
second equality holds since the maximal value of H(X)is
log |X|, and the last equality because the size of the reduced
input alphabet is |Xr|=nrow(W).
Remark 4: Ahlswede et al. [4], [43] stated the result in
Corollary 5 on the DI capacity of a DMC without constraints
(see Section IV. [4] and abstract of [43]), without providing
an explicit proof. A straightforward extension of the methods
in [48], using decoding territories, does not seem to yield the
desired result on the DI capacity. Thereafter, the proof has
remained an open problem.
Remark 5: An alternative expression for the DI capacity is
as follows,
CDI(W,L)= max
pX0∈P(Xr): E{φ(X0)}≤AH(X0),(29)
where X0is as in (28). As explained in Remark 3, one may
assume without loss of generality that the channel has distinct
rows. Under this assumption, the DI capacity formula reduces
to the formula in (1), i.e.,
CDI(W,L)= max
pX:E{φ(X)}≤AH(X),(30)
for a channel Wwith distinct rows and L(n, R)=2
nR.
To illustrate our results, we give the following example.
Example 1: Consider the binary symmetric channel (BSC),
Y=X+Zmod 2,(31)
where X=Y={0,1},Z∼Bernoulli(ε), with crossover
probability 0≤ε≤1
2. Suppose that the channel is subject to
a Hamming weight input constraint,
1
n
n

t=1
xt≤A, (32)
with φ(x)=x. Observe that for ε=1
2, the rows of the channel
matrix are identical. Hence, the reduced input alphabet consists
of one letter, and the DI capacity is zero (see Definition 3).
Fig. 2. The deterministic identification (DI) capacity of the BSC as a function
of the input constraint A. The dashed red line indicates the binary entropy
function, which is maximized in (33). The solid blue line indicates the DI
capacity.
Now, suppose that ε< 1
2. Then the rows of the channel
matrix W=1−εε
ε1−εare distinct, hence Wr=W.
By Theorem 4, the DI capacity is given by
CDI(W,L)=CDI(W)= max
0≤p≤AH2(p),(33)
since the channel input is binary, where H2(p)=−(1 −
p) log(1 −p)−plog(p)is the binary entropy function. There-
fore, the DI capacity of the BSC with Hamming weight
constraint is
CDI (W,L)=H2(A)if A<1
2
1if A≥1
2
,for L(n, R)=2
nR.
(34)
(See Figure 2). To show the direct part, set
X∼Bernoulli(A)if A< 1
2and X∼Bernoulli1
2,
otherwise. The converse part follows as the binary entropy
function H2(p)is strictly increasing on 0≤p≤1
2, attaining
its maximum value H2(1
2)=1, and strictly decreasing on
1
2<p≤1(see Figure 2). The geometric interpretation is
that the binary Hamming ball of radius np can be covered
with codewords. As the volume of the Hamming ball
is approximately 2nH2(p), one can achieve rates that are
arbitrarily close to H2(p). Without an input constraint, i.e.,
for A=1, we recover the result of JáJá [44],
CDI(W,L)=1.(35)
This example demonstrates that the DI capacity is discon-
tinuous in the channel statistics, as CDI (W,L)=1for ε< 1
2
and CDI(W,L)=0for ε=1
2.
C. Achievability Proof
Consider a DMC W. By Lemma 3 we can assume without
loss of generality that the channel matrix W:X→Yhas
distinct row vectors. To prove achievability of the DI capacity,
we combine methods and ideas from the work of JáJá [44] as
well as techniques by Ahlswede [48]. The analysis for the type
II error is based on ideas that are analogous to the combinatoric
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SALARISEDDIGH et al.: DETERMINISTIC IDENTIFICATION OVER CHANNELS WITH POWER CONSTRAINTS 7
analysis of Hamming distances in [44]. The codebook con-
struction is similar to that of Ahlswede’s coding scheme [48],
yet the decoder is significantly different. Nonetheless, the type-
class intersection lemma and the message-set analysis in [48]
are useful in our analysis for the type II error.
We extensively use the method of types [70, Ch. 2]. Here
a brief review of the definitions for type classes and δ-typical
sets is given. The type ˆ
Pxnof a given sequence xnis defined
as the empirical distribution ˆ
Pxn(a)=N(a|xn)/n for a∈X,
where N(a|xn)is the number of occurrences of the symbol
a∈Xin the sequence xn. The space of all types over X
of sequences of length nis denoted by Pn(X).Theδ-typical
set Tδ(pX)is defined as the set of sequences xn∈Xnsuch
that for every a∈X:|ˆ
Pxn(a)−pX(a)|≤δif pX(a)>0,
and ˆ
Pxn(a)=0if pX(a)=0. A type class is denoted by
T(ˆ
P)={xn:ˆ
Pxn=ˆ
P}. Similarly, a joint type is denoted
by ˆ
Pxn,yn(a, b)=N(a, b|xn,y
n)/n for (a, b)∈X×Y,where
N(a, b|xn,y
n)is the number of occurrences of the symbol pair
(a, b)in the sequence (xi,y
i)n
i=1, and as a conditional type by
ˆ
Pyn|xn(b|a)=N(a, b|xn,y
n)/N (a|xn). The conditional δ-
typical set Tδ(pY|X|xn)is defined as the set of sequences yn∈
Ynsuch that for every b∈Y:|ˆ
Pyn|xn(b|a)−pY|X(b|a)|≤δ
if pX,Y (a, b)>0,andpX,Y (a, b)=0if pX(a)=0.
1) The Codebook: First, we show that there exists a code
such that the codewords are separated by a distance of nε.Let
pX(x)be an input distribution on X, such that
E{φ(X)}=
x∈X
pX(x)φ(x)≤A−ε(δ)(36)
for X∼pX(x),whereε(δ)→0as δ→0. We may assume
without loss of generality that pXis a type, due to the entropy
continuity lemma [70, Lem. 2.7].
Lemma 6: Let R<H(X). Then, for sufficiently small
ε∈(0,1) and sufficiently large n, there exists a codebook
U∗={vi,i∈M}, which consists of |M| sequences in Xn,
such that the following hold:
1) All the codewords belong to the type class T(pX),
namely
vi∈T(pX)for all i∈M.(37)
2) The codewords are distanced by nε, i.e.,
dH(vi,v
j)≥nε for all i=j. (38)
3) The codebook size is at least 1
2·2nR,thatis,
|M| ≥ 2n(R−1
n).
Proof of Lemma 6: Denote
M2nR.(39)
Let U1,...,U
Mbe independent random sequences, each
uniformly distributed over the type class of pX,i.e.,
Pr (Ui=xn)=1
|T (pX)|xn∈T(pX),
0xn/∈T(pX).(40)
Next, define a new collection of sequences V1,...,V
Mas
follows,
Vi=Uiif dH(Ui,U
j)≥nε ∀i=j,
∅otherwise,(41)
where dH(·,·)denotes the Hamming distance, and ∅represents
an idle sequence of no interest. The assignment Vi=∅
is interpreted as “dropping the ith word Ui.” Consider the
following message set,

M={i:Vi=∅,i∈[[ M]] },(42)
corresponding to words that were not dropped, where we use
the notation 
Mto indicate that the set is random.
We show that even though we removed words from the
original collection {Ui}i∈[[ M]] (of size M), the rate decrease
can be made negligible. Following the lines of [48], we derive
an upper-bound on Pr(|
M| ≤ 1
2M)where 
Mdefined in (42)
is the operational message set. To this end, we will use the
following concentration lemma,
Lemma 7 (Also in [48]): Let A1,...,A
Kbe a sequence of
discrete random variables. Then,
Pr 1
K
K

i=1
Ai≥c≤
2−cK
K

i=1
max
ai−1
E2AiAi−1=ai−1.(43)
Now, define an indicator for dropping the ith word by
ˆ
Vi=1Vi=∅,
0Vi=∅,(44)
and notice the equivalence between the following events,

M≤1
2M=M

i=1
ˆ
Vi>1
2M.(45)
Observe that ˆ
Vi=1, if and only if Uiis inside an ε-
sphere of some other Uj. Namely, ˆ
Vi=1iff Ui∈
j=iSε(Uj).
The selection of codewords can be viewed as an iterative
procedure. Specifically, define
Ai=⎧
⎨
⎩
1Ui∈
j<iSε(Uj),
0otherwise,
(46)
Bi=⎧
⎨
⎩
1Ui∈
j>iSε(Uj),
0otherwise.
(47)
Now, since ˆ
Vi=1implies that either Ai=1or Bi=1,
it follows that the number of dropped messages is bounded by
M−
M=
M

i=1
ˆ
Vi
≤
M

i=1
Ai+
M

i=1
Bi.(48)
Consider the event that
M

i=1
ˆ
Vi>1
2M. (49)
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8IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 68, NO. 1, JANUARY 2022
If this holds, then the two sums in the right hand side of (48)
cannot be smaller than 1
4Mtogether, that is, either M
i=1 Ai≥
1
4M,orM
i=1 Bi≥1
4M, or both. Hence,
M

i=1
ˆ
Vi>1
2M
⊆M

i=1
Ai≥1
4M∪M

i=1
Bi≥1
4M,(50)
and by the union bound,
Pr M

i=1
ˆ
Vi>1
2M
≤Pr M

i=1
Ai≥1
4M+PrM

i=1
Bi≥1
4M
=2Pr
M

i=1
Ai≥1
4M,(51)
where the last line follows by symmetry, as the random
variables ¯
A=M
i=1 Aiand ¯
B=M
i=1 Bihave the same
probability distribution.
Next we apply Lemma 7,
Pr M

i=1
Ai≥1
4M
≤2−1
4M
M

i=1
max
ai−1
E2Ai|Ai−1=ai−1.(52)
Consider the conditional expectation above. Using the law
of total expectation, we can add conditioning on Ui−1as well,
i.e.
E2Ai|Ai−1=ai−1
=
ui−1
Pr(Ui−1=ui−1|Ai−1=ai−1)
·E(2Ai|Ui−1=ui−1,A
i−1=ai−1)
=
ui−1
Pr(Ui−1=ui−1|Ai−1=ai−1)
·E(2Ai|Ui−1=ui−1)
≤max
ui−1
E(2Ai|Ui−1=ui−1),(53)
where the second equality holds since Ai, is a deterministic
function of Ui−1(see (46)). Hence, by (52)-(53),
Pr M

i=1
Ai≥1
4M≤2−1
4M
M

i=1
max
ui−1
E2Ai|Ui−1=ui−1
=2
−1
4M
M

i=1
max
ui−11·Pr Ai=0|Ui−1=ui−1
+2·Pr Ai=1|Ui−1=ui−1
≤2−1
4M
M

i=1 1+2·max
ui−1Pr(Ai=1|Ui−1=ui−1).
(54)
We bound the probability term Pr(Ai=1|Ui−1=ui−1),
as follows. For a Hamming sphere of radius nε,
|Sε(xn)|≤n
nε·|X|
nε ≤2nθ(ε),(55)
for sufficiently large n,where
θ(ε)=H2(ε)+εlog |X|,(56)
tends to zero as ε→0. The first inequality holds by a
simple combinatoric argument. Namely, counting the number
of sequences with up to nε different entries compared to a
given xn,wehaven
nεoptional choices for the locations of
those entries, and |X| possible values for each of those entries.
The last inequality follows from Stirling’s approximation [81,
Example 11.1.3]. Hence,

