IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006 2903
Cutoff Rate Optimal Binary Inputs with
Imperfect CSI
Saswat Misra, Ananthram Swami, Senior Member, IEEE, and Lang Tong, Senior Member, IEEE
Abstract— We use the cutoff rate to study the optimal binary
input distributions for the Rayleigh flat-fading channel with
imperfect receiver channel state information (CSI). First, we
evaluate the cutoff rate and analyze the optimal binary input as
a function of the CSI quality and receiver SNR. Next, we study
the limiting distributions – BPSK and On-Off Keying (OOK) –
and derive an analytic design rule that allows adaptive switching
between these two as the receiver CSI changes. We establish the
virtues of a modulation scheme that employs only these limiting
distributions, rather than the full spectrum of binary inputs.
Finally, we use our results to design an adaptive modulation
scheme for Pilot Symbol Assisted Modulation systems. We show
that switching between just BPSK and equiprobable-OOK is
nearly optimal for moderate to large SNR, and that switching
between BPSK and generalized-OOK is nearly optimal for all
SNR.
Index Terms— Adaptive modulation, cutoff rate, correlated
fading, Doppler, imperfect CSI.
I. INTRODUCTION
BINARY input distributions are often assumed when
studying the reliable rates of communications systems,
either through channel capacity or other related metrics.
The widespread analysis of binary inputs follows from their
tractability and optimality, or near optimality, at low SNR
under varying amounts of receiver channel state information
[2], [34], [19]. We consider reliable rates (i.e., those for which
the probability of decoding error can be made arbitrarily small)
for communications over a discrete-time Rayleigh flat-fading
channel. We assume that the transmitter can select among
the class of binary input distributions, and that imperfect (or
partial) channel state information (CSI) is available at the
receiver.
When perfect receiver CSI is available, it is well known
that antipodal signaling (BPSK) maximizes the capacity of this
channel (among binary inputs). Conversely, without CSI at the
receiver, On-Off keying (OOK) has been shown to be capacity
achieving [2]. However, when only imperfect receiver CSI is
Manuscript received October 7, 2004; revised September 11, 2005; accepted
December 15, 2005. The associate editor coordinating the review of
this paper and approving it for publication was F. Daneshgaran. Parts of this
paper were presented at the 37th Annual Conference on Information Sciences
and Systems, Baltimore, March 2003, and the IEEE International Symposium
on Information Theory, Chicago, June 2004.
S. Misra is with the Army Research Laboratory, Adelphi, MD 20783 USA
and the School of Electrical and Computer Engineering, Cornell University,
Ithaca, NY 14853 USA (e-mail: smisra@arl.army.mil).
A. Swami is with the Army Research Laboratory, 2800 Powder Mill Rd.,
Adelphi, MD 20783-1197 USA (e-mail: aswami@arl.army.mil).
Lang Tong is with the School of Electrical and Computer Engineering,
Cornell University, 384 Rhodes Hall, 35 Rosina Drive, Ithaca, NY 14850
USA (e-mail: ltong@ece.cornell.edu).
Digital Object Identifier 10.1109/TWC.2006.04665
1536-1276/06$20.00 c○ 2006 IEEE
available, it is not clear as to which strategy, even among these
two, is optimal. In this paper, we investigate this intermediate
case.
We use the cutoff rate Ro to characterize reliable rates.
Cutoff rate analysis has been common since its reintroduction
in [26], and studies have been conducted for perfect receiver
CSI in [21] (independent fading), [23] and [27] (temporally
correlated fading), and for no CSI multiple-input multipleoutput
(MIMO) systems in [18]. The cutoff rate is a lower
bound on capacity that also provides a bound on the random
coding exponent (thereby characterizing the entire rate vs.
performance curve) via Pe ≤ 2−N(Ro−R) ,whereRis the
rate, and Pe the probability of decoding error for length
N codewords [13],[30, Sections I and II]. Although certain
encoding-decoding structures can achieve rates greater than
Ro (e.g., turbo coding with iterative decoding), the cutoff
rate remains a metric of interest for these systems, as well
as others [6]. For example, in sequential decoding, the cutoff
rate specifies the largest rate for which decoding complexity
remains finite [4]. The cutoff rate often leads to a tractable
analysis that often would not be possible through direct
evaluation of the random coding exponent or the capacity.
In Section II we introduce a Rayleigh fading channel with
imperfect receiver CSI and find the corresponding cutoff rate
under binary signaling. Our model has been abstracted away
from the particular estimator used. Rather, these mechanics
are absorbed into a single parameter, the normalized variance
of the channel estimate, termed the CSI quality. In Section III,
we analyze the optimal binary input as a function of the SNR
and CSI quality available at the receiver. We establish the
cutoff rate optimality of the limiting distributions – BPSK and
On-Off Keying – and find an analytic design rule that allows
adaptive switching between these two based on the receiver
CSI quality. We then establish the virtues of a modulation
scheme that employs only these limiting distributions, rather
than the full spectrum of binary inputs. In the remaining
sections, we introduce an explicit Pilot Symbol Assisted
Modulation (PSAM) front-end, and use it to illustrate how
results from Sections II and III can be applied to design
an adaptive modulation scheme. In Section IV we include
temporal correlation in the channel model, and find the cutoff
rate under a PSAM scheme with minimum mean square error
(MMSE) estimation. In Section IV-C, we discuss adaptive
modulation strategies and show that switching between just
BPSK and equiprobable-OOK nearly achieves optimal binary
signaling for moderate (≈ 0 dB) to large SNR. Further, we
show that switching between just BPSK and generalized-OOK
is nearly optimal for all SNR. Finally, we provide a summary
2904 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006
and discussion in Section V.
In this paper, we do not consider the design of higher order
inputs (which has been done in [2], [19], [10], and [34] for
the capacity) or optimal inputs when the channel is peakconstrained
(e.g., see [30], [31]). Here, we use the cutoff rate
to (i) study the behavior of the optimal binary inputs when
only imperfect CSI is available, and (ii) apply this analysis
to the design of a tractable adaptive modulation scheme for
PSAM based communications systems.
Notation and Definitions
We use the following notation and definitions: (a) x ∼
CN � μ, σ2� denotes a complex Gaussian random variable x
with mean μ and with independent real and imaginary parts,
each having variance σ2 /2, (b)|A| is the magnitude of the
complex number A, (c)E[.] is the expectation operator, and
(d) superscript “H” denotes complex conjugation.
II. SYSTEM MODEL
A. Channel Model
We consider single-user communications over a timevarying
Rayleigh flat-fading channel. The received signal yk
is given by
yk = √ Ehksk + nk, (1)
where k denotes discrete time, hk ∼ CN(0,σ2 h ) models
independent and identically distributed (i.i.d.) fading, E is
the average symbol energy used at the transmitter, and nk ∼
CN(0,σ 2 N
) models additive white Gaussian noise (AWGN).
The binary channel input sk ∈{A, −B} isassumedtobe
real-valued, without loss of generality, and subject to a unitenergy
constraint pA 2 +(1− p)B 2 =1,where0 ≤ p ≤ 1 is
the probability of transmitting A. Without loss of generality,
we assume that 1 ≤ A ≤∞and 0 ≤ B ≤ 1. We assume that
σ 2 N �=0and σ2 h �=0.
