Signal to Co-Channel Interference Ratio (SIR)

Cellular Networking and Mobile Radio Channel Characterization

2.2 The Crux of the Cellular Concept

2.2.3 Frequency Reuse and Radio Channel Multiplicity

2.2.3.2 Signal to Co-Channel Interference Ratio (SIR)

The objective of this section isfinding an approximate expression for signal to co-channel interference ratio (SIR) as a function of co-channel reuse ratioℓ. The approach follows that of Rappaport [4]. CCI is the primary disturbance that places limit on the capacity of cellular networks, therefore, understanding and tracking the quantities of SIR within a conceptual cell boundary is crucial at the initial stages of cellular network design.

It is well known that the average measured received signal strength in a radio channel decays with certain “power law” of transmitter receiver (TR) distance, as long as the receiving antenna is in the far-field region¹of

1 Far-field or Fraunhofer region of an antenna is defined by the distance range of 2D²=λ  d 1 from the transmitting antenna. D is the largest linear dimension of the antenna and λ is the wavelength of the radio wave.

the transmitting antenna [6]. For instance, signal average power in a free-space propagation environment attenuates proportional to the inverse of square of TR distance. The exponent in this power law is called propaga-tion path loss exponent and is shown byν. For free-space propagation ν ˆ 2, for outdoor land mobile environment path loss exponent is greater than 2. Since the propagation terrain is normally nonhomogeneous, the path loss exponent is not a constant throughout a cell footprint.

To facilitate the approximate computation of SIR, the following assumptions are made.

1) Cells have hexagonal shape.

2) The output power of transmit antenna of all base stations is equal.

3) Only interference from thefirst-tier co-channel cells is significant, that is, the CCI from second tiers and beyond are assumed to be negligible.

4) The propagation environment is homogeneous, that is, path loss exponentν has the same value throughout the coverage area.

Recognizing that in a hexagonal-shaped cellular layout there are exactly 6first-tier co-channel interfering cells, the SIR in the forward channel is given by

S I ˆ S

P⁶

kˆ1I_k

(2.6)

Here S is the effective received power and Ikis the interfering power from the kth co-channel cell in thefirst tier. For the worst-case scenario, that is, Table 2.1 SIR versus cluster size and CCRR, for ν ˆ 4.

i j N ℓ ˆ D=R SIR SIR (dB)

1 0 1 1.73 0.13 8.86

1 1 3 3.00 4.79 6.80

2 0 4 3.46 12.32 10.9

2 1 7 4.58 49.44 17

3 0 9 5.20 87.97 19.44

2 2 12 6.00 165.52 22

3 1 13 6.24 197.26 22.95

4 0 16 6.93 310.85 24.92

3 2 19 7.55 449.56 26.53

4 1 21 7.94 557.49 27.46

3 3 27 9.00 947.83 29.77

when mobile moving on the cell boundary, the SIR can be approximated by

S

I ˆ P=αR^ν P⁶

kˆ1P=αD^ν_k

ˆ R ^ν P⁶

kˆ1D_k^ν

ˆ 1

P⁶

kˆ1 Dk

R

  _ν (2.7)

Here P is the output power of the BS antenna, D_kis the distance between the MS and the BS of the kth co-channel cell, andα is the constant of proportionality.

As an example of worst-case approximate computation of SIR, consider Figure 2.4, which illustrates a part of a cellular layout for cluster size N= 7.

Equation 2.8 calculates an approximate measure for SIR in this sce-nario [4].

S

I ˆ R ^ν

2 D… R† ^ν‡ D R=2… † ^ν‡ D ‡ R=2… † ^ν‡ D ‡ R… † ^ν‡ D ^ν (2.8) As Equation 2.8 indicates, the distances between the MS and various centers of co-channel cells, shown by double-arrowed lines in Figure 2.5, are approximated in terms of D (the distance between neighboring co-channel cell centers) and R (the radius of each cell) as follows. Beginning from the thick solid line and moving to dashed lines counterclockwise, respectively,

Figure 2.5 A visual aid for approximate computation of SIR.

approximate distances are D‡R

2; D; D R; D R; D R

2; D ‡ R

The latter equation can be expressed in terms of co-channel reuse ratioℓ as given by Equation 2.9.

S

I ˆ 1

2…ℓ 1† ^ν‡ …ℓ 0:5† ^ν‡ …ℓ ‡ 0:5† ^ν‡ …ℓ ‡ 1† ^ν‡ ℓ ^ν (2.9) Table 2.1 presents approximate computed values of SIR, according to Equation 2.9, in direct and logarithmic scale in its two rightmost columns.

These computations are carried out for some practical values of CCRR, with the assumption that path loss exponentν ˆ 4.

Numerical Example 1

Cellular service is to be provided for a metropolitan district of area of 15,000 km². The resources and key cellular network parameters are given in the following list.