M

j=1 Sε(uj)≤M2nθ(ε)
=2
n(R+θ(ε)),(57)
for every given collection of sequences, u1,...,u
M∈T(pX).
Consider a random sequence ¯
Xnthat is uniformly distributed
over the type class T(pX), and statistically independent of
U1,...,U
M. We use this external sequence as an auxiliary in
the derivation below. Then,
Pr Ai=1Ui−1=ui−1
=Pr⎛
⎝Ui∈
j<i Sε(uj)⎞
⎠
=Pr⎛
⎝¯
Xn∈
j<i Sε(uj)⎞
⎠
≤Pr ⎧
⎨
⎩¯
Xn∈
M

j=1 Sε(uj)⎫
⎬
⎭.(58)
The first equality follows from the definition of Aiin (46)
and because U1,...,U
Mare statistically independent. The
second equality holds because Uiand ¯
Xnare both uniformly
distributed over the type class of pX. The inequality follows
as Pr(F1)≤Pr(F1∪F
2)for every pair F1,F2of proba-
bilistic events. Since ¯
Xnis uniformly distributed over T(pX),
we have
Pr ⎧
⎨
⎩¯
Xn∈
M

j=1 Sε(uj)⎫
⎬
⎭
=
xn∈T (pX)∩

M
j=1 Sε(uj)
1
|T (pX)|
=1
|T (pX)|·T(pX)∩
M

j=1 Sε(uj)
≤2n(R+θ(ε))
|T (pX)|
≤(n+1)
|X| ·2n(R+θ(ε))
2nH(X)
≤2−n(H(X)−R−2θ(ε)),(59)
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SALARISEDDIGH et al.: DETERMINISTIC IDENTIFICATION OVER CHANNELS WITH POWER CONSTRAINTS 9
for sufficiently large n, where the first inequality follows
from (57), and the second is due to standard type class
properties [81, Th. 11.1.3]. The last expression tends to zero
as n→∞, provided that
R<H(X)−3θ(ε).(60)
Together with (58)-(59), this implies
Pr Ai=1Ui−1=ui−1≤2−nθ(ε).(61)
Now plugging (61) into (54) yields
Pr M

i=1
Ai≥1
4M≤2−1
4M1+2·2−nθ(ε)M
=2−1
4+23
4·2−nθ(ε)M,(62)
for sufficiently large n,wehave23
4·2−nθ(ε)≤2−5hence,
2−1
4+23
4·2−nθ(ε)≤2−1
4+2
−5
=0.8721
<1.(63)
Thus we have a double exponential bound
Pr 
M≤1
2M≤2−α1M
=2
−α12nR ,(64)
for some α1>0. We deduce that there exists at least one
codebook with the desired properties. This completes the proof
of Lemma 6.
We continue to the main part of the achievability proof. Let
U∗={vi,i∈M}be a codebook of size 2n(R−1
n)as in
Lemma 6. Consider the following DI coding scheme for W.
a) Encoding: Given a message i∈Mat the sender,
transmit xn=vi.
b) Decoding: Let δ>0, such that δ→0as ε→0.Let
j∈Mbe the message that the decoder wishes to identify.
To do so, the decoder checks whether the channel output yn
belongs to the corresponding decoding set Djor not, where
Dj={yn:(vj,y
n)∈T
δ(pXW)}.(65)
Namely, given the channel output yn∈Y
n,if(vj,y
n)∈
Tδ(pXW), then the decoder declares that the message jwas
sent. On the other hand, if (vj,y
n)/∈T
δ(pXW), it declares
that jwas not sent.
2) Error Analysis: First, consider the error of type I, i.e.,
the event that Yn/∈D
i.Foreveryi∈M, the probability
of identification error of type I, Pe,1(i)=Pr((vi,Yn)/∈
Tδ(pXW)) tends to zero by standard type class considerations
[82, Th. 1.2].
We move to the error of type II, i.e., when Yn∈D
j
for j=i. To bound the probability of error Pe,2(i, j),
we use the conditional type-class intersection lemma, due to
Ahlswede [48], as stated below.
Lemma 8 (See [48, Lem. I1]): Let W:X→Ybe a
channel matrix of a DMC Wwith distinct rows. Then, for
every xn,x
n∈T
δ(pX)with dH(xn,x
n)≥nε,
|Tδ(pY|X|xn)∩T
δ(pY|X|xn)|
|Tδ(pY|X|xn)|≤2−nL(ε),(66)
with pY|X≡W, for sufficiently large nand some positive
function L(ε)>0which is independent of n.
Now, for short notation, denote the conditional δ-typical set
in Yn,givenxn∈T(pX),by
G(xn)≡T
δ(W|xn)={yn:(xn,y
n)∈T
δ(pXW)}.(67)
Then, for every i=j,
Pe,2(i, j)=Pr(Dj|xn=vi)
=
yn∈G(vj)
Wn(yn|vi)
=
yn∈G(vj)∩G(vi)
Wn(yn|vi)
+
yn∈G(vj)∩(G(vi))c
Wn(yn|vi).(68)
Observe that the second sum in the last line is bounded by
the probability Pr(Yn/∈T
δ(W|vi)|xn=vi), which in turn is
bounded by 2−α1(δ)nas before, and tends to zero as well.
To bound the first sum in (68), we first consider the
cardinality of the set that the sum acts upon (the domain).
We note that since viand vjbelong to the type class T(pX)by
the first property of Lemma 6, it follows that they also belong
to the δ-typical set, i.e., vi,v
j∈T
δ(pX). Further, according to
the second property of Lemma 6, every pair of codewords vi
and vjsatisfy dH(vi,v
j)≥nε. Finally, having assumed that
the rows of Waredistinct,wehavebyLemma8,
|G(vj)∩G(vi)|≤2−nL(ε)|G(vj)|
≤2n[H(Y|X)−L(ε)],(69)
where X∼pX, as we explained below. The second inequality
in (69) holds since the size of the conditional type class
G(xn)=Tδ(W|xn)is bounded by 2nH (Y|X)[70, Lem. 2.5],
as the type of viand vjis pX. Furthermore, by standard type
class properties [82, Th. 1.2],
Wn(yn|vi)≤2−n[H(Y|X)−δlog |Y|].(70)
Now by Equation (69) and (70),

yn∈G(vj)∩G(vi)
Wn(yn|vi)≤2−n[L(ε)−δlog |Y|],(71)
which tends to zero as n→∞for sufficiently small δ>0,
such that δlog |Y| <L(ε). Thus, by (68) and (71), the
probability of type II error is bounded by
Pe,2(i, j)≤2−nα2(ε,δ),(72)
for sufficiently large n,whereα2(ε, δ)=min{α1(δ),
L(ε)−δlog |Y|}. The proof follows by taking the limits
n→∞,andε,δ→0.
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10 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 68, NO. 1, JANUARY 2022
D. Converse Proof
To prove the converse part, we will use the following
observation. Let R>0be an achievable rate. We will
assume to the contrary that there exist two different messages
i1and i2that are represented by the same codeword, i.e.,
ui1=ui2=xn, and show that this leads to error probabilities
such that
Pe,1(i1)+Pe,2(i2,i
1)=1.(73)
Hence the assumption is false. The number of messages 2nR
is thus bounded by the size of the subset of input sequences
that satisfy the input constraint φn(xn)≤A. Then we notice
that the average cost of a codeword depends only on its type,
and hence this subset is in fact a union of type classes. This
also implies that we have a strong converse for the DI capacity.
Consider a sequence of (2nR,n,λ
(n)
1,λ
(n)
2)codes
(U(n),D(n))such that λ(n)
1and λ(n)
2tend to zero as
n→∞.
Lemma 9: Consider a sequence of codes as described above.
Then, given a sufficiently large n, the codebook U(n)satisfies
the following property. There cannot be two distinct messages
that are represented by the same codeword, i.e.,
i1=i2⇒ui1=ui2,(74)
where i1,i
2∈[[ 2 nR]] .
Proof: Assume to the contrary that there exist two mes-
sages i1and i2,wherei1=i2, such that
ui1=ui2=xn,(75)
for some xn∈X
n.Since(U(n),D(n))form a
(2nR,n,λ
(n)
1,λ
(n)
2)code, we have
Pe,1(i1)=Wn(Dc
i1|xn)≤λ(n)
1
Pe,2(i2,i
1)=Wn(Di1|xn)≤λ(n)
2.(76)
This leads to a contradiction as
1=Wn(Dc
i1|xn)+Wn(Di1|xn)
=Pe,1(i1)+Pe,2(i2,i
1)
≤λ(n)
1+λ(n)
2.(77)
Hence, the assumption is false, and i1and i2cannot have
the same codeword.
By Lemma 9, each message has a distinct codeword. Hence,
the number of messages is bounded by the number of input
sequences that satisfy the input constraint. That is, the size of
the codebook is upper-bounded as follows:
2nR ≤xn:1
n
n

t=1
φ(xt)≤A.(78)
Notice that the input cost of a given sequence xndepends
only on the type of the sequence, since
1
n
n