During each symbol interval, the receiver obtains imperfect
CSI in the form of a channel estimate, � hk, and so (1) can be
rewritten,
yk = √ E � hksk + √ E � hksk + nk,
where �hk = hk − �hk is the residual error in the channel
estimate. We assume that both the estimate and the residual
error are zero-mean Gaussian and independent, i.e., �hk ∼
CN(0, �σ 2 ), �hk ∼ CN(0, �σ 2 ), and �σ 2 + �σ 2 = σ2 h . MMSE
estimation schemes exist that satisfy these assumptions, and
we describe one such PSAM-based scheme in Section IV.
However, this information is not needed for the discussion
contained in this section or in Section III.
The receiver employs the soft decision ML decoder which
treats sk as the channel input and the pair (yk, �hk) as the
channel output. That is, letting s = (s1,...,sN ) denote
a transmitted codeword, and y = (y1,...,yN) and � h =
( �h1,..., �hN ) denote the observation and channel estimate
during the span of a codeword, the decision rule maximizes
the posteriori probability of the observation,
max
s ∈Q P ( y, � h | s),
where Q is the set of all possible length N input sequences.
Lastly, we will find it useful to define the CSI quality as the
normalized variance of the channel estimate at the receiver,
ω � �σ 2 /σ 2 h.
Note that ω =0denotes no CSI, while ω =1denotes perfect
CSI.
B. Cutoff Rate
The cutoff rate, measured in bits per channel use, is [26]
(also see [23] for time-selective fading channels with perfect
receiver CSI),
� � �
�
Q(sk)
Ro = − min
Q log 2
yk
� hk
×
sk∈{A,−B}
�
P (yk, � hk | sk)
� 2
d � hk dyk,
where Q(A) =p , Q(−B) =1− p ,andwhereP (yk, �hk|sk) is the probability distribution function (pdf) of the received
signal and channel estimate, conditioned upon the transmitted
signal. 1 We evaluate (2) in Appendix A and obtain
�
Ro = − min
1+2p (1 − p)
×
C(p,A,B) log 2
�� 1+κ (1 − ω) A 2 � 1+κ (1 − ω) B 2
1+ κ 2
� 1 − ω
2
�
(A2 + B2 )+ κ − 1
ω
2 AB
(2)
��
, (3)
where C(p, A, B) � {(p, A, B) : 0 ≤ p ≤ 1, 1 ≤ A <
∞, 0 ≤ B ≤ 1, pA 2 +(1− p)B 2 =1} is the constraint set
on the input. We have defined the received SNR as
κ � E σ2 h
σ2 .
N
III. OPTIMAL BINARY INPUTS
The optimal binary input (p∗ ,A∗ ,B∗ ) as a function of the
CSI quality ω and SNR κ is found from (3) through the
minimization
min p (1 − p)
C(p,A,B)
��
1+κ (1 − ω) A2 ×
� 1+κ (1 − ω) B2 1+ κ �
� �
ω
2 1 − 2 (A2 + B2 )+ κ − 1 . (4)
ω
2 AB
The behavior of the this input is shown in Fig. 1 parameterized
by ω.
Define the transitional CSI quality by
⋆
ω � 1 − 1
√ . (5)
3
Then the behavior of the optimal binary input is characterized
by the following statements:
R1. For small SNR (κ << 1): If the CSI quality is below
the ⋆ ω threshold (ω < ⋆ ω), then a solution resembling
1 Here, the term cutoff rate implies optimization over the inputs. Later, it
will imply certain fixed inputs.
MISRA et al.: CUTOFF RATE OPTIMAL BINARY INPUTS WITH IMPERFECT CSI 2905
Transmission Probability p*
Amplitude A*
10 1
10 0
ω = 0.0
ω = 0.3
ω = 1.0
ω = 0.4
ω = ω* − ε
ω = ω* + ε
ω = 0.5
ω = 0.0
ω = 0.1
ω = 0.2
ω = 0.3
ω = 0.4
ω = ω* − ε
ω = ω* + ε
ω = 0.5
ω = 0.6
ω = 0.7
ω = 0.8
ω = 0.9
ω = 1.0
ω = 0.9
−20 −15 −10 −5 0
SNR κ (dB)
5 10 15 20
0.5
0.45
0.4
ω = 1.0
(a) A ∗ vs. κ for several values of ω
ω = 0.6
0.35
ω = 0.5
ω = 0.0
0.3
ω = ω* + ε
ω = 0.1
ω = 0.2
0.25
ω = 0.3
ω = 0.4
0.2
ω = ω* − ε
0.15
ω = ω* − ε
ω = ω* + ε
ω = 0.5
0.1
ω = 0.4
ω = 0.6
ω = 0.7
ω = 0.8
0.05
ω = 0.0
ω = 0.9
ω = 1.0
0
−20 −15 −10 −5 0 5 10 15 20
SNR κ (dB)
(c) p ∗ vs. κ for several values of ω
Amplitude B*
Cutoff Rate R o
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
ω = 1.0
ω = ω* − ε
ω = ω* + ε
ω = 0.4
ω = 0.5
ω = 0.0
−20 −15 −10 −5 0 5 10 15 20
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
SNR κ (dB)
(b) B ∗ vs. κ for several values of ω
ω = 0.0
ω = 0.1
ω = 0.2
ω = 0.3
ω = 0.4
ω = ω* − ε
ω = ω* + ε
ω = 0.5
ω = 0.6
ω = 0.7
ω = 0.8
ω = 0.9
ω = 1.0
ω = 0.9
0
−20 −15 −10 −5 0 5 10 15 20
SNR κ (dB)
ω = 1.0
ω = 0.0
(d) Ro vs. κ for several values of ω
Fig. 1. The optimal binary input (p ∗ ,A ∗ ,B ∗ ) and cutoff rate Ro versus SNR κ for several values of CSI quality ω. The quantity ɛ � 0.01 is used to
illustrate behavior around the ω = ⋆ ω transition point discussed in the text.
OOK with large amplitude is optimal. As κ → 0,
limp→0 OOK(p) is optimal. 2 However, if the CSI quality
exceeds the ⋆ ω threshold (ω > ⋆ ω), the optimal distribution
is BPSK, A = B =1,p =1/2. Proofs are given in
Appendix B.
R2. For large SNR (κ >> 1): If ω< ⋆ ω, then from Fig. 1,
A ∗ decreases with SNR to the value √ 2.Ifω> ⋆ ω, A ∗
remains fixed at 1 before diverging from the BPSK solution
with increasing SNR. After divergence, A ∗ increases
in SNR, reaches a peak, and then also decreases to the
value √ 2.AsSNR→∞, OOK(1/2) is shown to be
optimal for any CSI quality (ω �= 1) in Appendix B.
R3. The optimal transmission probability satisfies p ∗ ≤ 1/2.
A sketch of the proof follows: let I1 � (p, A1, B) be an
arbitrary triple with p>1/2. We show that the � alternative
1−pB2 solution I2 � (1−p, A2, B), whereA2 =
1−p due
to the energy constraint results in a smaller value of (4).
2 Henceforth, we will use OOK(pθ) to denote the binary alphabet with
p = pθ, A =1/ √ pθ, andB =0.
That ⋆ ω =1−1/ √ 3 is the low SNR (κ → 0) transition value
is shown in Appendix B. However, the cutoff rate can be seen
to be well behaved around ω = ⋆ ω in Fig. 1(d). In Fig. 2, we
plot the optimal input parameterized by the SNR κ. Again,
the presence of the ⋆ ω threshold at low SNR is evident.
Having examined the general behavior of the optimal binary
input, we now focus on the limiting cases of OOK and BPSK.
