From 40a2ee7a20651c80557f94468706cb30b03ddfeb Mon Sep 17 00:00:00 2001 From: rzy0901 Date: Wed, 18 Oct 2023 21:55:36 +0800 Subject: [PATCH] Update channel.md --- content/post/channel.md | 75 +++++++++++++++++++++++++++++++---------- 1 file changed, 58 insertions(+), 17 deletions(-) diff --git a/content/post/channel.md b/content/post/channel.md index 0116f68..12163c5 100644 --- a/content/post/channel.md +++ b/content/post/channel.md @@ -55,10 +55,11 @@ typora-root-url: ../../static Substitute $k$ with $f$ at the left of the equation, and substitute $k$ with $f/\Delta f$ at the right side of the equation, we have: $$ - \begin{aligned} - H(t,f)&=\sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\frac{\mathrm{j2\pi f/\Delta f n}}{N}), \\ + \begin{align} + H(t,f)&=\sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\frac{\mathrm{j2\pi f/\Delta f n}}{N}), \nonumber \\ &\xlongequal[]{\Delta f =1/(N\Delta t)}\sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\mathrm{j2\pi f\tau_n(t)}). - \end{aligned} + \label{eq:CFR} + \end{align} $$ + Channel State Information (CSI): $H(t,f)$. (CSI is actually discrete time series of discrete CFR!) @@ -127,13 +128,15 @@ typora-root-url: ../../static \end{align} $$ - where $\phi_n(t)=2\pi f_\text{c} \tau_n(t)+\phi_{\text{D}_n}(t)$ denotes the phase of $n$-th path at time slot $t$, with $\phi_{\text{D}_n}(t)$ denotes the Doppler phase shift. + where $\phi_n(t)=2\pi f_\text{c} \tau_n(t)-\phi_{\text{D}_n}(t)$ denotes the phase of $n$-th path at time slot $t$, with $\phi_{\text{D}_n}(t)$ denotes the Doppler phase shift. + + + Doppler phase shift $\phi_{\text{D}_n}(t)$ is a function of Doppler frequency $f_{\text{D}_n}(t)$: $\phi_{\text{D}_n}(t)=\int_t 2\pi f_{\text{D}_n}(t) \mathrm d t$. + + + Doppler frequency shift $f_{\text{D}_n}(t)=v\cos(\theta(t))/\lambda$ with motion velocity $v$, angel of arrival relative to the direction of motion $\theta(t)$ and signal wavelength $\lambda$. - > + Doppler phase shift $\phi_{\text{D}_n}(t)$ is a function of Doppler frequency $f_{\text{D}_n}(t)$: $\phi_{\text{D}_n}(t)=\int_t 2\pi f_{\text{D}_n}(t) \mathrm d t$. - > - > + Doppler frequency shift $f_{\text{D}_n}(t)=v\cos(\theta(t))/\lambda$ with motion velocity $v$, angel of arrival relative to the direction of motion $\theta(t)$ and signal wavelength $\lambda$. + + When $\phi_{\text{D}_n}(t)=0$, equations $\eqref{eq:ch_real}$ and $\eqref{eq:ch_complex}$ are equivalent (See $\eqref{eq:received_relation}$). -## Received signal +### Received signal + Transmitted signal (See [definition](#definition)): @@ -146,9 +149,10 @@ typora-root-url: ../../static + Real: $$ \begin{align} - r(t) &= s(t) \otimes h(t,\tau) \nonumber \\ + r(t) &= s(t) \otimes h_\text{real}(t,\tau) \nonumber \\ &= s(t) \otimes \sum_{n=0}^{N(t)-1}\alpha_{n}(t)\delta(\tau-\tau_n(t)) \nonumber \\ - &=\sum_{n=0}^{N(t)-1}\alpha_{n}(t)s(\tau-\tau_n(t)). + &=\sum_{n=0}^{N(t)-1}\alpha_{n}(t)s(t-\tau_n(t)). + \label{eq:received_real} \end{align} $$ @@ -157,7 +161,9 @@ typora-root-url: ../../static \begin{align} u_r(t) &= u(t) \otimes h_\text{complex}(t,\tau)\nonumber \\ &=u(t) \otimes \sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\mathrm{j}\phi_n(t))\delta(\tau-\tau_n(t)) \nonumber \\ - &=\sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\mathrm{j}\phi_n(t))u(\tau-\tau_n(t)). + &=\sum_{n=0}^{N(t)-1}\alpha_n(t)u(t-\tau_n(t))\exp(-\mathrm{j}\phi_n(t)) \nonumber\\ + &=\sum_{n=0}^{N(t)-1}\alpha_n(t)u(t-\tau_n(t))\exp(-\mathrm{j}2\pi f_\text{c}\tau_n(t)+\phi_{\text{D}_n}(t)). + \label{eq:received_complex} \end{align} $$ @@ -165,14 +171,49 @@ typora-root-url: ../../static + Relationship (using $\eqref{eq:complex2real}$): $$ - r(t)=\text{Re}\left\{u(t)\exp(-\mathrm{j}2\pi f_\text{c}t)\right\}. + \begin{align} + r(t)&=\text{Re}\left\{u_r(t)\exp(\mathrm{j}2\pi f_\text{c}t)\right\} \nonumber \\ + &=\text{Re}\left\{\left[\sum_{n=0}^{N(t)-1}\alpha_n(t)u(t-\tau_n(t))\exp(-\mathrm{j}2\pi f_\text{c}\tau_n(t)+\phi_{\text{D}_n}(t))\right]\exp(\mathrm{j}2\pi f_\text{c}t)\right\} \nonumber \\ + &=\text{Re}\left\{\sum_{n=0}^{N(t)-1}\alpha_n(t)u(t-\tau_n(t))\exp(\mathrm{j}2\pi f_\text{c}(t-\tau_n(t))+\phi_{\text{D}_n}(t))\right\}. + \end{align} $$ + Now consdiering zero doppler shift that $\phi_{\text{D}_n}(t)=0$, the upper equation could be further reduced as follows: + $$ + \begin{align} + &r(t)=\text{Re}\left\{\sum_{n=0}^{N(t)-1}\alpha_n(t)u(t-\tau_n(t))\exp(\mathrm{j}2\pi f_\text{c}(t-\tau_n(t)))\right\} \nonumber \\ + &=\text{Re}\left\{\sum_{n=0}^{N(t)-1}\alpha_n(t)[s_\text{I}(t-\tau_n(t))+\mathrm{j}s_\text{Q}(t-\tau_n(t))]\left[\cos(2\pi f_\text{c}(t-\tau_n(t)))+\mathrm{j}\sin(2\pi f_\text{c}(t-\tau_n(t))))\right]\right\} \nonumber \\ + &=\sum_{n=0}^{N(t)-1}\alpha_n(t)[s_\text{I}(t-\tau_n(t))\cos(2\pi f_\text{c}(t-\tau_n(t)))-s_\text{Q}(t-\tau_n(t))\sin(2\pi f_\text{c}(t-\tau_n(t)))] \nonumber\\ + &=\sum_{n=0}^{N(t)-1}\alpha_{n}(t)s(t-\tau_n(t)). + \label{eq:received_relation} + \end{align} + $$ + # Wideband / Narrowband Channel Model -~~To be updated (DDL: before 2023.10.18)...~~ - - - - +### Narrowband Channel Model + +~~To be updated (DDL: before 2023.10.18)...~~ (Updated at 2023.10.18) + +When delay spread $T_\text{m}=\max_{i,j\in\{0,1,\ldots,N(t)-1\}}\{\tau_i-\tau_j\}$ and a manual signal period $T=1/B$ satisfies that $T_m \ll T$, we have $u(t-\tau_i)\approx u(t-\tau_0)\approx u(t)$. Equation $\eqref{eq:ch_complex}$ could be rewrote as: +$$ +\begin{align} +h_\text{nb}(t,\tau)&\approx\sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\mathrm{j}\phi_n(t))\delta(\tau-\tau_0). +\label{eq:ch_complex_nb} +\end{align} +$$ +Moreover, received signal could be rewrote as: +$$ +\begin{align} +u_r(t)&=u(t)\otimes h_\text{nb}(t,\tau) \nonumber \\ +&=\sum_{n}^{N(t)-1}\alpha_n(t)u(t-\tau_0)\exp(-\mathrm{j}\phi_n(t)) \nonumber \\ +&\approx \sum_{n}^{N(t)-1}\alpha_n(t)u(t)\exp(-\mathrm{j}\phi_n(t)) \nonumber \\ +&=u(t)\times \underbrace{\alpha_n(t)\exp(-\mathrm{j}\phi_n(t))}_{h_\text{nb}(t)} +\end{align} +$$ +An interesting finding is that received signal $u_r(t)$ can be expressed as a multiplication of $u(t)$ and $h_\text{nb}(t)$ approximately at narrowband (No need of convolution!), and $h_\text{nb}(t)$ has similar expression with its CFR $H(t,f)$ in $\eqref{eq:CFR}$ if Doppler phase $\phi_{\text{D}_n} = 0$. + +### Wideband Channel Model + +**COMING SOON** (Computer science-ers' trash fake open-source style :sweat_smile:: To be updated in the infinite future... That's why i dont like them. If u can not make ur project public for some reason, it dosen't matter, but please do not cheat us and waste our time. Just a joke here.)