**3. Achievable capacity limit for a single link with multipath fading**

In order to realize long distance coverages by single links with multipath effects in wireless mesh networks in rural areas, the IEEE 802.11a and IEEE 802.11n standards commodity devices can be used. This is because these devices are off-the-shelves, operate in multiple ISM channel bands and are affordable to the rural communities (Kyansanur & Vaidya, 2004). That is to say that only fewer radio interface cards at each node are needed than the number of non-overlapping frequency channels freely available. Kyasanur and Vaidya emphasized how expensive it could be to equip a node with one interface card for each frequency channel. The IEEE 802.11a standard, for example, offers 24+ non-overlapping channels and configuring a commensurate number of radio interface cards on each node might be unnecessary costly. As a result, many IEEE 802.11 interface cards can be switched from one channel to another, albeit at the cost of a switching delay. Moreover, the advantage of eliminating frame losses due to path-dependent (e.g., multipath fading effects), locationdependent (e.g., noise effects), and statistically independence between different receiving radios can be achieved by using multi-radio diversity principle (Miu, Balakrishnan & Koksal, 2007). The idea is that even when each individual reception of a data frame is erroneous, it might still be possible to combine the different versions to recover the correct version of the frame. In this study, the question to be addressed is that what is the capacity expression for single links with multipath effects in a rural based wireless mesh network. It is understood that most of previous studies solve capacity problem with simple channel models that may not reflect the true wireless channel conditions (Gupta & Kumar, 2000).

#### **3.1. IEEE 802.11a air interface**

154 Wireless Mesh Networks – Efficient Link Scheduling, Channel Assignment and Network Planning Strategies

links, nodes, and network failures in the area.

Nut Farm House

Wireless link

Mesh Client

Mesh Client

(Source: *http://wirelessafrica.meraka.org.za/*)

(Telkom) VSAT Gateway

Internet Sentech

Nut Farm House

**Figure 4.** Mesh network architecture at Peebles valley in South Africa

Traditionally, the PVM is endowed with VSAT link that provides the network at the clinic with 2 Gbits per month at a download rate of 256 kbps and an upload rate of 64 kbps (Johnson & Roux 2008). The clinic provides 400 Mbps per month available to the single radio mesh network. The single radio mesh has nine users (mesh routers) so that each user (mesh router) receives about 44.4 Mbps per month on average. This traffic bandwidth drops downstream the network from the satellite gateway to the terminal users. This is due to lack of single radio network resiliency against effects of wireless multipath. However, in this document we believe that the design of the HPNs making the BB4allTM architecture can be a suitable candidate for improved capacity in multipath environment (BelAir Networks, 2006). As a result high data rates as the network scales away from the satellite gateway can be realized in the PVM deployment. The HPNs utilize the multiplicity of the low cost radio devices and non-overlapping channels to improve capacity delivered across the network.

ACTS Clinic

Nurse House USAID Offices

ACTS Clinic accommodation flats

Mesh Client

Mesh Client

High School

ACTS Clinic administration blocks

**ACTS CLINIC**

Mesh Client

Hospice

Mesh Clients

Mesh Client

Peebles valley

Thus, the BB4allTM architecture constitutes a gateway connected to the internet via Sentech VSAT to the Peebles valley or ACTS clinic. Within the ACTS clinic there can be mesh

Auto-organizable connectivity against severe climatic conditions that commonly cause

Peebles valley

Wireless link

The standard IEEE 802.11a specifies an over-the-air interface between two wireless routers or between a wireless client and a router. It provides up to 54 Mbps in the 5 GHz frequency band and uses an orthogonal frequency division multiplexing (OFDM) encoding scheme.