A. OOK Cutoff Rate
The cutoff rate of OOK was derived for the AWGN channel
and hard decision ML decoder in [15]. Here, we derive the
OOK cutoff rate for the fading channel with imperfect CSI
and soft decision ML decoder. Consider first the no CSI case
(ω = 0). We find that OOK(p) modulation maximizes the
cutoff rate at all SNR κ (as shown in Appendix C), and so it
remains to determine p∗ . Setting A2 = 1
p , B =0,andω =0
in (4), p∗ is given by
⎡
⎤
p(1 − p) ⎣
− 1⎦
p ∗ = min
0<p< 1
2
�
1+κ 1
p
1+κ 1
2p
2906 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006
Amplitude A *
10 1
−5 dB
−10 dB
−15 dB
κ = −20 dB
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0 dB 5 dB
CSI Quality ω
10 dB
15 dB
(a) A ∗ vs. ω for several values of SNR κ
Fig. 2. The optimal binary input (A ∗ ,p ∗ )versus0 ≤ ω ≤ 1 for several values of SNR κ.
Transmission Probability p*
10 0
10 −1
ω = 1
ω = 0.75
ω = 0.5
ω = 0.25
ω = 0
p = 1/2
20 dB
10
−15 −10 −5 0 5 10 15 20 25 30
−2
SNR κ (dB)
(a)
Fig. 3. The optimal OOK probability p ∗ versus (a) SNR κ (dB) and (b) CSI quality.
�
= p : 0 ≤ p ≤ 1
2 , 2(κ +1)p4 � �
2 11κ
+
+ κ − 1 p
4 3
�
+ κ3 − 3κ2
�
− κ p
2 2 − κ3 p + κ3
4 =0
�
. (6)
Solving (6) yields p ∗ explicitly (as the valid root of the fourthorder
polynomial), and provides an easy characterization of
the optimal transmission probability (equivalently, the optimal
signal energy) as a function of the SNR.
At low SNR (κ ≪ 1), the transmission probability is linear
in SNR, i.e., p ∗ = β κ, where
β � (19 + 3√33) 2
3 − 2(19 + 3 √ 33) 1
3 +4
6(19 + 3 √ 33) 1
=0.419 .... (7)
3
Returning now to the case of partial CSI, (3) yields the
OOK cutoff rate for arbitrary ω as
�
1+2p(1 − p)
Ro,K = − min
0<p<1 log 2
Transmission Probability p*
Transmission Probability p *
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
10 −1
10 −2
10 −3
κ = 20 dB
15 dB
10 dB
5 dB
0 dB
−5 dB
−10 dB
−15 dB
−20 dB
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
CSI Quality ω
(b) p ∗ vs. ω for several values of SNR κ
κ = 10 dB
κ = 0 dB
κ = −10 dB
κ = −20 dB
κ = −30 dB
0 0.2 0.4 0.6 0.8 1
CSI Quality ω
(b)
⎛�
1+κ(1 − ω)
× ⎝
1
⎞ �
p
− 1⎠
. (8)
1+κ (2 − w) 1
4p
Analytic maximization of (8) over p leads to a high-order
polynomial that does not have an explicit solution as a function
of κ and ω. Weplotp ∗ as a function of κ in Fig. 3(a) and as a
function of ω in Figure 3(b). It can be verified that as κ →∞,
p ∗ → 1/2, and that as κ → 0,p ∗ → 0. The transmission
probability p ∗ is seen to be non-monotonic in ω at low SNR. A
second-order Taylor series expansion of the expression within
the curly brackets in (8) at κ =0, yields (ω �= 0)
p ∗ =argmin 0<p<1 −κ(1 − p) � 4 pω+ κ(3ω2 − 6ω +2) �
,
16p
from which we get
p ∗ √ �
κ −2+6ω − 3ω2 =
2
w
for ω> ⋆ ω, (9)
MISRA et al.: CUTOFF RATE OPTIMAL BINARY INPUTS WITH IMPERFECT CSI 2907
Cutoff Rate R o,K
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Full CSI, ω = 1
ω = 0.8
No CSI, ω = 0
0
−10 −5 0 5 10 15 20 25 30
SNR κ (dB)
Fig. 4. The OOK cutoff rate Ro,K vs. SNR κ (dB) for no CSI (ω =0),
perfect CSI (ω =1) and imperfect CSI (ω =0.8), when p = p ∗ (solid lines,
from (8)) and p =1/2 (dashed lines, from (10)).
3 A third-order Taylor expansion yields an expression for p ∗ that is valid
for all ω; it is omitted for brevity.
Cutoff Rate R o,B
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
ω = 1
0
−10 −5 0 5 10 15 20
ω = 0
25 30
SNR κ (dB)
ω = 0.8
ω = 0.6
ω = 0.4
ω = 0.2
Fig. 5. The BPSK cutoff rate Ro,B vs. SNR κ (dB) for different values of
the CSI quality ω.
which is decreasing for ω ∈ ( � 2/3, 1). 3 As κ increases, the
amplitude A∗ decreases (since A2 = 1
p ). From Fig. 3, we see
that for fixed κ, the amplitude A∗ is a decreasing function
of ω. For moderate to large values of κ, letting p = 1/2
is a reasonable approximation to p∗ . Using p = 1/2, the
equiprobable-OOK cutoff rate is found to be
�
� �
1 1 1+2κ(1 − ω)
Ro,K = − log2 +
2 2 1+κ � 1 − w
��
� . (10)
2
In Fig. 4, we plot Ro,K for p = p∗ (solid lines) and p =1/2
(dashed lines) for no CSI (ω =0), perfect CSI (ω =1), and
imperfect CSI (ω =0.8). Note that the cutoff rate of both
OOK( 1
2 ) and OOK(p∗ ) approaches 1 at high SNR for any ω.
B. BPSK Cutoff Rate
Consider the case of perfect CSI. We let ω =1in (4), which
leads readily to the BPSK solution, A = B, and p =1/2. For
arbitrary ω, the cutoff rate of BPSK is
� � ��
1 1 1+κ(1 − ω)
Ro,B = − log2 + . (11)
2 2
1+κ
Fig. 5 shows the cutoff rate as a function of the received
SNR κ and the CSI quality ω. Note that the CSI quality
imposes an asymptotic ceiling on Ro,B, and that at high SNR,
the cutoff rate saturates to
�
Ro,B = − log2 1 − ω
�
.
2
As expected, the cutoff rate is zero when there is no CSI.
To study the relative impact of imperfect CSI on BPSK and
OOK, it is instructive to consider the statistics of yk under the
two hypotheses:
yk| � �√
hk,sk ∼ CN E �hksk, σ 2 N (1 + s2 where sk ∈ {−1, 1} for BPSK and sk ∈ {0,
�
k κ(1 − ω)) ,
√ 2} for
OOK(1/2). When the SNR is large enough, i.e., κ ≫ 1
1−ω ,
the channel estimation error dominates, and it is easy to verify
that the BPSK performance saturates.
Thus, we expect that OOK is optimal at large κ, andthat
BPSK is optimal for small κ. Next, we quantify the SNR at
which one should switch from BPSK to OOK as a function
of estimator quality ω.
