**6. Numerical results**

292 Wireless Communications and Networks – Recent Advances

The channel inversion and truncated inversion policies use codes designed for AWGN channels, and are therefore the least complex to implement, but in severe fading conditions

The truncated channel inversion policy inverts the channel fading only above a fixed cutoff fade depth *γ*0. The capacity with this truncated channel inversion and fixed rate policy

0

log 1 1 1 , !

0 0

*p i*

*CB f i*

1 0

*e pp*

0 0

*CB f L i*

*p i*

1 0

 

( )!

10-15 terms to be summed in order to achieve accuracy at the 5th significant digit.

 

over Weibull fading channels when MRC diversity is applied in the form of:

(1 ) 1,

 

*e p Lp*

*L pL*

 

( )!

 

log 1 1 1 , !

*L k*

Convergence of infinite series expressions in (1.42) and (1.43) is rapid, since we need about

After substituting (1.6) into (1.41) we can obtained expression for the CIFR channel capacity

(1 ) 1,

*p*

<sup>2</sup> log 1 1 / 1 

After substituting (1.2) into (1.40) we can obtain expression for the CIFR channel capacity

1

*kL L k*

0

1

*k k*

0

. (1.42)

(1.43)

*out tifr CB p d P* . (1.41)

*ei i*

*i*

*k*

 

*e iL i*

*i*

 

1

*L k*

1

 

  **5. Truncated channel inversion with fixed rate** 

<*C*>*tifr*/*B* is derived in [8]:

**5.1** *κ***-***μ* **fading channels** 

3

*f*

4

*f*

0

*tifr*

*MRC*

0

**5.2 Weibull fading channels** 

*p*

*p p*

over *κ*-*μ* fading channel in the following form:

2 3

*tifr k*

 

Case when MRC diversity is applied can be modelled by:

2 4

 

*pp p*

*p L*

they exhibit large capacity losses relative to the other techniques.

In order to discuss usage of diversity techniques and adaptation policies and to show the effects of various system parameters on obtained channel capacity, numerically obtained results are graphically presented.

In Figs. 1.1 and 1.8 channel capacity without diversity, *<C>ora* given by (1.22), for the cases when *κ*-*μ* and Weibull fading are affecting channels, for various system parameters are plotted against *γ* . These figures also display the capacity per unit bandwidth of an AWGN channel, *CAWGN* given by:

$$C\_{AWGN} = B \log\_2(1+\mathcal{Y})\,. \tag{1.45}$$

Considering obtained results, with respect that *CAWGN* = 3.46 dB for average received SNR of 10dB we find that depending of fading parameters of *κ*-*μ* and Weibull distribution, channel capacity could be reduced up to 30 %. From Fig. 1.1 we can see that channel capacity is less reduced for the cases when fading severity parameter *μ*, and dominant/scattered components power ratio *κ*, have higher values, since for smaller *κ* and *μ* values the dynamics in the channel is larger. Also from Fig. 1.8 we can observe that channel capacity is less reduced in the areas where Weibull fading parameter *β* has higher values.

Figures 1.2-1.4,1.6 show the channel capacity per unit bandwidth as a function of for the different adaptation policies with MRC diversity over *κ*-*μ* fading channels. It can be seen that as the number of combining branches increases the fading influence is progressively reduced, so the channel capacity improves remarkably. However, as *L* increases, all capacities of the various policies converge to the capacity of an array of *L* independent AWGN channels, given by:

$$C\_{\text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\tiny \text{\text}}}}}}}^{\text{\text{\textdegree C}}} = B \log\_2(1 + L\gamma) \tag{1.46}$$

Thus, in practice it is not possible to entirely eliminate the effects of fading through space diversity since the number of diversity branches is limited. Also considering downlink (base station to mobile) implementation, we found that mobile receivers are generally constrained in size and power.

In Fig. 1.5 comparison of the channel capacity per unit bandwidth with CIFR adaptation policy, when SC and MRC diversity techniques are applied at the reception is shown. As expected, better performances are obtained when MRC reception over *κ*-*μ* fading channels is applied.

Figure 1.7 shows the calculated channel capacity per unit bandwidth as a function of for different adaptation policies. From this figure we can see that the OPRA protocol yields a small increase in capacity over constant transmit power adaptation and this small increase in capacity diminishes as increases. However, greater improvement is obtained in going from complete to truncated channel inversion policy. Truncated channel inversion policy provides better diversity gain compared to complete channel inversion varying any of parameters.

Fig. 1.1 Average channel capacity per unit bandwidth for a κ-μ fading and an AWGN channel versus average received SNR.

