SIGNAL AND SYSTEM

The period of the signal $x(t) = \cos 60\pi t + \sin 50\pi t$ is

The value of the following integral $x(t) = \int_{-\infty}^{\infty} e^{-\alpha t^2} \cdot \delta(t-10) dt$ is

Which of the following is causal?

Which of the following is linear?

The Fourier transform of the function shown below [Diagram: Step function at -1 and 1] is

The inverse Laplace transform of $X(s) = \frac{1}{s(s+2)}$ is

z-transform of convolution of two signals is equal to the —— of their z-transform.

Which one of the following represents the impulse response of a system is defined by $H(z) = z^{-m}$?

A system with input $x(t)$ and output $y(t)$ is described by the relation $y(t) = tx(t)$. The system is

The step response of the system whose impulse response $h(t) = tu(t)$ is given by

Define z-transform. What is/are its application(s)? Find z-transform and ROC of the following signal :

$x(n) = [3(3)^n - 4(2)^n] u(n)$

Determine all possible signals $x(n)$ associated with z-transform

$X(z) = \frac{5z^{-1}}{(1 - 2z^{-1})(1 - 3z^{-1})}$

Explain the conditions under which any periodic waveform can be expressed using Fourier series.

Find trigonometric Fourier series representation of the triangular wave shown below : [Diagram: Triangular wave]

Define Fourier transform for a periodic signal. What are the conditions required for existence of Fourier transform?

Find Fourier transform of— (i) $x(t) = e^{-at} u(t)$; (ii) $x(t) = e^{-3t} [u(t+2) - u(t-3)]$.

Define Laplace transform. What is region of convergence? What is the necessary condition for existence of the Laplace transform? What is the difference between Laplace transform and Fourier transform?

Find Laplace transform and ROC of the signal

$x(t) = e^{-at} u(t) + e^{-bt} u(-t)$

Define convolution sum.

Find the convolution of $x(t)$ and $h(t)$ :

$x(t) = 1, 0 \le t < 2; = 0, \text{otherwise}$ $h(t) = 1, 0 \le t \le 3; = 0, \text{otherwise}$

(i) Define Discrete Time Fourier Transform (DTFT). What is the condition for the existence of DTFT? Does Fourier transform of sequence $x(n) = 3^n u(n)$ exist? If not, why? (ii) Find Fourier transform of the following sequence :

$x(n) = \delta(n+2) - \delta(n-2)$

Find 4-point DFT of the following sequence :

$x(n) = \sin \frac{n\pi}{2}$

For the given mechanical system, draw the electrical analogous circuit using f-v (force-voltage) and f-i (force-current) analogies : [Diagram: Mechanical system with masses $M_1, M_2, M_3$, springs $k_1, k_2, k_3$, dashpot $B_1$, force $F$]

Write short notes on any two of the following : (a) Energy signal and power signal (b) Classification of system (c) Analogous system (d) Fast Fourier Transform (FFT)

What is the fundamental period $T$ of the signal $x(t) = 4 \cos 5\pi t$?

Which of the following systems is time-invariant?

The system $y(t) = e^{x(t)}$ is

The system $y(t) = tx(t)$ is

A good measure of similarity between two signals $x_1(t)$ and $x_2(t)$ is

If $x(t)$ is odd, then its Fourier series coefficients must be

The Fourier transform of odd signal is

The inverse Laplace transform of the function $y(s) = \frac{s+5}{(s+1)(s+3)}$ is

The number of complex multiplications required to calculate $N$-point DFT using radix-2 DIT-FFT algorithm is

The region of convergence of the z-transform of a unit step function is

Define z-transform. Which type of system is studied using z-transform? Find the z-transform and region of convergence (ROC) for the signal $x(n) = -b^n u(-n-1)$.

Find the inverse of z-transform of the following :

$X(z) = \frac{\frac{1}{4} z^{-1}}{(1 - \frac{1}{2} z^{-1})(1 - \frac{1}{4} z^{-1})}$, ROC: $|z| > \frac{1}{2}$

What are Dirichlet conditions?

Find the trigonometric Fourier series for the periodic signal $x(t)$ as shown in the figure below : [Diagram: Sawtooth wave]

Define Fourier transform for a periodic signal. Explain briefly how Fourier transform is different from Fourier series. Can we find the Fourier transform of $x(t) = e^{2t} u(t)$? If not, why?

Find the Fourier transform of : (i) $\text{sgn}(t)$ (ii) $u(t)$

Define Laplace transform. Find out the relation between Fourier transform and Laplace transform. What is the difference between Laplace transform and Fourier transform?

Find the Laplace transform and ROC of the signal $x(t) = e^{-3t} u(t) + e^{-2t} u(t)$.

Define convolution sum.

Determine convolution of the following sequence : $x(n) = 2\delta(n+1) - \delta(n) + \delta(n-1) + 3\delta(n-2)$ $h(n) = 3\delta(n-1) + 4\delta(n-2) + 2\delta(n-3)$

If $x(n) = x_1(n) * x_2(n)$, where $x_1(n) = \left(\frac{1}{3}\right)^n u(n)$ $x_2(n) = \left(\frac{1}{5}\right)^n u(n)$ find $X(z)$ using convolution property for z-transform.

Define discrete Fourier series. What is the condition for the existence of Discrete Time Fourier Transform? Does DTFT of the sequence $x(n) = 2^n u(n)$ exist?

Find the Fourier transform of $x(n) = u(n-k)$.

What do you mean by analogous system?

Draw force-voltage (f-v) and force-current (f-i) analogous circuits of the mechanical system shown in the figure below : [Diagram: Mechanical system with masses $M_1, M_2, M_3$, springs $k_1, k_2, k_3$, friction $B_1$, force $F$]

Write short notes on any two of the following : (a) Causal and noncausal signals (b) Bounded input bounded output (BIBO) stability criterion (c) Cross correlation (d) Relationship between s-plane and z-plane