C. Transitional SNR
We confine our interest to BPSK (optimal for perfect CSI)
and OOK (optimal for no CSI) and provide the transitional
SNR ⋆
κ, above which OOK is optimal, and below which BPSK
is optimal. This result provides an initial characterization of
the intermediate region where imperfect CSI is available, and
provides an analytic basis for an adaptive modulation scheme
in which the transmitter can select between OOK and BPSK
based on the SNRκand CSI quality ω available at the
receiver as described in subsequent sections. For OOK(1/2),
the transitional SNR ⋆
κ is found by equating (11) and (10) and
solving for κ. Doing so yields the solution
�
⋆
κ(ω) = κ : κ ≥ 0, (2 − ω)2 (1 − ω) 2
κ
4
3 (1 − ω)
−
2
�
�
× (10 − 3ω)ω − 4 κ2 � �
�
2 13ω
+
− 5ω +1 κ − ω =0 ,
4
for which the explicit solution is
⋆
κ(ω) = (a + b)1/3 +(a− b) 1/3 − 2 � 4 − 10ω +3ω2� 3(2 − ω) 2 ,
(1 − ω)
(12)
with the definitions
a � 81ω 6 − 468ω 5 + 828ω 4 − 640ω 3 + 624ω 2 − 192ω +64,
b � 6 √ 3(ω − 2) 2 w 2� 61ω4 − 208ω3 + 168ω2 − 64ω +16.
The transitional SNR ⋆
κ depends on the CSI quality, and is
shown in Fig. 6 (dashed line). At the end points, ω = {0, 1},
2908 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006
Transitional SNR κ* (dB)
40
30
20
10
0
−10
−20
−30
−40
OOK(1/2)
OOK(p*)
On−Off Keying
ω = ω*
−50
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
CSI Quality ω
BPSK
Fig. 6. The transitional SNR ⋆
κ above which OOK is optimal and below
which BPSK is optimal, for: equiprobable-OOK and BPSK (dashed line) and
OOK(p∗ ) and BPSK (solid line).
⋆
we find the anticipated results: κ (0) = 0, implying that
equiprobable-OOK is preferred to BPSK at any SNR when
⋆
no CSI is available, and limω→1 κ(ω) =∞, implying that
BPSK is preferred to equiprobable-OOK when perfect CSI is
available.
In Fig. 6, we repeat the threshold curve for OOK(p∗ ) (solid
line). To find this region we equate (11) and (8) and solve
for ⋆
κ numerically. As expected, optimizing over p results in
OOK(p) being preferred to BPSK over a wider range of SNR
for fixed ω. Interestingly, we find that there is a threshold
value of CSI below which BPSK is not useful. A low SNR
analysis once again reveals this value to be ⋆ ω=1− 1/ √ 3. 4
D. Adaptive Modulation Based on CSI Quality
The performance of the adaptive modulation schemes described
in Section III-C is shown in Fig. 7. As upper and
lower bounds, we plot the cutoff rate of optimal binary
signaling (determined from (4)), and for the BPSK–only and
OOK( 1
2 )–only schemes when κ =0dB. Each curve has been
normalized by the cutoff rate of optimal binary signaling.
The OOK( 1
2 )–BPSK scheme simply traces out the best of
the BPSK and OOK cutoff rates. The BPSK–only scheme
performs arbitrarily poorly for small ω (as expected due to
its saturating behavior at large SNR, see Fig. 5), while the
OOK( 1)–only
scheme is seen to be suboptimal by up to ∼ 40
2
percent for large values of ω. In contrast, the OOK(p∗ )–BPSK
scheme performs nearly as well as optimal binary signaling
over the entire range of ω. To understand this behavior, we
partition the (κ,ω) plane into three regions in Fig. 8: (a) the
region where BPSK is within one-percent of optimal, (b) the
region where OOK(p∗ ) is within one-percent of optimal, and
(c) the remaining region. 5 Over most of the (κ,ω) plane, we
4 This reoccurrence of ⋆ ω is to be expected. Earlier in this section we found
that, among all binary inputs, only OOK(p ∗ ) or BPSK is optimal at low SNR.
5 While BPSK is optimal over a (κ,ω) region, OOK is optimal only on
a line (from Fig. 1). This necessitates the use of optimality regions in the
figure.
1
0.8
0.6
0.4
0.2
OOK(p*)−BPSK
OOK(1/2)−BPSK
BPSK−only
Optimal Input
OOK(1/2)−only
0
0 0.2 0.4 0.6 0.8 1
CSI Quality ω
Fig. 7. The normalized cutoff rate for the Rayleigh flat-fading channel with
imperfect CSI as a function of the CSI quality ω, and for five different binary
transmission strategies with κ =0dB.
SNR (dB)
40
30
20
10
0
−10
−20
−30
−40
OOK(p*)
OTHER
ω = ω*
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
CSI quality ω
BPSK
Fig. 8. Partitioning of the (SNR(κ), CSI(ω)) plane into three regions: (a)
where BPSK is within one-percent of optimal, (b) where OOK(p ∗ ) is within
one-percent of optimal, and (c) the remaining region.
see that either BPSK or OOK(p ∗ ) is indeed nearly optimal.
Comparing Fig. 8 to Fig. 6, we find that BPSK retains nearly
its entire optimality region even when arbitrary binary inputs
are admitted. In contrast, OOK(p ∗ ) loses a portion of its
optimality region in this scenario. In particular, the region
loosely described by {(κ,ω):−20 dB ≤ κ ≤ 10 dB, 0.2 ≤
ω ≤ ⋆ ω} is now allocated to region (c).
E. Sensitivity Analysis
We study the sensitivity of the cutoff rate to the binary input
used for four limiting cases: (a) large κ (κ =30dB), large ω
(ω =0.95) (b)smallκ (κ = −10 dB), large ω, (c)largeκ,
small ω (ω =0.1), and (d) small κ, smallω.
Large ω. For large κ, the cutoff rate is sensitive to the choice
of p, but not to the choice of B. For example, with B =1,
MISRA et al.: CUTOFF RATE OPTIMAL BINARY INPUTS WITH IMPERFECT CSI 2909
the cutoff rate increases more than 300 percent as p increases
from 0.1 to 0.5. On the other hand, with p =0.4, thecutoff
rate increases by only 4 percent as B varies from 0 to 1. For
small κ, the cutoff rate is sensitive to choice of both p and
B. For example, with B =1, the cutoff rate increases by 280
percent as p increases from 0.1 to 0.5. With p =0.5, the
cutoff rate increases by a factor of approximately 200 percent
as B varies from 0 to 1.
Small ω. At both small and large κ, the cutoff rate is
sensitive to the choice of B when p is chosen optimally. When
p is chosen suboptimally, sensitivity decreases. For example, at
small κ the cutoff rate increases by 400 percent as B increases
from 0 to 1. Atlargeκ, the increase is 800 percent. In each
case, sensitivity to B diminishes if p is chosen suboptimally.
Overall, it is clear that optimization can provide large gains
in the cutoff rate.
IV. ADAPTIVE MODULATION SCHEME FOR PSAM
In this section we consider a temporally correlated flatfading
channel and apply the results derived in Section III, in
particular those of Section III-C, to the design of an adaptive
modulation scheme for PSAM-based communications. We
then analyze the performance of the proposed scheme relative
to some clear alternatives.
In PSAM [8], [32], known pilot symbols are multiplexed
with data symbols for transmission through the communications
channel. At the receiver, knowledge of these pilots is
used to form channel estimates, which aid the detection of
the data both directly (by modifying the detection rule based
on the channel estimate) and indirectly (e.g., by allowing
for estimate-directed modulation, power control, and media
access). In general, there is no guarantee that PSAM-based
approaches are optimal, and PSAM has been shown to be
suboptimal when the channel coherence time is small and/or
the SNR small from various perspectives [17], [25], [2], [11].
Nevertheless, the technique is of great practical significance.
In addition to providing implementable receiver structures,
PSAM facilitates accurate timing and synchronization. PSAM
has been incorporated into many commercial and Military
standards, and optimized approaches to PSAM have been
studied from the perspectives of frequency and timing offset
estimation [22], [14], bit-error rate (BER) [8], [11], [7], [12],
and the channel capacity or its bounds [17], [3], [24], [29].
