Advanced Wireless Networks - 4G Technologies

738 NETWORK INFORMATION THEORY

19.1.6 Approximations

If we assume that the channel estimation is perfect (εa = εθ = 0) the parameter Cmkl becomes Ckl = 2(1 − 2εm ) − 1 = 1 − 4εm . For DPSK modulation εm = (1/2)exp(−y/2) where y is signal-to-noise ratio and for CPSK εm = (1/2)erfc(√y). So Ckl = 1 − 2e−y for DPSK, and Ckl = 1 − 2 erfc(√y) for CPSK. For large y, Ckl 1 and for small y in DPSK

where Yb is given by Equation (19.14). Bearing in mind that Yb depends on Ckl the whole equation can be solved through an iterative procedure starting up with an initial value of Ckl = 0, m, k, l. Similar approximations can be obtained for σθ2 and εa . From practical point of view an attractive solution could be a scheme that would estimate and cancel only the strongest interference (e.g. successive interference cancellation schemes [2]).

19.1.7 Outage probability

The previous section already completely defines the simulation scenario for the system performance analysis. For the numerical analysis further assumptions and specifications are necessary. First of all we need the channel model. The exponential multipath intensity profile (MIP) is widely used analytical model realized as a tapped delay line [2]. It is very flexible in modeling different propagation scenarios. The decay of the profile and the number of taps in the model can vary. Averaged power coefficients in the multipath intensity profile are

where λ is the decay parameter of the profile. Power coefficients should be normalized as

L−1 α0e−λl = 1. For λ = 0 the profile will be flat. The number of resolvable paths depends

l=0

on the channel chip rate and this must be taken into account. We will start from Equation (19.14) and look for the average system performance for ρ2 = 1/G where G = W/Rb is the system processing gain and W is the system bandwidth (chip rate). The average signal to noise ratio will be expressed as

Now, if we accept some quality of transmission, BER = 10−e, that can be achieved with the given SNR = Y0, then with the equivalent average interference density η0 = Ioic + Ioin + ηth SNR will be

To evaluate the outage probability Pout we need to evaluate [2]

where δ is given as

δ = r(L0)G − ηW Y0 S

It can be shown that this outage probability can be represented by Gaussian integral

δ − mg

Pout = Q (19.35)

σg

where mg and σg are the mean value and the standard variance respectively of the overall interference. From Equation (19.32) we have for the system capacity K with ideal system components

Owing to imperfections in the operation of the rake receiver and interference canceller, this capacity will be reduced to

where r(L0) = r0(L0) f (α) − r(L0) f0(α) and f (α) = f (α) − f0(α).

19.1.7.1 Performance example

In this section we present some numerical results for illustration purposes. The results are obtained for a channel with double exponential (space and delay) profile with decay factors λs and λt . Graphical results are presented with: solid line, 4 × 4 rake; dashed line, 4 × 1 rake. In the case of a 3 dB approximation of the real antenna, the beam forming is approximated by the rectangular shape of the antenna pattern in the range ϕ3dB = 30◦ (for ρc = 1), and uniform distribution of ψkl and ϕkl ([0, π ], [0, 2 π ], respectively) , λs = λt = 0 if not specified otherwise, Y0 = 2, L = 4, interference margin SNR = (20 α0/alfa mean)−1.

The users are uniformly distributed in the sphere (ϕ, ψ ) with ϕ, ψ (0, 360◦). The results can be easily scaled down for a more realistic scenario where ϕ (0, 360◦) and ψ (ψmin, ψmax). The canceling efficiency for maximum capacity is calculated by assuming no estimation errors in Equation (19.27). When estimation errors are included, canceling efficiencies are given by Equation (19.27). Carrier phase tracking error variance is assumed to be σθ2 = 1/SNRL. For MRC the amplitude estimation error is approximated from Equation (19.30) to follow εa = 1/4 SNRL. For real antennas the amplitude patterns are specified in Figure 19.2. The base station amplitude antenna pattern is given as

740 NETWORK INFORMATION THEORY

and four examples cover a relatively wide range of shapes, from a wide lobe for ρc = 1 to a rather narrow lobe for ρc = 12. At mobile stations an omnidirectional antenna pattern A(ϕ, ψ ) = 1 is assumed. Capacity curves, defined as the number of users with data rate R = chip rate/G = 4.096/G, for different antenna patterns, are shown in Figure 19.3. The highest capacity is obtained for ρc = 12 because with the narrowest lobe the spatial division multiple access effect is the most effective. With increased receiver velocity the capacity will be reduced due to the increased effect of imperfections. A comparison of the systems using ideal and real antennas is shown in Figure 19.4. Figure 19.4 presents the system capacity vs G. In general higher G means more users in the network and more MAI resulting in more impact of imperfections. A 4 × 4 rake performs better for lower G but for higher G (more users) it deteriorates faster. The degradation is more severe for higher receiver velocities.

