If $X(f)$ is the Fourier transform of $x(t)$, then what is the Fourier transform of $x(2t)$?

Which of the following statements is true?

Find the fundamental period of $x(t) = \sin(5\pi t) + \cos(7\pi t)$.

If a system's impulse response $h(t) = \delta(t) - \delta(t-1)$, the system is:

Which system property ensures that the output depends only on past and present inputs?

Which of the following is an example of a deterministic signal?

The impulse response of an LTI system is given by $h(t) = e^{-2t}u(t)$. Find the system response to an input $x(t) = e^{-t}u(t)$.

The step response of an LTI system is obtained by:

The Laplace transform of $e^{-at}u(t)$ is:

If a signal contains frequency components up to 5 kHz, what should be the minimum sampling rate?

Find the inverse Laplace transform of: $F(s) = \frac{s(s+2)}{(s+1)(s+3)(s+5)}$

Given a system with a difference equation $y[n] - 0.5y[n-1] = x[n]$, determine its stability.

Find the state space model of the system represented by the differential equation: $\frac{d^3y}{dt^3} + 8\frac{d^2y}{dt^2} + 9\frac{dy}{dt} = 3x(t)$.

A system is described by the equation: $y(t) = 3x(t) + 2$. Check whether the system is linear or not.

Compute the poles and zeros of $F(s) = \frac{s+1}{s^3 + 7s^2 + 10s + 18}$.

Consider a system defined by $y(t) = tx(t)$. Determine whether it is linear and shift-invariant.

Compute the output $y[n]$ of an LTI system with impulse response $h[n] = \{1, 2, 1\}$ and input $x[n] = \{2, 1, 3\}$ using convolution.

Compute the energy and power of the signal $x(t) = e^{-t}u(t)$.

Find the Fourier transform of $x(t) = e^{-2|t|}$.

Find the Laplace Transform of $x(t) = e^{-t}u(t)$ and determine its region of convergence.

Given $x(t) = \sin(10t) + \cos(15t)$, determine whether it is periodic and find the fundamental period.

Compute the z-transform of $x[n] = -b^n u[-n-1]$.

Find the step response of a system with impulse response $h(t) = e^{-3t}u(t)$.

Determine the even and odd components of the signal $x(t) = t^2 + 3t + 2$.

Discuss the properties of a system with respect to linearity, causality, and stability.

Define Fourier series and explain its importance in signal representation.

State and explain the Sampling Theorem and its significance.

State and explain Parseval's Theorem with its application.

If $ X(f) $ is the Fourier transform of $ x(t) $, then what is the Fourier transform of $ x(2t) $?

Which of the following statements is true?

Find the fundamental period of $ x(t) = \sin(5\pi t) + \cos(7\pi t) $.

If a system's impulse response is $ h(t) = \delta(t) + \delta(t-1) $, the system is:

Which system property ensures that the output depends only on past and present inputs?

Which of the following is an example of a deterministic signal?

The impulse response of an LTI system is given by $ h(t) = e^{-2t}u(t) $. Find the system response to an input $ x(t) = e^{-t}u(t) $.

The step response of an LTI system is obtained by:

The Laplace transform of $ e^{-at}u(t) $ is:

If a signal contains frequency components up to 5 kHz, what should be the minimum sampling rate?

Find the inverse Laplace transform of: $ F(s) = \frac{s(s+2)}{(s+1)(s+3)(s+5)} $.

Given a system with a difference equation $ y[n] - 0.5y[n-1] = x[n] $, determine its stability.

Find the state space model of the system represented by the differential equation: $ \frac{d^3y}{dt^3} + 8\frac{d^2y}{dt^2} + 9\frac{dy}{dt} = 3x(t) $.

A system is described by the equation: $ y(t) = 3x(t) + 2 $. Check whether the system is linear or not.

Compute the poles and zeros of $ F(s) = \frac{s+1}{s^3 + 7s^2 + 10s + 18} $.

Consider a system defined by $ y(t) = tx(t) $. Determine whether it is linear and shift-invariant.

Compute the output $ y[n] $ of an LTI system with impulse response $ h[n] = \{1, 2, 1\} $ and input $ x[n] = \{2, 1, 3\} $ using convolution.

Compute the energy and power of the signal $ x(t) = e^{-t}u(t) $.

Find the Fourier transform of $ x(t) = e^{-2|t|} $.

Find the Laplace Transform of $ x(t) = e^{-t}u(t) $ and determine its region of convergence.

Given $ x(t) = \sin(10t) + \cos(15t) $, determine whether it is periodic and find the fundamental period.

Compute the z-transform of $ x[n] = -b^n u[n-1] $.

Find the step response of a system with impulse response $ h(t) = e^{-3t}u(t) $.

Determine the even and odd components of the signal $ x(t) = t^2 + 3t + 2 $.

