Mixing
A maximization problem in the paper “Maximum Ratio Transmission”.
$begingroup$
On page 1459 in [1], there is a maximization problem:
$$
max_{mathbf{g}} gamma, tag{1}
$$
where $mathbf{g} = [g_1, ..., g_L]$ is an $1 times L$ complex vector. $L$ is a positive integer.
$$
gamma = frac{a^2}{mathbf{g g}^H} gamma_0 = frac{a^2 gamma_0}{sum_{p=1}^L |g_p|^2}.
$$
$a$ is the normalization factor which is required to be
$$
a = |mathbf{g H}| = left( sum_{p=1}^L sum_{q=1}^L g_p g_q^* sum_{k=1}^K h_{pk} h_{qk}^* right)^{1/2},
$$
where $mathbf{H}$ is an $L times K$ complex matrix where the entry $h_{pk}$ are statistically independent and identical. $g_p$ and $g_q$ are $p$-th and $q$-th elements of the vector $mathbf{g}$, respectively. The superscript $^*$ denotes the complex conjugate. $K$ is a positive integer. $h_{pk}$ and $h_{qk}$ are the $(p,k)$ and $(q,k)$ entries of the matrix $mathbf{H}$, respectively. $gamma_0$ is a positive constant.
The author of [1] wrote
... the condition that $|g_1| = |g_2| = ... = |g_L|$ has to be satisfied for the maximum value of the SNR.
Here SNR means signal-to-noise ratio $gamma$. I do not understand why $|g_1| = |g_2| = ... = |g_L|$. Could you please tell me? Thank you in advance.
Reference
[1] T. K. Y. Lo, "Maximum ratio transmission," 1999 IEEE International Conference on Communications (Cat. No. 99CH36311), Vancouver, BC, 1999, pp. 1310-1314 vol.2.
doi: 10.1109/ICC.1999.765552
keywords: {error statistics;antenna arrays;receiving antennas;transmitting antennas;diversity reception;radiocommunication;radiofrequency interference;maximum ratio transmission;wireless communications;multiple antennas;transmission;reception;maximum ratio combining;SNR;cross-correlation;channel vectors;error probability;Transmitting antennas;Diversity reception;Fading;Transmitters;Wireless communication;Receiving antennas;Antennas and propagation;Delay;Analytical models;Error probability},
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=765552&isnumber=16571
optimization norm
asked Jan 29 at 9:21
Wei-Cheng LiuWei-Cheng Liu
7414
$endgroup$
add a comment |
$begingroup$
On page 1459 in [1], there is a maximization problem:
$$
max_{mathbf{g}} gamma, tag{1}
$$
where $mathbf{g} = [g_1, ..., g_L]$ is an $1 times L$ complex vector. $L$ is a positive integer.
$$
gamma = frac{a^2}{mathbf{g g}^H} gamma_0 = frac{a^2 gamma_0}{sum_{p=1}^L |g_p|^2}.
$$
$a$ is the normalization factor which is required to be
$$
a = |mathbf{g H}| = left( sum_{p=1}^L sum_{q=1}^L g_p g_q^* sum_{k=1}^K h_{pk} h_{qk}^* right)^{1/2},
$$
where $mathbf{H}$ is an $L times K$ complex matrix where the entry $h_{pk}$ are statistically independent and identical. $g_p$ and $g_q$ are $p$-th and $q$-th elements of the vector $mathbf{g}$, respectively. The superscript $^*$ denotes the complex conjugate. $K$ is a positive integer. $h_{pk}$ and $h_{qk}$ are the $(p,k)$ and $(q,k)$ entries of the matrix $mathbf{H}$, respectively. $gamma_0$ is a positive constant.
The author of [1] wrote
... the condition that $|g_1| = |g_2| = ... = |g_L|$ has to be satisfied for the maximum value of the SNR.
Here SNR means signal-to-noise ratio $gamma$. I do not understand why $|g_1| = |g_2| = ... = |g_L|$. Could you please tell me? Thank you in advance.
Reference
[1] T. K. Y. Lo, "Maximum ratio transmission," 1999 IEEE International Conference on Communications (Cat. No. 99CH36311), Vancouver, BC, 1999, pp. 1310-1314 vol.2.
doi: 10.1109/ICC.1999.765552
keywords: {error statistics;antenna arrays;receiving antennas;transmitting antennas;diversity reception;radiocommunication;radiofrequency interference;maximum ratio transmission;wireless communications;multiple antennas;transmission;reception;maximum ratio combining;SNR;cross-correlation;channel vectors;error probability;Transmitting antennas;Diversity reception;Fading;Transmitters;Wireless communication;Receiving antennas;Antennas and propagation;Delay;Analytical models;Error probability},
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=765552&isnumber=16571
optimization norm
asked Jan 29 at 9:21
Wei-Cheng LiuWei-Cheng Liu
7414
$endgroup$
add a comment |
$begingroup$
On page 1459 in [1], there is a maximization problem:
$$
max_{mathbf{g}} gamma, tag{1}
$$
where $mathbf{g} = [g_1, ..., g_L]$ is an $1 times L$ complex vector. $L$ is a positive integer.
