这篇博客深入解析了Wishart矩阵迹的期望值在多天线通信中的应用，包括迹迹和迹平方的期望，以及矩阵逆和二次逆迹的期望计算过程。通过Alozano和Verdu等论文的推导，展示了在低功率和容量分析中的关键数学技巧。
几篇论文里关于 Wishart 矩阵迹的期望的一些推导过程。
有几个地方还有疑问。
Multiple-Antenna Channel Hardening and Its Implications for Rate Feedback and Scheduling Bertrand M. Hochwald, Member, IEEE, Thomas L. Marzetta, Fellow, IEEE, and Vahid Tarokh, Member, IEEE
附录 lemma A
E{trW}=kKE{λi}=KE{trW2}=kK(K+k))E{λi2}=K(K+k))E{(trW)2}=kK(kK+1))E{λiλj}=K(K−1))\begin{aligned} &\mathbf{E}\left \{ \mathrm{tr}W\right \}=kK \quad &\mathbf{E}\left \{ \lambda _i\right \}=K \\ &\mathbf{E}\left \{ \mathrm{tr}W^2\right \}=kK(K+k)) \quad & \mathbf{E}\left \{ \lambda _i^2\right\}=K(K+k)) \\ & \mathbf{E}\left \{ ( \mathrm{tr}W)^2\right \}=kK(kK+1)) \quad &\mathbf{E}\left \{ \lambda _i\lambda_j\right\}=K(K-1)) \\ \end{aligned}E{trW}=kKE{trW2}=kK(K+k))E{(trW)2}=kK(kK+1))E{λi}=KE{λi2}=K(K+k))E{λiλj}=K(K−1))
E{tr(W2)}=E{∑i=1k(∑j=1kWijWji)}=E{∑i=1k(Wii)2+∑i,j=1,i≠jk(WijWji)}=E{∑i=1k(∑p=1K∣hip∣2)2+∑i≠j(∑p=1K∣hip∣2∣hjp∣2)}=k(2K+K(K−1))+k(k−1)K=kK(K+k)\begin{aligned} \mathbf{E}\left \{ \mathrm{tr}(W^2)\right \}&=\mathbf{E}\left \{ \sum _{i=1}^k\left(\sum_{j=1}^k W_{ij}W_{ji}\right)\right \}\\ &=\mathbf{E}\left \{\sum_{i=1}^k \left( W_{ii}\right)^2+\sum_{i,j=1,i\neq j}^k \left( W_{ij}W_{ji}\right)\right \}\\ &=\mathbf{E}\left \{\sum_{i=1}^k \left(\sum_{p=1}^K \left | h_{ip} \right |^2 \right)^2+\sum_{i\neq j} \left( \sum_{p=1}^K\left | h_{ip} \right |^2\left | h_{jp} \right |^2\right)\right \} \\ &=k(2K+K(K-1))+k(k-1)K\\ &=kK(K+k ) \end{aligned}E{tr(W2)}=E{i=1∑k(j=1∑kWijWji)}=E⎩⎨⎧i=1∑k(Wii)2+i,j=1,i=j∑k(WijWji)⎭⎬⎫=E⎩⎨⎧i=1∑k(p=1∑K∣hip∣2)2+i=j∑(p=1∑K∣hip∣2∣hjp∣2)⎭⎬⎫=k(2K+K(K−1))+k(k−1)K=kK(K+k)
