Capacity of κ-μ Shadowed Fading Channels

Algorithm 1

Implementation of the bivariate Meijer 𝐺-function in Mathematica software (*modification of the EGBMGF [18] to conform the 𝜅-𝜇 shadowed model*).

Clear All;

S[{a1st_, b1st_}, {a2s_, b2s_}, {a3t_, b3t_}, 𝜇S_, mS_, {zs_, zt_}]:= Module[{},

(*Function Φ₁(s + t)*)

Past = Function[u, Product[Gamma[1 − a1st[[1, n]] − u], {n, 1, Length[a1st[[1]]]}]];

Qast = Function[u, Product[Gamma[a1st[[2, n]] + u], {n, 1, Length[a1st[[2]]]}]];

Pbst = Function[u, Product[Gamma[b1st[[1, n]] + u], {n, 1, Length[b1st[[1]]]}]];

Qbst = Function[u, Product[Gamma[1 − b1st[[2, n]] − u], {n, 1, Length[b1st[[2]]]}]];

Mst = Function[u, Past[u] * Pbst[u]/(Qast[u] Qbst[u])];

(*Function Φ₂(s)*)

Pas = Function[u, Product[Gamma[1 − a2s[[1, n]] − u], {n, 1, Length[a2s[[1]]]}]];

Qas = Function[u, Product[Gamma[a2s[[2, n]] + u], {n, 1, Length[a2s[[2]]]}]];

Pbs = Function[u, Product[Gamma[b2s[[1, n]] + u], {n, 1, Length[b2s[[1]]]}]];

Qbs = Function[u, Product[Gamma[1 − b2s[[2, n]] − u], {n, 1, Length[b2s[[2]]]}]];

Ms = Function[u, Pas[u] * Pbs[u]/(Qas[u] Qbs[u])];

(*Function Φ₃(t)*)

Pat = Function[u, Product[Gamma[1 − a3t[[1, n]] − u], {n, 1, Length[a3t[[1]]]}]];

Qat = Function[u, Product[Gamma[a3t[[2, n]] + u], {n, 1, Length[a3t[[2]]]}]];

Pbt = Function[u, Product[Gamma[b3t[[1, n]] + u], {n, 1, Length[b3t[[1]]]}]];

Qbt = Function[u, Product[Gamma[1 − b3t[[2, n]] − u], {n, 1, Length[b3t[[2]]]}]];

Mt = Function[u, Pat[u] * Pbt[u]/(Qat[u] Qbt[u])];

MT = Function[u, v, Mst[u + v] * Ms[u] * Mt[v]];

(*Countour limiters*)

Zs = zs; (*x*)

Zt = zt; (*y*)

W = 50;

Wp = 10; (*When it is necessary to apply integration by parts. Only for t variable*)

Rs = −1/4;

If[(𝜇S − mS) > 0,

 Rt = (𝜇S − mS)/2;

 (*Final Evaluation*)

 Print[“Numerical Integration:”];

 value = 1/(2𝜋I)² NIntegrate[MT[s, t] Zs−𝑠 Zt−𝑡, {s, Rs – I * W, Rs + I * W}, {t, Rt – I * W, Rt + I * W}],

 PosPoleT = Ceiling[Abs[𝜇S − mS]] + (𝜇S − mS);

 NegPoleT = Floor[Abs[𝜇S − mS]] + (𝜇S − mS);

 Rt1 = PosPoleT/2;

 Rt2 = NegPoleT/2;

  (*Final Evaluation*)

 Print[“Numerical Integration by parts:”];

 valIni = 1/(2𝜋I)² NIntegrate[MT[s, t] Zs−𝑠 Zt−𝑡, {s, Rs – I * W, Rs + I * W}, {t, Rt1 – I * W, Rt1 – I * Wp}];

 nIter = Abs[Floor[𝜇S − mS]];

 val1T = Sum[1/(2𝜋I)² NIntegrate[MT[s, t] Zs−𝑠 Zt−𝑡, {s, Rs – I * W, Rs + I * W}, {t, (Rt1 − n + 1) – I * Wp,

   (Rt1 − n + 1) + I * Wp}], {n, 1, nIter}];

 val2T = Sum[1/(2𝜋I)² NIntegrate[MT[s, t] Zs−𝑠 Zt−𝑡, {s, Rs – I * W, Rs + I * W}, {t, (Rt1 − n + 1) + I * Wp,

   (Rt2 − n + 1) + I * Wp}], {n, 1, nIter}];

 val3T = Sum[1/(2𝜋I)² NIntegrate[MT[s, t] Zs−𝑠 Zt−𝑡, {s, Rs – I * W, Rs + I * W}, {t, (Rt2 − n + 1) + I * Wp,

   (Rt2 − n + 1) – I * Wp}], {n, 1, nIter}];

 val4T = Sum[1/(2𝜋I)² NIntegrate[MT[s, t] Zs−𝑠 Zt−𝑡, {s, Rs – I * W, Rs + I * W}, {t, (Rt2 − n + 1) – I * Wp,

   (Rt1 − n) – I * Wp}], {n, 1, nIter}];

 valFin = 1/(2𝜋I)² NIntegrate[MT[s, t] Zs−𝑠 Zt−𝑡, {s, Rs – I * W, Rs + I * W}, {t, (Rt1 − nIter) – I * Wp,

   (Rt1 − nIter) + I * W}];

 value = valIni + valFin + val1T + val2T + val3T + val4T];

 (*Returning back the value*)

 Return[value];

];

(*End of MEGBMGF*)