A. System Model
We generalize the Rayleigh fading channel of (1) to include
temporal correlation. The observation equation is
yk = √ Ehksk + nk,
where hk ∼ CN � 0,σ2 �
h now exhibits temporal correlation
described by the normalized correlation function Rh(τ) �
1
σ2 E
h
� hkhH �
k+τ . We assume that training is sent with period
T at times k = mT, m ∈Zand that smT =+1. In each
data slot mT + ℓ (1 ≤ ℓ ≤ T −1), an MMSE estimate of the
channel �hmT +ℓ is made at the receiver using some subset N
of past and future training symbol observations, so that
�hmT +ℓ = E[hmT +ℓ|{ynT }, n∈N ⊆Z], (13)
1 ≤ ℓ ≤ T−1, m∈Z. The system equation in the mth frame,
i.e., mT ≤ k ≤ (m +1)T− 1, isthen
�√
EhmT + nmT , (pilot),
yk = √ �
E �hmT
+ℓ + � �
hmT +ℓ smT +ℓ + nmT +ℓ (data).
(14)
The use of an MMSE estimator implies that the estimate
�hmT +ℓ and the estimation error �hmT +ℓ are zero-mean,
jointly Gaussian, and independent with variances �σ 2 ℓ and
σ2 h − �σ2 ℓ , respectively; �hmT +ℓ ∼ CN � 0, �σ 2 �
ℓ and �hmT +ℓ ∼
CN � 0,σ2 h − �σ2 �
ℓ . To characterize the partial CSI provided by
the estimator ℓ slots from the last-pilot, we define the CSI
quality in the ℓth slot
ωℓ � �σ 2 ℓ /σ 2 h, (15)
for ℓ =1,...,T−1. The CSI quality ωℓ, 0 ≤ ωℓ ≤ 1, captures
the impact of the channel correlation Rh(τ), estimator N ,and
SNR κ on the statistical quality of channel estimates at the
receiver. The variance of any estimator can be found by noting
that, �hmT +ℓ is the expected value of one Gaussian vector
conditioned upon another. The CSI quality is then readily
obtained via (15).
Given the periodic nature of the training, it is natural to
let the binary signaling scheme vary from data slot to data
slot, with period T . Therefore, defining [k] � k mod T ,we
allow the input sk to be selected from a real-valued binary
signal set S [k] = {A [k], −B [k]} subject to a unit averageenergy
constraint: p [k]A2 [k] +(1− p [k])B2 [k] =1,wherep [k]
is the probability of transmitting A [k] (note that S0 = {+1}).
We assume that A [k] and B [k] are real-valued, and that 1 ≤
A [k] < ∞ and 0 ≤ B [k] ≤ 1. Finally, we assume that
codewords can occur in integers multiples of a frame length,
i.e., N = n(T −1),n =1, 2,..., and are decoded using the
ML decoder which treats s1,...,sT−1 as the channel input
and the pair ( �h1,..., �hT −1 ; y1,...,yT−1) as the channel
output.
We consider a system in which perfect interleaving [5], [23],
[27] is performed at the transmitter and channel estimation is
performed before deinterleaving at the receiver. The system
equation under interleaving is still given by (14), but now
hk ∼CN � 0,σ2 �
h and nk ∼CN � 0,σ2 �
N are i.i.d. sequences
representing the interleaved channel and noise. Interleaving
implies that �hk and �hk are independent sequences in k and
with respect to each other. However, the marginal statistics of
the channel estimate and estimation error are preserved, i.e.,
�hmT +ℓ ∼ CN � 0, �σ 2 �
ℓ and �hmT +ℓ ∼ CN � 0,σ2 h − �σ2 �
ℓ .
B. Cutoff Rate
In [28], we have previously studied the cutoff rate of
a PSAM communications system. Under the assumption of
perfect interleaving, and by modification of the proof in [28]
to the case of generalized binary inputs, the cutoff rate of the
system of Section IV-A can be seen to be given by
�
1+2pℓ (1−pℓ)
Ro = − 1
T� −1
min
T
Cℓ(p,A,B)
ℓ=1
log2 2910 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006
��
1+κ (1 − ωℓ) A
×
2 �
ℓ 1+κ (1 − ωℓ) B2 ℓ
1+ κ � � ωℓ
2 1 − 2 (A2 ℓ + B2 ωℓ
ℓ )+κ2 AℓBℓ
��
− 1 ,
(16)
where Cℓ(p, A, B) � {(pℓ,Aℓ,Bℓ) :0≤ pℓ ≤ 1, 1 ≤ Aℓ <
=1} is the constraint set
∞, 0 ≤ Bℓ ≤ 1, pℓA2 ℓ +(1−pℓ)B2 ℓ
on the ℓth input.
Comparing the cutoff rate of the i.i.d. channel (3) to that of a
PSAM system operating over the temporally correlated fading
channel under interleaving (16), it is clear that the latter can be
interpreted as consisting of T−1 parallel data-channels, where
the ℓth channel consists of all data slots occurring ℓ positions
after the most recent pilot. The ℓth (1 ≤ ℓ ≤ T − 1) term
in the sum of (16) represents the cutoff rate in one of T −1
data channels, with CSI quality ωℓ and SNR κ. Therefore,
letting ω = ωℓ, we can apply the analysis given in Section III
on a per-channel basis. This motivates design of a PSAM
system in which the optimal binary distribution (A∗ ℓ ,B∗ ℓ ,p∗ℓ )
is used in each data channel. Next, we combine the cutoff rate
(16) with the optimal input analysis of Section III to design
adaptive modulation schemes in which the transmitter selects
the modulation in each data slot based on the partial CSI ωℓ
and SNR κ at the receiver.
used otherwise. Denote the cutoff rate of this system
by RHYB1. This scheme is implemented through the
analytic switching rule derived in (12).
C2. The OOK(p ∗ )–BPSK adaptive system in which
generalized-OOK is used in each sub-channel where
it is preferred to BPSK, and where BPSK is used
otherwise. Denote this cutoff rate by RHYB2. This
scheme is implemented using the solid curve shown
in Fig. 6. In practice, this curve can be implemented
in hardware at low cost.
As benchmarks, we compare the performance of our
schemes to:
C3. Optimal binary signaling, in which each data slot is
assigned the cutoff rate optimal binary input as determined
from (4). This will be computed numerically.
Denote the cutoff rate of this system by RBIN. This
scheme provides an upper bound on the performance
of C1 and C2.
C4. The BPSK–only system. Denote the cutoff rate of
this system by RBPSK. This scheme provides a lower
C. Adaptive Modulation Scheme
Adaptive transmission techniques for fading channels have
been well studied in the literature (e.g., see [7], [16], [9],
and the references therein). Typically, a subset of the key
transmission parameters – power, rate, modulation shape and
size, and bandwidth – is adapted based on some instantaneous
measure of the channel quality, which may determined by
the fading, noise, or interference level at the receiver. This
knowledge is typically provided to the transmitter via a feedback
link, which introduces its own noise and/or delay to the
process. However, when PSAM is employed over continuously
time-varying fading channels, the transmitter need not adapt
to instantaneous channel quality measurements, since it can
adapt instead to the statistical quality of the channel estimates
– which varies with the estimate’s position relative to pilot
symbols. Further, if the transmitter has knowledge of the
channel Doppler spectra, Rh(τ), it can compute this statistical
quality without requiring explicit feedback.
