List of top Signals and Systems Questions asked in GATE EE

Let a causal LTI system be governed by the following differential equation \[ y(t) + \frac{1}{4} \frac{dy}{dt} = 2x(t), \] where $ x(t) $ and $ y(t) $ are the input and output respectively. Its impulse response is

Let an input $ x(t) = 2\sin(10\pi t) + 5\cos(15\pi t) + 7\sin(42\pi t) + 4\cos(45\pi t) $ be passed through an LTI system having an impulse response, \[ h(t) = 2\left(\frac{\sin(10\pi t)}{\pi t}\right) \cos(40\pi t). \] The output of the system is

An LTI system is shown in the figure where \[ G(s) = \frac{100}{s^2 + 0.1s + 10} \] The steady state output of the system, to the input $ r(t) $, is given as \[ y(t) = a + b \sin(10t + \theta) \] The values of 'a' and 'b' will be

If only 5% of the supplied power to a cable reaches the output terminal, the power loss in the cable, in decibels, is ________. (round off to nearest integer)

If the input $x(t)$ and output $y(t)$ of a system are related as $y(t) = \max(0, x(t))$, then the system is

A full-wave rectified sinusoid is clipped at $\omega t = \frac{\pi}{4}$ and $\frac{3\pi}{4}$. The ratio of the RMS value of the full-wave rectified waveform to the RMS value of the clipped waveform is $\underline{\hspace{2cm}}$. (Round off to 2 decimal places.)

A continuous-time signal $x(t)$ is defined as $x(t)=0$ for $|t|>1$, and $x(t)=1-|t|$ for $|t|\le 1$. Let its Fourier transform be $X(\omega)=\int_{-\infty}^{\infty} x(t)e^{-j\omega t}\,dt$. The maximum magnitude of $X(\omega)$ is $\underline{\hspace{2cm}}$.

Let $f(t)$ be an even function. The Fourier transform is $F(\omega)=\int_{-\infty}^\infty f(t)e^{-j\omega t}dt$. Suppose $\frac{dF(\omega)}{d\omega} = -\omega F(\omega)$ for all $\omega$, and $F(0)=1$. Then

The causal signal with z-transform $z^{2}(z-a)^{-2}$ is ($u[n]$ is the unit step signal)

A counter uses three D flip-flops generating the Gray code sequence 000, 001, 011, 010, 110, 111, 101, 100 repeatedly. The bits are in $Q_2 Q_1 Q_0$ format. The combinational logic expression for $D_1$ is $\underline{\hspace{1cm}}$.

A 16-bit synchronous binary up-counter is clocked with frequency $f_{CLK}$. The two most significant bits are OR-ed to form output Y. Y remains high for 24 ms. The clock frequency $f_{CLK}$ is $\underline{\hspace{1cm}}$ MHz. (Round off to 2 decimal places.)

Two discrete-time linear time-invariant systems with impulse responses $h_1[n] = \delta[n-1] + \delta[n+1]$ and $h_2[n] = \delta[n] + \delta[n-1]$ are connected in cascade. The impulse response of the cascaded system is