This implies that a frequency selective channel is the most suitable approach to model the IEEE 802.11a air interface (Tse & Viswanath, 2005). This is because a frequency selective channel perfectly captures effects of multipath on signal propagation (i.e., due to terrain irregularity and tree foliage). The OFDM scheme is basically the preferred method to the frequency hopping spread spectrum (FHSS) or direct sequence spread spectrum (DSSS) schemes due to its robust performance over multipath. In this context, the IEEE 802.11a radio interface cards (Intini, 2000) make use of OFDM to provide high capacity over parallel wireless channels. In their definition, Tse and Viswanath (2005) states that a parallel channel is a channel which consists of a set of non-interfering sub-channels, each of which is corrupted by independent additive white Gaussian noise (AWGN).

To obtain the capacity over single link wireless medium, we assume each *m*th sub-channel of a parrallel channel is allocated a waterfilling power *mp* such that the average power constraint *P* is still met on each input OFDM symbol to the multipath channel. Also consider that the AWGN power level to a parallel channel is *N*0 and the co-channel interference caused by neighbouring transmissions is denoted as *I* . These parameters may be held constant in practice considering that most rural network applications are characterized by constant and low interference levels (Ismael et al., 2008). Then, the maximum capacity per every OFDM symbol of a reliable communication over *Mc* parallel streams or subcarriers is given by:

$$R\_{M\_c} = \sum\_{m=0}^{M\_c - 1} \log\_2 \left( 1 + \frac{p\_m \left| \tilde{h}\_m \right|^2}{\left( N\_0 + I \right)} \right) \text{ bits/OFDM symbol,} \tag{1}$$

whereby the achievable capacity per link in bits/s/Hz for each parallel stream is written as

$$R\_{multiply} = R\_{M\_c} / M\_{c'} \text{ bits/s/Hz.} \tag{2}$$

Achievable Capacity Limit of High Performance Nodes for Wireless Mesh Networks 157

*antenna antennaTxT antennaRxV* (5)

is the path loss exponent. We denote *antenna* as the

, the known Fourier Transform

 

From (4), *P* is the maximum power allowed per sub-carrier, *L* is the number of paths

combined antenna gain which is simply the product of the transmitter and the receiver antenna gains, 0 *d* is the reference distance (Abhayawardhana et al., 2007). The combined

Inserting the result in (5) into the expression in (4) reveals that the higher the combined antenna gain, the higher the achievable link capacity. The improved antenna gain is the main attractive feature that the HPN based BB4allTM architecture offers to the conventional standards (Makitla, Makan & Roux, 2010). From (3), to view effects of frequency

1 1

*j lm j lf h h Hf h M W*

where *f* 0,*W* . It should be deduced from the exponential relation that lowering *f* increases the gain in (6) that in turn increases the capacity limit in (4). Suppose we let that between any two mesh nodes directly connected there exists a clear line of sight (LOS) as it is the usual case in a mesh network. Then, the following multipath channel simplification

> 1 1 2 exp . *L L m l l l l c j lm h h <sup>h</sup>*

 

Based on these simplication, it is worthwhile noting that effects of multipath often produce inter-symbol interference (ISI), signal attentuation and multipath echoes. This leads to significant capacity drops. Fortunately, the OFDM communication exploits these channel diversity to improve capacity. Therefore, joint OFDM and HPN structural configurations

In the case of IEEE 802.11n air interface, the model of the wireless channel is characterized by antenna arrays with LOS and reflected paths as shown in Figure 5. The difference with IEEE 802.11a air interface is that multiple antennas are required at both transceivers when constructing the IEEE 802.11n HPNs. In this way, the LOS and reflected paths present wireless channel diversity that multi-input multi-output (MIMO) techniques need to exploit for channel capacity enhancement. In particular, if the direct path is denoted as *path 1* and the reflected path is denoted as *path 2*, then the channel **H** is given by the principle of superpo-

*M* 

(7)

*L L m l l l l c*

0 0

2 2 exp exp ,

 (6)

/ *<sup>c</sup> f nW M* on the time invariant wireless channel *mh*

can be utilised for capacity improvement in rural based networks.

associated to each sub-carrier and

antenna gain is thus, expressed as:

(Bracewell, 1986) can be invoked:

**3.2. IEEE 802.11n air interface** 

sition (Franceschetti et al, 2009):

can be made:

