DISCLAIMER: This note is for reference only. I am not 100% sure of the accuracy of my note. If you find any mistakes/typos, feel free to contact me.
This note records some interesting issues that I met in wireless channel modeling 😄.

CIR, CFR (CSI) Conversion

Channel Impulse Response (CIR): \(h(t,\tau)\); Channel Frequency Response (CFR): \(H(t,f)\).
- \(t\): time domain; \(\tau\): delay domain; \(f\): frequency domain.
- Discrete Fourier Transform (DFT), Inverse Discrete Fourier Transform (IDFT) formula:
  \[\begin{equation} \begin{cases} X[k]=\sum_{n=0}^{N-1}x[n]\exp\left(-\frac{\mathrm{j}2\pi kn}{N}\right)\\=\sum_{n=-\infty}^{\infty}x[n]\exp(-\mathrm j \omega n)|_{\omega=2\pi k/N}=X(e^{\mathrm j\omega})|_{\omega=2\pi k/N}\\ x[n]=\frac{1}{N}\sum_{k=0}^{N-1}X[k]\exp\left(\frac{\mathrm j 2\pi kn}{N}\right) \end{cases}. \label{eq:DFT} \end{equation} \]
- Simple proof (Considering CIR/CFR at one time slot \(t\)):
  Note that \(f_n=n\Delta f\), \(\tau_n=n\Delta t\), \(\Delta f\Delta t=1/N\). (Take a look at above DFT equations or time/frequency settings in MATLAB Fast Fourier Transform (fft) documentation example codes. )
  We have CIR in below form:
  \[h(t,\tau)=\sum_{n=0}^{N(t)-1}\alpha_n(t)\delta(\tau-\tau_n(t))\rightarrow \mathbf{h}=[\alpha_0(t),\ldots,\alpha_{N-1}(t)]. \]
  Do DFT at \(k\) using \(\eqref{eq:DFT}\):
  \[H(t,k)=\sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\frac{\mathrm{j2\pi kn}}{N}). \]
  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:
  \[\begin{align} H(t,f)&=\sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\frac{\mathrm{j2\pi f/\Delta f n}}{N}), \nonumber \\ &\xlongequal[]{\Delta f =1/(N\Delta t)}\sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\mathrm{j2\pi f\tau_n(t)}). \label{eq:CFR} \end{align} \]
Channel State Information (CSI): \(H(t,f)\). (CSI is actually discrete time series of discrete CFR!)
- \(t\): CSI packet timestamp.
- \(f\): OFDM subcarrier frequency.
- Sometimes there is an extra space dimension in CSI Matrix which refers to Tx and Rx antenna pairs.
一个靠谱的知乎问题：在通信专业里的时域，频域，空域，角域到底都有怎样的联系呢？

Real Bandpass Signals, Equivalent Complex Baseband (Lowpass) Signals Conversion

Definition

Real bandpass signal:
\[\begin{align} s(t)=s_\text{I}(t)\cos(2\pi f_\text{c} t) -s_\text{Q}(2\pi f_\text{c} t). \end{align} \]
- \(s_\text{I}(t)\): lowpass in phase component.
- \(s_\text{Q}(t)\): lowpass quadrature component.
Many involved signals in wireless communication are always bandpass signal with carrier frequency \(f_\text{c}\) and bandwidth \(2B\), with \(2B \ll f_\text{c}\).
Equivalent complex baseband (lowpass) signal (real bandpass \(\rightarrow\) complex baseband):
\[\begin{align} u(t)=s_\text{I}(t)+\mathrm{j}s_\text{Q}(t). \end{align} \]
Complex baseband \(\rightarrow\) real bandpass:
\[\begin{align} \text{Real bandpass signal} &= \text{Re}\{\text{Complex baseband signal}\times\exp(\mathrm j2\pi f_\text{c}t)\}, \nonumber \\ s(t) &= \text{Re}\{u(t)\times\exp(\mathrm j2\pi f_\text{c}t)\}. \label{eq:complex2real} \end{align} \]

Two types of CIR: Real Bandpass Channel, Equivalent Complex Baseband (Lowpass) Channel

Channel impulse response

Real bandpass channel:
\[\begin{align} h_\text{real}(t,\tau)=\sum_{n=0}^{N(t)-1}\alpha_{n}(t)\delta(\tau-\tau_n(t)), \label{eq:ch_real} \end{align} \]
where:
- \(t\) and \(\tau\): time domain and delay domain.
- \(N(t)\): 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:
\[\begin{align} h_\text{complex}(t,\tau)=\sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\mathrm{j}\phi_n(t))\delta(\tau-\tau_n(t)), \label{eq:ch_complex} \end{align} \]
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.
- 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