If $X(f)$ is the Fourier transform of $x(t)$, then what is the Fourier transform of $ x(2t) $?

Which of the following statements is true?

Find the fundamental period of $x(t) = \sin(5\pi t) + \cos(7\pi t)$.

If a system's impulse response is $h(t) = \delta(t) + \delta(t-1)$, the system is:

Which system property ensures that the output depends only on past and present inputs?

Which of the following is an example of a deterministic signal?

The impulse response of an LTI system is given by $h(t) = e^{-2t}u(t)$. Find the system response to an input $x(t) = e^{-t}u(t)$.

The step response of an LTI system is obtained by:

The Laplace transform of $e^{-at}u(t)$ is:

If a signal contains frequency components up to 5 kHz, what should be the minimum sampling rate?

Find the inverse Laplace transform of: $F(s) = \frac{s(s+2)}{(s+1)(s+3)(s+5)}$.

Given a system with a difference equation $y[n] - 0.5y[n-1] = x[n]$, determine its stability.

Find the state space model of the system represented by the differential equation: $ \frac{d^3y}{dt^3} + 8\frac{d^2y}{dt^2} + 9\frac{dy}{dt} = 3x(t) $.

A system is described by the equation: $y(t) = 3x(t) + 2$. Check whether the system is linear or not.

Compute the poles and zeros of $F(s) = \frac{s+1}{s^3 + 7s^2 + 10s + 18}$.

Consider a system defined by $y(t) = tx(t)$. Determine whether it is linear and shift-invariant.

Compute the output $y[n]$ of an LTI system with impulse response $ h[n] = {1, 2, 1} $ and input $x[n] = \{2, 1, 3\}$ using convolution.

Compute the energy and power of the signal $x(t) = e^{-t}u(t)$.

Find the Fourier transform of $x(t) = e^{-2|t|}$.

Find the Laplace Transform of $x(t) = e^{-t}u(t)$ and determine its region of convergence.

Given $x(t) = \sin(10t) + \cos(15t)$, determine whether it is periodic and find the fundamental period.

Compute the z-transform of $x[n] = -b^n u[n-1]$.

Find the step response of a system with impulse response $h(t) = e^{-3t}u(t)$.

Determine the even and odd components of the signal $x(t) = t^2 + 3t + 2$.

• Discuss the properties of a system with respect to linearity, causality, and stability.
• Define Fourier series and explain its importance in signal representation.
• State and explain the Sampling Theorem and its significance.
• State and explain Parseval's Theorem with its application.

The minimum sampling rate to avoid aliasing for $ x(t) = 5 \cos(400\pi t) $ is:

Which of the following system is memory less?

The ROC of $ x(t) = e^{-2t}u(t) + e^{-3t}u(t) $ is:

Time period of: $ x(t) = 3 \cos(20t+5) + \sin(8t-3) $ is:

The power of the signal: $ x(t) = \cos(t) $ is:

The Fourier transform exponential signal $ f(t) = e^{-at}u(t), a > 0 $ is:

$ x(5t) $ is:

Which one of the following is an example of a bounded signal?

The simplified valve of $ X(n) = \sum_{n=-5}^{5} \sin(2n)\delta(n+7) $ is:

If $ X(S) = \frac{4S+1}{S^2+6S+3} $, then initial value of $ x(0) $ will be:

Find the inverse Z-transform of $ X(Z) = \frac{1}{1+3Z^{-1}+2Z^{-2}} $, $ ROC: |Z| > 2 $.

For the system $ y(t) = 12x(t) + 7 $, check whether the system is (i) time variant/time-invariant (ii) causal/non-causal (iii) linear/non-linear.

Find the even and odd components of the sequence $ X(n) = 5\delta(n+4) + 4\delta(n+3) + 3\delta(n+2) + \delta(n+1) $.

Determine the power of the signal $ x(t) = e^{j\alpha t}\cos(\omega_o t) $.

Find the Fourier transform of $ x(t) = \frac{1}{a^2+t^2} $.

Find the time response of LTI system with impulse response $ h(t) = 2u(t) - 2u(t-3) $ & input is $ x(t) = 8u(t) - 8u(t-5) $.

Sketch the signal $ x(-4t-3) $ as shown in figure.

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

Compute the DFT of $ x(n) = \{0, 1, 2, 4\} $.

Compute the output of the following signals whose impulse response and input are given by $ h(t) = e^{-at}u(t) $; $ x(t) = e^{at}u(-t), a>0 $ respectively.

Find the Laplace Transform of signal in the figure.

Calculate the fundamental period of $ x(t) = 1 + \sin(\frac{2\pi}{3}t)\cos(\frac{4\pi}{5}t) $.

Determine the Nyquist sampling rate and Nyquist sampling intervals for the signal $ x(t) = \text{sinc}(100\pi t)\text{sinc}(200\pi t) $.