Resulting from (2), the link capacity of a propagating OFDM signal over a wireless multipath channel is expanded in terms of the exponential function of the channel gain:

$$\begin{split} R\_{\text{OFDM}/\text{multiplets}} &= \frac{1}{M\_c} \sum\_{m=1}^{M\_c} \log\_2 \left( 1 + \frac{p\_m}{\left( N\_0 + I \right)} \times \left| \tilde{h}\_m \right|^2 \right), \text{bits/s/Hz} \\ R\_{\text{OFDM}/\text{multiply}} &= \frac{1}{M\_c} \sum\_{m=1}^{M\_c} \log\_2 \left( 1 + \frac{p\_m}{\left( N\_0 + I \right)} \times \left| \sum\_{l=1}^L h\_l \exp \left( -\frac{j2\pi lm}{M\_c} \right) \right|^2 \right), \text{bits/s/Hz} \end{split} \tag{3}$$

The achievable capacity of IEEE 802.11a air interface in terms of antenna gains (each antenna system for each radio interface), the range distances, the path loss exponent, the path multiplicity, and over the total bandwidth, *W* is defined as:

$$R\_{\rm OFDM/multiply} \approx \,\,\,\,\,\,\,\text{W}\log\_2\left(1 + \frac{P}{\left(N\_0 + I\right)} \times L^2 \times \frac{\mathbf{K}\_{\rm antenna} d\_0^\alpha}{d^\alpha}\right)\,,\,\,\text{bits/s.}\tag{4}$$

From (4), *P* is the maximum power allowed per sub-carrier, *L* is the number of paths associated to each sub-carrier and is the path loss exponent. We denote *antenna* as the combined antenna gain which is simply the product of the transmitter and the receiver antenna gains, 0 *d* is the reference distance (Abhayawardhana et al., 2007). The combined antenna gain is thus, expressed as:

$$\mathbf{K}\_{antemna} = \mathbf{K}\_{antemnaTxT} \times \mathbf{K}\_{antemnaRxV} \tag{5}$$

Inserting the result in (5) into the expression in (4) reveals that the higher the combined antenna gain, the higher the achievable link capacity. The improved antenna gain is the main attractive feature that the HPN based BB4allTM architecture offers to the conventional standards (Makitla, Makan & Roux, 2010). From (3), to view effects of frequency / *<sup>c</sup> f nW M* on the time invariant wireless channel *mh* , the known Fourier Transform (Bracewell, 1986) can be invoked:

$$\tilde{h}\_m = \sum\_{l=0}^{L-1} h\_l \exp\left(-\frac{j2\pi lm}{M\_c}\right) \Leftrightarrow H(f) = \sum\_{l=0}^{L-1} h\_l \exp\left(-\frac{j2\pi lf}{W}\right) \tag{6}$$

where *f* 0,*W* . It should be deduced from the exponential relation that lowering *f* increases the gain in (6) that in turn increases the capacity limit in (4). Suppose we let that between any two mesh nodes directly connected there exists a clear line of sight (LOS) as it is the usual case in a mesh network. Then, the following multipath channel simplification can be made:

$$\left| \tilde{h}\_m \right| = \left| \sum\_{l=1}^L h\_l \exp \left( -\frac{j2\pi lm}{\mathcal{M}\_c} \right) \right| = \sum\_{l=1}^L \left| h\_l \right|. \tag{7}$$

Based on these simplication, it is worthwhile noting that effects of multipath often produce inter-symbol interference (ISI), signal attentuation and multipath echoes. This leads to significant capacity drops. Fortunately, the OFDM communication exploits these channel diversity to improve capacity. Therefore, joint OFDM and HPN structural configurations can be utilised for capacity improvement in rural based networks.

#### **3.2. IEEE 802.11n air interface**

156 Wireless Mesh Networks – Efficient Link Scheduling, Channel Assignment and Network Planning Strategies

rupted by independent additive white Gaussian noise (AWGN).

