Recent questions tagged signals-and-systems

Let $H(z)$ be the $z-$ transform of a real-valued discrete-time signal $h[n].$ If $P(z) = H(z) H(\frac{1}{z})$ has a zero at $z= \frac{1}{2}+\frac{1}{2}j,$ and $P(z)$ has...

Consider a six-point decimation-in-time Fast Fourier Transform $(FFT)$ algorithm, for which the signal-flow graph corresponding to $X $ is shown in the figure. Let $W_{6}...

Let the state-space representation of an LTI system be $x(t)=A x(t)+B u(t), y(t)=Cx(t)+du(t)$ where $A,B,C$ are matrices, $d$ is a scalar, $u(t)$ is the input to the syst...

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

A discrete-time signal $x[n] = \delta[n – 3] + 2 \delta[n – 5]$ has $z$-transform $X(z)$. If $Y(z) = X(-z)$ is the $z$-transform of another signal $y[n]$, then $y[n] = x[...

A wide sense stationary random process $X(t)$ passes through the LTI system shown in the figure. If the autocorrelation function of $X(t)$ is $R_X(\tau)$, then the autoco...

A first-order low-pass filter of time constant $T$ is excited with different input signals (with zero initial conditions up to $t = 0$). Match the excitation signals $X, ...

Consider the signal $$x[n] = 6 \delta[n + 2] + 3 \delta[n + 1] + 8 \delta[n] + 7 \delta[n - 1] + 4 \delta[n - 2]$$ If $X(e^{jw})$ is the discrete-time Fourier transform o...

Suppose $x[n]$ is an absolutely summable discrete-time signal. Its $z$-transform is a rational function with two poles and two zeroes. The poles are at $z = \pm 2j.$ Whic...

Let $\widetilde{x}[n] = 1 + \cos\left(\dfrac{\pi n}{8}\right)$ be a periodic signal with period $16.$ Its DFS coefficients are defined by $a_{k} = \displaystyle{}\dfrac{1...

The characteristic equation of an LTI system is given by $F(s) = s^{5} + 2s^{4} + 3s^{3} + 6s^{2} – 4s – 8 = 0.$ The number of roots that lie strictly in the left half $s...

The magnitude and phase of the complex Fourier series coefficients ܽ$a_{k}$ of a periodic signal $x(t)$ are shown in the figure. Choose the correct statement from the fou...

Input $x(t)$ and output $y(t)$ of an LTI system are related by the differential equation $y’’(t) – y’(t) – 6y(t) = x(t).$ If the system is neither causal nor stable, the ...

The result of the convolution $x(-t) * \delta (-t-t_0)$ is$x(t+t_0)$$x(t-t_0)$$x(-t+t_0)$$x(-t – t_0)$

The waveform of a periodic signal $x(t)$ is shown in the figure.A signal $g(t)$ is defined by $g(t) = x \big( \frac{t-1}{2} \big)$. The average power of $g(t)$ is _______...

In the system shown in Figure (a), $m(t)$ is a low-pass signal with bandwidth $W$ Hz. The frequency response of the band-pass filter $H(f)$ is shown in Figure (b). If it ...

A Fourier transform pair is given by $$\left ( \frac{2}{3} \right ) \: u[n+3] \overset{FT}{\Leftrightarrow} \frac{Ae^{-j6 \pi f}}{1- (\frac{2}{3} ) e^{-j2 \pi f}}$$ where...

A stable linear time variant (LTI) system has a transfer function $H(s) = \frac{1}{s^2+s-6}$. To make this system casual it needs to be cascaded with another LTI system ...

A casual LTI system has zero initial conditions and impulse response $h(t)$. Its input $x(t)$ and output $y(t)$ are related through the linear constant-coefficient differ...

In the root locus plot shown in the figure, the pole/zero marks and the arrows have been removed. Which one of the following transfer functions has this root locus? ...

An unforced linear time invariant (LTI) system is represented by $$\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0& -2 \end{bmatrix} \begin{bmatrix...

The transfer function $\dfrac{V_{2}(s)}{V_{1}(s)}$ of the circuit shown below is$\frac{0.5s+1}{s+1} \\ $$\frac{3s+6}{s+2} \\ $$\frac{s+2}{s+1} \\ $$\frac{s+1}{s+2}$

For a periodic signal $v(t) = 30\sin100\:t + 10\cos300\:t + 6\sin(500\:t+\pi/4),$ the fundamental frequency in $rad/s$ is$100$$300$$500$$1500$

Two systems with impulse responses $h_{1}(t)$ and $h_{2}(t)$ are connected in cascade. Then the overall impulse response of the cascaded system is given by product of $...

Let $y[n]$ denote the convolution of $h[n]$ and $g[n]$, where $h[n]=(\frac{1}{2})^nu[n]$ and $g[n]$ is a casual sequence. If $y[0]=1$ and $y =\frac{1}{2}$, then $g $ equa...

The state variable description of an LTI system is given by$$\begin{pmatrix} \dot{x_1}\\ \dot{x_2}\\ \dot{x_3} \end{pmatrix}=\begin{pmatrix} 0 & a_1 & 0\\ 0 & 0 & a_2\\a_...

The Fourier transform of a signal $h(t)$ is $H(j\omega)=(2\cos\omega)(\sin2\omega)/\omega$. The value of $h(0)$ is$\frac{1}{4}$$\frac{1}{2}$$1$$2$

A band limited low-pass signal $x(t)$ of bandwidth $5\:kHz$ is sampled at a sampling rate $f_{s}$.The signal $x(t)$ is reconstructed using the reconstruction filter $H(f)...

Let $\text{x(t)}$ be a periodic function with period $\text{T = 10}$.The Fourier series coefficients for this series are denoted by $a_{k},$ that is$$x\left ( t \right )=...

Consider the parallel combination of two LTI systems shown in the figure.The impulse responses of the systems are $$ \begin{array} {} h_1(t)=2\delta (t+2)-3\delta (t+1) \...