Questions & Answers of Linear Time-Invariant (LTI) Systems

•  (A) $e^{j{\omega }_{0}t}u\left(t\right)$
•  (B) $\cos \left({\omega }_{0}t\right)$
•  (C) $e^{j{\omega }_{0}t}$
•  (D) $\sin \left({\omega }_{0}t\right)$

$\sum\limits_{k=1}^3a_k\;\cos\left(k\omega_0t\right),\;\mathrm{where}\;a_k\neq0\;,\omega_0\neq0.$

•  (A)  $ \sum\limits_{k=1}^3b_k\;\cos\left(k\omega_0t+\phi_k\right),\;\mathrm{where}\;b_k\neq a_k\;,\forall\;k $
•  (B)  $ \sum\limits_{k=1}^4b_k\;\cos\left(k\omega_0t+\phi_k\right),\;\mathrm{where}\;b_k\neq 0\;,\forall\;k$
•  (C)  $ \sum\limits_{k=1}^3a_k\;\cos\left(k\omega_0t+\phi_k\right) $
•  (D)  $ \sum\limits_{k=1}^2a_k\;\cos\left(k\omega_0t+\phi_k\right) $

•  (A)  X→R, Y→Q, Z→P
•  (B)  X→Q, Y→P, Z→R
•  (C)  X→R, Y→P, Z→Q
•  (D)  X→P, Y→R, Z→Q

If the signal $x\left(t\right)=\frac{\sin \left(t\right)}{\mathrm{\pi t}}*\frac{\sin \left(t\right)}{\mathrm{\pi t}}$ with *  denoting the convolution operation, then x(t) is equal to

•  (A)  $\frac{\sin \left(t\right)}{\mathrm{\pi t}}$
•  (B)  $\frac{\sin \left(2t\right)}{2\mathrm{\pi t}}$
•  (C)  $\frac{2\sin \left(t\right)}{\mathrm{\pi t}}$
•  (D)  ${\left(\frac{\sin \left(t\right)}{\mathrm{\pi t}}\right)}^2$

The result of the convolution $ \mathrm x(-\mathrm t)\;\ast\;\mathrm\delta(-\mathrm t-{\mathrm t}_0) $ is

•  (A)  $ \mathrm x(\mathrm t+{\mathrm t}_0) $
•  (B)  $ \mathrm x(\mathrm t-{\mathrm t}_0) $
•  (C)  $ \mathrm x(-\mathrm t+{\mathrm t}_0) $
•  (D)  $\mathrm x(-\mathrm t-{\mathrm t}_0) $

The impulse response of an LTI system can be obtained by

•  (A)  differentiating the unit ramp response
•  (B)  differentiating the unit step response
•  (C) integrating the unit ramp response
•  (D)  integrating the unit step response

A realization of a stable discrete time system is shown in the figure. If the system is excited by a unit step sequence input $x\left[n\right]$, the response $y\left[n\right]$ is

•  (A)  $4{\left(-\frac{1}{3}\right)}^{n}u\left[n\right]-5{\left(-\frac{2}{3}\right)}^{n}u\left[n\right]$
•  (B)  $5{\left(-\frac{2}{3}\right)}^{n}u\left[n\right]-3{\left(-\frac{1}{3}\right)}^{n}u\left[n\right]$
•  (C)  $5{\left(\frac{1}{3}\right)}^{n}u\left[n\right]-5{\left(\frac{2}{3}\right)}^{n}u\left[n\right]$
•  (D)  $5{\left(\frac{2}{3}\right)}^{n}u\left[n\right]-5{\left(\frac{1}{3}\right)}^{n}u\left[n\right]$

The complex envelope of the bandpass signal $x\left(t\right)=-\sqrt{2}\left(\frac{\sin \left(\pi t/5\right)}{\pi t/5}\right)\sin (\pi t-\frac{\pi }{4})$,centered about $f=\frac{1}{2}\mathrm{Hz}$, is

•  (A) $\left(\frac{\sin \left(\pi t/5\right)}{\pi t/5}\right)e^{j\frac{\pi }{4}}$
•  (B) $\left(\frac{\sin \left(\pi t/5\right)}{\pi t/5}\right)e^{-j\frac{\pi }{4}}$
•  (C) $\sqrt{2}\left(\frac{\sin \left(\pi t/5\right)}{\pi t/5}\right)e^{j\frac{\pi }{4}}$
•  (D) $\sqrt{2}\left(\frac{\sin \left(\pi t/5\right)}{\pi t/5}\right)e^{-j\frac{\pi }{4}}$

A continuous, linear time-invariant filter has an impulse response h(t) described by

$h\left(t\right)=\left\{\begin{array}{ll}3& for 0\le t\le 3\\ 0& otherwise\end{array}\right.$

When a constant input of value 5 is applied to this filter, the steady state output is_____.