1

*p h <sup>R</sup>*

*m*

*c*

*OFDM multipath m c m*

*M*

*<sup>p</sup> R h*

*c*

*OFDM multipath l*

2 0 0

*<sup>M</sup> m m*

streams or subcarriers is given by:

*c*

/ 2

/ 2

plicity, and over the total bandwidth, *W* is defined as:

*OFDM multipath*

/ 2

*RW L*

*M*

This implies that a frequency selective channel is the most suitable approach to model the IEEE 802.11a air interface (Tse & Viswanath, 2005). This is because a frequency selective channel perfectly captures effects of multipath on signal propagation (i.e., due to terrain irregularity and tree foliage). The OFDM scheme is basically the preferred method to the frequency hopping spread spectrum (FHSS) or direct sequence spread spectrum (DSSS) schemes due to its robust performance over multipath. In this context, the IEEE 802.11a radio interface cards (Intini, 2000) make use of OFDM to provide high capacity over parallel wireless channels. In their definition, Tse and Viswanath (2005) states that a parallel channel is a channel which consists of a set of non-interfering sub-channels, each of which is cor-

To obtain the capacity over single link wireless medium, we assume each *m*th sub-channel of a parrallel channel is allocated a waterfilling power *mp* such that the average power constraint *P* is still met on each input OFDM symbol to the multipath channel. Also consider that the AWGN power level to a parallel channel is *N*0 and the co-channel interference caused by neighbouring transmissions is denoted as *I* . These parameters may be held constant in practice considering that most rural network applications are characterized by constant and low interference levels (Ismael et al., 2008). Then, the maximum capacity per every OFDM symbol of a reliable communication over *Mc* parallel

whereby the achievable capacity per link in bits/s/Hz for each parallel stream is written as

/ , bits / s / Hz. *multipath M c <sup>c</sup>*

Resulting from (2), the link capacity of a propagating OFDM signal over a wireless multi-

<sup>1</sup> log 1 , bits / s / Hz

*m*

*M M N I*

*c c m l*

The achievable capacity of IEEE 802.11a air interface in terms of antenna gains (each antenna system for each radio interface), the range distances, the path loss exponent, the path multi-

0

1 1 0

*<sup>M</sup> <sup>L</sup> <sup>m</sup>*

path channel is expanded in terms of the exponential function of the channel gain:

1 0

*<sup>p</sup> j lm R h*

*M N I*

*N I*

2

log 1 , bits / OFDM symbol, *<sup>c</sup>*

2

<sup>1</sup> <sup>2</sup> log 1 exp , bits / s / Hz

*R RM* (2)

2

2 0

log 1 , bits / s. *antenna*

*N I d*

*P d*

(1)

(3)

(4)

In the case of IEEE 802.11n air interface, the model of the wireless channel is characterized by antenna arrays with LOS and reflected paths as shown in Figure 5. The difference with IEEE 802.11a air interface is that multiple antennas are required at both transceivers when constructing the IEEE 802.11n HPNs. In this way, the LOS and reflected paths present wireless channel diversity that multi-input multi-output (MIMO) techniques need to exploit for channel capacity enhancement. In particular, if the direct path is denoted as *path 1* and the reflected path is denoted as *path 2*, then the channel **H** is given by the principle of superposition (Franceschetti et al, 2009):

$$\mathbf{H} = a\_1^b \mathbf{e}\_\mathbf{r} \left(\boldsymbol{\Omega}\_{r1}\right) \mathbf{e}\_\mathbf{t} \left(\boldsymbol{\Omega}\_{t1}\right)^\ast + a\_2^b \mathbf{e}\_r \left(\boldsymbol{\Omega}\_{r2}\right) \mathbf{e}\_\mathbf{r} \left(\boldsymbol{\Omega}\_{t2}\right)^\ast, \text{for } i = 1, 2 \tag{8}$$

Achievable Capacity Limit of High Performance Nodes for Wireless Mesh Networks 159

 