t=1
φ(xt)=
a∈X
ˆ
Pxn(a)φ(a)
=Eφ(X),(79)
where the random variable Xis distributed according to the
type of xn, i.e., pX=ˆ
Pxn. Therefore, the subset on the right
hand side of (78) can be written as a union of type classes:
xn:1
n
n

t=1
φ(xt)≤A
=
pX∈Pn(X):
E{φ(X)}≤A
T(pX)
≤|P
n(X)|max
pX∈Pn(X):
E{φ(X)}≤A
|T (pX)|
≤|P
n(X)|·2nH(X)
≤2n(H(X)+αn)
≤2n(CDI (W)+αn),(80)
where αn→0as n→∞,wherePn(X)denotes the space
of all types over Xof sequences of length n. The second
inequality holds since the size of a type class T(pX)is
bounded by |T (pX)|≤2nH(X)[81, Th. 11.1.3]. The third
inequality holds since the number of types on Xis polynomial
in n[81, Th. 11.1.1]. Thus, by (78) and (80), the code rate is
bounded by R≤CDI (W)+αn, which completes the proof
of Theorem 4.
IV. FADING CHANNELS
In this section we consider Gaussian channels with either
fast fading or slow fading. We will see that the capacity
characterization is inherently different in the sense that for the
Gaussian channel, the code size scales L(n, R)=2
nlog(n)R=
nnR. We note that the scale of the DI capacity can be viewed
as a special case of a tetration function, as 2n=nn=
2nlog(n)[83], [84]. To prove this property, we establish lower
and upper bounds in this scale, both positive and finite. As a
consequence, it follows that the capacity is infinite in the
exponential scale L(n, R)=2
nR and zero in the double
exponential scale L(n, R)=2
2nR .
A. Fading Channels
Consider the Gaussian channel Gfast with fast fading, spec-
ified by the input-output relation
Y=G◦x+Z,(81)
where Gis a random sequence of fading coefficients and
Zis an additive white Gaussian process (see Figure 3).
Specifically, Gis a sequence of i.i.d. continuous random
variables ∼fGwith finite moments, while the noise sequence
Zis i.i.d. ∼N(0,σ
2
Z). It is assumed that the noise sequence
Zand the sequence of fading coefficients Gare statistically
independent, and that the values of the fading coefficients
belong to a bounded set G, either countable or uncountable.
The transmission power is limited to x2≤nA.
Similarly, the Gaussian channel Gslow with slow fading is
specified by the input-output relation
Yt=Gxt+Zt,(82)
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SALARISEDDIGH et al.: DETERMINISTIC IDENTIFICATION OVER CHANNELS WITH POWER CONSTRAINTS 11
Fig. 3. Deterministic identification over fading channels. For fast fading,
G=(Gt)∞
t=1 is a sequence of i.i.d. fading coefficients ∼fG.Forslow
fading, the fading sequence remains constant throughout the transmission
block, i.e., Gt=G.
where Gis a continuous random variable ∼fG(g). Suppose
that the values of Gbelong to a set G,andthatGhas finite
expectation and finite variance var(G)>0with additive
white Gaussian noise, i.e., where the noise sequence Zis i.i.d.
∼N(0,σ
2
Z). The transmission power is limited to x2≤nA.
B. Coding for Fading Channels
A code for the Gaussian channel with fast fading is defined
below.
Definition 4: An (L(n, R),n)DI code with channel side
information (CSI) at the decoder for a Gaussian channel Gfa st
under input constraint A, assuming L(n, R)is an integer,
is defined as a system (U,D)which consists of a codebook
U={ui}i∈[[ L(n,R)]],U⊂Rn, such that
ui2≤nA, for all i∈[[ L(n, R)]],(83)
and a collection of decoding regions D=
{Di,g}i∈[[ L(n,R)]],g∈Gnwith L(n,R)
i=1 Di,g⊂RnGiven
a message i∈[[ L(n, R)]], the encoder transmits ui.The
decoder’s aim is to answer the following question: Was a
desired message jsent or not? There are two types of errors
that may occur: Rejecting the true message, or accepting a
false message. Those are referred to as type I and type II
errors, respectively.
The error probabilities of the identification code (U,D)are
given by
Pe,1(i)=1−$Gn
fG(g)%$Di,g
fZ(y−g◦ui)dy&dg,
(84)
Pe,2(i, j)=$Gn
fG(g)%$Dj,g
fZ(y−g◦ui)dy&dg,(85)
with fZ(z)= 1
(2πσ2
Z)n/2e−z2/2σ2
Z(see Figure 3). An
(L(n, R),n,λ
1,λ
2)DI code further satisfies
Pe,1(i)≤λ1,(86)
Pe,2(i, j)≤λ2,(87)
for all i, j ∈[[ L(n, R)]], such that i=j.ArateR>0is called
achievable if for every λ1,λ
2>0and sufficiently large n,
there exists an (L(n, R),n,λ
1,λ
2)) DI code. The operational
DI capacity in the L-scale is defined as the supremum of
achievable rates, and will be denoted by CDI(Gfast ,L).
Coding for the Gaussian channel with slow fading is defined
as in the compound channel model, considering the worst-case
channel. Thus, the error is maximized over the set of values of
the fading coefficients. A code for the Gaussian channel with
slow fading is defined in a similar manner as in Definition 4.
However, the errors are defined with a supremum over the
values of the fading coefficient G∈G, namely,
Pe,1(i)=sup
g∈G '1−$Di,g n

