假設發送信號為
\[x\left( t \right) = {\mathop{\rm Re}\nolimits} \left\{ {s\left( t \right){e^{j2\pi {f_c}t}}} \right\} \]
其中 ${s\left( t \right)} $ 是 \({x\left( t \right)}\) 的等效基帶信號，\(f_{c}\) 是載波頻率。忽略噪聲，則接收信號是直射信號和所有可分辨多徑分量之和
\[y\left( t \right) = {\mathop{\rm Re}\nolimits} \left\{ {\sum\limits_{i = 0}^{N\left( t \right)} {{\beta_i}\left( t \right)s\left( {t - {\tau _i}\left( t \right)} \right){e^{j2\pi {f_c }\left( {t - {\tau _i}\left( t \right)} \right) }}} } \right\} \]
其中可分辨多徑數目為 \({N\left( t \right)}\)，\(i=0\) 時對應着直射信號。各徑的長度為 \({{d_i}\left( t \right)}\)，則對應的時延為 \({\tau _i}\left( t \right) = \frac{{{d_i}\left( t \right)}}{c}\). \({{\beta_i}\left( t \right)}\) 表示大尺度衰落系數。
令 \({\phi _i}\left( t \right) = 2\pi {f_c}{\tau _i}\left( t \right)\)，則可以把接收信號簡化為
\[\begin{aligned}y(t) &=\operatorname{Re}\left\{\left[\sum_{i=0}^{N(t)} \beta_{i}(t) e^{-j \phi_{i}(t)} s\left(t-\tau_{i}(t)\right)\right] e^{j 2 \pi f_{c} t}\right\} \\&=\operatorname{Re}\left\{r(t) e^{j 2 \pi f_{c} t}\right\}\end{aligned} \]
其中
\[r(t)={\sum\limits_{i = 0}^{N\left( t \right)} {{\beta_i}\left( t \right){e^{ -j{\phi _i}\left( t \right)}}s\left( {t - {\tau _i}\left( t \right)} \right){}} } \]
這部分可以看作是接收信號 \(y\left( t \right)\) 的等效基帶信號.由於 \({{\beta_i}\left( t \right)}\) 取決於路徑損耗和陰影衰落，而 \({{\phi _i}\left( t \right)}\) 取決於時延，則一般可假設這兩個隨機過程是相互獨立的。
等效基帶輸入信號 \(s\left( t \right)\) 與時變信道的等效基帶沖激響應 \(h\left( {t,\tau } \right)\) 卷積再上變頻到載波頻率 \(f_{c}\) 即可得到接收信號 \(y\left( t \right)\). 這里直接寫出沖激響應函數的表達式「\(r(t)=s(t)*h(t,\tau)\)」
\[\begin{aligned} h\left( {t,\tau } \right) &= \sum\limits_{i = 0}^{N\left( t \right)} {{\beta_i}\left( t \right){e^{- j{\phi _i}\left( t \right)}}\delta \left( {t - {\tau _i}\left( t \right)} \right)} \\&= \sum\limits_{i = 0}^{N\left( t \right)} \alpha_i(t) \delta \left( {t - {\tau _i}\left( t \right)} \right)\end{aligned} \]
通常把\(\alpha_i(t)={\beta_i}\left( t \right){e^{ j{\phi _i}\left( t \right)}}\)當成信道衰落系數，注意這是基帶的信道。
當接收端移動的時候，令\({{\theta _i}\left( t \right)}\) 表示第 \(i\) 徑信道信號的到達方向與接收機運動方向的夾角，此時接收信號可以寫為
\[\begin{aligned} y\left( t \right) &= {\mathop{\rm Re}\nolimits} \left\{ {\sum\limits_{i = 0}^{N\left( t \right)} {{\beta_i}\left( t \right)s\left( {t - {\tau _i} \left( t \right)-\frac{v\mathrm{cos}\theta_i(t)\cdot t}{C}} \right){e^{j2\pi {f_c }\left( {t - {\tau _i}\left( t \right)}-\frac{v\mathrm{cos}\theta_i(t)\cdot t}{C} \right) }}} } \right\} \\ &=\operatorname{Re}\left\{\left[\sum_{i=0}^{N(t)} \beta_{i}(t) e^{-j \phi_{i}(t)} e^{-j2\pi f_m \mathrm{cos} \theta_i(t)\cdot t} s\left(t-\tau_{i}(t)-\frac{v\mathrm{cos}\theta_i(t)\cdot t}{C}\right)\right] e^{j 2 \pi f_{c} t}\right\} \\ &\approx \operatorname{Re}\left\{\left[\sum_{i=0}^{N(t)} \alpha_{i}(t) e^{-j2\pi f_m \mathrm{cos} \theta_i(t)\cdot t} s\left(t-\tau_{i}(t)\right)\right] e^{j 2 \pi f_{c} t}\right\} \\\end{aligned} \]
其中\(f_m=\frac{vf_c}{C}=\frac{v}{\lambda}\)稱為最大多普勒頻移。
此時等效基帶信道沖激響應可以寫為
\[\begin{aligned} h\left( {t,\tau } \right) &= \sum\limits_{i = 0}^{N\left( t \right)} \alpha_i(t)e^{-j2\pi f_m \mathrm{cos} \theta_i(t)\cdot t} \delta \left( {t - {\tau _i}\left( t \right)} \right)\end{aligned} \]