**H H** (11)

*L n* (10)

then the matrix **H** is of rank 2 holds. That is, the maximum number of independent rows or columns of the matrix is 2. Based on the defined condition, we let *t tt L n* and *r rr L n* whereby *<sup>t</sup> L* and *<sup>r</sup> L* are normalized lengths of transmit and receive arrays, respectively. As a subsequence, the implication is explained as follows. When the number of antennas in each HPN is increased for any fixed normalized length of arrays, the factor denoted by will decrease and from the modulo operation, the directional cosine denoted by will increase proportionately. This might cause ill-conditioned **H** with impossible inverse. To make **H** well-conditioned so that its inverse can be computed, the angular

2 1 2 1 cos cos , and cos cos , . *t t t t tt r r r r rr*

Moreover, in order to view the influence of multipath, **H** is re-written as **'' ' H HH** , where

**<sup>H</sup>** is a 2 by *<sup>t</sup> <sup>n</sup>* matrix while **'' <sup>H</sup>** is an *<sup>r</sup> <sup>n</sup>* by 2 matrix. Consequently, the capacity limit of

log 1 , bits / s / Hz

**H**

*b bb b*

log 1 2 2 ,bits / s / Hz,

log 1 , bits / s / Hz

2 2

**'' '**

2

2

,

*<sup>a</sup>* **H ee** (12)

2

(13)

*i r ri t ti*

the channel with LOS and reflected paths in an HPN based mesh link is given as:

2 1 12 2

*no of multipaths*

*i*

*multipaths*

*i*

*SINR a a a a*

 2 2 2 2 11 22 , , *b b t r t r a a nn a a nn*

Suppose fading channels are assumed, then the single link of IEEE 802.11n based HPNs can be characterized by stochastic channel behaviours. Statistical MIMO channel models are adopted to capture the key properties that enable spatial multiplexing (Tse & Viswanath, 2005). For instance, given an arbitrary number of physical paths between the transmitter and

*b*

generality, the capacity limit of the MIMO multipath fading channel can similarly be written as:

2

log 1 , bits / s / Hz

**e e**

log 1 , bits / s / Hz

**H**

*b j i r ri t ti*

*SINR a*

*<sup>i</sup> a* , **e***<sup>r</sup>* and **e***<sup>t</sup>* take the definition provided in (9). From (11) and without loss of

separations *t* and *r* should satisfy the following:

 *L n*

Re 2

*LoS flect j*

*R SINR*

*LoS flect j*

*R SINR*

*j*

Re 2

the receiver, the channel matrix **H** may be written as:

2

*multipath j*

*R SINR*

2

**'**

where

where *<sup>b</sup>*

with, 2 exp *i b i i tr c j d a a nn* , where *<sup>t</sup> n* is the number of transmit antennas, *<sup>r</sup> n* is the

number of receive antennas and *<sup>c</sup>* is the wavelength of the pass-band transmitted signal. The distance between the transmit antenna 1 and recieve antenna 1 along path *i* is denoted by *<sup>i</sup> d* . Figure 5 illustrates transmit and receive antenna arrays that is seperated by a concatenation of two channel **' H** and **'' H** with virtual relays A and B. According to expression (8) and Figure 5, the unit spatial signature in the directional cosine (i.e., cos ) is defined as follows:

$$\mathbf{e}\_{r}(\cdot) = \frac{1}{\sqrt{n\_{r}}} \begin{pmatrix} 1 \\ \exp\{-j2\pi\Delta\_{r}\Omega\} \\ \exp\{-j2\pi\Delta\_{r}\Omega\} \\ \vdots \\ \exp\{-j2\pi\Delta\_{r}\Omega\} \\ \end{pmatrix} \begin{pmatrix} 1 \\ \exp\{-j2\pi\Delta\_{t}\Omega\} \\ \exp\{-j2\pi\Delta\_{t}\Omega\} \\ \exp\{-j2\pi\Delta\_{t}\Omega\} \\ \vdots \\ \exp\{-j2\pi(n\_{r}-1)\Delta\_{t}\Omega\} \\ \end{pmatrix} \tag{9}$$

