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.