Consider now the capacity of the channel whose model is shown in Figure 1.29. In our considerations we follow the work of Goldsmith and Varaiya (1997). The channel input signal is band-limited toB Hz. The channel is modeled by a multiplier that performs signal amplification by√
g(i), whereiis the current timing instant. Let us assume thatg(i) (g(i)≥0) is a sample function of the stationary and ergodic random process characterized
Encoder Power control
Receive
filter Decoder Channel estimator g(i) n(i)
Message Code
sequence
Signal
g(i)^ w^ w
Figure 1.29 System model with flat fading channel, channel estimation and feedback channel (dashed line) (Goldsmith and Varaiya 1997IEEE 1997)
by a unit mean and a given probability density function. Additive white Gaussian noise with the power density N0/2 is added to the signal that is modified by the channel coefficient√
g(i). The time varying coefficient√
g(i)models a situation often appearing on radio channels, in which the received signal level is varying in time and the whole signal spectrum is basically attenuated in the same way. We say that it is a flat fading channel and the transmitted signal is the subject offlat fading.7At the receiver, the input filter limits the bandwidth of the received signal toB Hz, the channel estimator determines the current value of the channel coefficient g(i), and the decoder decodes the received codeword. The dashed line in Figure 1.29 denotes the feedback channel from the receiver to the transmitter, which allows for selection of the appropriate transmitted power level and the particular coding scheme. Let us denote the mean power of the transmitted signal asP. Thus, the SNR at the output of the receive filter is
γ (i)=P g(i)/(N0B) (1.128)
At theith moment the channel is practically flat with the bandwidth limited toB Hz. In this case its capacity is given by formula (1.111), i.e. for a given value ofγ it is equal to
Cγ =Blog(1+γ ) (1.129)
Let the probability distribution of the SNR γ be p(γ ). In practice, it refers to the processg(i). In this case the channel capacity can be understood as an ensemble average of the capacityCγ, i.e.
C=
$
γ
Cγp(γ )dγ =
$
γ
Blog(1+γ )p(γ )dγ (1.130)
One can show that the capacity defined by formula (1.130) is lower than the capacity of a flat channel band-limited toB Hz with the SNR equal to the average SNR of P /(N0B).
So far we have presented the formula for the capacity of a flat fading channel when the input signal has a constant power equal to P. One can state the following problem:
How should we select the transmitted signal power with respect to the current value of the SNR,γ, at the given mean signal powerP, in order to maximize the capacity given by the formula
C(P )=max
P (γ )
$
γ
Blog
"
1+P (γ )γ P
#
p(γ )dγ (1.131)
The constraint for the choice of the transmitted signal power is its mean power, which is expressed by the formula
$
γ
P (γ )p(γ )dγ =P (1.132)
7The channel model is calledselective fadingif in some parts of the passband substantial attenuation is introduced.
Let us note that if the ratioγ is given by formula (1.128), then the expressionP (γ )γ /P = P (γ )g(i)/(N0B)determines the current value of the SNR. As shown by Goldsmith and Varaiya (1997), there exists a channel coding scheme that achieves efficiencyR < C(P ) with a sufficiently small codeword detection error probability when the mean input signal power P is applied. In contrast, the probability of erroneous codeword decoding of the channel code applied in the considered channel with the efficiency R > C(P ) is higher than zero.