**Figure 5.** A MIMO channel with a direct path 1 and a reflected path 2. The channel is a concatenation of two channels H' and H'' with virtual relays A and B

From (9), the notation is the angle of incidence of the LOS onto the receive antenna array and *a* is the signal attenuation. In a reasonable sense, the condition that as long as

$$
\Omega\_{t1} \neq \Omega\_{t2} \mod \frac{1}{\Delta\_t} \text{ and } \Omega\_{r1} \neq \Omega\_{r2} \mod \frac{1}{\Delta\_r}
$$

then the matrix **H** is of rank 2 holds. That is, the maximum number of independent rows or columns of the matrix is 2. Based on the defined condition, we let *t tt L n* and *r rr L n* whereby *<sup>t</sup> L* and *<sup>r</sup> L* are normalized lengths of transmit and receive arrays, respectively. As a subsequence, the implication is explained as follows. When the number of antennas in each HPN is increased for any fixed normalized length of arrays, the factor denoted by will decrease and from the modulo operation, the directional cosine denoted by will increase proportionately. This might cause ill-conditioned **H** with impossible inverse. To make **H** well-conditioned so that its inverse can be computed, the angular separations *t* and *r* should satisfy the following:

$$
\Omega\_t = \cos\phi\_{t2} - \cos\phi\_{t1'}, \ L\_t = \mathfrak{n}\_t \Delta\_t \text{ and } \Omega\_r = \cos\phi\_{r2} - \cos\phi\_{r1'}, \ L\_r = \mathfrak{n}\_r \Delta\_r. \tag{10}
$$

Moreover, in order to view the influence of multipath, **H** is re-written as **'' ' H HH** , where **' <sup>H</sup>** is a 2 by *<sup>t</sup> <sup>n</sup>* matrix while **'' <sup>H</sup>** is an *<sup>r</sup> <sup>n</sup>* by 2 matrix. Consequently, the capacity limit of the channel with LOS and reflected paths in an HPN based mesh link is given as:

$$\begin{aligned} \,^1R\_{LoS+Refact} &= \log\_2\left(1+\text{SINR}\_f \times \left\|\mathbf{H}\right\|^2\right), \text{ bits/s/Hz} \\ \,^1R\_{LoS+Refact} &= \log\_2\left(1+\text{SINR}\_f \times \left\|\mathbf{H}^\top \mathbf{H}\right\|^2\right), \text{bits/s/Hz} \\ &= \log\_2\left(1+\text{SINR}\_f \times 2 \times \left(\left(a\_1^b\right)^2 + 2a\_1^b a\_2^b + \left(a\_2^b\right)^2\right)\right), \text{bits/s/Hz} \end{aligned} \tag{11}$$

where

158 Wireless Mesh Networks – Efficient Link Scheduling, Channel Assignment and Network Planning Strategies

2

*c*

 

two channels H' and H'' with virtual relays A and B

From (9), the notation

*r t*

*t*2

*t*1

 

*i*

exp

) is defined as follows:

*j d a a nn*

number of receive antennas and *<sup>c</sup>*

with,

cos

*b*

*i i tr*

11 1 22 2 ,for 1, 2 *b b*

The distance between the transmit antenna 1 and recieve antenna 1 along path *i* is denoted by *<sup>i</sup> d* . Figure 5 illustrates transmit and receive antenna arrays that is seperated by a concatenation of two channel **' H** and **'' H** with virtual relays A and B. According to expression (8) and Figure 5, the unit spatial signature in the directional cosine (i.e.,

1 1

. . . .

*r t*

*n n*

exp 2 exp 2 exp 2 2 exp 2 2 1 1 . . ,. . ,

*j j j j*

**e e** (9)

*r t r t*

 

exp 2 1 exp 2 1

**Figure 5.** A MIMO channel with a direct path 1 and a reflected path 2. The channel is a concatenation of

and *a* is the signal attenuation. In a reasonable sense, the condition that as long as

1 2 1 2 1 1 mod and mod *t t r r*

*j n j n*

*r r t t*

is the angle of incidence of the LOS onto the receive antenna array

*r*1

*i d*

*r*2

*t r*

*r t rr t aa i* **He e e e r t <sup>r</sup>** (8)

, where *<sup>t</sup> n* is the number of transmit antennas, *<sup>r</sup> n* is the

is the wavelength of the pass-band transmitted signal.