Transmitted signal (See definition):
- Real: \(s(t)=s_\text{I}(t)\cos(2\pi f_\text{c}t)-s_\text{Q}(t)\cos(2\pi f_\text{c}t)\).
- Complex baseband: \(u(t)=s_\text{I}(t)+\mathrm{j}s_\text{Q}(t)\).
Received signal:
- Real:
  \[\begin{align} r(t) &= s(t) \otimes h_\text{real}(t,\tau) \nonumber \\ &= s(t) \otimes \sum_{n=0}^{N(t)-1}\alpha_{n}(t)\delta(\tau-\tau_n(t)) \nonumber \\ &=\sum_{n=0}^{N(t)-1}\alpha_{n}(t)s(t-\tau_n(t)). \label{eq:received_real} \end{align} \]
- Complex baseband:
  \[\begin{align} u_r(t) &= u(t) \otimes h_\text{complex}(t,\tau)\nonumber \\ &=u(t) \otimes \sum_{n=0}^{N(t)-1}\alpha_n(t)\exp(-\mathrm{j}\phi_n(t))\delta(\tau-\tau_n(t)) \nonumber \\ &=\sum_{n=0}^{N(t)-1}\alpha_n(t)u(t-\tau_n(t))\exp(-\mathrm{j}\phi_n(t)) \nonumber\\ &=\sum_{n=0}^{N(t)-1}\alpha_n(t)u(t-\tau_n(t))\exp(-\mathrm{j}2\pi f_\text{c}\tau_n(t)+\phi_{\text{D}_n}(t)). \label{eq:received_complex} \end{align} \]
  Here, \(\otimes\) denotes the convolution operation.
Relationship (using \(\eqref{eq:complex2real}\)):
\[\begin{align} r(t)&=\text{Re}\left\{u_r(t)\exp(\mathrm{j}2\pi f_\text{c}t)\right\} \nonumber \\ &=\text{Re}\left\{\left[\sum_{n=0}^{N(t)-1}\alpha_n(t)u(t-\tau_n(t))\exp(-\mathrm{j}2\pi f_\text{c}\tau_n(t)+\phi_{\text{D}_n}(t))\right]\exp(\mathrm{j}2\pi f_\text{c}t)\right\} \nonumber \\ &=\text{Re}\left\{\sum_{n=0}^{N(t)-1}\alpha_n(t)u(t-\tau_n(t))\exp(\mathrm{j}2\pi f_\text{c}(t-\tau_n(t))+\phi_{\text{D}_n}(t))\right\}. \end{align} \]
Now consdiering zero doppler shift that \(\phi_{\text{D}_n}(t)=0\), the upper equation could be further reduced as follows:
\[\begin{align} &r(t)=\text{Re}\left\{\sum_{n=0}^{N(t)-1}\alpha_n(t)u(t-\tau_n(t))\exp(\mathrm{j}2\pi f_\text{c}(t-\tau_n(t)))\right\} \nonumber \\ &=\text{Re}\left\{\sum_{n=0}^{N(t)-1}\alpha_n(t)[s_\text{I}(t-\tau_n(t))+\mathrm{j}s_\text{Q}(t-\tau_n(t))]\left[\cos(2\pi f_\text{c}(t-\tau_n(t)))+\mathrm{j}\sin(2\pi f_\text{c}(t-\tau_n(t))))\right]\right\} \nonumber \\ &=\sum_{n=0}^{N(t)-1}\alpha_n(t)[s_\text{I}(t-\tau_n(t))\cos(2\pi f_\text{c}(t-\tau_n(t)))-s_\text{Q}(t-\tau_n(t))\sin(2\pi f_\text{c}(t-\tau_n(t)))] \nonumber\\ &=\sum_{n=0}^{N(t)-1}\alpha_{n}(t)s(t-\tau_n(t)). \label{eq:received_relation} \end{align} \]
How about considering channel with doppler phase shift \(\phi_{\text{D}_n}(t) \neq 0\)?
(To be updated after searching some articles…)

Wideband / Narrowband Channel Model

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 \(\phi_{\text{D}_n} = 0\).

Wireless Channel Trouble-Shooting

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