Compute the state transition matrix $ \Phi(t) $ for the system represented by state equation: $ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -2 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $.

The minimum sampling rate to avoid aliasing for $ x(t) = 5 \cos(400\pi t) $ is:

Which of the following system is memory less?

The ROC of $ x(t) = e^{-2t}u(t) + e^{-3t}u(t) $ is:

Time period of: $ x(t) = 3 \cos(20t+5) + \sin(8t-3) $ is:

The power of the signal: $ x(t) = \cos(t) $ is:

The Fourier transform exponential signal $ f(t) = e^{-at}u(t), a > 0 $ is:

$ x(5t) $ is:

Which one of the following is an example of a bounded signal?

The simplified valve of $ X(n) = \sum_{n=-5}^{5} \sin(2n)\delta(n+7) $ is:

If $ X(S) = \frac{4S+1}{S^2+6S+3} $, then initial value of $ x(0) $ will be:

Find the inverse Z-transform of $ X(Z) = \frac{1}{1+3Z^{-1}+2Z^{-2}} $, $ ROC: |Z| > 2 $.

For the system $ y(t) = 12x(t) + 7 $, check whether the system is (i) time variant/time-invariant (ii) causal/non-causal (iii) linear/non-linear.

Find the even and odd components of the sequence $ X(n) = 5\delta(n+4) + 4\delta(n+3) + 3\delta(n+2) + \delta(n+1) $.

Determine the power of the signal $ x(t) = e^{j\alpha t}\cos(\omega_o t) $.

Find the Fourier transform of $ x(t) = \frac{1}{a^2+t^2} $.

Find the time response of LTI system with impulse response $ h(t) = 2u(t) - 2u(t-3) $ & input is $ x(t) = 8u(t) - 8u(t-5) $.

Sketch the signal $ x(-4t-3) $ as shown in figure.

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

Compute the DFT of $ x(n) = \\{0, 1, 2, 4\\} $.

Compute the output of the following signals whose impulse response and input are given by $ h(t) = e^{-at}u(t) $; $ x(t) = e^{at}u(-t), a>0 $ respectively.

Find the Laplace Transform of signal in the figure.

Calculate the fundamental period of $ x(t) = 1 + \sin(\frac{2\pi}{3}t)\cos(\frac{4\pi}{5}t) $.

Determine the Nyquist sampling rate and Nyquist sampling intervals for the signal $ x(t) = \text{sinc}(100\pi t)\text{sinc}(200\pi t) $.

Compute the state transition matrix $ \Phi(t) $ for the system represented by state equation: $ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -2 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $.

Determine the fundamental period of the signal $ x[n]=1+e^{j\frac{4\pi n}{7}}-e^{j\frac{2\pi n}{5}} $

Check whether the signal $ x(t)=2e^{j(t+\frac{\pi}{4})}u(t) $ is periodic or not. If periodic, then compute the periodicity.

Find the convolution of two signals $ x_{1}(t)=e^{-t^{2}} $ and $ x_{2}(t)=3t^{2} $

Let $ X(e^{j\omega}) $ be the DTFT of $ x[n] $ prove that $ X(e^{j0})=\sum_{n=-\infty}^{\infty}x[n] $

The step response of an LTI system when the impulse response $ h(n) $ is unit step $ u(n) $ is _________.

Use the convolution property of Laplace transform to determine $ y(t)=e^{at}u(t)*e^{bt}u(t) $ where symbols have their usual meanings.

Determine the Laplace transform of the signal $ x(t)=\cos^{3}(3t)u(t) $.

If $ X(z)\leftarrow\underline{z}\rightarrow x[n] $ with ROC: R, then prove that $ Z\{x[-n]\}=X(z^{-1}) $ with ROC: $ \frac{1}{R} $

List down the properties of region of convergence (ROC).

Determine the conditions on the sampling interval $ T_{s} $ so that $ x(t)=\cos(\pi t)+3\sin(2\pi t)+\sin(4\pi t) $ is uniquely represented by the discrete-time sequence $ x[n]=x(nT_{s}) $

Consider a causal LTI system that is represented by the difference equation $ y[n]-\frac{3}{4}y[n-1]+\frac{1}{8}y[n-2]=2x[n] $. Find the frequency response $ H(e^{j\omega}) $ and the impulse response $ h[n] $ of the system.

Find the inverse DTFT of $ X(e^{j\omega})=\delta(\omega), -\pi < \omega < \pi $

Find the Fourier transform of $ x(n)=a^{|n|}, |a|< 1 $

Find the Z-transform of $ x(n)=\begin{cases}(0.5)^{n}u(n),&n>0\\ (0.25)^{-n},&n < 0\end{cases} $

Find the inverse Z-transform of $ X(z)=\frac{1-\frac{1}{4}z^{-1}}{1-\frac{1}{9}z^{-1}} $ ROC: $ |z|>\frac{1}{3} $

Comment on the causality of the system whose transfer function is given by $ H(z)=\frac{3-4z^{-1}}{1-3.5z^{-1}+1.5z^{-2}}, |z|>3 $

Derive the condition for BIBO stability.