In [1], this methodology is proposed and used to characterize
PSAM-based achievable rates for the Gauss-Markov
correlation model. An adaptive binary modulation scheme
was proposed in which the transmitter selects the achievable
rate optimal binary input in each data slot. It was shown
that adaptive modulation provides significant gains in the
achievable rates relative to static modulation when the channel
fading is slower and/or the estimator more accurate. Here,
we consider an adaptive binary modulation scheme based on
the cutoff rate. Our aim is to compare the performance of
the simple two-distribution modulation techniques derived in
Section III-C to optimal binary signaling (from Section III).
Specifically, we propose:
C1. The OOK( 1
bound on the performance of C1 and C2.
C5. The OOK(
2 )–BPSK adaptive system in which
equiprobable-OOK is used in each sub-channel
where it is preferred to BPSK, and where BPSK is
1
1
2 )–only system which uses OOK( 2 )in
each sub-channel. Denote the cutoff rate of this
system by ROOK. This scheme also provides a lower
bound on the performance of C1 and C2.
We show that our adaptive modulation scheme, based on
switching between just two inputs6 , captures the optimality
of scheme C3 over a wide range of SNR, while requiring a
fraction of the complexity.
Simulation. We consider two estimators (see [28],[1]): The
causal (1, 0) estimator N = {m}, forwhich
ω (1,0)
ℓ = � � 2
Rh(ℓ) � � κ
1+κ ,
and the non-causal (1, 1) estimator N = {m, m+1},forwhich
ω (1,1)
(κ
ℓ =
2 �
+ κ) |Rh(ℓ)| 2 �
2
+ |Rh(T−ℓ)|
(κ +1) 2 − κ2 |Rh(T )| 2
2κ
−
2 �
Re Rh(ℓ) Rh(T−ℓ) R∗ �
h (T )
(κ +1) 2 − κ2 |Rh(T )| 2 ,
where Re {.} denotes the real part. We assume that the channel
correlation is described by the well-known Jakes model [20],
for which Rh(τ) =Jo(2πfDTDτ), whereJo(.) is the zerothorder
Bessel function of the first kind, and where fDTD is the
normalized Doppler spread. We let fDTD =1/50,andT =7.
Fig. 9(a) plots the cutoff rate versus SNR (dB) for each of
the schemes C1-C5 for the (1, 0) estimator (each curve has
been normalized by the cutoff rate of optimal binary signaling,
scheme C3). Note that:
R4. For small SNR, BPSK outperforms OOK( 1
2 ). At high
SNR, the reverse is true.
R5. The performance of the OOK(p∗ )–BPSK adaptive strategy
is nearly identical to that of optimal binary signaling.
6Note that this switching may be oscillatory, e.g., producing an BPSK–
OOK–BPSK or OOK–BPSK–OOK behavior, if ωℓ is non-monotonic in ℓ.
This may be the case if the channel correlation Rh(τ) is non-monotonic
and/or a noncausal estimator is used.
MISRA et al.: CUTOFF RATE OPTIMAL BINARY INPUTS WITH IMPERFECT CSI 2911
Normalized Cutoff Rate
1
0.9
OOK(1/2)−BPSK
0.8
0.7
0.6
0.5
0.4
0.3
OOK(p*)−BPSK
OOK(1/2)−only
BPSK−only
−10 −5 0 5 10 15 20 25 30
SNR κ (dB)
(a)
Normalized Cutoff Rate
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
OOK(p*)−BPSK
OOK(1/2)−BPSK
BPSK−only
OOK(1/2)−only
−10 −5 0 5 10 15 20 25 30
SNR κ (dB)
Fig. 9. The (normalized) cutoff rate of PSAM when using the (a) (1, 0) or (b) (1, 1) estimator for five different binary transmission strategies. Parameters:
Rh(τ) =Jo(2πfDTDτ), fDTD =1/50, andT =7. All curves are normalized to the range [0, 1] by the optimal binary input, which is not plotted
explicitly.
Therefore, using only two types of inputs, BPSK and
the OOK family, is nearly optimal.
R6. Limited to the OOK( 1
2 )–BPSK scheme, performance is
nearly identical to optimal binary signaling for moderate
to high SNR. This implies that nearly optimal transmission
can be achieved even under transmitter peakto-average
power ratio (PAPR) constraints, simply by
switching between two constellations when the SNR is
moderate to large (in this example, k ≥ 2 dB).
In Fig. 9(b) we repeat the analysis for the (1, 1) estimator.
This estimator provides at least the CSI quality of the (1, 0)
estimator, implying that BPSK will be preferred to OOK in at
least as many data slots. 7 Here, we find that BPSK is preferred
to OOK in every data slot. Note that for SNR greater than −2
dB, the optimality of scheme C3 is captured by the simple
)–BPSK adaptive scheme.
OOK( 1
2
V. DISCUSSION
We have studied cutoff rate optimal binary inputs for the
Rayleigh flat-fading channel with imperfect receiver CSI.
First, we evaluated the cutoff rate for i.i.d. fading (3), and
analyzed the optimal binary input as a function of the CSI
quality and SNR at the receiver (summarized in remarks R1-
R3). It was seen that there exists a CSI quality threshold,
⋆
ω, which characterizes the phase transition in the optimal
input versus the CSI quality at low SNR. Next, we considered
the limiting distributions – BPSK and OOK. Under OOK,
Equations (6), (7), (9) show that the cutoff rate provides a
simple characterization of the probability versus location of
the non-zero mass point as a function the CSI quality and
SNR. We derived a transitional SNR ⋆
κ (see (12) and Fig. 6)
that enables adaptive switching between these distributions
based on the CSI quality, and have shown that this scheme is
nearly optimal. Next, we applied our results to adaptive modulation
design in PSAM communications over a temporally
7 A similar conclusion holds for any estimator that provides uniformly as
good or better CSI quality than a simpler one.
(b)
correlated channel. We have shown that switching between
just BPSK and equiprobable-OOK nearly achieves optimal
binary signaling at moderate (≈ 0 dB) to large SNR, and
that switching between BPSK and generalized-OOK is nearly
optimal at all SNR.
While explicit expressions for the optimal transmission
probability, such as (6), (7), and (9), are not readily available
in a capacity analysis, similar trends have been observed [2].
Explicit expressions for the transitional SNR or CSI quality as
defined here do not readily follow for the mutual information,
and may not exist (given the sometimes drastically different
behavior of these two metrics, e.g., see [27]). As has been
previously noted in various contexts for the capacity, we have
found an extreme form of “peaky" OOK to be cutoff rate
optimal at low SNR when no CSI is available (and as long as
the CSI remains below the ⋆ ω threshold).
We have restricted our attention to binary inputs. However,
even at low SNR, binary inputs are not second-order optimal
[33], and a study of optimal higher order inputs for imperfect
CSI may be of interest.
APPENDIX
A. DERIVATION OF THE CUTOFF RATE, EQUATION (3)
Define S � {A, −B} and omit the subscript k for brevity.
Starting from (2), we get
� � �
� �
Q(s) P (y,
y �h s∈S
� �2 h | s) d�hdy = ��
�� �
Q(v)Q(w)E�h P (y|v,
v,w∈ S
y
�h)P (y|w, � �
h)dy .
Note that y|v, � �√E �
h ∼ CN
� 2 2 2 hv, E�σ v + σN and similarly
for y|w, �h. Thus, we get
� �
P (y|v,
y
�h)P (y|w, � �
h) dy =exp−
1 E|
4
�h| 2 (v − w) 2
�
�
+ σ2 N
E�σ 2 � v 2 +w 2
2
2912 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006
� �
E�σ 2v2 + σ2 N E�σ 2w2 + σ2 N
×
.
