PDF of Sum of Gamma Variates

The gamma random variables in (3.11) have a nice structure in the sense that theα parameter for all of them is the same and is equal to¹₂. This arises due to the assumption that θ[n] is zero-mean Gaussian. All the more, the random variablesZilare correlated. This is because Δθ[i, l] is constructed from a set of Ncrandom variables {θ[n]}^N_n=0^c−1which in turn can be described using a finite set of independent Gaussian random vari-ables. In Publication I, the PDF of a sum of correlated gamma random variables is derived using the Moschopoulos technique [95]. The PDF de-rived in Publication I is a generalization of the result of [96] which is applicable only for full-rank covariance matrix of the gamma variables.

In (3.11), the gamma variables have a rank-deficient covariance matrix.

The following theorem summarizes the result.

Theorem 3.3.1. Let {Zn}^N_n=1 be a set of N correlated gamma variates (Zn∼ G(α, βn)) with normalized covariance matrix Mzof any rankR ≤ N.

where{λn}^R_n=1are the ordered eigenvalues of the matrix PBP^TΔ with λ1

being the minimum. The P and Δ matrices are obtained from eigenvalue decomposition of Mxwhich is related to Mzas

(Mx)_ij=

A careful observation of (3.16) provides a nice interpretation of the PDF of Y : Firstly, we note that the parenthesis term in (3.16) represents a gamma distributed PDF; and thus, the PDF ofY is expressed as a weighted sum of gamma distributed PDFs with weightsζk.

PDF of ICI Power under the Wiener Model

Theorem 3.3.1 can be used to determine the PDF ofY in (3.11). In Pub-lication I, the parameters of the PDF of (3.16) for a Wiener phase noise model are derived. Of interest and importance is to relate the behavior of the PDF with the ratioρ = ^f_f^3dB_sub which is a measure of the level of inter-carrier interference. The behavior of the PDF in (3.16) is mainly dictated by the parameters R and λ1 which is the smallest eigenvalue. In Pub-lication I, for a Wiener phase noise model, it is shown thatR = Nc− 1

sub. From (3.22), we see thatλ1increases linearly withρ and, thus, we can expect a broadening in the PDF ofY . This is illustrated in Fig. 3.1, where we plot the PDF ofY for different values of Ncwhile keeping the system bandwidth andf3dB fixed. Since, the bandwidth is kept constant, varyingNcimplies varyingfsub, and hence,ρ also varies.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Marker Lines represent the simulated PDF Solid Lines represent the analytical PDF

y/(10⁻³)

Figure 3.1. PDF of the ICI power for different values ofNc. The bandwidth of the OFDM system is set to625 kHz and f3dB= 200 Hz.

PDF of ICI Power under the PARMA Model

For modeling phase noise in PLL-based devices, a discrete PARMA model is used which consists of a set of parallel auto-regressive-moving average filters. The resulting phase noise is given by

θ[n] =

where θp[n] is the output from the pth parallel filter (with impulse re-sponsehp[n]) corresponding to zero-mean, unit-variance white Gaussian inputs denoted by p[n]. We refer the reader to Chapter 2.4.3, where the relation between the filter coefficients and PLL device parameters is given. Utilizing this model, the expression for the gamma variablesZil

can be derived and is given by Zil∼ G

Since the diagonal matrix B, defined in (3.20), is composed of elements βil= 2

0 0.005 0.01 0.015 0.02 0.025 0

50 100 150 200 250 300

- *R

s= 50 kΩ - oR

s= 25 kΩ Nc = 1024, f

sub = 8.9 kHz

Nc = 2048, f

sub = 4.5 kHz

Nc = 4096, f

sub = 2.2 kHz

y

pY(y)

Figure 3.2. Comparison between simulated and analytical PDF plots ofY for different subcarrier spacingsfsuband phase noise PSD bandwidths obtained by vary-ing the loop filter resistanceRs. The solid lines represent the analytical PDF while the marker lines represent the simulated histograms. The OFDM sys-tem bandwidth is9.14 MHz. The PARMA filter parameters are given in Pub-lication II.

andλ1is the eigenvalue of PBP^TΔ, we have that

λ1∝ σ_η²ilp. (3.27)

In Fig. 3.2, we plot the PDF of the ICI power for a PARMA phase noise model. The PARMA filter coefficients are obtained assuming a charge-pump PLL device. Such a device is shown in Fig. 2.17. The filter coeffi-cients are obtained as described in Chapter 2.4.3. The phase noise band-width is controlled by varying the loop filter resistanceRs of the charge-pump PLL. An increase inRscauses the PDF to spread over higher values of magnitude as seen in the figure. This behavior can also be explained us-ing the PDF expression of (3.16). Firstly, the second term in (3.25) can be interpreted as the correlation between the impulse response coefficients of thepth parallel filter hp[j]. Thus, for fast-varying phase noise processes (large values ofRs), we can expect less correlation between the coefficients ofhp[j] and, thus, the second term is large which essentially results in a large value forλ1in (3.27). This effectively renders the PDF ofY towards higher values of magnitude.

Second-Order Statistics of ICI Power: The mean and variance of the ICI power can derived analytically using the PDF of (3.16). The mean

is evaluated as follows:

. The integral above is of the form [97],

 _∞

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.