# Slow AdaptiveM-QAM under Third-Party Received Signal Constraints in Shadowing Environments

Authors: Yan Li; Shalinee Kishore
Journal: Research Letters in Communications
 (2008)

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## Abstract

Motivated by recent interest in dynamic spectrum sharing between primary and secondary spectrum
users, we investigate slow adaptive quadrature amplitude modulation (QAM) under third-party received power constraints. The optimal rate and power adaptation schemes are derived to maximize the average spectral efficiency (SE) for the secondary users. Additionally, closed-form expressions of the achievable SE are presented for correlated lognormal shadow fading environments. Our analytical and numerical results underscore the importance of exploiting shadowing correlation in the proposed adaptive schemes.


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## Body

## 1. Introduction

Opportunistic
secondary access of licensed spectrum bands permits secondary users to transmit
on these spectral bands as long as the primary licensee's interference
temperature is below its tolerable level. This spectrum sharing approach has
motivated recent studies of the channel capacity with constrained
received-signal power at a third-party receiver [1, 2]. Figure 1 shows the
associated system model in which a secondary transmitter communicates with a
secondary receiver over a licensed-band, resulting in interference to a primary
user. The channel gain between the transmitter and the primary receiver is
denoted as g;
the channel gain between this transmitter and the secondary receiver is denoted
as h.
In [1], authors assume
additive white Gaussian noise (AWGN) channels, where g and h are deterministic. In [2], authors assume flat fading
channels, where the instantaneous values of g and h are known at the transmitter. This letter
presents a communication-theoretic study of a similar system model but assuming
the transmitter is only aware of slow fading conditions. We assume
discrete-time block fading channels with stationary and ergodic time-varying
power gains g=g¯rg and h=h¯rh; rg and rh are (unitmean) fast small-scale fading
coefficients, and g¯ and h¯ are slow large-scale fading coefficients. Due
to (bandwidth, delay) limitations of the feedback channel, we assume only h¯ and its correlation with g¯ are reliably known at the transmitter. The
correlation between g¯ and h¯ can be fed back by a band manager [3], which coordinates the two
parties. The transmitter then uses this information to apply a slow adaptive
quadrature amplitude modulation (QAM) technique [4, 5]. We derive the optimal rate and power adaptation
policies that maximize the average spectral efficiency (SE) for secondary
users. Under optimal adaptation, we present closed-form expressions for
achievable SE in correlated lognormal shadowing environments. The results
underscore the importance of knowing/estimating shadow fading correlations in
the proposed adaptive scheme.Figure 1
System model.


## 2. Optimal Power and Rate Adaptation

Assuming unit power AWGN at the primary and secondary
receivers, the instantaneous BER forM-QAM can be approximated as [6, 7](1)BERin(h)≈c1exp(−c2hPM−1),
where P is the transmit power, M is the QAM constellation size. c1 and c2 are two parameters optimized via curve
fitting. Assuming Nakagami-m fading, the short-term average BER over fast
fading can be calculated as(2)BERst(h¯)=∫0∞BERin(h¯rh)frh(rh)drh=∫0∞BERin(h¯rh)mmrhm−1Γ(m)exp(−mrh)drh=c1(c2h¯P(h¯)m(M(h¯)−1)+1)−m,
where P(h¯) and M(h¯) are two transmission parameters adaptive to h¯.
We consider here continuous rate adaptation, where the set of signal
constellations is unrestricted [7, 8],
that is, our results will provide an upper bound if only a discrete finite set
of constellations is available. It is worth noting that in [5], exact average BER
expression has been derived for discrete rate QAM with diversity reception over
flat fading and shadowing channels. We do not apply this accurate expression in
our analysis here due to the intractability of noninvertible integrals.
Instead, we employ the approximation above as it provides analytical means to derive
power and rate adaptation schemes for the system under study.Given the short-term average BER constraintBERst=BER⋆,
the optimal rate adaptation policy must satisfy(3)M(h¯)=1+Kh¯P(h¯),where(4)K:=c2m[(c1/BER⋆)1/m−1].Furthermore, the optimal power
adaptation policy for maximal long-term average SE is the solution
to(5)maxP(h¯)𝔼{log2(1+Kh¯P(h¯))}subjectto𝔼{g¯rgP(h¯)}≤PI,P(h¯)≥0,where PI is the maximal average interference power
tolerable at the primary receiver.For this problem, we can show the following.Case 1 (whenh¯/𝔼{g¯∣h¯} is not a constant).
The optimal
power allocation policy is(6)P⋆(h¯)=1K𝔼{g¯∣h¯}(1κ0−1h¯/𝔼{g¯∣h¯})+︸waterfillingoverh¯/𝔼{g¯∣h¯},where (•)+=max(0,•), and κ0 is the solution of 𝔼{𝔼{g¯∣h¯}P⋆(h¯)}=PI.
Note that κ0 is the cutoff ratio of h¯/E{g¯∣h¯} and(7)P⋆(h¯){≥0,ifh¯/E{g¯∣h¯}≥κ0,=0,otherwise.P⋆(h¯) is the product of a water-filling solution
over h¯/𝔼{g¯∣h¯} and (K𝔼{g¯∣h¯})−1.
Since 𝔼{g¯∣h¯} is a function of h¯, P⋆(h¯) is not a pure water-filling solution in
general. However, when g¯ and h¯ are independent such that 𝔼{g¯|h¯}=E{g¯},
(6) reduces to the classic water-filling solution.
Using the optimal power and rate adaptation scheme,
the long-term average SE can be expressed as(8)SE⋆=∫h¯/𝔼{g¯∣h¯}≥κ0log2(h¯κ0𝔼{g¯∣h¯})fh¯(h¯)dh¯.Case 2 (whenh¯/𝔼{g¯∣h¯} is equal to a constant).
In this case, the optimal power allocation policy is
channel inversion:(9)P⋆(h¯)=RPIh¯,and the average SE
becomes
(10)SE⋆=log2(1+KRPI).