t=1
fZ(yt−gui,t)dy(,
(88)
Pe,2(i, j)=sup
g∈G '$Dj,g n

t=1
fZ(yt−gui,t)dy(.(89)
The capacity of the Gaussian channel with slow fading is
denoted by CDI (Gslow,L).
C. Main Results–Fading Channels
We determine the coding scale for the DI capacity of
Gaussian channels with fading. Before we give our results,
we make the following observation. Recall that we use the
notation of L2≺L1for a coding scale L1that dominates
L2(see Definition 1). The following property readily follows
from the definition. Suppose that the capacity in a given scale
is positive yet finite, i.e., 0<CDI (Gfast ,L
0)<∞for a given
L0. Then, for every L−≺L0,
CDI(Gfast ,L
−)=∞,(90)
and for every L+L0,
CDI(Gfast ,L
+)=0.(91)
Our DI capacity theorem for the Gaussian channel with fast
fading is stated below.
Theorem 10: Assume that the fading coefficients are pos-
itive and bounded away from zero, i.e., 0/∈cl(G).TheDI
capacity of the Gaussian channel Gfas t with fast fading in the
2nlog(n)-scale, i.e., for L(n, R)=2
(nlog n)Ris bounded by
1
4≤CDI(Gfast ,L)≤1.(92)
Hence, the DI capacity is infinite in the exponential scale
and zero in the double exponential scale, i.e.,
CDI(Gfast ,L)=∞for L(n, R)=2
nR,
0for L(n, R)=2
2nR .(93)
The proofs for the lower and upper bounds in the first
part of Theorem 10 are given in Subsection IV-D and
Subsection IV-E, respectively. The second part of the theo-
rem is a direct consequence of the observation given at the
beginning of this subsection (see (90)-(91)).
Remark 6: Observe that Theorem 10 assumes that the
fading coefficients are positive and bounded away from zero,
i.e., 0/∈cl(G). In practice, however, communication with
fading may involve small gain that can be close to zero. For
instance, if the fading distribution fGis Gaussian, then the
probability Pr(|Gt|<ε)is positive for arbitrarily small ε>0.
Hence, Gaussian fading does not meet the assumption in our
theorem. Unfortunately, the case where the fading coefficients
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12 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 68, NO. 1, JANUARY 2022
can be arbitrarily close (or equal) to zero remains unsolved.
We give a rough explanation of how we use the assumption
in the analysis. In the achievability proof in Subsection IV-D,
we assume that there exists γ>0such that Gt>γfor all
twith probability 1. The codebook is constructed such that
the codewords are distanced from each other by √nεn. Then,
in the analysis for the type II error, we consider an error event
of the form
Z2≤n(σ2
Z+2δn)−G◦(ui−uj)2,(94)
(see (120)). Given our assumption, we have
G◦(ui−uj)2≥γ2ui−uj2. By choosing εn=3δn
γ2,
the error event in (94) implies
Z2≤n(σ2
Z+2δn−γ2εn)=n(σ2
Z−δn),(95)
or equivalently,
1
n
i=1
Z2
i−σ2
Z≤−δn.(96)
As the random sequence Z2
iis i.i.d. with E(Z2
i)=σ2
Z,
we show that this probability tends to zero using large devia-
tions arguments. Further details are given in Subsection IV-D.
Our DI capacity theorem for the Gaussian channel with slow
fading is stated below.
Theorem 11: The DI capacity of the Gaussian channel Gslow
with slow fading in the 2nlog(n)-scale is bounded by
1
4≤CDI(Gslow,L)≤1if 0/∈cl(G)
CDI(Gslow,L)=0 if 0∈cl(G),
for L(n, R)=2
nlog(n)R.(97)
Hence, the DI capacity is infinite in the exponential scale,
if 0/∈cl(G), and zero in the double exponential scale, i.e.,
CDI(Gslow,L)=0if 0∈cl(G)
∞if 0/∈cl(G),
for L(n, R)=2
nR,(98)
CDI (Gslow,L)=0,for L(n, R)=2
2nR .(99)
The derivation of the result above is similar to that of
the proof for fast fading in Section IV-C. For completeness,
we give the proof of Theorem 11 in Appendix B.
D. Lower Bound (Achievability Proof for Theorem 10)
Consider the Gaussian channel Gfast with fast fading.
We show that the DI capacity is bounded by CDI (Gfast ,L)≥
1
4for L(n, R)=2
nlog(n)R. Achievability is established using
a dense packing arrangement and a simple distance-decoder.
A DI code for the Gaussian channel Gfast with fast fading is
constructed as follows. Consider the normalized input-output
relation,
¯
Y=G◦¯
x+¯
Z,(100)
where the noise sequence ¯
Zis i.i.d. ∼N0,σ2
Z
n,andan
input power constraint
¯
x≤√A, (101)
Fig. 4. Illustration of a sphere packing, where small spheres of radius
r0=√εncover a bigger sphere of radius r1=√A−√εn.Thesmall
spheres are disjoint from each other and have a non-empty intersection with
the big sphere. Some of the small spheres, marked in gray, are not entirely
contained within the bigger sphere, and yet they are considered to be a part
of the packing arrangement. As we assign a codeword to each small sphere
center, the norm of a codeword is bounded by √Aas required.
with ¯
x=1
√nx,¯
Z=1
√nZ,and ¯
Y=1
√nY. Assuming
0/∈cl(G), there exists a positive number γsuch that
|Gt|>γ, (102)
for all twith probability 1.
1) Codebook Construction: We use a packing arrangement
of non-overlapping hyper-spheres of radius √εnin a hyper-
sphere of radius (√A−√εn), with
εn=A
n1
2(1−b),(103)
where b>0is arbitrarily small.
Let Sdenote a sphere packing, i.e., an arrangement of L
non-overlapping spheres Sui(n, r0),i∈[[ L(n, R)]] that cover a
bigger sphere S0(n, r1), with r1>r
0. As opposed to standard
sphere packing coding techniques, the small spheres are not
necessarily entirely contained within the bigger sphere. That
is, we only require that the spheres are disjoint from each
other and have a non-empty intersection with S0(n, r1).See
illustration in Figure 4. The packing density Δn(S)is defined
as the fraction of the big sphere volume Vol (S0(n, r1)) that
is covered by the small spheres, i.e.
Δn(S)Vo l S0(n, r1)∩L
i=1 Sui(n, r0)
Vo l (S0(n, r1)) ,(104)
[85, see Ch. 1]. A sphere packing is called saturated if no
spheres can be added to the arrangement without overlap.
We use a packing argument that has a similar flavor as in the
Minkowski–Hlawka theorem in lattice theory [85]. We use the
property that there exists an arrangement L
i=1 Sui(n, √εn)of
non-overlapping spheres inside S0(n, √A)with a density of
Δn(S)≥2−n[78, Lem. 2.1]. Specifically, consider a satu-
rated packing arrangement of L(n, R)=2
nlog(n)Rspheres
of radius r0=√εncovering the big sphere S0(n, r1=
√A−√εn), i.e., such that no spheres can be added without
overlap. Then, for such an arrangement, there cannot be a
point in the big sphere S0(n, r1)with a distance of more than
2r0from all sphere centers. Otherwise, a new sphere could
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SALARISEDDIGH et al.: DETERMINISTIC IDENTIFICATION OVER CHANNELS WITH POWER CONSTRAINTS 13
be added. As a consequence, if we double the radius of each
sphere, the 2r0-radius spheres cover the whole sphere of radius
r1. In general, the volume of a hyper-sphere of radius ris
given by
Vo l (Sx(n, r)) = πn
2
Γ(n
2+1) ·rn,(105)
(see Eq. (16) in [85]). Hence, doubling the radius multiplies
the volume by 2n. Since the 2r0-radius spheres cover the
entire sphere of radius r1, it follows that the original r0-radius
packing has a density of at least 2−n, i.e.,
Δn(S)≥2−n.(106)
We assign a codeword to the center uiof each small
sphere. The codewords satisfy the input constraint as
ui≤r0+r1=√A. Since the small spheres have the
same volume, the total number of spheres is bounded from
below by
L=
Vo l L
i=1 Sui(n, r0)
Vo l (Su1(n, r0))
≥
Vo l S0(n, r1)∩L
i=1 Sui(n, r0)
Vo l (Su1(n, r0))
=Δn(S)·Vo l (S0(n, r1)))
Vo l (Su1(n, r0))
≥2−n·Vo l (S0(n, r1)))
Vo l (Su1(n, r0))
=2
−n·rn
1
rn
0
,(107)
where the second equality is due to (104), the inequality
that follows holds by (106), and the last equality follows
from (105). That is, the codebook size satisfies
L(n, R)=2
nlog(n)R
≥2−n·√A−√εn
√εnn
.(108)
Hence,
R≥1
log(n)log √A−√εn
√εn−1
log(n)
=1
log(n)log n1
4(1−b)−1−1
log(n)
≥1
log(n)log n1
4(1−b)−1−1
log(n)
=1
4(1 −b)−2
log(n),(109)
which tends to 1
4when n→∞and b→0, where the second
inequality holds, since log(t−1) ≥log(t)−1for t≥2.
2) Encoding: Given a message i∈[[ L(n, R)]],
transmit ¯
x=¯
ui.
3) Decoding: Let
δn=γ2εn
3=γ2A
3n1
2(1−b).(110)
To identify whether a message j∈[[ L(n, R)]] was sent,
given the sequence g, the decoder checks whether the channel
output ¯
ybelongs to the following decoding set,
Dj,g=¯
y∈Rn:¯
y−g◦¯
uj≤)σ2
Z+δn.(111)
4) Error Analysis: Consider the type I error, i.e., when the
transmitter sends ¯
ui,yet ¯
Y/∈D
i,G.Foreveryi∈[[ L(n, R)]],
the type I error probability is bounded by
Pe,1(i)=Pr*
*¯
Y−G◦¯
ui*
*2>σ
2
Z+δn¯
x=¯
ui
=Pr*
*¯
Z*
*2>σ
2
Z+δn
=Prn

t=1
¯
Zt
2>σ
2
Z+δn
≤Pr n

t=1
¯
Zt
2>σ
2
Z+δn
≤3σ4
Z
nδ2
n
=27σ4
Z
nbA2γ4
≤λ1,(112)
for sufficiently large nand arbitrarily small λ1>0,where
the second inequality follows by Chebyshev’s inequality, and
since the fourth moment of a Gaussian variable V∼N(0,σ
2
V)
is E{V4}=3σ4
V.
Next we address the type II error, i.e., when ¯
Y∈D
j,G
while the transmitter sent ¯
ui. Then, for every i, j ∈[[ L(n, R)]],
where i=j, the type II error probability is given by
Pe,2(i, j)=Pr*
*¯
Y−G◦¯
uj*
*2≤σ2
Z+δn¯
x=¯
ui
=Pr*
*G◦(¯
ui−¯
uj)+¯
Z*
*2≤σ2
Z+δn.
(113)
Observe that the square norm can be expressed as
*
*G◦(¯
ui−¯
uj)+¯
Z*
*2=G◦(¯
ui−¯
uj)2+*
*¯
Z*
*2
+2
n

t=1
Gt(¯ui,t −¯uj,t)¯
Zt.(114)
Then, define the event
E0=
n

t=1
Gt(¯ui,t −¯uj,t )¯
Zt>δn
2.(115)
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14 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 68, NO. 1, JANUARY 2022
By Chebyshev’s inequality, the probability of this event
vanishes,
Pr(E0)≤n
t=1(¯ui,t −¯uj,t)2E{G2
t}E{¯
Z2
t}
δn
22
=σ2
Z(σ2
G+μ2
G)n
t=1(¯ui,t −¯uj,t)2
nδn
22
=4σ2
Z(σ2
G+μ2
G)¯
ui−¯
uj2
nδ2
n
,(116)
where the first inequality holds since the sequences {¯
Zt}and
{Gt}are i.i.d. ∼N0,σ2
Z
nand ∼fGwith E{Gt}=μG
and E{G2
t}=σ2
G+μ2
G. By the triangle inequality,
¯
ui−¯
uj2≤(¯
ui+¯
uj)2
≤(√A+√A)2
=4A, (117)
hence
Pr(E0)≤16Aσ2
Z(σ2
G+μ2
G)
nδ2
n
=144σ2
Z(σ2
G+μ2
G)
γ4Anb
≤η1,(118)
for sufficiently large n,whereη1>0is arbitrarily small.
Furthermore, observe that given the complementary event Ec
0,
we have 2n
t=1 Gt(¯ui,t −¯uj,t )¯
Zt≥−δn, hence
*
*G◦(¯
ui−¯
uj)+¯
Z*
*2≤σ2
Z+δn⇒
G◦(¯
ui−¯
uj)2+*
*¯
Z*
*2≤σ2
Z+2δn,(119)
(see (114)). Therefore, applying the law of total probability,
we have
Pe,2(i, j)
(a)
=Pr
*
*G◦(¯
ui−¯
uj)+¯
Z*
*2≤σ2
Z+δnand

n

t=1
Gt(¯ui,t −¯uj,t)¯
Zt>δn
2
+Pr*
*G◦(¯
ui−¯
uj)+¯
Z*
*2≤σ2
Z+δnand

n

t=1
Gt(¯ui,t −¯uj,t)¯
Zt≤δn
2
(b)
≤Pr(E0)+Pr*
*G◦(¯
ui−¯
uj)+¯
Z*
*2≤σ2
Z+δnand
2
n

t=1
Gt(¯ui,t −¯uj,t)¯
Zt≥−δn
(c)
≤Pr(E0)+PrG◦(¯
ui−¯
uj)2+*
*¯
Z*
*2≤σ2
Z+2δn
(d)
≤Pr G◦(¯
ui−¯
uj)2+*
*¯
Z*
*2≤σ2
Z+2δn+η1,
(120)
where (a)follows by applying the law of total probability
to (113), (b)follows from (115), (c)holds by (119), and (d)
by (118).
Based on the codebook construction, each codeword is
surrounded by a sphere of radius √εn, which implies that
¯
ui−¯
uj≥√εn.(121)
Then, by (102),
G◦(¯
ui−¯
uj)2≥γ2¯
ui−¯
uj2≥γ2εn,(122)
where γis the minimal value in G. Hence, according to (120),
Pe,2(i, j)≤Pr *
*¯
Z*
*2≤σ2
Z+2δn−γ2εn+η1
≤Pr *
*¯
Z*
*2≤σ2
Z−δn+η1,(123)
where the last line holds, since 2δn−γ2εn=−δnby (110).
Therefore, by Chebyshev’s inequality,
Pe,2(i, j)≤Pr n