E{(trW)2}=E{(∑i=1kWii)2}=E{∑i=1k(Wii)2+∑i,j=1,i≠jk(WiiWjj)}=E{∑i=1k(∑p=1K∣hip∣2)2+∑i≠j(∑p=1K∣hip∣2∑p′=1K∣hjp′∣2)}=k(2K+K(K−1))+k(k−1)KK=k2K2+kK\begin{aligned} \mathbf{E}\left \{ ( \mathrm{tr}W)^2\right \}&=\mathbf{E}\left \{ \left( \sum _{i=1}^k W_{ii}\right)^2\right \}\\ &=\mathbf{E}\left \{\sum_{i=1}^k \left( W_{ii}\right)^2+\sum_{i,j=1,i\neq j}^k \left( W_{ii}W_{jj}\right)\right \}\\ &=\mathbf{E}\left \{\sum_{i=1}^k \left(\sum_{p=1}^K \left | h_{ip} \right |^2 \right)^2+\sum_{i\neq j} \left( \sum_{p=1}^K\left | h_{ip} \right |^2\sum_{p'=1}^K\left | h_{jp'} \right |^2\right)\right \} \\ &=k(2K+K(K-1))+k(k-1)KK\\ &=k^2K^2+kK \end{aligned}E{(trW)2}=E⎩⎨⎧(i=1∑kWii)2⎭⎬⎫=E⎩⎨⎧i=1∑k(Wii)2+i,j=1,i=j∑k(WiiWjj)⎭⎬⎫=E⎩⎨⎧i=1∑k(p=1∑K∣hip∣2)2+i=j∑⎝⎛p=1∑K∣hip∣2p′=1∑K∣hjp′∣2⎠⎞⎭⎬⎫=k(2K+K(K−1))+k(k−1)KK=k2K2+kK
A lozano, A M Tulino, S Verdu, Multiple-Antenna Capacity in the Low-Power Regime, IEEE Trans. Inform. Theory, vol. 49, pp. 2527–2544, Oct. 2003.
Lemma 6:
E{tr(W−1)}=nm−n, (m>n)E{tr(W−2)}=mn(m−n)3−(m−n),E{tr2(W−1)}=nm−n(m(m−n)2−1+n−1m−n+1),\begin{aligned} \mathbf{E}\left \{ \mathrm{tr}\left (W^{-1} \right )\right \}&=\frac{n}{m-n}, \quad \ (m>n) \\ \mathbf{E}\left \{ \mathrm{tr}\left (W^{-2} \right )\right \}&=\frac{mn}{(m-n)^3-(m-n)}, \\ \mathbf{E}\left \{ \mathrm{tr}^2\left ( W^{-1} \right )\right \}&=\frac{n}{m-n}\left ( \frac{m}{(m-n)^2-1}+\frac{n-1}{m-n+1}\right ), \\ \end{aligned}E{tr(W−1)}E{tr(W−2)}E{tr2(W−1)}=m−nn, (m>n)=(m−n)3−(m−n)mn,=m−nn((m−n)2−1m+m−n+1n−1),
proof:
(1) E{tr(W−1)}=nE[1λ]\mathbf{E}\left \{ \mathrm{tr}\left (W^{-1} \right )\right \}=n E \left [\frac{1}{\lambda} \right ]E{tr(W−1)}=nE[λ1]
Eigenvalue marginal density:
fλ(z)=1n∑k=0n−1k!(Lkm−n(z))2(k+m−n)!zm−ne−zf_\lambda(z)=\frac{1}{n}\sum_{k=0}^{n-1}\frac{k!(L_k^{m-n}(z))^2}{(k+m-n)!}z^{m-n}e^{-z}fλ(z)=n1k=0∑n−1(k+m−n)!k!(Lkm−n(z))2zm−ne−z
E[1λ]=∫0∞fλ(z)zdz=1m−n?\begin{aligned} E \left [\frac{1}{\lambda} \right ]&=\int_{0}^{\infty}\frac{f_\lambda(z)}{z}dz\\ &=\frac{1}{m-n} \quad ? \end{aligned}E[λ1]=∫0∞zfλ(z)dz=m−n1?