## 3. Lognormal Shadowing

In contrast to
the small-scale fading effects, which are typically independent from one
terminal to another, shadowing effects tend to be correlated over much larger
distances. For example, [9] presents shadowing cross correlation between several
base stations and a mobile terminal in a 1900 MHz GSM system; the resulting
correlations range from −0.34 to 0.43.For correlated lognormal shadowing, we can writeg¯=eXg and h¯=eXh,
where Xg and Xh are jointly Gaussian distributed with means μg and μh,
variances σg2 and σh2,
and a correlation coefficient ρ.
Using this model, we can write
(11)𝔼{g¯∣h¯}=h¯(σg/σh)ρexp[μg−σgσhρμh+(1−ρ2)σg22].Based on (11), we observe that
the optimal power and rate adaptation scheme depends on the relationship
between σh and ρσg,
which dictates if 𝔼{g¯∣h¯} is a linear function of h¯.
We consider all three possible functional relationships between 𝔼{g¯∣h¯} and h¯. Substituting (11) into the analytical
expressions presented in Section 2 and resorting to commonly used
manipulations, we have the following.Case 1 (whenσh>ρσg).
The probability of service (i.e., the complement of
the outage probability) is(12)Pr(h¯𝔼{g¯∣h¯}≥κ0)=Pr(h¯≥exp[μh+ψ(κ0)σh])=Q[ψ(κ0)],where Q(•) is the Gaussian right-tail probability
function, and(13)ψ(κ0):=1σh−ρσg[logκ0+μg−μh+(1−ρ2)σg22].The achievable average SE
is(14)SE⋆=log2(e)(σh−ρσg)×{12πexp[−ψ2(κ0)2]−ψ(κ0)Q[ψ(κ0)]},where κ0 is solved from(15)KPI=K𝔼{g¯rgP⋆(h¯)}=K𝔼{𝔼{g¯∣h¯}P⋆(h¯)}=1κ0Q[ψ(κ0)]−exp(σg2+σh2−2ρσgσh2+μg−μh)×Q[ψ(κ0)+σh−ρσg].Case 2 (whenσh=ρσg).
In this
case,(16)h¯E{g¯∣h¯}=exp(μh−μg+σh2−σg22):=R.Thus, the optimal adaptation
policy is channel inversion, and the achievable SE can be obtained by
substituting (16) into (10).Case 3 (whenσh<ρσg).
The probability
of service is(17)Pr(h¯𝔼{g¯∣h¯}≥κ0)=Pr(h¯≤exp[μh+ψ(κ0)σh])=1−Q[ψ(κ0)].The achievable average SE
is(18)SE⋆=log2(e)(ρσg−σh)×{12πexp[−ψ2(κ0)2]+ψ(κ0){1−Q[ψ(κ0)]}},where κ0 is solved from(19)KPI=1κ0{1−Q[ψ(κ0)]}−exp(σg2+σh2−2ρσgσh2+μg−μh)×{1−Q[ψ(κ0)+σh−ρσg]}.


## 4. Numerical Results

In our
numerical results, we assumeBER⋆=10−3,
the instantaneous BER approximation parameters c1=0.2 and c2=1.6 [7], the tolerable interference-to-noise ratio (INR) at
the primary receiver is 0 dB, the fast fading is Rayleigh distributed,
and the slow large-scale fading conditions are lognormal. In Figure 2, we plot
the average SE versus the path loss difference μh,dB−μg,dB,
assuming that σg,dB=σh,dB=8 dB. Note that μ{g,h},dB=10log10(e)μ{g,h} and σ{g,h},dB=10log10(e)σ{g,h}. g¯ and h¯ are assumed correlated with (a) ρ=0, −0.3, and −0.7;
and (b) ρ=0, 0.3, 0.8, and 1.
For negative correlations, the average SE increases as ρ decreases. This is because (1) higher h¯ implies lower g¯,
providing better opportunity for secondary transmissions; and (2) smaller ρ (or larger |ρ|) implies heavier correlation, implying h¯ gives more complete information on g¯ and enables better adaptation over the
time-varying channel. As reference, we also plot the average SE achieved using
the variable-rate constant-power adaptation scheme. We observe that by adapting
the transmit power, we can achieve much higher average SE when the correlation
is heavy. When no correlation exists, the increase in SE, achieved via classic
water-filling solution, is not significant compared with the constant-power
scheme. For positive correlations, we do not see a clear trend between the
average SE and ρ due to two contrary effects. On the one hand, g¯ and h¯ are varying statistically along the same
direction; thus, a large ρ limits good transmission opportunities for the
secondary system. On the other hand, heavier correlation provides more complete
channel information, aiding in channel adaptation. We can observe that when the
path-loss difference is small, the average SE increases as ρ decreases, which means that the first effect
is dominant. However, the second effect becomes dominant as the path-loss
difference increases, for example, when μh,dB−μg,dB is greater than 18 dB, the average SE for ρ=1 (achieved by the channel inversion) becomes
the highest amongst all curves.Average SE versus the path loss differenceμh,dB−μg,dB. σg,dB=σh,dB=8 dB.
(a)
Negative correlation(b)
Possitive correlation


## 5. Conclusions

We investigate
slow adaptive QAM technique under the third-party received power constraints.
Optimal rate and power adaptations are derived to maximize average SE for
secondary users. We present closed-form expressions of achievable SE for
correlated lognormal shadowing environments. Results underscore the importance
of exploiting shadowing correlation in proposed adaptive schemes.



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*Source: 121383-2008-06-12.xml*