t=1
¯
Z2
t−σ2
Z≤−δn+η1
≤n
t=1 var(¯
Z2
t)
δ2
n
+η1
≤n
t=1 E{¯
Z4
t}
δ2
n
+η1
=
n·3σ2
Z
n2
δ2
n
+η1
=27σ4
Z
γ4A2nb+η1
≤λ2,(124)
for sufficiently large n,whereλ2is arbitrarily small.
We have thus shown that for every λ1,λ
2>0and
sufficiently large n, there exists a (2nlog(n)R,n,λ
1,λ
2)code.
As we take the limits of n→∞,andthenb→0,the
lower bound on the achievable rate tends to 1
4, by (109). This
completes the achievability proof for Theorem 10.
E. Upper Bound (Converse Proof for Theorem 10)
We show that the capacity is bounded by CDI (Gfas t ,L)≤1.
We note that in the converse proof, we do not normalize the
sequences. Suppose that Ris an achievable rate in the L-scale
for the Gaussian channel with fast fading. Consider a sequence
of (L(n, R),n,λ
(n)
1,λ
(n)
2)codes (U(n),D(n))such that λ(n)
1
and λ(n)
2tend to zero as n→∞.
We begin with the following lemma.
Lemma 12: Consider a sequence of codes as described
above. Let b>0be an arbitrarily small constant that does
not depend on n. Then there exists n0(b), such that for all
n>n
0(b), every pair of codewords in the codebook U(n)are
distanced by at least √nεn,i.e.,
ui1−ui2≥√nεn,(125a)
where
εn=A
n2(1+b),(125b)
for all i1,i
2∈[[ L(n, R)]] such that i1=i2.
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SALARISEDDIGH et al.: DETERMINISTIC IDENTIFICATION OVER CHANNELS WITH POWER CONSTRAINTS 15
Proof: Fix λ1and λ2.Letκ, θ, ζ > 0be arbitrarily small.
Assume to the contrary that there exist two messages i1and
i2,wherei1=i2, such that
ui1−ui2<√nεn=αn,(126)
where
αn=√A
n1
2(1+2b).(127)
Observe that
EG◦(ui1−ui2)2=
n

t=1
EG2
t(ui1,t −ui2,t)2
=EG2ui1−ui22,(128)
and consider the subsets
Ai1,i2={g∈G
n:g◦(ui1−ui2)>δ
n},(129)
Fi2=y∈Rn:y−g◦ui,2≤)n(σ2
Z+ζ),
(130)
where
δn=√A
n1
2(1+b).(131)
By Markov’s inequality, the probability that the fading
sequence Gbelongs to this set is bounded by
Pr (G∈A
i1,i2)=Pr(G◦(ui1−ui2)2>δ
2
n)
(a)
≤E{G2}ui1−ui22
δ2
n
(b)
≤E{G2}α2
n
δ2
n
=E{G2}
nb
≤κ, (132)
for sufficiently large nwhere (a)holds since the sequence
{Gt}n
t=1 is i.i.d. and (b)is due to (126).
Then, observe that
1−Pe,1(i1)
=$Gn
fG(g)%$Di1,g
fZ(y−g◦ui1)dy&dg
≤$Ac
i1,i2
fG(g)%$Di1,g
fZ(y−g◦ui1)dy&dg
+Pr(G∈A
i1,i2)
≤$Ac
i1,i2
fG(g)%$Di1,g
fZ(y−g◦ui1)dy&dg+κ. (133)
Hence,
1−κ−Pe,1(i1)
≤$Ac
i1,i2
fG(g)%$Di1,g
fZ(y−g◦ui1)dy&dg
=$Ac
i1,i2
fG(g)%$Di1,g∩Fi2
fZ(y−g◦ui1)dy
+$Di1,g∩Fc
i2
fZ(y−g◦ui1)dy&dg
≤$Ac
i1,i2
fG(g)%$Fi2
fZ(y−g◦ui1)dy
+$Fc
i2
fZ(y−g◦ui1)dy&dg.(134)
Consider the second integral, where y−g◦ui,2>
+n(σ2
Z+ζ)(see (130)). Then, by the triangle inequality
y−g◦ui,1≥y−g◦ui,2−g◦(ui,1−ui,2)
>)n(σ2
Z+ζ)−g◦(ui,1−ui,2)
≥)n(σ2
Z+ζ)−δn,(135)
for every g∈A
c
i1,i2(see (126)). For sufficiently large n,this
implies
y−g◦ui,1≥)n(σ2
Z+η),(136)
for η<ζ
2.Thatis,Fi2implies y−g◦ui,1≥
+n(σ2
Z+η).
We deduce that for every g∈A
c
i1,i2, the subset of output
sequences y∈Ynsuch that y−g◦ui,2>+n(σ2
Z+ζ)is
included within the subset of sequences with y−g◦ui,1>
+n(σ2
Z+η). Hence, the second integral in the right hand side
of (134) is bounded by
$

y:y−g◦ui,1>√n(σ2
Z+η)

fZ(y−g◦ui1)dy
=$

z:z>√n(σ2
Z+η)

fZ(z)dz
=Pr(Z2−nσ2
Z>nη)
≤3σ4
Z
nη2
≤κ, (137)
for large n, where the first inequality is due to Chebyshev’s
inequality, followed by the substitution of z≡y−g◦ui1.
Thus, by (134),
1−2κ−Pe,1(i1)
≤$Ac
i1,i2
fG(g)%$Di1,g∩Fi2
fZ(y−g◦ui1)dy&dg.(138)
Now, we can focus on g∈A
c
i1,i2and y∈F
i2, such that
y−g◦ui,2≤)n(σ2
Z+ζ).(139)
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16 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 68, NO. 1, JANUARY 2022
Observe that
fZ(y−g◦ui1)−fZ(y−g◦ui2)
=fZ(y−g◦ui1)%1−e−1
2σ2
Z