E�σ 2 � v2 +w2 �
2 + σ2 N
Following (2), we take the expectation of the above with
respect to � h ∼CN(0, �σ 2 ), yielding
� σ 2 N + �σ 2 Ev 2� σ 2 N + �σ2 Ew 2
σ2 N + �σ2 E � v2 +w2 �
1
2 + 4 �σ2 2 .
E (v − w)
Dividing the numerator and denominator by σ2 N and substituting
the result into (2) yields
Ro = − min
C(p,A,B) log2 [1 + 2p (1 − p)
� �
1+κ (1 − ω) A2 ×
� 1+κ (1 − ω) B2 1+ 1
2 κ (1 − ω)(A2 + B2 )+ 1
��
− 1 ,
4 κ ω (A + B)2
(17)
where C(p, A, B) is the constraint set on the input. Simple
algebraic manipulation yields (3).
B. PROOF OF OPTIMAL INPUTS -SECTION III
Proof of R1. Low SNR analysis (κ << 1):
Assume ω �= 1(if ω =1, the solution to (4) is easily seen
to be BPSK). Retaining the first two terms in a Taylor series
expansion of (4) about κ =0,weget
max
C(p,A,B) JL � p (1 − p)(A + B) 2
�
�
1
× ω + κ
4 (A − B)2 (3ω 2 ��
−6ω+2)− ωAB . (18)
† If ω< ⋆ ω. Consider the case where B =0.ThelowSNR
cost function (18) becomes
�
JL =(1−p) 1+ κ
�
φ(w)
,
4 p
where φ(w) � 3ω2 − 6ω +2,andwherewehaveusedthe
energy constraint. The cost function becomes arbitrarily large
if p → 0, with pA2 =1, provided φ(ω) > 0. Therefore, at low
SNR and for ω< ⋆ ω, limp→0 OOK(p) is the optimal input.
† If ω> ⋆ ω. It follows that 3ω2−6ω+2 < 0. A consequence
is that the optimal A in (18) must be finite. If not, JL will take
on an arbitrarily large negative value. Since A is finite and κ
small, we omit the κ term in (18). Removing other irrelevant
terms, we seek
max
C(p,A,B) p(1 − p)(A + B)2 . (19)
The solution to (19) is any input of the form Cp0 � � � �
=
1−p0
p0
p0,
,
where p0 ∈ (0, 1/2]. Next, using Cp0 as
p0
1−p0
a candidate set of possible solutions, we increase κ slightly,
to consider the κ term and determine � which p0 ∈
�
(0, 1/2]
1−p0
p0
maximizes (18) when A =
and B =
1−p0 .
min
C(p,A,B) JH � p(1 − p)
� √ �
1 − wAγB(κ)
× √ �� � � − 1 ,
κ2
ω 1 − 2 (A2 + B2 )+ωAB
where γB(κ) ={1, if B =0, B � κ(1 − ω), if B �= 0}.
Note that JH √ ≥−p(1 − p) ≥−1/4, with equality for A =
1
2, B =0,andp =1/2. Therefore, OOK( 2 ) is optimal as
κ →∞.
C. ON-OFF KEYING OPTIMAL FOR NO CSI - SECTION III
Setting ω =0, we seek to minimize (4) with the constraint
set C(p, A, B). Let p = p0 ≤ 1
2 be fixed, and let x =
B2 . Using the energy constraint, the minimization problem
becomes
min f(x) =
0≤x≤1
��κ+p0 � � � 2
+ x
(2p0−1)κ+κ
− x p0
p0
min
0≤x≤1
2κ2 1−p0
p0
� �
p0+κ/2
+ x p0
κ � � . (20)
2p0−1
2
p0
It can be verified that ∂f(x)/∂x ≥ 0 for x ∈ [0, 1],
implying that (20) is minimized for x = B2 =0. Therefore,
when ω =0, an On-Off keying solution is optimal.
REFERENCES
Substituting Cp0 into the above, and removing irrelevant terms
(note that p(1 − p)(A + B) 2 = AB =1for solutions in Cp0),
we must solve
��
p
max
C(p,A,B) 1 − p −
� �2
1 − p
(3ω
p
2 − 6ω +2),
which is maximized for p0 =1/2 (since 3ω2 − 6ω +2< 0).
Therefore, the optimal input distribution is C1/2 =( 1
2 , 1, 1),
or BPSK.
p0
Proof of R2. High SNR Analysis (κ →∞):
Again, assume that ω �= 1. At high SNR, the minimization
problem (4) becomes
[1] I. Abou-Fayçal, M. Médard, and U. Madhow, “Binary adaptive coded
pilot symbol assisted modulation over Rayleigh fading channels without
feedback,” IEEE Trans. Commun., vol. 53, no. 6, pp. 1036–1046, June
2005.
[2] I. Abou-Fayçal, M. Trott, and S. Shamai. “The capacity of discrete-time
memoryless Rayleigh-fading channels,” IEEE Trans. Inform. Theory, vol.
47, no. 4, pp. 1290–1301, May 2001.
[3] S. Adireddy, L. Tong, and H. Viswanathan, “Optimal placement of
training for frequency selective block-fading channels,” IEEE Trans.
Inform. Theory, vol. 48, no. 8, pp. 2338–2353, Aug. 2002.
[4] E. Arikan, “An upper bound on the cutoff rate of sequential decoding,”
IEEE Trans. Inform. Theory, vol. 34, no. 1, pp. 55–63, Jan. 1988.
[5] J. Baltersee, G. Fock, and H. Meyr, “An information theoretic foundation
of synchronized detection,” IEEE Trans. Commun., vol. 49, no. 12, pp.
2115–2123, Dec. 2001.
[6] E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: Informationtheoretic
and communications aspects,” IEEE Trans. Inform. Theory, vol.
44, no. 6, pp. 2619–2692, Oct. 1998.
[7] X. Cai and G. Giannakis, “Adapative PSAM accounting for channel
estimation and prediction errors,” IEEE Trans. Wireless Commun., vol.
4, no. 1, pp. 24–256, Jan. 2005.
[8] J. K. Cavers, “An analysis of pilot symbol assisted modulation for
Rayleigh fading channels [mobile radio],” IEEE Trans. Veh. Technol.,
vol. 40, no. 4, pp. 686–693, Nov. 1991.
[9] J. K. Cavers, “Variable-rate transmission for Rayleigh fading channels,”
IEEE Trans. Commun., vol. 20, no. 2, pp. 15–22, Feb. 1972.
[10] R.-R. Chen, B. Hajek, R. Koetter, and U. Madhow “On fixed input
distributions for noncoherent communication over high SNR Rayleigh
fading channels,” IEEE Trans. Inform. Theory, vol. 50, no. 12, pp. 3390–
3396, Dec. 2004.
[11] M. Dong, L. Tong, and B. Sadler, “Optimal insertion of pilot symbols
for transmissions over time-varying flat fading channels,” IEEE Trans.
Signal Processing, vol. 52, no. 5, pp. 1403–1418, May 2004.
[12] X. Dong and L. Xiao, “Symbol error probability of two-dimensional
signaling in Ricean fading with imperfect channel estimation,” IEEE
Trans. Veh. Technol., vol. 54, no. 2, Mar. 2005.
MISRA et al.: CUTOFF RATE OPTIMAL BINARY INPUTS WITH IMPERFECT CSI 2913
[13] R. Gallager, Information Theory and Reliable Communication. New
York: John Wiley and Sons, 1968.
