From b0b67fc2f9abddb8046655b0b3c35077f5dda741 Mon Sep 17 00:00:00 2001 From: rzy0901 Date: Tue, 10 Oct 2023 14:41:59 +0800 Subject: [PATCH] fix typos --- content/post/channel.md | 14 +++++++------- static/channel.assets/channel.png | Bin 87214 -> 88851 bytes 2 files changed, 7 insertions(+), 7 deletions(-) diff --git a/content/post/channel.md b/content/post/channel.md index d677002..ea88ea9 100644 --- a/content/post/channel.md +++ b/content/post/channel.md @@ -41,19 +41,19 @@ typora-root-url: ../../static We have CIR in below form: $$ - h(\tau)=\sum_{i=0}^{N-1}\alpha_0\delta(\tau-\tau_i)\rightarrow \mathbf{h}=[\alpha_0,\ldots,\alpha_{N-1}]. + h(t,\tau)=\sum_{n=0}^{N(t)-1}\alpha_n(t)\delta(\tau-\tau_n(t))\rightarrow \mathbf{h}=[\alpha_0(t),\ldots,\alpha_{N-1}(t)]. $$ Do DFT at $k$: $$ - H(k)=\sum_{n=0}^{N-1}\alpha_n\exp(-\frac{\mathrm{j2\pi kn}}{N}). + H(t,k)=\sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\frac{\mathrm{j2\pi kn}}{N}). $$ 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: $$ \begin{aligned} - H(f)&=\sum_{n=0}^{N-1}\alpha_n\exp(-\frac{\mathrm{j2\pi f/\Delta f n}}{N}), \\ - &\xlongequal[]{\Delta f =1/(N\Delta t)}\sum_{n=0}^{N-1}\alpha_n\exp(-\mathrm{j2\pi f\tau_n}). + H(t,f)&=\sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\frac{\mathrm{j2\pi f/\Delta f n}}{N}), \\ + &\xlongequal[]{\Delta f =1/(N\Delta t)}\sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\mathrm{j2\pi f\tau_n(t)}). \end{aligned} $$ @@ -96,18 +96,18 @@ typora-root-url: ../../static + Real bandpass channel: $$ - h(t,\tau)=\sum_{n=0}^{N-1}\alpha_{n}(t)\delta(\tau-\tau_n(t)), + h(t,\tau)=\sum_{n=0}^{N(t)-1}\alpha_{n}(t)\delta(\tau-\tau_n(t)), $$ where: + $t$ and $\tau$: time domain and delay domain. - + $N$: number of multipaths. + + $N(t)$: number of multipaths at time slot $t$. + $\alpha_n(t)$ and $\tau_n(t)$: path loss (amplitude) and delay for $n$-th path at time slot $t$. + Equivalent complex baseband (lowpass) channel: $$ - h(t,\tau)=\sum_{n=0}^{N-1}\alpha_n(t)\exp(-\mathrm{j}\phi_n(t))\delta(\tau-\tau_n(t)), + h(t,\tau)=\sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\mathrm{j}\phi_n(t))\delta(\tau-\tau_n(t)), $$ 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 shif t. diff --git a/static/channel.assets/channel.png b/static/channel.assets/channel.png index 9073fbb399a1295bacc425f8276fed5da3cc8b95..146a83789e71f72e3b3f291999ad56d4855e4553 100755 GIT binary patch literal 88851 zcmeFYhf~vA*ESqPL_|SBLAr{FsE~t7?_G&B=^#Og2pF1#-cb}eNRy`0!O)8cp@jr& zR3U&6dQnIMp#%a+B#`iib044Yp84ke2i|!y15Dt@T6?c@t+lSb6K`&6bmExsF#rH? z;`S~5y8r;UF95(ceDnzGH-awRVAc;d*j*!CK+S;YGV9G@Hysll0H7|Jd+z}U>phq6 zEqfRMASZe7#TE?L3IqW5;kWg5tb)O-4CkdswizUDpR2C$zm4I2?=9sgegm!ahGNGK zoqEk3))g_m-gtiAaN@A8-pL7lm4{b{{bN5^cwR`16BBXpR2iI^KZ4>Kjwlq=?aDUn 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