y−g◦ui22−y−g◦ui12

&.
(140)
By the triangle inequality,
y−g◦ui1≤y−g◦ui2+g◦(ui1−ui2).(141)
Taking the square of both sides, we have
y−g◦ui12
≤y−g◦ui22+g◦(ui2−ui1)2
+2y−g◦ui2·g◦(ui2−ui1)
≤y−g◦ui22+δ2
n+2δn)n(σ2
Z+ζ)
=y−g◦ui22+δ2
n+2+A(σ2
Z+ζ)
nb
2
,(142)
where the last inequality follows from the definition of Ai1,i2
in (129), and due to (139). Thus, for sufficiently large n,
y−g◦ui12−y−g◦ui22≤θ. (143)
Hence,
fZ(y−g◦ui1)−fZ(y−g◦ui2)
≤fZ(y−g◦ui1)(1 −e−1
2σ2
Z
θ)
≤κfZ(y−g◦ui1),(144)
for sufficiently small θ>0, such that 1−e−1
2σ2
Z
θ≤κ.Now
by (138), we get
λ1+λ2
≥Pe,1(i1)+Pe,2(i2,i
1)
≥1−2κ
−$Ac
i1,i2
fG(g)%$Di1,g∩Fi2
fZ(y−g◦ui1)dy&dg
+$Gn
fG(g)%$Di1,g
fZ(y−g◦ui2)dy&dg
≥1−2κ
−$Ac
i1,i2
fG(g)%$Di1,g∩Fi2
fZ(y−g◦ui1)dy&dg
+$Ac
i1,i2
fG(g)%$Di1,g∩Fi2
fZ(y−g◦ui2)dy&dg
≥1−2κ−$Ac
i1,i2
fG(g)%$Di1,g∩Fi2
(fZ(y−g◦ui1)
−fZ(y−g◦ui2))dy&dg.(145)
Hence, by (144),
λ1+λ2≥1−2κ
−κ$Ac
i1,i2
fG(g)%$Di1,g∩Fi2
fZ(y−g◦ui1)dy&dg
≥1−3κ, (146)
which leads to a contradiction for sufficiently small κsuch that
3κ<1−λ1−λ2. This completes the proof of Lemma 12.
By Lemma 12, we can define an arrangement of non-
overlapping spheres Sui(n, √nεn)of radius √nεncentered
at the codewords ui. Since the codewords all belong to a
sphere S0(n, √nA)of radius √nA centered at the origin,
it follows that the number of packed spheres, i.e., the number
of codewords 2nlog(n)R, is bounded by
2nlog(n)R≤Vo l (S0(n, √nA +√nεn))
Vo l (Sui(n, √nεn))
=√A+√εn
√εnn
.(147)
Thus,
R≤1
log nlog √A+√εn
√εn
=log 1+n1+b
log n
=log n1+b1+ 1
n1+b
log n
=(1 + b)logn+log1+ 1
n1+b
log n
=1+b+log 1+ 1
n1+b
log n,(148)
which tends to 1+bas n→∞. Therefore, R≤1+b.Now,
since b>0is arbitrarily small, an achievable rate must satisfy
R≤1. This completes the proof of Theorem 10.
V. S UMMARY AND DISCUSSION
We have established the deterministic identification (DI)
capacity of a DMC subject to an input constraint and have
developed lower and upper bounds on the DI capacity of
Gaussian channels with fast fading and slow fading in the
scale L(n, R)=2
nlog(n)R=nnR,wherenis the blocklength.
We have thus established that the super-exponential scale nnR
is the appropriate scale for the DI capacity of the Gaussian
channel. That is, the DI capacity can only be positive and finite
in this coding scale. This scale is sharply different from the
usual scales in the transmission and randomized-identification
settings, where the code size scales exponentially and double
exponentially, respectively. Different non-standard scales are
also observed in other communication models, such as covert
identification [86], [87], where the identification message is of
size 22√nR . For the DMC, the DI capacity formula is given
in terms of the entropy of the reduced channel input (see
Definition 3), and does not depend on the specific transition
probabilities corresponding to the reduced input alphabet.
We have mentioned molecular communication as a moti-
vating application of this study. Recently, there have been
significant advances in molecular communication for complex
nano-networks. The interconnection of nanothings with the
Internet is known as the Internet of NanoThings (IoNT)
and is the basis for various future healthcare and military
applications [88]. Furthermore, the concept of the Internet of
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SALARISEDDIGH et al.: DETERMINISTIC IDENTIFICATION OVER CHANNELS WITH POWER CONSTRAINTS 17
Bio-NanoThings (IoBNT) has been introduced in [89], where
nanothings are biological cells that are created using tools from
synthetic biology and nanotechnology. For the communication
between cells, molecular communication is well suited, since
the natural exchange of information between cells is already
based on this paradigm. Molecular communication in cells is
based on signal pathways (chains of chemical reactions) that
process information that is modulated into chemical charac-
teristics, such as molecule concentration. Identification is of
interest for these applications. In the case of Bio-NanoThings,
it is uncertain whether natural biological processes can be
controlled or reinforced by local randomness at this level.
Therefore, for the design of synthetic IoNT, or for the analysis
and utilization of IoBNT, identification with deterministic
encoding is more appropriate.
Gaussian channels have an entirely different and unusual
behaviour. We have considered deterministic identification for
Gaussian channels with fast fading and slow fading, where CSI
is available at the decoder. The DI capacity of the standard
memoryless Gaussian channel Y=X+Zis infinite in the
exponential scale, i.e.,
CDI(G,L)=∞,for L(n, R)=2
nR,(149)
as we have recently observed in [2] and [77]. However, for
a finite blocklength n, the number of codewords must be
finite. Thereby, the meaning of the infinite capacity result
is that the number of messages scales super-exponentially.
This raises the question: What is the true order of the code
size. In mathematical terms, what is the scale L(n, R)for
which the DI capacity is positive yet finite. To answer this
question, we have shown that the number of messages scales
as 2nlog(n)R. As a consequence, we have deduced that the
DI capacity of a Gaussian channel with fast fading is infinite
in the exponential scale, and zero in the double exponential
scale, regardless of the channel noise. For slow fading, the DI
capacity in the exponential scale is infinite, unless the fading
gain can be zero or arbitrarily close to zero (with positive
probability), in which case the DI capacity is zero. Note,
however, that this scale is comparably lower than the double
exponential scale of RI coding.
Next, we compare and discuss different results from the
literature on the DI capacity. For the double exponential scale,
or equivalently, when the rate is defined as R=1
nlog log (#
of messages), the DI capacity is
CDI (W,L)=0; for L(n, R)=2
2nR ,(150)
since the code size of DI codes scales only exponentially
in block length. On the other hand, as observed by Bracher
and Lapidoth [33], if one considers an average error criterion
instead of the maximal error, then the double exponential per-
formance of randomized-encoder codes can also be achieved
using deterministic codes.
By providing a detailed proof for the DI capacity theorem
with and without an input constraint, we have filled the
gap in the previous analysis [4], [43] as well. In particular,
in [43], Ahlswede and Cai asserted that the DI capacity for a
compound channel is given by
CDI(Wcompound,L)=max
pX
min
sH(ˆ
X(s)) ;
for L(n, R)=2
nR,(151)
where s∈Sis the channel state, and the map ˆ
X(s)is induced
from Xby a partition of the input alphabet to equivalent
classes as specified in [43, Sec. I.F]. This result immediately
yields Corollary 5, since the DMC is a special case of a
compound channel with a single state value. Indeed, taking
|S| =1 and considering the reduced channel Wr(see Defini-
tion 3), it can be readily shown that ˆ
X(s)=X. Nonetheless,
a significant part of the proof in [43] is missing. At the
beginning of Sec. VII in [43], the following claim is given: “It
was shown in [A’80] that for any channel ˜
V:X→Ywithout
two identical rows, any u1,u
2,ε> 0, sufficiently large nand
any U⊂X
nsuch that for all u, u∈U,d
H(u, u)>nε,
there exists a family of subsets of Yn,sayDu,u ∈U,such
that ˜
Vn(Du|u)>1−u1and ˜
Vn(Du|u)<u
2for all u=u,
where dHis the Hamming distance.”, where [A’80] refers to
a paper by Ahlswede [48] on the arbitrarily varying channel,
and does not include identification.
Alternatively, one may consider the ε-capacity for a fixed
0<ε<1.ArateRis called ε-achievable in an
L-scale if there exists an (L(n, R),n,ε,ε)code for sufficiently
large n(see Definition 2). The DI ε-capacity Cε
DI(W,L)is
then defined as the supremum of ε-achievable rates. As the
RI capacity in the double exponential scale has a strong
converse [4], [80], [90], for L(n, R)=Ldouble(n, R)and
0<ε<1
2,
Cε
RI (W,L
double)=CRI (W,L
double)=max
pX
I(X;Y).(152)
Based on Subsection III-D, a strong converse holds for the
DI capacity as well, hence
Cε
DI(W,L
double)=CDI (W,L
double)=0.(153)
On the other hand, for ε≥1
2
Cε
DI(W,L
double)=Cε
RI (W,L
double)=∞.(154)
To understand (154), suppose ε> 1
2, and consider an
arbitrary set of codewords with a stochastic decoder that makes
a decision for the identification hypothesis by flipping a fair
coin [4]. Both error probabilities of type I and II equal 1
2,
and are thus smaller than ε. Similarly, for the Gaussian
channel, [30], [39],
Cε
RI (G,L
double)=CRI (G,L
double)= 1
2log 1+ A
σ2
Z,
for 0<ε<1
2,(155)
Cε
DI(G,L
double)=Cε
RI (G,L
double)=∞,for ε≥1
2.
(156)
Based on our result in Theorem 10, with Gt≡1for all t,
we have that
1
4≤CDI(G,L
∗)≤Cε
DI(G,L
∗)≤1,(157)
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18 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 68, NO. 1, JANUARY 2022
with
L∗(n, R)=2
nlog(n)R=nnR.(158)
Hence, in the double exponential scale,
Cε
DI(G,L
double)=CDI (G,L
double)=0.(159)
APPENDIX A
PROOF OF LEMMA 3
Let Wbe a given DMC, with a stochastic matrix W:
X→Yand its reduced version Wr:Xr→Yas defined in
Definition 3. Observe that the capacity of the original channel
is lower bounded by that of the reduced channel, i.e.,
CDI (W,L)≥CDI(Wr,L),(160)
since every code for Wrcan also be used for W. Hence,
it remains to be shown that
CDI (Wr,L)≥CDI(W,L).(161)
Assume without loss of generality that the input alphabet
of the original channel Wis given by X={1,2,···,|X|}.
Let σ:X→X
rdenote the projection of the input alphabet
onto the equivalent classes,
σ[x]=z()iff x∈X().(162)
Now let (U,D)be an (L(n, R),n,λ
1,λ
2)code for W.Then
the type I probability of error can be expressed as
PW
e,1(i)= 
yn/∈Di
Wn(yn|ui)= 
yn/∈Di
n

t=1
W(y(t)|ui(t)),
(163)
where we use the notation yn=y(t)n
t=1 and ui=
ui(t)n
t=1.Nextwedefineacode(,
U,D)for the channel Wr,
where the codebook consists of the following codewords,
˜ui=σ[ui(t)]n
t=1.(164)
Now recall that we have defined the equivalence classes
such that input letters in the same equivalence class correspond
to identical rows in the channel matrix W(see Definition 3).
Thus, by definition,
Wr(y|σ[x]) = W(y|x),(165)
for all x∈Xand y∈Y. Hence, the error probability of type
I for the reduced channel Wrsatisfies
PWr
e,1(i)(a)
=
yn/∈Di
Wr(yn|˜ui)
(b)
=
yn/∈Di
n

t=1
Wry(t)|σ[ui(t)]
(c)
=
yn/∈Di
n

t=1
W(y(t)|ui(t))
(d)
=
yn/∈Di
Wn(yn|ui)
(e)
=PW
e,1(i),(166)
for all i,where(a)and (e)are due to (9); (b)and (d)hold
since the channel is memoryless, and (c)follows from (165).
By the same considerations, we also have PWr
e,2(i, j)=
PW
e,2(i, j)for all j=i. That is, the error probabilities of
the code (,
U,D)are the same as those of the original code
for W. Therefore, the code constructed above for Wris
also an (L(n, R),n,λ
1,λ
2)code, and the proof of Lemma 3
follows.
APPENDIX B
PROOF OF THEOREM 11
Consider the Gaussian channel Gslow with slow fading.
When 0∈cl(G), it immediately follows that the DI capacity is
zero. To see this, observe that if 0∈cl(G), then by (88)-(89),
Pe,1(i)+Pe,2(j, i)
=sup
g∈G '1−$Di,g n

t=1
fZ(yt−gui,t)dy(
+sup
g∈G '$Dj,g n

t=1
fZ(yt−gui,t)dy(
≥'1−$Di,g n

t=1
fZ(yt−gui,t)dy(g=0
+'$Di,g n

t=1
fZ(yt−guj,t)dy(g=0
=1.(167)
Hence, in the sequel we suppose that 0/∈cl(G).
A. Lower Bound (Achievability Proof)
We show here that the DI capacity of the Gaussian channel
with slow fading can be achieved using a dense packing
arrangement and a simple distance-decoder. A DI code for
the Gaussian channel Gslow with slow fading is constructed as
follows. Since the decoder can normalize the output symbols
by 1
√n, we have an equivalent input-output relation,
¯
Yt=G¯xt+¯
Zt,(168)
where G∼fG, and the noise sequence ¯
Zis i.i.d.
∼N0,σ2
Z
n, under an input power constraint
¯
x≤√A, (169)
with ¯
x=1
√nx,¯
Z=1
√nZ,and ¯
Y=1
√nY.
1) Codebook Construction: As in our achievability proof
for the fast fading setting (see Subsection IV-D.1), we use
a packing arrangement of non-overlapping hyper-spheres of
radius √εnover a hyper-sphere of radius (√A−√εn), with
εn=A
n1
2(1−b).(170)
As observed in Subsection IV-D, there exists an arrangement
2nlog(n)R