 

 

 

 

$$\left(a\_1^b\right)^2 = a\_1^2 n\_t n\_{r'} \left(a\_2^b\right)^2 = a\_2^2 n\_t n\_{r'} \dots$$

Suppose fading channels are assumed, then the single link of IEEE 802.11n based HPNs can be characterized by stochastic channel behaviours. Statistical MIMO channel models are adopted to capture the key properties that enable spatial multiplexing (Tse & Viswanath, 2005). For instance, given an arbitrary number of physical paths between the transmitter and the receiver, the channel matrix **H** may be written as:

$$\mathbf{H} = \sum\_{i}^{no\ of\ multipliers} a\_i^b \mathbf{e}\_r \left(\boldsymbol{\Omega}\_{ri}\right) \mathbf{e}\_t \left(\boldsymbol{\Omega}\_{ti}\right)^\*,\tag{12}$$

where *<sup>b</sup> <sup>i</sup> a* , **e***<sup>r</sup>* and **e***<sup>t</sup>* take the definition provided in (9). From (11) and without loss of generality, the capacity limit of the MIMO multipath fading channel can similarly be written as:

$$\begin{aligned} R\_{multipath} &= \log\_2\left(1 + \text{SINR}\_j \times \left\|\mathbf{H}\right\|^2\right) \text{ bits/s/Hz} \\ &= \log\_2\left(1 + \text{SINR}\_j \times \left\|\sum\_{i}^{multiths} a\_i^b \mathbf{e}\_r\left(\boldsymbol{\Omega}\_{ri}\right) \mathbf{e}\_t\left(\boldsymbol{\Omega}\_{ti}\right)^\*\right\|^2\right) \text{ bits/s/Hz} \end{aligned} \tag{13}$$

In conclusion, if the number of physical paths is two then the expression of capacity over multipath phenomenon (13) simply reduces to the expression (11) of the direct and reflected paths. Clearly from expression (13), one notes that increasing the multiplicity of paths of a single wireless link and the number of antennas at each HPN in (11) or (13) will increase the capacity limit of the wireless mesh links, depicted in (4) due to MIMO technology benefits.

Achievable Capacity Limit of High Performance Nodes for Wireless Mesh Networks 161

*n*

, is given as *mc nR*

be uniform with distribution regularity

,

 

*m*

collision avoidance (CSMA/CA) in IEEE 802.11 standards (IEEE 802.11 standard working group, 1999). In this topology setting, the regular placement of nodes on a fixed area will be termed as an arbitrary network. That is, the location of nodes and traffic patterns can be controlled as introduced by Gupta and Kumar (2000). Controlling nodes' placement locations and the traffic patterns makes the derived capacity bounds to be viewed as *the best case* capacity bounds with results remaining applicable to any network. As introduced by Gupta and Kumar, the aggregate *end to end* network throughput over a given flow or a set of flows is measured in terms of "bit-meters/sec". That is the network is said to transport one "bitmeter/sec" when one bit has been transported across a distance of one meter in one second.

*Theorem 1:* The E2E upper bound on capacity of a statically assigned channel network of

*Proof:* For the best case capacity limit, let's assume that multiple interfaces of HPNs receive and transmit on interference free channels. This assumption is reasonable with the HPNs that transmit directionally but receive and ensure connectivity omnidirectionally. As the number of channels is much larger than the number of interfaces. Thus, given that each HPN has a constant radio range, the spatial reuse is considered to be proportional to the

type *m c*, -arbitrary regular placement of nodes when, *<sup>c</sup>*

equals to one (i.e., probability equals to one) throught the deployment area.

physical area of the network. Let the node density

**Figure 6.** Regular placement of HPNs in a 5 km x 5 km

bit-meters/sec.