Fig. 1.2 Power and rate adaptation policy capacity per unit bandwidth over κ-μ fading channels, for various values of diversity order.

from complete to truncated channel inversion policy. Truncated channel inversion policy provides better diversity gain compared to complete channel inversion varying any of

0 2 4 6 8 10 12 14 16 18

Fig. 1.1 Average channel capacity per unit bandwidth for a κ-μ fading and an AWGN

power and rate adaptation policy

Average received SNR [dB]

0 2 4 6 8 10 12 14 16 18

*L*=1 *L*=2 *L*=3 *L*=4 *L*=5 *L*=6

Average received SNR (dB)

Fig. 1.2 Power and rate adaptation policy capacity per unit bandwidth over κ-μ fading

increases. However, greater improvement is obtained in going

in capacity diminishes as

parameters.

1

channels, for various values of diversity order.

= 2.2, *k* = 6dB

Capacity per unit bandwidth [bits/sec/Hz]

channel versus average received SNR.

2

3

4

Capacity per unit bandwith <C>/B [Bits/Sec/Hz]

5

6

7

 AWGN fading channels =1, =1 =1, =2 =2, =1 =2, =2

Fig. 1.3 ORA policy capacity per unit bandwidth over κ-μ fading channels, for various values of MRC diversity order.

Fig. 1.4 CIFR policy capacity per unit bandwidth over κ-μ fading channels, for various values of MRC diversity order.

Similar results are presented considering channels affected by Weibull fading. Figures 1.9- 1.12 show the channel capacity per unit bandwidth as a function of for the different adaptation policies with *L*-branch MRC diversity applied. Comparison of adaptation policies is presented at Fig. 1.13.

Fig. 1.5 CIFR policy capacity per unit bandwidth over κ-μ fading channels, for MRC and SC diversity techniques various orders.

Fig. 1.6 TIFR policy capacity per unit bandwidth over κ-μ fading channels, for various values of MRC diversity order.

CIFR adaptation policy no diversity SC L=2 SC L=3 MRC L=2 MRC L=3

0 2 4 6 8 10 12 14 16 18

Average received SNR [dB]

0 2 4 6 8 10 12 14 16 18

Average received SNR [dB]

Fig. 1.6 TIFR policy capacity per unit bandwidth over κ-μ fading channels, for various

 = 6 dB = 2

Fig. 1.5 CIFR policy capacity per unit bandwidth over κ-μ fading channels, for MRC and SC

TIFR adaptation policy no diversity MRC L=2 MRC L=3 MRC L=4 MRC L=5 MRC L=6

 = 6 dB = 2

Capacity per unit bandwith <C>/B [Bits/Sec/Hz]

values of MRC diversity order.

Capacity per unit bandwith <C>/B [Bits/Sec/Hz]

diversity techniques various orders.

Fig. 1.7 Comparison of adaptation policies over MRC diversity reception in the presence of κ-μ fading.

Fig. 1.8 Average channel capacity per unit bandwidth for a Weibull fading for various values of system parameters and an AWGN channel versus average received SNR [dB].

Fig. 1.9 ORPA policy capacity per unit bandwidth over Weibull fading channels, for various values of MRC diversity order.

Fig. 1.10 ORA policy capacity per unit bandwidth over Weibull fading channels, for various values of MRC diversity order.

4 6 8 10 12 14 16 18

Average received SNR [dB]

0 2 4 6 8 10 12 14 16 18

Average received SNR [dB]

Fig. 1.10 ORA policy capacity per unit bandwidth over Weibull fading channels, for various

= 2.5

Fig. 1.9 ORPA policy capacity per unit bandwidth over Weibull fading channels, for various

0

0

values of MRC diversity order.

1

2

3

4

Capacity per unit bandwith <C>/B [Bits/Sec/Hz]

5

6

7

ORA adaptation plocy no diversity MRC L=2 MRC L=3 MRC L=4 MRC L=5 MRC L=6

values of MRC diversity order.

1

= 2.5

OPRA adaptation plocy no diversity MRC L=2 MRC L=3 MRC L=4

2

3

4

Capacity per unit bandwith <C>/B [Bits/Sec/Hz]

5

6

7

Fig. 1.11 CIFR policy capacity per unit bandwidth over Weibull fading channels, for various values of MRC diversity order.

Fig. 1.12 TIFR policy capacity per unit bandwidth over Weibull fading channels, for various values of MRC diversity order.

Fig. 1.13 Comparison of adaptation policies over MRC diversity reception in the presence of Weibull fading.

The nested infinite sums in (1.38) and (1.39), as can be seen from Table 1, for dual and triple branch diversity case, converge for any value of the parameters *κ*, *μ* and . As it is shown in this Table 1, the number of the terms need to be summed to achieve a desired accuracy, depends strongly on these parameters and it increases as these parameter values increase.


Table 1. Number of terms that need to be summed in (1.38) and (1.39) to achieve accuracy at the specified significant digit for some values of system parameters.