fλ(z)=1n∑k=0n−1k!(Lkm−n(z))2(k+m−n)!zm−ne−zf_\lambda(z)=\frac{1}{n}\sum_{k=0}^{n-1}\frac{k!(L_k^{m-n}(z))^2}{(k+m-n)!}z^{m-n}e^{-z}fλ(z)=n1k=0∑n−1(k+m−n)!k!(Lkm−n(z))2zm−ne−z
∫0∞fλ(z)zdz=∫0∞1n∑k=0n−1k!(Lkm−n(z))2(k+m−n)!zm−n−1e−zdz=∫0∞1n∑k=0n−1k!(Lkm−n(z))2(k+m−n)!e−zdzm−nm−n=−1m−n∫0∞1n∑k=0n−1k!(k+m−n)!zm−nd(Lkm−n(z))2e−z=1m−n∫0∞1n∑k=0n−1k!(k+m−n)!((Lkm−n(z))2−2Lkm−n(Lkm−n(z))′)e−zzm−ndz=1m−n(1−∫0∞1n∑k=0n−1k!(k+m−n)!(2Lkm−n(Lkm−n(z))′)e−zzm−ndz)\begin{aligned} \int_0^{\infty}\frac{f_\lambda(z)}{z}dz&=\int_0^{\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{k!(L_k^{m-n}(z))^2}{(k+m-n)!}z^{m-n-1}e^{-z}dz\\ &=\int_0^{\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{k!(L_k^{m-n}(z))^2}{(k+m-n)!}e^{-z}\frac{dz^{m-n}}{m-n}\\ &=\frac {-1}{m-n} \int_0^{\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{k!}{(k+m-n)!}z^{m-n}d(L_k^{m-n}(z))^2e^{-z}\\ &=\frac {1}{m-n} \int_0^{\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{k!}{(k+m-n)!}\left ((L_k^{m-n}(z))^2-2L_k^{m-n}(L_k^{m-n}(z))' \right )e^{-z}z^{m-n}dz\\ &=\frac {1}{m-n}\left(1- \int_0^{\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{k!}{(k+m-n)!}\left (2L_k^{m-n}(L_k^{m-n}(z))' \right )e^{-z}z^{m-n}dz \right )\\ \end{aligned}∫0∞zfλ(z)dz=∫0∞n1k=0∑n−1(k+m−n)!k!(Lkm−n(z))2zm−n−1e−zdz=∫0∞n1k=0∑n−1(k+m−n)!k!(Lkm−n(z))2e−zm−ndzm−n=m−n−1∫0∞n1k=0∑n−1(k+m−n)!k!zm−nd(Lkm−n(z))2e−z=m−n1∫0∞n1k=0∑n−1(k+m−n)!k!((Lkm−n(z))2−2Lkm−n(Lkm−n(z))′)e−zzm−ndz=m−n1(1−∫0∞n1k=0∑n−1(k+m−n)!k!(2Lkm−n(Lkm−n(z))′)e−zzm−ndz)
∫0∞1n∑k=0n−1k!(k+m−n)!(Lkm−n(Lkm−n(z))′)e−zzm−ndz=0?\int_0^{\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{k!}{(k+m-n)!}\left (L_k^{m-n}(L_k^{m-n}(z))' \right )e^{-z}z^{m-n}dz=0 \quad ?∫0∞n1k=0∑n−1(k+m−n)!k!(Lkm−n(Lkm−n(z))′)e−zzm−ndz=0?