Figure 19.4 Capacity vs processing gain for EGC. a, vrec = 200 km/h, b, vrec = 150 km/h, c, vrec = 100 km/h; d, vrec = 50 km/h. Solid line: 4 × 4 rake; dashed line: 4 × 1 rake; A real antenna pattern of circular array at base station; B, 3 dB aproximation (ψ3dB = 30◦) of the real antenna pattern; ρc = 1; N = 4, λs = λt = 0, Y0 = 2, L = 4, SNR = (20*α0/alpha mean)−1.

Figure 19.5 represents the same results as a function of the receiver velocity. The system sensitivity function defined by Equation (19.38) is shown in Figure 19.6. Sensitivity equal to 1 means that all capacity has been lost due to imperfections. Figure 19.6 demonstrates that very high values for the system sensitivity, even in the range close to 0.9, can be expected if a large number of users (low data rate corresponding to high G) are in the network.

In this section we have presented a systematic analytical framework for the capacity evaluation of an advanced CDMA network. This approach provides a relatively simple way to specify the required quality of a number of system components. This includes multiple access interference canceller and rake receiver, taking into account all their imperfections. The

742 NETWORK INFORMATION THEORY

Figure 19.5 Capacity vs the receiver velocity for EGC; solid lines, 4 × 4 rake; dashed

lines, 4 × 1 rake. A, real antenna; B, 3 dB approximation of the real antenna. ρc = 1; N = 4, λt = λs = 0, Y0 = 2, L = 4, SNR = (20 α0/alpha mean)−1.

(a) a, G = 256; b, G = 160; (b) a, G = 80; b, G = 48; c, G = 40.

system performance measure is the network sensitivity function representing the relative losses in capacity due to all imperfections in the system implementation. Some numerical examples are presented for illustration purposes. These results are obtained for a channel with double exponential (space and delay) profile. It was shown that for the receiver velocity 100–200 km/h as much as 70–90 % of the system capacity can be lost due to the imperfections of the three-dimensional rake receiver and interference cancellation operation. variety of results are presented for different channel decay factor, fading rate and number of rake fingers. In general, under ideal conditions, the system capacity is increased if the number of fingers is increased. At the same time one should be aware that the system sensitivity

744 NETWORK INFORMATION THEORY

an arbitrary rate, so that the traffic pattern is arbitrary too. Each node can choose an arbitrary range or power level for each transmission.

To define when a transmission is received successfully by its intended recipient we will allow for two possible models for successful reception of a transmission over one hop, called the protocol model and the physical model. Let Xi denote the location of a node; we will also use Xi to refer to the node itself.

19.2.1.1 The protocol model

Suppose node Xi transmits over the mth subchannel to a node X j . In this case transmission is successfully received by node X j if |Xk − X j | ≥ (1 + )|Xi − X j |. In this case a guard zone > 0 is specified by the protocol to prevent a neighboring node from transmitting on the same subchannel at the same time.

19.2.1.2 The physical model

Let {Xk ; k T } be the subset of nodes simultaneously transmitting at some time instant over a certain subchannel. Let Pk be the power level chosen by node Xk for k T . In this case the transmission from a node Xi , i T , is successfully received by a node X j if S/I ≥ β , where

S = P/ Xi − X j |α

and

I = N + P/ Xi − X j |α

k T k=i

This models a situation where a minimum signal-to-interference ratio (SIR) of β is necessary for successful receptions, the ambient noise power level is N, and signal power decays with distance r as 1/rα . For a model outside a small neighborhood of the transmitter, α > 2.

