Average Capacity

0 x^v−1e^−μxdx = μ^−vΓ(v). (3.29) Applying (3.29) in (3.28), the final result for the mean is given by

Y = Kλ¯ 1

The variance of the ICI power is given by σ²_Y = E! Substituting the PDF ofY in (3.31) and making use of (3.29), we obtain

σ²_Y =Kλ²1 From (3.30) and (3.32), we see that the mean and variance depends on the phase noise process through λ1 which is in direct proportion to the level of phase noise. For fast-varying phase noise processesλ1takes higher values of magnitude which consequently imply larger values for the mean and variance of the ICI power.

3.3.4 Average Capacity

Channel capacity is a measure of the number of bits that can be transmit-ted through the channel with a very small error probability. In addition to SINR, SEP and BEP, channel capacity is a standard performance met-ric used in assessing the performance of a communication system. For an AWGN channel, the channel capacity is a function of the SNR or SINR in our case and is given by

Cj = log₂(1 + Υ_j), (3.33)

where Υ_j is the SINR seen by thejth subcarrier.

The capacity in (3.33) is applicable only under certain assumptions: The total additive noise must be Gaussian. In our case, the additive noise

given in (3.1), is the ICI plus the receiver Gaussian noise. As shown in [83], for slow-varying phase noise processes, the ICI is not Gaussian, thereby, the effective noise is non-Gaussian in general. Thus, in a strict information-theoretic sense, Cj is not the average capacity. However, it is the mutual-information assuming a Gaussian input alphabet for the transmitted symbolssj. This is seen as follows: For a fixed realization of the channel and phase noise, the ICI plus receiver noise is Gaussian dis-tributed. The SINR for this realization is given by Υ_j of (3.7) and, thus, the capacity for this Gaussian alphabet is obtained using (3.33). Such an approach of evaluating the channel capacity was also utilized in [98] for an OFDM link impaired by IQ-imbalance and in [92] for an OFDM link impaired by IQ-imbalance and phase noise. We shall, thus, refer toCj of (3.33) as the capacity of (3.1) assuming a Gaussian input alphabet.

In Publications I and II, closed-form expressions of the average capacity are derived. It is obtained as follows: FirstCjis averaged over the PDF of Y in (3.16), and the result is given by

C¯j= log₂(1 +γj)− K

∞ k=0

ζklog₂(1 +bkγj), (3.34)

where γj = ^g_σ^jσ₂^s2

w denotes the instantaneous signal-to-noise ratio (SNR) withgjbeing the realization of the random variableGj =|Hj|²and

bk=Γ(R/2 + k + 1)λ1

Γ(R/2 + k) . (3.35)

The parametersK and ζkare defined in (3.16). Equation (3.34) is the ca-pacity for a given realization of the channelgj and has a nice interpreta-tion: The first term represents the capacity without phase noise while the second term arises because of phase noise. Without any phase noise, the second term is zero sinceλ1= 0. However, in the presence of phase noise, we have a non-zero contribution from the second term, and the overall effect is a net-reduction from the phase noise-free case.

The average capacity, denoted by ¯C¯j, is obtained by averaging ¯Cj over the PDF ofgj. Assuming a Rayleigh fading channel which implies thatgj

is exponentially distributed, the final expression for the average capacity, derived in Publications I and II, is given by

¯¯

-5 0 5 10 15 20 25 30 35 0

2 4 6 8 10 12

R_s = 5 kΩ

R_s = 25 kΩ R_s = 50 kΩ AWGN, no phase noise

Fixed channel

Rayleigh fading channel

γj,¯γj[dB]

¯Cj,

¯ ¯C[bit/s/Hz]j

Figure 3.3. Capacity plots of ¯Cjand ¯¯Cjfor different phase noise PSD bandwidths obtained by varying the loop filter resistanceRsin the PARMA phase noise model.

The markers and solid/dashed lines represent the simulations and analytical plots, respectively. The PARMA filter parameters are given in Publication II.

in (3.36) is the average capacity for a Rayleigh fading channel without phase noise, while due to phase noise, we have the non-zero second term in (3.36), and the result is a net-reduction from the Rayleigh fading capacity.

Figure 3.3 shows plots of the capacities of (3.34) and (3.36). The OFDM bandwidth is 9.14 MHz with Nc= 4096. Channel is Rayleigh faded with 50 taps of exponential power delay profile with coherence bandwidth set to 400kHz. The phase noise process used to generate these figures is of the discrete PARMA type. The phase noise bandwidth is controlled by varying the loop filter resistanceRs, and it increases with increase inRs. As seen from Fig. 3.3, the presence of phase noise causes a reduction in the capac-ity when compared to the fixed channel case or Rayleigh fading case. This behavior of the capacity in (3.34) and (3.36) w.r.t. phase noise is through the parameterλ1. In general,λ1is directly proportional to the phase noise bandwidth (see Section 3.3.3). As this bandwidth increases, for example in the PARMA model by varyingRs,λ1increases and, hence, the second terms in (3.34) and (3.36) increase, thereby, resulting in a larger reduction from the fixed channel and Rayleigh fading capacities.

Equations (3.34) and (3.36) represents the capacity for a particular (j)th subcarrier. The net throughput of the system is obtained by summing the capacities over all subcarriers and dividing the result over the OFDM symbol duration which is (Nc+Ncp)Ts, whereNcpis the cyclic prefix length

6 7 8 9 10 11 12 13 14 15 16

Figure 3.4. Net throughput plots of (3.37) for different phase noise PSD bandwidths ob-tained by varying the loop filter resistanceRs. The markers and solid/dashed lines represent the simulations and analytical plots, respectively. The PARMA filter parameters are given in Publication II.

andTsis the sampling period. For the fixed channel case, we consider the realization wherein{γj}^N_j=0^c−1are equal and, for the fading case,{¯γj}^N_j=0^c−1 are set to the same average SNR. The net throughput is given by

C =¯ of the loss in efficiency due to cyclic prefix.

Figure 3.4 shows the net throughput as a function ofNcfor a fixedNcp= 64. In the absence of phase noise, ¯C and ¯¯ C (shown by the black curves) increase withNc and saturates to a particular value for Nc  1. This is because η → 1 for Nc  Ncp, i.e., the efficiency can be improved by choosing a large value of Nc in comparison with Ncp. However, in the presence of phase noise, we see that there is an optimalNcfor which ¯C and ¯C are maximum. This can be explained as follows: For a fixed phase¯ noise bandwidth, we know that ¯Cjand ¯C¯jdecrease whenNcis increased.

Thus, we have two conflicting scenarios, where ¯C and ¯¯C increase with η and decrease with ¯Cj and ¯C¯j, simultaneously. Thus, we could expect ¯C and

¯¯

C to increase with Ncup to a maximum as long asfsubis large enough to cause the ICI to be small. Beyond this maximum, iffsubis decreased then the resulting ICI causes ¯Cjand ¯¯Cjto decrease much faster compared with the increase inη as evidenced in Fig. 3.4.