東京大学 情報理工学系研究科 創造情報学専攻 2010年8月実施 筆記試験 第3問

标签：

• Tokyo-University
• Signal-Processing
• Inner-Product-and-Orthogonality
• Code-Division-Multiple-Access

Author

itsuitsuki

Description

(1) As shown in figure 1, four periodical signals $S_i(t)(i=1,2,3,4)$ with a period of $4\Delta t$ are denoted as a sequence of digital signals $\boldsymbol{w}_i=(w_{i1},w_{i2},w_{i3},w_{i4})^T (i=1,2,3,4)$. Each element $w_{ik} (k=1,2,3,4)$ takes on values $+1$ or $-1$. Calculate a $4\times4$ matrix $W$ which consists of each element; $W_{ij}=\frac{1}{4}\boldsymbol{w}_i^T\boldsymbol{w}_j\text{ }(i,j=1,2,3,4)$ by using $\boldsymbol{w}_i$ shown in figure 1. T denotes transpose.

(2) Consider a method of communication shown in figure 2, where a signal period for 1 bit is $4\Delta t$. The transmitting signal is generated by multiplying the original signal $v_i(t)(i=1,2,3,4)$ by the signal $S_i(t)$ defined in (1). The demodulated signal is generated by multiplying the receiving signal by the signal $S_i(t)$. Suppose that the synchronization between the original signal $v_i(t)$ and the signal $S_i(t)$ and the synchronization between the receiving signal and the signal $S_i(t)$ are realized by using an appropriate method. In addition, suppose that there is no delay in the communication line and no effect of noise and attenuation of signals. Answer the following problems.

1. Show that the demodulated signal matches to the original signal.
2. When the transmitting signal generated by multiplying the original signal $v_i(t)$ by the signal $S_i(t)$ is received by the method, write the signal obtained by multiplying the receiving signal by the signal $S_j(t) (i\neq j)$.

(3) Consider an optical sensing system using concurrently four light sources shown in figure 3. If the same constant lighting levels are used for the light sources, it is difficult to separate the light sources at the photo detectors, so that concurrent sensing cannot be realized. Therefore consider a method for using concurrently four light sources by separating only the signal of the $i$-th light source at the $i$-th photo detector by using an appropriate signal for the $i$-th light source $(i=1,2,3,4)$.

1. Show a detailed method for separating the signal of $i$-th light source at the $i$-th photo detector from the signal of other light sources. Suppose that the synchronizations at the modulation and the demodulation are realized by using an appropriate method. In addition, suppose that there is no delay in the communication line and no effect of noise and attenuation of signals.
2. If there is attenuation of the signals in the above situation, show a detailed method for separating the signal of each light source.

题目描述

1. 如图 1，四个周期信号 $S_i(t)$（$i=1,2,3,4$）的周期均为 $4\Delta t$，分别用数字序列

   $$\boldsymbol w_i=(w_{i1},w_{i2},w_{i3},w_{i4})^T$$

   表示，每个 $w_{ik}\in\{+1,-1\}$。根据图 1 给出的各 $\boldsymbol w_i$，计算 $4\times4$ 矩阵 $W$，其中

   $$W_{ij}=\frac14\boldsymbol w_i^T\boldsymbol w_j\qquad(i,j=1,2,3,4),$$

   上标 $T$ 表示转置。

2. 考虑图 2 的通信方法，每比特信号周期为 $4\Delta t$。发送端将原信号 $v_i(t)$ 与第 1 问的 $S_i(t)$ 相乘生成发送信号；接收端再把接收信号与 $S_i(t)$ 相乘生成解调信号。假设原信号与 $S_i(t)$、接收信号与 $S_i(t)$ 均已适当同步，信道无延迟、噪声和衰减。

   1. 证明解调信号与原信号一致。
   2. 若接收到的是 $v_i(t)S_i(t)$，写出改与 $S_j(t)$（$i\ne j$）相乘后得到的信号。

3. 图 3 的光学传感系统同时使用四个光源。若光源均以恒定亮度点亮，光电探测器无法区分来源，因而不能并行测量。现为第 $i$ 个光源选取适当信号，使第 $i$ 个探测器只分离出该光源的成分。

   1. 详细说明如何把第 $i$ 个光源信号与其他光源分离。假设调制、解调已适当同步，传播无延迟、噪声和信号衰减。
   2. 若存在信号衰减，详细说明如何仍然分离各光源信号。

题目中的序列波形、通信框图和光学系统结构沿用原文图 1～3。