Let us find the rule that should govern the selection of the power levelP (γ )of the chan- nel input signal depending on parameterγ, so that the channel capacityC(P )described by formula (1.131) for the constraint (1.132) is maximized. For this purpose we apply once more a similar procedure to that applied in derivation of the optimum input signal power density spectrum that maximizes the capacity of the channel with transfer function H (f ). As previously, let us apply Theorem 1.12.1. In the current case, the integrated function of the maximized integral and the integrated function of the constraint are of the form
F (γ , P (γ ))=Blog2
"
1+P (γ )γ P
# p(γ )
ϕ(γ , P (γ ))=P (γ )p(γ ) (1.133)
The optimum value of the applied input powerP (γ )results from solution of the following equation
∂F (γ , P (γ ))
∂P (γ ) +λ∂ϕ(γ , P (γ ))
∂P (γ ) =0 (1.134)
Calculation of equation (1.134) by applying (1.133) gives the following dependence Bp(γ )log2eã P
P +P (γ )γ ã γ
P +λp(γ )=0 (1.135) The coefficientγ is selected from the range in which p(γ ) >0, therefore the equation from which we deriveP (γ ) has a simpler form
Blog2eã P
P +P (γ )γ ã γ
P +λ=0 (1.136)
Applying the following substitution
γ0= − λP
Blog2e (1.137)
after simple calculations we achieve the following result
P (γ )=
P
1 γ0 − 1
γ
forγ ≥γ0
0 forγ < γ0
(1.138)
Let us analyze the meaning of (1.138). Forγ ≥γ0, the transmitted power should increase with increase of the mean SNR γ. However, if this ratio falls below a certain threshold valueγ0, we should abandon transmission of the signal. The value ofγ0results from the established limit for the mean power (1.132) and is a solution of the equation
$∞
γ0
P 1
γ0
− 1 γ
p(γ )dγ =P (1.139)
In turn, after simple calculations, applying formula (1.138) in expression (1.131) we obtain the following result
C(P )=
$∞
γ0
Blog2 γ
γ0
p(γ )dγ (1.140)
For comparison let us consider a situation in which the transmitter applies the knowl- edge about the channel attenuation in a nonoptimal way, namely it transmits the signal with a higher power if the channel attenuates the signal more. This means that the transmitter power is selected according to the rule
P (γ )=Pσ
γ (1.141)
where σ is the mean value of the SNR and constraint (1.132) holds. Thus, the constant σ results from this constraint, which has the form
$
γ
Pσ
γdγ =P i.e. σ = 1
E[1/γ] (1.142)
and the channel capacity is
C(P )=Blog2(1+σ )=Blog2
1+ 1
E[1/γ]
(1.143) At the end of this section consider the case in which there is no feedback channel that could be used to transmit data related to the channel estimated at the receiver to the transmitter. This time the knowledge about the channel can be used only by the receiver.
The knowledge of the channel coefficient √
g(i) allows us to equalize the level of the received signal, i.e. multiply it by 1/√
g(i). Thus, the received signal power is constant and equal toP, whereas the instantaneous noise power isBN0/g(i). Therefore, the SNR is γ =P g(i)/(BN0)and it is the same as in (1.128). We conclude that in this case the channel capacity is also described by formula (1.130).
Figure 1.30 cited after Goldsmith and Varaiya (1997) presents examples of the capacity curves, normalized with respect to the channel bandwidth B, as a function of the mean SNR in a dB scale for the log-normal probability density function of the channel coefficient
5 10 15 20 25 30 0
2 4 6 8 10 12 14
Mean SNR [dB]
C/B [bit /s /Hz]
1 2
3 4
Figure 1.30 Capacity per spectum unit of the channel with log-normal fading (σγ =8 dB): (1) system with AWGN flat channel, (2) system with the optimum use of the channel state information at the transmitter and receiver, (3) system with the optimum use of the channel state information at the receiver only, (4) system with inversion of the power level at the transmitter. Reproduced by permission of IEEE (Goldsmith and Varaiya 1997IEEE 1997)
√g(i)=q. This probability density function is described by the formula
p(q)=
√ 1
2π σqqexp
−
lnq−mq
2
2σq2
forq≥0
0 forq <0
(1.144)
Knowing the rules of transformation of random variables, on the basis of probability density functionp(q)one can easily receive the probability density function of the variable γ =P q2/(BN0). The normalized capacity of the flat AWGN channel provides a reference curve in Figure 1.30.
From analysis of the curves shown in Figure 1.30 we can see that, in general, fading decreases channel capacity because the flat AWGN channel has the highest capacity.
The next channel, as far as quality is concerned, is the one in which optimum power control is performed at the transmitter and the channel state information is used both by the transmitter and receiver. The capacity of the channel for which the channel state information is applied exclusively at the receiver by compensating the channel attenuation is only slightly lower. Finally, the lowest capacity is achieved when the transmitted signal power is increased if the channel attenuation increases. Goldsmith and Varaiya (1997) present similar results for other probability density functionsp(q), however the described tendencies are basically preserved.