i=1 Sui(n, √εn),
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SALARISEDDIGH et al.: DETERMINISTIC IDENTIFICATION OVER CHANNELS WITH POWER CONSTRAINTS 19
over S0(n, √A−√εn)with a density of Δn≥2−n[78,
Lem. 2.1]. We assign a codeword to the center of each small
sphere ui. Since the small spheres have the same volume, the
total number of spheres, i.e., the codebook size, satisfies
2nlog(n)R=
Vo l 2nlog(n)R
i=1 Sui(n, √εn)
Vo l (Su1(n, √εn))
≥2−n·Vo l (S0(n, √A−√εn))
Vo l (Su1(n, √εn))
=2
−n·√A−√εn
√εnn
,(171)
in a similar manner as in Subsection IV-D.1, hence,
R≥1
4(1 −b)−2
log(n),(172)
which tends to 1
4when n→∞and b→0.
2) Encoding: Given a message i∈[[ L(n, R)]],transmit
¯
x=¯
ui.
3) Decoding: Let
δn=γ2εn
3=Aγ2
3n1
2(1−b).(173)
To identify whether a message j∈[[ L(n, R)]] was sent,
given the fading coefficient g, the decoder checks whether the
channel output ¯
ybelongs to the following decoding set,
Dj,g =¯
y∈Rn:
n

t=1
(¯yt−g¯uj,t)2≤)σ2
Z+δn.
(174)
4) Error Analysis: Consider the type I error, i.e., when the
transmitter sends ¯
ui,yet ¯
Y/∈D
i,G.Foreveryi∈[[ L(n, R)]],
the type I error probability is bounded by
Pe,1(i)=sup
g∈G
Pr n

t=1
(¯
Yt−G¯ui,t)2>σ
2
Z+δn¯
x=(¯ui,t)n
t=1,G=g
=Prn

t=1
¯
Zt
2−σ2
Z>δ
n
≤3σ4
Z
nδ2
n
=27σ4
Z
A2γ4nb
≤λ1,(175)
for sufficiently large nand arbitrarily small λ1>0,where
the second equality holds since Gand ¯
Zare statistically
independent, and the first inequality holds by Chebyshev’s
inequality and since the fourth moment of a Gaussian variable
V∼N(0,σ
2
V)is E{V4}=3σ4
V.
Next we address the type II error, i.e., when ¯
Y∈D
j,G
while the transmitter sent ¯
ui. Then, for every i, j ∈[[ L(n, R)]],
where i=j, the type II error probability is given by
Pe,2(i, j)=sup
g∈G -Pe,2i, j g.,(176)
where we have defined
Pe,2i, j g
≡Pr n

t=1
(¯
Yt−G¯uj,t)2≤σ2
Z+δn¯
x=(¯ui,t )n
t=1,G =g
=Prn

t=1
(g(¯ui,t −¯uj,t)+ ¯
Zt)2≤σ2
Z+δn(177)
for g∈G, as the fading coefficient Gand the noise vector
¯
Zare statistically independent. Now we bound the probability
within the square brackets.
We divide into two cases. First, consider g∈Gsuch
that g(¯
ui−¯
uj)>2+σ2
Z+δn. Therefore, by the reverse
triangle inequality, a−b≥|a−b|,wehave
/
0
0
1
n

t=1 (g(¯ui,t −¯uj,t)) + ¯
Zt2≥g(¯
ui−¯
uj)−*
*¯
Z*
*
≥2)σ2
Z+δn−*
*¯
Z*
*.
(178)
Hence, for every gsuch that g(¯
ui−¯
uj)>2+σ2
Z+δn,
we can bound the type II error probability by
Pe,2i, j g≤Pr *
*¯
Z*
*≥)σ2
Z+δn
=Prn

t=1
¯
Zt
2>σ
2
Z+δn
≤3σ4
Z
nδ2
n
=27σ4
Z
nbA2γ4
≤λ2,(179)
for sufficiently large nand arbitrarily small λ2>0,wherethe
second inequality follows by Chebyshev inequality.
Now, we turn to the second case, i.e., when
g(¯
ui−¯
uj)≤2)σ2
Z+δn.(180)
Observe that for every given g∈G,
n

t=1
(g(¯ui,t −¯uj,t )+ ¯
Zt)2
=
n

t=1
g2(¯ui,t −¯uj,t )2+
n

t=1
¯
Z2
t+2
n

t=1
g(¯ui,t −¯uj,t )Zt.
(181)
Then define the event
E0(g)=
n

t=1
g(¯ui,t −¯uj,t)¯
Zt>δn
2.(182)
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20 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 68, NO. 1, JANUARY 2022
By Chebyshev’s inequality, the probability of this event
vanishes,
Pr (E0(g)) ≤g2n
t=1(¯ui,t −¯uj,t)2E{¯
Z2
t}
δn
22
=4σ2
Zg(¯
ui−¯
uj)2
nδ2
n
≤16σ2
Zσ2
Z+δn
nδ2
n
≤τ0,(183)
for sufficiently large nand arbitrarily small τ0>0,where
the first inequality holds since the sequence {¯
Zt}is i.i.d.
∼N0,σ2
Z
n, and the second inequality follows from (180).
Furthermore, observe that given the complementary event
Ec
0(g),wehave2n
t=1 g(¯ui,t −¯uj,t)¯
Zt≥−δn, hence
n

t=1
(g(¯ui,t −¯uj,t )+ ¯
Zt)2≤σ2
Z+δn⇒
n

t=1
g2(¯ui,t −¯uj,t)2+
n

t=1
¯
Z2
t≤σ2
Z+2δn,(184)
(see (181)). Therefore, applying the law of total probability,
we have
Pe,2i, j g
=Prn

t=1
(g(¯ui,t −¯uj,t)+ ¯
Zt)2≤σ2
Z+δnand

n

t=1
g(¯ui,t −¯uj,t )¯
Zt>δn
2
+Prn

t=1
(g(¯ui,t −¯uj,t )+ ¯
Zt)2≤σ2
Z+δnand

n

t=1
g(¯ui,t −¯uj,t )¯
Zt≤δn
2
(a)
≤Pr(E0(g))
+Prn

t=1
(g(¯ui,t −¯uj,t )+ ¯
Zt)2≤σ2
Z+δnand
2
n

t=1
G(¯ui,t −¯uj,t)¯
Zt≥−δn
(b)
≤Pr(E0(g))
+Prn

t=1
g2(¯ui,t −¯uj,t)2+
n

t=1
¯
Z2
t≤σ2
Z+2δn
(c)
≤Pr g2¯
ui−¯
uj2+*
*¯
Z*
*2≤σ2
Z+2δn+τ0,(185)
where (a)follows from (182), (b)holds by (184), and (c)
by (183). Based on the codebook construction, each codeword
is surrounded by a sphere of radius √εn, which implies that
¯
ui−¯
uj≥√εn.(186)
Thus, we have
g2¯
ui−¯
uj2≥γ2εn,(187)
where γis the minimal value in G. Hence, according to (185),
Pe,2i, j g≤Pr *
*¯
Z*
*2≤σ2
Z+2δn−γ2εn+τ0
=Pr*
*¯
Z*
*2−σ2
Z≤−δn+τ0,(188)
(see (173)). Therefore, by Chebyshev’s inequality,
Pe,2i, j g≤Pr n

t=1
¯
Z2
t−σ2
Z≤−δn+τ0
≤n
t=1 var(¯
Z2
t)
δ2
n
+τ0
≤n
t=1 E{¯
Z4
t}
δ2
n
+τ0
=
n·3σ2
Z
n2
δ2
n
+τ0
=3σ4
Z
nδ2
n
+τ0
=27σ4
Z
nbA2γ4+τ0
≤λ2,(189)
for sufficiently large n. Based on (179) and (189), we have
Pe,2i, j g≤λ2for all g∈G. Hence, the type II error
probability satisfies Pe,2(i, j )≤λ2(see (176)).
We have thus shown that for every λ1,λ
2>0and
sufficiently large n, there exists a (2nlog(n)R,n,λ
1,λ
2)code.
As we take the limits of n→∞,andthenb→0,the
lower bound on the achievable rate tends to 1
4, by (172). This
completes the achievability proof for Theorem 11.
B. Upper Bound (Converse Proof)
Suppose that Ris an achievable rate in the L-scale for the
Gaussian channel with slow fading. Consider a sequence of
(L(n, R),n,λ
(n)
1,λ
(n)
2)codes (U(n),D(n)), such that λ(n)
1and
λ(n)
2tend to zero as n→∞. We begin with the following
lemma.
Lemma 13: Consider a sequence of codes as described
above. Let b>0be an arbitrarily small constant that does
not depend on n. Then there exists n0(b), such that for all
n>n
0(b), every pair of codewords in the codebook U(n)are
distanced by at least √nεn,i.e.,
ui1−ui2≥√nεn,(190a)
where
εn=A
n2(1+b),(190b)
for all i1,i
2∈[[ L(n, R)]], such that i1=i2.
Proof: Fix λ1and λ2.Letκ, θ, ζ > 0be arbitrarily small.
Assume to the contrary that there exist two messages i1and
i2,wherei1=i2, such that
ui1−ui2<√nεn=αn,(191)
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SALARISEDDIGH et al.: DETERMINISTIC IDENTIFICATION OVER CHANNELS WITH POWER CONSTRAINTS 21
where
αn≡√A
n1
2(1+2b).(192)
Observe that for every g∈G,
$Di1,g n

t=1
fZ(yt−gui1,t)dy
=$y∈Di1,g :
n

t=1
(yt−gui2,t)2≤n(σ2
Z+ζ)n

t=1
fZ(yt−gui1,t)dy
+$y∈Di1,g :
n

t=1
(yt−gui2,t)2>n(σ2
Z+ζ)n

t=1
fZ(yt−gui1,t)dy
≤$n

t=1
(yt−gui2,t)2≤n(σ2
Z+ζ)n

t=1
fZ(yt−gui1,t)dy
+$n

t=1
(yt−gui2,t)2>n(σ2
Z+ζ)n

t=1
fZ(yt−gui1,t)dy.
(193)
Consider the second integral, for which y−g◦ui,2>
+n(σ2
Z+ζ), where we denote
g≡(g,g,...,g).(194)
Then, by the triangle inequality,
y−g◦ui,1≥y−g◦ui,2−g◦(ui,1−ui,2)
=y−g◦ui,2−gui,1−ui,2
>)n(σ2
Z+ζ)−gui,1−ui,2
≥)n(σ2
Z+ζ)−gαn.(195)
This implies
y−g◦ui,1≥)n(σ2
Z+η),(196)
for η<ζ
2.Thatis,y−g◦ui,2≥+n(σ2
Z+ζ)implies
y−g◦ui,1≥+n(σ2
Z+η).
We deduce that the second integral in the right hand side
of (193) is bounded by
$

