Signals and Systems - Electronics and Communication (ECE)
A comprehensive test on Signals and Systems for Electronics and Communication Engineering, covering Fourier transforms and series, system properties, z-transforms, impulse responses, discrete-time signals, and filtering concepts.
Questions
Two discrete time systems with impulse responses h1[n] = $\delta$[n -1] and h2[n] = $\delta$[n - 2] are connected in cascade. The overall impulse response of the cascaded system is
- $\delta$[n - 1] + $\delta$[n - 2]
- $\delta$[n - 4]
- $\delta$[n - 3]
- $\delta$[n - 1] $\delta$[n - 2]
The 4-point Discrete Fourier Transform (DFT) of a discrete time sequence {1, 0, 2, 3} is
- [0, -2 + 2j, 2, - 2 - 2j]
- [2, 2 + 2j, 6, 2 - 2j]
- [6, 1 - 3j, 2, 1 + 3j]
- [6, - 1 + 3j, 0, - 1 - 3j]
A sequence x(n) with the z-transform X(z) = z4 + z2 -2z + 2-3z-4 is applied as an input to a linear, time-invariant system with the impulse response h(n) = 2$\delta$(n-3) where
$\delta$(n) = $
\begin{cases}
1, n = 0 \\
0, otherwise
\end{cases}
$
The output at n = 4 is
- -6
- zero
- 2
- -4
The input x(t) and output y(t) of a system are related as y(t) = $
\oint_\infty x(\tau) cos(3 \tau) \ d\tau
$the system is
- time - invariant and stable
- stable and not time-invariant
- time -invariant and not stable
- not time-invariant and not stable
A linear, time - invariant, causal continuous time system has a rational transfer function with simple poles at s = - 2 and s = - 4 and one simple zero at s = - 1.
A unit step u (t) is applied at the input of the system. At steady state, the output has constant value of 1. The impulse response of this system is
- [exp (- 2t) + exp (- 4t)] u (t)
- [- 4 exp (- 2t) - 12 exp (- 4t) - exp (- t)] u (t)
- [- 4 exp (- 2t) + 12 exp (- 4t) u (t)
- [- 0.5 exp (- 2t) + 1.5 exp (- 4t)] u (t)
For a signal x(t), the Fourier transform is X(f). Then the inverse Fourier transform of X(3f + 2) is given by
- $ \dfrac{1}{2} \times (\dfrac{1}{2}) e^{j3\pi t} $
- $ \dfrac{1}{3} \times (\dfrac{1}{3}) e^{-j4\pi t} $
- $ 3 \times (3t) e^{-j4\pi t}$
- x(3t + 2)
Consider the sequence| x[n] = [– 4 – j51 + j25]. The conjugate anti-symmetric part of the sequence is
- [– 4 – j2.5,j2, 4 – |j25]
- [– j2.5, 1, j25]
- [– j2.5, j2, 0]
- [– 4, 1, 4]
The system under consideration is an RC low-pass filter (RC-LPF) with R = 1.0 k$\Omega$and C = 1.0 $\delta$F.
Let tg(f) be the group delay function of the given RC-LPF and f2 = 100 Hz. Then tg(f2) in ms, is
- 0.717
- 7.17
- 71.7
- 4.505
The power in the signal s(t) = 8 cos $\left( 20\pi t - \dfrac{\pi}{2} \right)$ + 4 sin $(15 \pi t)$ is
- 40
- 41
- 42
- 82
A 5-point sequence x (n) is given as X [–3] = 1, x [–2] = 1, x [–1] = 0, x [0] = 5, x [1] = 1. If x($e^{j\omega}$) denotes the discrete – time fourier transform of x [n], what is the value of $\displaystyle \int_{-\pi}^\pi x(e^{j\omega})$$d\omega$?
- 5
- 10$\pi$
- 16$\pi$
- 5 + j10$\pi$
If the unit step response of a network is $(1 - e^{-\omega t})$, then its unit impulse response is
- $\alpha e^{-\omega t}$
- $\alpha^{-t} e^{-\omega t}$
- $(1 - \alpha^{-1}) e^{-\omega t}$
- $(1 - \alpha) e^{-\omega t}$
Choose the function $f(t); -\infty \lt 1 \lt +\infty$for which a Fourier series cannot be defined.
- 3 sin (25t)
- 4 cos (20t + 3) + 2sin (10t)
- exp (-|t|) sin(25t)
- 1
The system under consideration is an RC low-pass filter (RC-LPF) with R = 1.0 k$\Omega$and C = 1.0$\mu$F.
Let H(f) denote the frequency response of the RC-LPF. Let f1 be the highest frequency such that 0$\le$|f| $\le$f1, $\dfrac{ | H(f_1) | }{H(0)}$$\ge$ 0.95. Then f1 (in Hz) is
- 327.8
- 163.9
- 52.2
- 104.4
The trigonometric Fourier series for the waveform f(t) shown below contains
- only cosine terms and zero value for the dc component
- only cosine terms and a positive value for the dc component
- only cosine terms and a negative value for the dc component
- only sine terms and a negative for the dc component
The impulse response h [n] of a linear time-invariant system is given by h[n]= u[n+3] + u [n-2)-2n[n-7] where u[n] is the unit step sequence. The above system is
- stable but not causal
- stable and causal
- causal but unstable
- unstable and not causal
If the region of convergence of x1 [n] + x2 [n] is $\dfrac{1}{3} \lt |z| \lt \dfrac{2}{3}$, then the region of convergence of xn [n] - x2 [n] includes
- $\dfrac{1}{3} \lt |z| \lt 3$
- $\dfrac{2}{3} \lt |z| \lt 3$
- $\dfrac{3}{2} \lt |z| \lt 3$
- $\dfrac{1}{3} \lt |z| \lt \dfrac{2}{3}$
Let x(t) be the input to a linear, time-invariant system. The required output is 4x (t-2). The transfer function of the system should be
- 4 ej4$\pi$f
- 2 e-j8$\pi$f
- 4 e-j4$\pi$f
- 2 ej8$\pi$f
A function is given by f (t) = sin2 t + cos 2t. Which of the following is true?
- f has frequency components at 0 and $\dfrac{1}{2\pi}$Hz.
- f has frequency components at 0 and $\dfrac{1}{\pi}$Hz.
- f has frequency components at $\dfrac{1}{2\pi}$ and $\dfrac{1}{\pi}$Hz.
- f has frequency components at $\dfrac{0.1}{2\pi}$ and $\dfrac{1}{\pi}$Hz.
The differential equation 100$\dfrac{d^2 y}{dt^2}$- 20$\dfrac{dy}{dt}$ + y = x(t) describes a system with an input x(t) and an output y(t). The system, which is initially relaxed, is excited by a unit step input. The output y(t) can be represented by the waveform
In the system shown below, x (t ) = (sin t)u (t). In steady-sate, the response y (t) will be
- $\dfrac{1}{\sqrt 2} sin \left( t - \dfrac{\pi}{4} \right)$
- $\dfrac{1}{\sqrt 2} sin \left( t - \dfrac{\pi}{4} \right)$
- $\dfrac{1}{\sqrt 2} e^{-t} sint$
- sin t - cos t