$$
gamma = frac{a^2}{mathbf{g g}^H} gamma_0 = frac{a^2 gamma_0}{sum_{p=1}^L |g_p|^2}.
$$
$a$ is the normalization factor which is required to be
$$
a = |mathbf{g H}| = left( sum_{p=1}^L sum_{q=1}^L g_p g_q^* sum_{k=1}^K h_{pk} h_{qk}^* right)^{1/2},
$$
where $mathbf{H}$ is an $L times K$ complex matrix where the entry $h_{pk}$ are statistically independent and identical. $g_p$ and $g_q$ are $p$-th and $q$-th elements of the vector $mathbf{g}$, respectively. The superscript $^*$ denotes the complex conjugate. $K$ is a positive integer. $h_{pk}$ and $h_{qk}$ are the $(p,k)$ and $(q,k)$ entries of the matrix $mathbf{H}$, respectively. $gamma_0$ is a positive constant.
The author of [1] wrote
... the condition that $|g_1| = |g_2| = ... = |g_L|$ has to be satisfied for the maximum value of the SNR.
Here SNR means signal-to-noise ratio $gamma$. I do not understand why $|g_1| = |g_2| = ... = |g_L|$. Could you please tell me? Thank you in advance.
Reference
[1] T. K. Y. Lo, "Maximum ratio transmission," 1999 IEEE International Conference on Communications (Cat. No. 99CH36311), Vancouver, BC, 1999, pp. 1310-1314 vol.2.
doi: 10.1109/ICC.1999.765552
keywords: {error statistics;antenna arrays;receiving antennas;transmitting antennas;diversity reception;radiocommunication;radiofrequency interference;maximum ratio transmission;wireless communications;multiple antennas;transmission;reception;maximum ratio combining;SNR;cross-correlation;channel vectors;error probability;Transmitting antennas;Diversity reception;Fading;Transmitters;Wireless communication;Receiving antennas;Antennas and propagation;Delay;Analytical models;Error probability},
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=765552&isnumber=16571
optimization norm
asked Jan 29 at 9:21
Wei-Cheng LiuWei-Cheng Liu
7414
$endgroup$
On page 1459 in [1], there is a maximization problem:
$$
max_{mathbf{g}} gamma, tag{1}
$$
where $mathbf{g} = [g_1, ..., g_L]$ is an $1 times L$ complex vector. $L$ is a positive integer.
$$
gamma = frac{a^2}{mathbf{g g}^H} gamma_0 = frac{a^2 gamma_0}{sum_{p=1}^L |g_p|^2}.
$$
$a$ is the normalization factor which is required to be
$$
a = |mathbf{g H}| = left( sum_{p=1}^L sum_{q=1}^L g_p g_q^* sum_{k=1}^K h_{pk} h_{qk}^* right)^{1/2},
$$
where $mathbf{H}$ is an $L times K$ complex matrix where the entry $h_{pk}$ are statistically independent and identical. $g_p$ and $g_q$ are $p$-th and $q$-th elements of the vector $mathbf{g}$, respectively. The superscript $^*$ denotes the complex conjugate. $K$ is a positive integer. $h_{pk}$ and $h_{qk}$ are the $(p,k)$ and $(q,k)$ entries of the matrix $mathbf{H}$, respectively. $gamma_0$ is a positive constant.
The author of [1] wrote
... the condition that $|g_1| = |g_2| = ... = |g_L|$ has to be satisfied for the maximum value of the SNR.
Here SNR means signal-to-noise ratio $gamma$. I do not understand why $|g_1| = |g_2| = ... = |g_L|$. Could you please tell me? Thank you in advance.
Reference
[1] T. K. Y. Lo, "Maximum ratio transmission," 1999 IEEE International Conference on Communications (Cat. No. 99CH36311), Vancouver, BC, 1999, pp. 1310-1314 vol.2.
doi: 10.1109/ICC.1999.765552
keywords: {error statistics;antenna arrays;receiving antennas;transmitting antennas;diversity reception;radiocommunication;radiofrequency interference;maximum ratio transmission;wireless communications;multiple antennas;transmission;reception;maximum ratio combining;SNR;cross-correlation;channel vectors;error probability;Transmitting antennas;Diversity reception;Fading;Transmitters;Wireless communication;Receiving antennas;Antennas and propagation;Delay;Analytical models;Error probability},
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=765552&isnumber=16571
optimization norm
optimization norm
asked Jan 29 at 9:21
Wei-Cheng LiuWei-Cheng Liu
7414
asked Jan 29 at 9:21
Wei-Cheng LiuWei-Cheng Liu
7414
asked Jan 29 at 9:21
Wei-Cheng LiuWei-Cheng Liu
7414
asked Jan 29 at 9:21
Wei-Cheng LiuWei-Cheng Liu
7414
asked Jan 29 at 9:21
Wei-Cheng LiuWei-Cheng Liu
7414
7414
add a comment |
add a comment |
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