y:y−g◦ui,1>√n(σ2
Z+η)

fZ(y−g◦ui1)dy
=$

z:z>√n(σ2
Z+η)

fZ(z)dz
=Pr(Z2−nσ2
Z>nη)
≤3σ4
Z
nη2
≤κ, (197)
for large n, where the first inequality is due to Chebyshev’s
inequality, followed by the substitution of z≡y−g◦ui1.
Thus, by (193),
$Di1,g n

t=1
fZ(yt−gui1,t)dy
≤$y∈Di1,g:
y−g◦ui,2≤√n(σ2
Z+ζ)fZ(y−g◦ui1)dy+κ. (198)
Now, we can focus on y∈Rnsuch that
y−g◦ui,2≤)n(σ2
Z+ζ).(199)
Then,
fZ(y−g◦ui1)−fZ(y−g◦ui2)
=fZ(y−g◦ui1)%1−e−1
2σ2
Z

y−g◦ui22−y−g◦ui12

&.
(200)
By the triangle inequality,
y−g◦ui1≤y−g◦ui2+gui1−ui2.(201)
Taking the square of both sides, we have
y−g◦ui12
≤y−g◦ui22+g2ui2−ui12
+2y−g◦ui2·gui2−ui1
≤y−g◦ui22+g2α2
n+2gαn)n(σ2
Z+ζ)
=y−g◦ui22+g2α2
n+2g+A(σ2
Z+ζ)
nb,(202)
where the second inequality follows from (191) and (199), and
the equality is due to (192). Thus, for sufficiently large n,
y−g◦ui12−y−g◦ui22≤θ. (203)
Hence,
fZ(y−g◦ui1)−fZ(y−g◦ui2)
≤fZ(y−g◦ui1)1−e−1
2σ2
Z
θ
≤κfZ(y−g◦ui1),(204)
for sufficiently small θ>0, such that 1−e−1
2σ2
Z
θ≤κ.Now
by (198), we get,
λ1+λ2
≥Pe,1(i1)+Pe,2(i2,i
1)
=sup
g∈G '1−$Di1,g n

t=1
fZ(yt−gui1,t)dy(
+sup
g∈G '$Di1,g n

t=1
fZ(yt−gui2,t)dy(
≥sup
g∈G 21−$Di1,g n

t=1
fZ(yt−gui1,t)dy
+$Di1,g n

t=1
fZ(yt−gui2,t)dy3
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22 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 68, NO. 1, JANUARY 2022
≥sup
g∈G %1−$y∈Di1,g:
y−g◦ui,2≤√n(σ2
Z+ζ)fZ(y−g◦ui1)dy−κ
+$Di1,g
fZ(y−g◦ui2)dy&
≥sup
g∈G %1−κ−$y∈Di1,g:
y−g◦ui,2≤√n(σ2
Z+ζ)fZ(y−g◦ui1)dy
+$y∈Di1,g:
y−g◦ui,2≤√n(σ2
Z+ζ)fZ(y−g◦ui2)dy&
=sup
g∈G %1−κ−$y∈Di1,g:
y−g◦ui,2≤√n(σ2
Z+ζ)(fZ(y−g◦ui1)
−fZ(y−g◦ui2))dy&.(205)
Hence, by (204),
λ1+λ2
≥1−κ
−κ·inf
g∈G ⎡
⎢
⎢
⎣$y∈Di1,g:
y−g◦ui,2≤√n(σ2
Z+ζ)fZ(y−g◦ui1)dy⎤
⎥
⎥
⎦
≥1−2κ, (206)
which leads to a contradiction for sufficiently small κ,such
that 2κ<1−λ1−λ2. This completes the proof of
Lemma 13.
By Lemma 13 we can define an arrangement of non-
overlapping spheres Sui(n, √nεn)of radius √nεncentered
at the codewords ui. Since the codewords all belong to a
sphere S0(n, √nA)of radius √nA centered at the origin,
it follows that the number of packed spheres, i.e., the number
of codewords 2nlog(n)R, is bounded by
2nlog(n)R≤Vo l (S0(n, √nA +√nεn))
Vo l (Sui(n, √nεn))
=√A+√εn
√εnn
.(207)
Thus,
R≤1
log nlog √A+√εn
√εn
=1+b+log 1+ 1
n1+b
log n,(208)
by the same arguments as in the proof for the Gaussian channel
with fast fading (see (148)), which tends to 1+bas n→∞.
Now, since b>0is arbitrarily small, an achievable rate must
satisfy R≤1. This completes the proof of Theorem 11.
ACKNOWLEDGMENT
The authors gratefully thank Andreas Winter (Universitat
Autònoma de Barcelona) for useful discussions concerning
the scaling of the DI capacity. They also thank Ning Cai
(ShanghaiTech University) for a discussion on the DI capacity
for DMCs. Finally, they also thank Robert Schober (Friedrich
Alexander University) for discussions and questions about the
application of identification theory in molecular communica-
tions, and the anonymous reviewers for helpful comments.
Holger Boche thanks Marios Kountouris for discussions on the
talk and Kai Börner and Dimitar Kroushkov for applications
of deterministic identification.
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Mohammad J. Salariseddigh received the B.Sc. degree in electrical engineer-
ing from the K. N. Toosi University of Technology, Tehran, Iran, in 2014, the
M.Sc. degree in communication and multimedia engineering from Friedrich-
Alexander-Universität Erlangen-Nürnberg, Germany, in 2018. He is currently
pursuing the Ph.D. degree with the Institute for Communications Engineering,
Technische Universität München, Germany. He is also working on a joint
research project NewCom—Post Shannon Communication sponsored by the
Federal Ministry of Education and Research with partners from Technische
Universität München and Technische Universität Berlin. His current research
interests include information theory, and coding and identification theory.
Uzi Pereg (Member, IEEE) was born in Jerusalem, Israel, in 1989. He received
the B.Sc. degree (summa cum laude) in electrical engineering from the
Azrieli College of Engineering, Jerusalem, in 2011, and the M.Sc. and Ph.D.
degrees from the Technion—Israel Institute of Technology, Haifa, Israel,
in 2015 and 2019, respectively. Currently, he is a Post-Doctoral Researcher
with the Institute for Communications Engineering, Technical University of
Munich, Munich, Germany. In 2020, he joined the Theory Group, German
Federal Government (BMBF) Project for the design and analysis of quantum
communication and repeater systems. His research interests are in the areas
of quantum communications, information theory, and coding theory. He was a
recipient of the 2018 Pearl Award for Outstanding Research Work in the field
of communications, the 2018 KLA-Tencor Award for Excellent Conference
Paper, the 2018–2019 Viterbi Fellowship, and the 2020–2021 Israel CHE
Fellowship for Quantum Science and Technology.
Holger Boche (Fellow, IEEE) received the Dr.rer.nat. degree in pure mathe-
matics from the Technische Universität Berlin, Berlin, Germany, in 1998, and
the Dipl.Ing. degree in electrical engineering, the degree in mathematics, and
the Dr.Ing. degree in electrical engineering from the Technische Universität
Dresden, Dresden, Germany, in 1990, 1992, and 1994, respectively. He is
currently with the Institute of Theoretical Information Technology, Technische
Universität München, Munich, Germany; the Munich Center for Quantum
Science and Technology (MCQST), Munich; and the Cyber Security in the
Age of Large-Scale Adversaries—Exzellenzcluster (CASA), Ruhr Universität
Bochum, Bochum, Germany. From 1994 to 1997, he was involved in
postgraduate studies in mathematics with the Friedrich-Schiller Universität
Jena, Jena, Germany. In 1997, he joined the Heinrich-HertzInstitut (HHI)
für Nachrichtentechnik Berlin, Berlin. In 2002, he was a Full Professor
of mobile communication networks with the Institute for Communications
Systems, Technische Universität Berlin. In 2003, he became the Director of
the Fraunhofer German-Sino Laboratory for Mobile Communications, Berlin,
and the Director of HHI in 2004. He was a Visiting Professor with ETH
Zurich, Zürich, Switzerland, in Winter 2004 and 2006, and KTH Stockholm,
Stockholm, Sweden, in Summer 2005. Since 2010, he has been with the
Institute of Theoretical Information Technology and a Full Professor with
the Technische Universität München. He was elected as a member of the
German Academy of Sciences Leopoldina in 2008 and the Berlin Brandenburg
Academy of Sciences and Humanities in 2009. Since 2014, he has been a
member and an Honorary Fellow of the TUM Institute for Advanced Study,
Munich. He is a member of the IEEE Signal Processing Society SPCOM
and the SPTM Technical Committee. He was a recipient of the Research
Award Technische Kommunikation from the Alcatel SEL Foundation in 2003,
the Innovation Award from the Vodafone Foundation in 2006, the Gottfried
Wilhelm Leibniz Prize from the German Research Foundation in 2008, and
the 2007 IEEE Signal Processing Society Best Paper Award, and a co-recipient
of the 2006 IEEE Signal Processing Society Best Paper Award.
Christian Deppe (Member, IEEE) received the Dipl.Math. and Dr.Math.
degrees in mathematics from Universität Bielefeld, Bielefeld, Germany,
in 1996 and 1998, respectively. He was a Research and Teaching Assistant
with the Fakultät für Mathematik, Universität Bielefeld, from 1998 to 2010.
From 2011 to 2013, he was the Project Leader of the project Sicherheit und
Robustheit des Quanten-Repeaters of the Federal Ministry of Education and
Research, Fakultät für Mathematik, Universität Bielefeld. In 2014, he was
supported by the DFG Project at the Institute of Theoretical Information
Technology, Technische Universität München. In 2015, he had a temporary
professorship with the Fakultät für Mathematik und Informatik, Friedrich-
Schiller Universität Jena. He was the Project Leader of the project Abhör-
sichere Kommunikation über Quanten-Repeater of the Federal Ministry of
Education and Research, Fakultät für Mathematik, Universität Bielefeld. Since
2018, he has been with the Department of Communications Engineering,
Technische Universität München.
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