(Lkm−n(z))′=−Lk−1m−n+1(z),(see 8.981, p1439)(L_k^{m-n}(z))'=-L_{k-1}^{m-n+1}(z), \quad (\textrm {see 8.981, p1439})(Lkm−n(z))′=−Lk−1m−n+1(z),(see 8.981, p1439)
Laguerre Polynomials:
∫0∞(Lkm−nLk′m−n(z))e−zzm−ndz=0,k≠k′\int_0^{\infty}\left (L_k^{m-n}L_{k'}^{m-n}(z) \right )e^{-z}z^{m-n}dz=0 \quad , \quad k\neq k'∫0∞(Lkm−nLk′m−n(z))e−zzm−ndz=0,k=k′
Lka=1k!exx−adkdxk(e−xxa+k)=1k!exx−a(∑i=0k(ki)dk−idxk−ie−xdidxixa+k)=1k!∑i=0k(ki)(−1)k−ie−x(a+k)!(a+k−i)!xk−i\begin{aligned} L_k^{a}&=\frac{1}{k!}e^xx^{-a}\frac{d^k}{dx^k}\left ( e^{-x}x^{a+k} \right )\\ &=\frac{1}{k!}e^xx^{-a}\left (\sum_{i=0}^{k} \binom{k}{i}\frac{d^{k-i}}{dx^{k-i}}e^{-x}\frac{d^i}{dx^i}x^{a+k} \right )\\ &=\frac{1}{k!}\sum_{i=0}^{k} \binom{k}{i}(-1)^{k-i}e^{-x}\frac{(a+k)!}{(a+k-i)!}x^{k-i}\\ \end{aligned}Lka=k!1exx−adxkdk(e−xxa+k)=k!1exx−a(i=0∑k(ik)dxk−idk−ie−xdxidixa+k)=k!1i=0∑k(ik)(−1)k−ie−x(a+k−i)!(a+k)!xk−i
(2)
E{tr(W−2)}=nE[1λ2]\mathbf{E}\left \{ \mathrm{tr}\left (W^{-2} \right )\right \}=n E \left [\frac{1}{\lambda^2} \right ]E{tr(W−2)}=nE[λ21]
E[1λ2]=∫0∞fλ(z)z2dz=mn(m−n)3−(m−n)?\begin{aligned} E \left [\frac{1}{\lambda^2} \right ]&=\int_{0}^{\infty}\frac{f_\lambda(z)}{z^2}dz\\ &=\frac{mn}{(m-n)^3-(m-n)} \quad ? \end{aligned}E[λ21]=∫0∞z2fλ(z)dz=(m−n)3−(m−n)mn?
∫0∞fλ(z)z2dz=∫0∞1n∑k=0n−1k!(Lkm−n(z))2(k+m−n)!zm−n−2e−zdz\begin{aligned} \int_0^{\infty}\frac{f_\lambda(z)}{z^2}dz=\int_0^{\infty}\frac{1}{n}\sum_{k=0}^{n-1}\frac{k!(L_k^{m-n}(z))^2}{(k+m-n)!}z^{m-n-2}e^{-z}dz \end{aligned}∫0∞z2fλ(z)dz=∫0∞n1k=0∑n−1(k+m−n)!k!(Lkm−n(z))2zm−n−2e−zdz
∫0∞(Lkm−n(z))2zm−n−2e−zdz=∫0∞(Lkm−n(z))2e−zdzm−n−1m−n−1=−∫0∞zm−n−1m−n−1d(Lkm−n(z))2e−z=∫0∞zm−n−1m−n−1((Lkm−n(z))2+(Lkm−n(z)Lk−1m−n+1(z))e−zdz\begin{aligned} \int_0^{\infty}(L_k^{m-n}(z))^2z^{m-n-2}e^{-z}dz&=\int_0^{\infty}(L_k^{m-n}(z))^2e^{-z}\frac{dz^{m-n-1}}{m-n-1} \\&=-\int_0^{\infty}\frac{z^{m-n-1}}{m-n-1}d(L_k^{m-n}(z))^2e^{-z}\\ &=\int_0^{\infty}\frac{z^{m-n-1}}{m-n-1}\left ((L_k^{m-n}(z))^2+(L_k^{m-n}(z)L_{k-1}^{m-n+1}(z) \right )e^{-z}dz\\ \end{aligned}∫0∞(Lkm−n(z))2zm−n−2e−zdz=∫0∞(Lkm−n(z))2e−zm−n−1dzm−n−1=−∫0∞m−n−1zm−n−1d(Lkm−n(z))2e−z=∫0∞m−n−1zm−n−1((Lkm−n(z))2+(Lkm−n(z)Lk−1m−n+1(z))e−zdz