19.2.1.3 The transport capacity of arbitrary networks

In this contest we say that the network transports one bit-meter (b-m) when 1 b has been transported a distance of 1 m towards its destination. (We do not give multiple credit for the same bit carried from one source to several different destinations as in the multicast or broadcast cases.) This sum of products of bits and the distances over which they are carried is a valuable indicator of a network’s transport capacity CT. (It should be noted that, when the area of the domain is A square meters rather than the normalized 1 m2, then all the transport capacity results presented below should be scaled by √ A.) By using the notation: f (n) = [g(n)] when f (n) = O[g(n)] as well as g(n) = O[ f (n)], we will show later in the section that the transport capacity of an arbitrary network under the protocol model is CT = (W √n) b-m/s if the nodes are optimally placed, the traffic pattern is optimally chosen, and if the range of each transmission is chosen optimally. An upper bound is

C = √(8/π )(W/ )√n b-m/s for every arbitrary network for all spatial and temporal

T √ √

scheduling strategies, while CT = W n/[(1 + 2 ) ( n + 8π )] bit-meters per second (for

n a multiple of four) can be achieved when the nodes and traffic patterns are appropriately chosen, and the ranges and schedules of transmissions are appropriately chosen.

If this transport capacity were to be equally divided between all the n nodes, then each node would obtain (W/√n) b-m/s. If, further, each source has its destination about the same distance of 1 m away, then each node would obtain a throughput capacity of (W/√n) b/s.

The upper bound on transport capacity does not depend on the transmissions being omnidirectional, as implied by Equation (19.1), but only on the presence of some dispersion in the neighborhood of the receiver. It will be shown later in the section that, for the physical model, cW √n b-m/s is feasible, while c W nα−1/α b-m/s is not, for appropriate c, c . Specifically,

It is suspected that that an upper bound of order (W √n) b-m/s may actually hold. In the special case where the ratio Pmax/ Pmin between the maximum and minimum powers that transmitters can employ is bounded above by β, then an upper bound is in fact

Both bounds suggest that transport capacity improves when α is larger, i.e. when the signal power decays more rapidly with distance.

19.2.2 Random networks

In this case, n nodes are randomly located, i.e. independently and uniformly distributed, either on the surface S2 of a three-dimensional sphere of area 1 m2, or in a disk of area 1 m2 in the plane. The purpose in studying S2 is to separate edge effects from other phenomena. Each node has a randomly chosen destination to which it wishes to send λ(n) b/s. The destination for each node is independently chosen as the node nearest to a randomly located point, i.e. uniformly and independently distributed. (Thus destinations are on the order of 1 m away on average.) All transmissions employ the same nominal range or power (homogeneous nodes). As for arbitrary networks, both a protocol model and a physical model are considered.

19.2.2.1 The protocol model

All nodes employ a common range r for all their transmissions. When node Xi transmits to a node X j over the mth subchannel, this transmission is successfully received by X j if |Xi − X j | ≤ r and for every other node Xk simultaneously transmitting over the same subchannel |Xk − X j | ≥ (1 + )r.

746 NETWORK INFORMATION THEORY

19.2.2.2 The physical model

All nodes choose a common power level P for all their transmissions. Let {Xk ; k T } be the subset of nodes simultaneously transmitting at some time instant over a certain subchannel. A transmission from a node Xi , i T , is successfully received by a node X j if S/I ≥ β , where

S = P/ Xi − X j |α

and

I = N + P/ Xi − X j |α

k T

k=i

19.2.2.3The throughput capacity of random networks

The throughput is defined in the usual manner as the time average of the number of bits per second that can be transmitted by every node to its destination.

19.2.2.4 Feasible throughput

A throughput of λ(n) b/s for each node is feasible if there is a spatial and temporal scheme for scheduling transmissions, such that by operating the network in a multihop fashion and buffering at intermediate nodes when awaiting transmission, every node can send λ(n) b/s on average to its chosen destination node. That is, there is a T < ∞ such that in every time interval [(i − 1)T, iT] every node can send T λ(n) b to its corresponding destination node.

19.2.2.5 The throughput capacity of random wireless networks

We say that the throughput capacity of the class of random networks is of order [ f (n)] b/s if there are deterministic constants c > 0 and c < +∞ such that

lim Prob [λ(n) = c f (n) is feasible] = l

n→∞

lim inf Prob [λ(n) = c f (n) is feasible] < l

n→ ∞

It will be shown in the next section that, in the case of both the surface of the sphere and a planar disk, the order of the throughput capacity is λ(n) = [W/√(n log n)] b/s for the protocol model. For the upper bound for some c ,