Introduction Of C T And D T Signals – GATE EE Signals And Systems Practice Questions & PYQs | BattleExam | BattleExam

Q5GATE 2024MCQ

Consider the discrete-time system $T_{1}$ and $T_{2}$ defined as follows:

\begin{aligned} & \left\{T_{1} x\right\}[n]=x[0]+x[1]+\ldots+x[n] \\ & \left\{T_{2} x\right\}[n]=x[0]+\frac{1}{2} x[1]+\cdots+\frac{1}{2^{n}} x[n] \end{aligned}

Which one of the following statement is true?

🚀 Solve in Practice Mode

📖 Explanation

\begin{aligned} \left\{T_{1} x\right\}[n] & =x[0]+x[1]+\ldots+x[n] \\ \left\{T_{1} x\right\}[n]^{-} & =\sum_{K=0}^{n} x[K] \end{aligned}

Let,

\begin{aligned} x[K] & =u[K] \\ \mid x[K] & =|u[K]| \\ \left|\left\{T_{1} x\right\}[n]\right| & =\left|\sum_{K=0}^{n} u[K]\right|=\sum_{K=0}^{n} 1=n \end{aligned}

$\mid\left[\left.T_{1} x(n)\right|_{n=\infty }=\infty \right.$ system is not BIBO stable

\begin{aligned} \left\{T_{2} x\right\}[n] & =x[0]+\frac{1}{2} x[1]+\cdots+\frac{1}{2^{n}} x[n] \\ & =\sum_{K=0}^{n}\left(\frac{1}{2}\right)^{K} x(K) \end{aligned}

Let,

\begin{aligned} x(K) & =u[K] \\ |x[K]| & =\mid u[K] \\ \left\{T_{2} x\right\}[n] & =\sum_{K=0}^{n}\left|\left(\frac{1}{2}\right)^{K} u(K)\right| \\ \left\{T_{2} x\right\}[n] & =\sum_{K=0}^{n}\left(\frac{1}{2}\right)^{K} \\ \mid\left[\left.T_{2} x(n)\right|_{n=\infty }\right. & =\sum_{K=0}^{\infty }\left(\frac{1}{2}\right)^{K}=\frac{1}{1-\frac{1}{2}}=2 \text { stable } \end{aligned}

$T_{1}$ is not BIBO stable but $T_{2}$ is $\mathrm{BIBO}$ stable.