∫0∞zm−n−1m−n−1(Lkm−n(z)Lk−1m−n+1(z))e−zdz=∫0∞dzm−n(m−n)(m−n−1)(Lkm−n(z)Lk−1m−n+1(z))e−z=−∫0∞zm−n(m−n)(m−n−1)d(Lkm−n(z)Lk−1m−n+1(z))e−z=∫0∞zm−n(m−n)(m−n−1)(Lkm−n(z)Lk−1m−n+1(z)−Lkm−n(z)Lk−2m−n+2(z)−Lk−1m−n+1(z)Lk−1m−n+1(z))e−zdz=∫0∞zm−n(m−n)(m−n−1)(−Lk−1m−n+1(z)Lk−1m−n+1(z))e−zdz=∫0∞zm−n(m−n)(m−n−1)(−∑t=0k−1Ltm−n(z)Ltm−n(z))e−zdz\begin{aligned} \int_0^{\infty}\frac{z^{m-n-1}}{m-n-1}\left (L_k^{m-n}(z)L_{k-1}^{m-n+1}(z) \right )e^{-z}dz&=\int_0^{\infty}\frac{dz^{m-n}}{(m-n)(m-n-1)}\left (L_k^{m-n}(z)L_{k-1}^{m-n+1}(z) \right )e^{-z}\\ &=-\int_0^{\infty}\frac{z^{m-n}}{(m-n)(m-n-1)}d\left (L_k^{m-n}(z)L_{k-1}^{m-n+1}(z) \right )e^{-z}\\ &=\int_0^{\infty}\frac{z^{m-n}}{(m-n)(m-n-1)}\left (L_k^{m-n}(z)L_{k-1}^{m-n+1}(z) -L_k^{m-n}(z)L_{k-2}^{m-n+2}(z)-L_{k-1}^{m-n+1}(z)L_{k-1}^{m-n+1}(z)\right )e^{-z}dz\\ &=\int_0^{\infty}\frac{z^{m-n}}{(m-n)(m-n-1)}\left (-L_{k-1}^{m-n+1}(z)L_{k-1}^{m-n+1}(z)\right )e^{-z}dz\\ &=\int_0^{\infty}\frac{z^{m-n}}{(m-n)(m-n-1)}\left (-\sum_{t=0}^{k-1}L_{t}^{m-n}(z)L_{t}^{m-n}(z)\right )e^{-z}dz\end{aligned}∫0∞m−n−1zm−n−1(Lkm−n(z)Lk−1m−n+1(z))e−zdz=∫0∞(m−n)(m−n−1)dzm−n(Lkm−n(z)Lk−1m−n+1(z))e−z=−∫0∞(m−n)(m−n−1)zm−nd(Lkm−n(z)Lk−1m−n+1(z))e−z=∫0∞(m−n)(m−n−1)zm−n(Lkm−n(z)Lk−1m−n+1(z)−Lkm−n(z)Lk−2m−n+2(z)−Lk−1m−n+1(z)Lk−1m−n+1(z))e−zdz=∫0∞(m−n)(m−n−1)zm−n(−Lk−1m−n+1(z)Lk−1m−n+1(z))e−zdz=∫0∞(m−n)(m−n−1)zm−n(−t=0∑k−1Ltm−n(z)Ltm−n(z))e−zdz
S. Verdú and S. Shamai (Shitz}, “Spectral efficiency of CDMA with random spreading,” IEEE Trans. inform. Theory, vol. 45, pp. 622–640, Mar. 1999.
B. Hochwald, T. Marzetta, and B. Hassibi, “Space–time autocoding,” IEEE Trans. Inform. Theory, vol. 47, pp. 2761–2781, Nov. 2001
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