[14] M. Garcia and J. Paez-Borrallo, “Tracking of time misalignments for
OFDM systems in multipath fading channels,” IEEE Trans. Consumer
Electron., vol. 48, no. 4, pp. 982–989, Nov. 2002.
[15] J. M. Geist, “The cutoff rate for on-off keying,” IEEE Trans. Commun.,
vol. 39, no. 8, pp. 1179–1181, Aug. 1991.
[16] D. Goeckel, “Adaptive coding for time-varying channels using outdated
fading estimates,” IEEE Trans. Commun., vol. 47, no. 6, pp. 844–855,
June 1999.
[17] B. Hassibi and B. Hochwald, “How much training is needed in multipleantenna
wireless links?,” IEEE Trans. Inform. Theory, vol. 49, no. 4, pp.
951–963, Apr. 2003.
[18] A. O. Hero and T. L. Marzetta, “Cutoff rate and signal design for the
quasi-static Rayleigh fading space-time channel,” IEEE Trans. Inform.
Theory, vol. 47, no. 6, pp. 2400–2416, Sep. 2001.
[19] J. Huang and S. Meyn, “Characterization and computation of optimal
distributions for channel coding,” in Proc. 37th Annual Conf. Information
Sciences Syst., Mar. 2003.
[20] W. C. Jakes, Jr., Microwave Mobile Communication. New York: Wiley,
1974.
[21] S. Jamali, and T. Le-Ngoc, Coded-Modulation Techniques for Fading
Channels. Norwell, MA: Kluwer Publishers, 1994.
[22] W. Kuo and M. P. Fitz, “Frequency offset compensation of pilot symbol
assisted modulation in frequency flat fading,” IEEE Trans. Commun., vol.
45, no. 11, pp. 1412–1416, Nov. 1997.
[23] K. Leeuwin-Boulle and J. C. Belfiore, “The cutoff rate of time-correlated
fading channels,” IEEE Trans. Inform. Theory, vol. 39, no. 2, pp. 612–
617, Mar. 1993.
[24] X. Ma, G. Giannakis, and S. Ohno, “Optimal training for block transmissions
over doubly selective wireless fading channels,” IEEE Trans.
Signal Processing, vol. 51, no. 5, pp. 1351–1366, May 2003.
[25] T. Marzetta and B. Hochwald, “Capacity of a mobile multiple-antenna
communication link in Rayleigh flat fading,” IEEE Trans. Inform. Theory,
vol. 45, no. 1, pp. 139–157, Jan. 1999.
[26] J. Massey, “Coding and moulation in digital communications,” in Proc.
Int. Zurich Seminar Digital Commun., pp. E2(1)-E2(4), Mar. 1974.
[27] R. McEliece and W. Stark, “Channels with block interference,” IEEE
Trans. Inform. Theory, vol. 30, no. 1, pp. 44–53, Jan. 1984.
[28] S. Misra, A. Swami, and L. Tong, “Optimal training for time-selective
wireless fading channels using cutoff rate,” EURASIP J. Applied Signal
Processing, vol. 2006, Article ID 47245, 2006.
[29] S. Ohno and G. Giannakis, “Capacity maximizing MMSE-optimal
pilots for wireless OFDM over frequency-selective block Rayeligh-fading
channels,” IEEE Trans. Inform. Theory, vol 50, no. 9, pp. 2138–2145,
Sep. 2004.
[30] A. Saleh and J. Salz, “On the computational cutoff rate, R0, forthe
peak-power-limited Gaussian channel,” IEEE Trans. Commun., vol. 35,
no. 1, pp. 13–20, Jan. 1987.
[31] S. Shamai and I. Bar-David, “The capacity of average and peak-powerlimited
quadrature Gaussian channels,” IEEE Trans. Inform. Theory, vol.
41, no. 4, pp. 1060–1071, July 1995.
[32] L. Tong, B. Sadler, and M. Dong, “Pilot-assisted wireless transmissions,”
IEEE Signal Processing Mag., pp. 12–25, Nov. 2004.
[33] S. Verdú, “Spectral efficiency in the wideband regime,” IEEE Trans.
Inform. Theory, vol. 48, no. 6, pp. 1319–1343, June 2002.
[34] S. Verdú, “On channel capacity per unit cost,” IEEE Trans. Inform.
Theory, vol. 36, no. 5, pp. 1019–1030, Sep. 1990.
Saswat Misra was born in College Park, MD, USA
in 1978. He received the B.S. in electrical engineering
from the University of Maryland at College Park
in 2000 and the M.S. in electrical engineering from
the University of Illinois at Urbana-Champaign in
2002. Since 2002, he has been a Research Scientist
at the Army Research Laboratory (ARL) in Adelphi,
MD in the Communications and Network Systems
division. Mr. Misra has previously worked on optimal
training design for wireless communication
systems; an area in which he has published several
papers and holds two patents (pending). Since Fall 2005, he has been a Ph.D.
candidate at Cornell University. He is currently studying routing and security
issues in wireless military networks.
Ananthram Swami (SM’96) is a Senior Research
Scientist and Fellow of the Army Research Laboratory,
Adelphi, MD, USA, where he works in the
broad area of signal processing for communications.
He received the B.S. degree from the Indian Institute
of Technology, Bombay, India; the M.S. degree from
Rice University, Houston, TX; and the Ph.D. degree
from the University of Southern California, Los
Angeles, all in electrical engineering.
Dr. Swami has held positions with Unocal Corporation,
the University of Southern California, CS-
3, and Malgudi Systems. He was a Statistical Consultant to the California
Lottery, and has held visiting faculty positions at INP, Toulouse, France. Dr.
Swami is chair of the IEEE Signal Processing Society’s TC on Signal Processing
for Communications, an associate editor of the IEEE TRANSACTIONS
ON WIRELESS COMMUNICATIONS, and of the IEEE TRANSACTIONS ON
SIGNALPROCESSING. He was co-organizer and co-chair of a 1999 ASA-IMA
Workshop on Heavy-Tailed Phenomena, and of a 2002 ONR/NSF/ARO/CTA
Workshop on “Future Challenges to Wireless Communications and Networking.”
He was co-guest editor of a 2004 special issue of the IEEE Signal
Processing Magazine (SPM) on “Signal Processing for Networking.” He is
also co-guest editor of an upcoming IEEE SPM special issue on “Distributed
Signal Processing in Sensor Networks,” an EURASIP JASP special issue
on “Reliable Communications over Rapidly Time-varying Channels,” and an
EURASIP JWCN special issue on “Wireless Mobile Ad Hoc Networks.”
Lang Tong (F’05) is a Professor in the School
of Electrical and Computer Engineering, Cornell
University, Ithaca, NY, USA. He received the B.E.
degree from Tsinghua University, Beijing, China,
in 1985, and M.S. and Ph.D. degrees in electrical
engineering in 1987 and 1991, respectively, from
the University of Notre Dame, Notre Dame, Indiana.
He was a Postdoctoral Research Affiliate at the Information
Systems Laboratory, Stanford University
in 1991. He was also the 2001 Cor Wit Visiting
Professor at the Delft University of Technology.
Dr. Tong received the Young Investigator Award from the Office of Naval
Research in 1996, the Outstanding Young Author Award from the IEEE
Circuits and Systems Society, the 2004 Best Paper Award (with M. Dong)
from the IEEE Signal Processing Society, and the 2005 Leonard G. Abraham
Prize Paper Award (with P. Venkitasubramaniam and S. Adireddy) from the
IEEE Communications Society. His areas of interest include statistical signal
processing, wireless communications, communication networks and sensor
networks, and information theory.