1) Total available bandwidth: 27 MHz

2) Required bandwidth for traffic and control channels in each direction:

30 kHz

3) Bandwidth dedicated to control and signaling channels: 3 MHz 4) Cell radius 12 Km

5) Path loss exponentν ˆ 4

For four cluster sizes of 4, 7, 12, and 19, determine the following:

1) Number of traffic channels per cell 2) Number of control channels per cell 3) The cellular network capacity 4) Worst-case SIR

Preliminary Computations a_cˆ 3 ffiffiffi

p3

R²_c=2 ˆ 3 ffiffiffi p3

…12†²=2 ≅ 374:12 km² Cell area ncˆ 15000=374:12 ≅ 40 Total number of required cells

…BW†_T:C:ˆ 27 3 ˆ 24 MHz Bandwidth available for traffic channels MTˆ 24,000=2  30 ˆ 400Totalnumberofavailableduplextrafficchannels MCˆ 3000=2  30 ˆ 50 Total number of available duplex control channels Solution for N = 4

q_T ˆ MT=N ˆ 400=4 ˆ 100 (Equation 2.1) Total number of available duplex voice channels per cell

ℜ ˆ 40=4 ˆ 10 Number of clusters

ΓT ˆ ℜ
qT
N ˆ 10  100  4 ˆ 4000 (Equation 2.2) Network voice capacity

ℓ  ffiffiffiffiffiffiffi p3N

ˆ ffiffiffiffiffi p12

≅ 3:46 (Equation 2.5) Co-channel reuse ratio

SIR≅ 12:32 ≅ 10:9 dB (Equation 2.9 or Table 2.2) worst-case signal to co-channel interference ratio

ncc ˆ 50=4 ≅ 13 Number of control channels per cell Solution for N = 7

q_T ˆ MT=N ˆ 400=7 ≅ 57 (Equation 2.1) Total number of available duplex traffic channels per cell

ℜ ˆ 40=7 ≅ 6 Number of clusters

ΓTˆ ℜ
qT
N ≅ 6  57  7 ≅ 2394 (Equation 2.2) Network traffic capacity

ℓ  ffiffiffiffiffiffiffi p3N

ˆ ffiffiffiffiffi p21

≅ 4:58 (Equation 2.5) Co-channel reuse ratio

SIR≅ 49:56 ≅ 17 dB (Equation 2.9 or Table 2.2) worst-case signal to co-channel interference ratio

ncc ˆ 50=7 ≅ 7 Number of control channels per cell Solution for N = 12

q_T ˆ MT=N ˆ 400=12 ≅ 34 (Equation 2.1) Total number of available duplex traffic channels per cell

ℜ ˆ 40=12 ≅ 3 Number of clusters

ΓTˆ ℜ
q_T
N ≅ 3  34  12 ≅ 1224 (Equation 2.2) Network traffic capacity

ℓ  ffiffiffiffiffiffiffi p3N

ˆ ffiffiffiffiffi p36

ˆ 6 (Equation 2.5) Co-channel reuse ratio

SIR≅ 165:57 ≅ 22 dB (Equation 2.9 or Table 2.2) worst-case signal to co-channel interference ratio

n_cc ˆ 50=12 ≅ 4 Number of control channels per cell Solution for N = 19

q_T ˆ MT=N ˆ 400=19 ≅ 21 (Equation 2.1) Total number of available duplex traffic channels per cell

ℜ ˆ 40=19 ≅ 2 Number of clusters

ΓTˆ ℜ
q_T
N ≅ 2  21  19 ≅ 798 (Equation 2.2) Network traffic capacity

Table 2.2 The key results of Example 1.

N ΓT SIR(dB)

4 4000 10.9

7 2394 17

12 1224 22

19 798 26.53

ℓ  ffiffiffiffiffiffiffi p3N

ˆ ffiffiffiffiffi p57

ˆ 7:55 (Equation 2.5) Co-channel reuse ratio

SIR≅ 449:56 ≅ 26:53 dB (Equation 2.9 or Table 2.2) worst-case signal to co-channel interference ratio

nccˆ 50=19 ≅ 3 Number of control channels per cell Results and Conclusions

Table 2.2 summarizes the key results of this example. As the table indicates, if cell size remains fixed, with an increase in cluster size, capacity decreases whereas SIR increases. This example serves to provide a general understanding on relationship between three key parameters of cellular networks, namely, cluster size N, capacityΓT(defined as the total number of available traffic channels in the network for a fixed cell size), and SIR.

However, a number of important practical issues remain to be addressed, some of which are listed below.

1) How many users can a network, such as the one in example 1, accommodate (this is also known as network capacity)?

2) What is the overall quality of service (QoS) in this network? We have one indication about the QoS expressed by SIR values, but what is the overall QoS and grade of service (GoS) in this network?

3) Should the cell size be assumed to befixed at the outset of the cellular network design, or should it be considered a parameter at the disposal of the network designer?

4) How do we determine the actual area in which a given BS provides coverage, that is, how does one determine the cell footprint? The cell footprint depends not only on the transceiver antenna height and output power, but also on the nature of surrounding terrain, that is, how cluttered the area is, whether it is an urban area or suburban area or rural area, whether it is hilly orflat or the combination of the two, and so on.

5) What are the levels of power that are needed for each BS and what is the relationship between the BS output power and cell footprints?

6) Given a cell and its footprint, what percentage of the area inside the cell footprint provides acceptable QoS, and what percentage of the cell footprint is in blackout conditions? In other words, where are the blind spots in the cell footprint?

Some of these questions will be addressed in Section 2.2.4.