For a function g(t), it is given that ${\int }_{-\infty }^{+\infty }g\left(t\right)e^{-jwt}dt=\omega e^{-2{\omega }^2}$ for any real value $\omega$.
If $y\left(t\right)={\int }_{-\infty }^{t}g\left(t\right)d\tau ,$, then ${\int }_{-\infty }^{t}y\left(t\right)dt$ is

•  (A)   0
•  (B) $-j$
•  (C) $-\frac{j}{2}$
•  (D) $\frac{j}{2}$

Consider a discrete-time signal

$x\left[n\right]=\left\{\begin{array}{ll}n& for 0\le n\le 10\\ 0& otherwise\end{array}\right..$

If y[n] is the convolution of x[n] with itself, the value of y[4] is _________.

The input $-3e^{2t}u\left(t\right)$, where $u\left(t\right)$ is the unit step function, is applied to a system with transfer function $\frac{s-2}{s+3}$. If the initial value of the output is −2, then the value of the output at steady state is _______.

The sequence $x\left[n\right]=0.5^{n}u\left[n\right],$ where $u\left[n\right]$ is the unit step sequence, is convolved with itself to obtain $y\left[n\right]$. Then ${\sum }_{n=-\infty }^{+\infty }y\left[n\right]$ is _______.

The DFT of a vector $\left[\begin{array}{cccc}a& b& c& d\end{array}\right]$ is the vector $\left[\begin{array}{cccc}\alpha & \beta & \gamma & \delta \end{array}\right]$ . Consider the product

$\left[\begin{array}{cccc}p& q& r& s\end{array}\right]$=$\left[\begin{array}{cccc}a& b& c& d\end{array}\right]$$\begin{array}{}\\ \\ \\ \end{array}\left[\begin{array}{cccc}a& b& c& d\\ d& a& b& c\\ c& d& a& b\\ b& c& d& a\end{array}\right]$

The DFT of the vector $\left[\begin{array}{cccc}p& q& r& s\end{array}\right]$ is a scaled version of

•  (A) $\left[\begin{array}{cccc}{\alpha }^2& {\beta }^2& {\gamma }^2& {\delta }^2\end{array}\right]$
•  (B) $\left[\sqrt{\mathrm{\alpha <!--\; a\; -->}}\sqrt{\mathrm{\beta <!--\; ß\; -->}}\sqrt{\mathrm{\gamma <!--\; ?\; -->}}\sqrt{\mathrm{\delta <!--\; d\; -->}}\right]$
•  (C) $\left[\begin{array}{cccc}\alpha +\beta & \beta +\delta & \delta +\gamma & \gamma +\alpha \end{array}\right]$
•  (D) $\left[\begin{array}{cccc}\alpha & \beta & \gamma & \delta \end{array}\right]$

Let y[n] denote the convolution of h[n] and g[n], where h[n]=(1/2)ⁿ u[n] and g[n] is a causal sequence. If y[0] = 1 and y[1] = 1/2, then g[1] equals

•  (A) 0
•  (B) 1/2
•  (C) 1
•  (D) 3/2

A system is defined by its impulse response h(n) = 2ⁿ u(n − 2). The system is

•  (A) stable and causal
•  (B) causal but not stable
•  (C) stable but not causal
•  (D) unstable and non-causal

The impulse response $h\left(t\right)$ of a linear time-invariant continuous time system is described by $h\left(t\right)=exp\left(\alpha t\right)u\left(t\right)+exp\left(\beta t\right)u\left(-t\right)$, where $u\left(t\right)$ denotes the unit step function, and $\alpha$ and $\beta$ are real constants. This system is stable if

•  (A) $\alpha$ is positive and $\beta$ is positive
•  (B) $\alpha$ is negative and $\beta$ is negative
•  (C) $\alpha$ is positive and $\beta$ is negative
•  (D) $\alpha$ is negative and $\beta$ is positive

A discrete time linear shift-invariant system has an impulse response h[n] with h[0]=1, h[1]=-1. h[2]=-2, and zero otherwise. The system is given an input sequence x[n] with x[0]=x[2]=1, and zero otherwise. The number of nonzero samples in the outp ut sequence y[n], and the value of y[2] are, respectively

•  (A) 5, 2
•  (B) 6, 2
•  (C) 6, 1
•  (D) 5, 3