Consider a continuous-time system with input $ x(t) $ and output $ y(t) $ is related by $ y(t)=x(\sin(t)) $. Is the system (i) causal and (ii) linear?

Sketch the signal $ x(t)=\delta(\cos t) $.

Find even and odd components of signal $ x(t)=e^{-2t}\cos(t) $.

Compute the Fourier transform of the rectangular pulse train signal shown in Fig. 1.

Compute the step response of the LTI system $ H(s)=\frac{6(s+1)}{s(s+3)}; Re\{s\}>0 $

State and prove Parseval's theorem.

A system is defined as $ y(n)=x(n^{2}) $. Check whether the system is linear or non-linear, time-varying or time-invariant, causal or non-causal and memory less or memory type.

Compute the convolution $ y[n]=x[n]*h[n] $ where $ x[n]=u(n-1) $ and $ h[n]=\alpha^{n}u(n-1) $

Compute the Nyquist sampling rate for the signal $ g(t)=10\cos(50\pi)\cos^{2}(150\pi t) $

Consider the causal difference equation $ y[n]-0.8y[n-1]=2x[n] $ where the input signal is $ x[n]=(\frac{1}{2})^{n}u(n) $ with $ y[-1]=0 $. Find the output response $ y[n] $.

Compute the inverse Z-transform of $ X(z)=\frac{z}{(1-0.5z^{-1})}; |z|<0.5 $ using the power series expansion method. Find the signal $ x[n] $.

Consider the stable LTI system defined by its transfer function $ H(z)=\frac{z^{2}+z-2}{z^{2}+z+0.5} $. Sketch the pole-zero plot for this transfer function and give its ROC. Is the system causal? Sketch the direct form realization of this system.

Let $ x[n] $ be an arbitrary function with even and odd parts as $ x_{e}[n] $, $ x_{o}[n] $, respectively. Show that $ \sum_{n=-\infty}^{\infty}x^{2}[n]=\sum_{n=-\infty}^{\infty}x_{e}^{2}[n]+\sum_{n=-\infty}^{\infty}x_{o}^{2}[n] $

Perform the convolution operation between $ x[n]=\{0,0,0,0,(2),-3,1,0,0\} $ and $ h[n]=\{0,0,0,1,(2),2,0,0,0\} $ using graphical method.

The signal $ x(t) $ is shown in Fig. 2. Sketch the signals for $ \alpha=\frac{1}{2} $ and $ \alpha=2 $.

Determine the fundamental period of the signal $ x[n]=1+e^{j\frac{4\pi n}{7}}-e^{j\frac{2\pi n}{5}} $

Check whether the signal $ x(t)=2e^{j(t+\frac{\pi}{4})}u(t) $ is periodic or not. If periodic, then compute the periodicity.

Find the convolution of two signals $ x_{1}(t)=e^{-t^{2}} $ and $ x_{2}(t)=3t^{2} $

Let $ X(e^{j\omega}) $ be the DTFT of $ x[n] $ prove that $ X(e^{j0})=\sum_{n=-\infty}^{\infty}x[n] $

The step response of an LTI system when the impulse response $ h(n) $ is unit step $ u(n) $ is _________.

Use the convolution property of Laplace transform to determine $ y(t)=e^{at}u(t)*e^{bt}u(t) $ where symbols have their usual meanings.

Determine the Laplace transform of the signal $ x(t)=\cos^{3}(3t)u(t) $.

If $ X(z)\leftarrow\underline{z}\rightarrow x[n] $ with ROC: R, then prove that $ Z\\{x[-n]\\}=X(z^{-1}) $ with ROC: $ \frac{1}{R} $

List down the properties of region of convergence (ROC).

Determine the conditions on the sampling interval $ T_{s} $ so that $ x(t)=\cos(\pi t)+3\sin(2\pi t)+\sin(4\pi t) $ is uniquely represented by the discrete-time sequence $ x[n]=x(nT_{s}) $

Consider a causal LTI system that is represented by the difference equation $ y[n]-\frac{3}{4}y[n-1]+\frac{1}{8}y[n-2]=2x[n] $. Find the frequency response $ H(e^{j\omega}) $ and the impulse response $ h[n] $ of the system.

Find the inverse DTFT of $ X(e^{j\omega})=\delta(\omega), -\pi < \omega < \pi $

Find the Fourier transform of $ x(n)=a^{|n|}, |a|< 1 $

Find the Z-transform of $ x(n)=\begin{cases}(0.5)^{n}u(n),&n>0\\ (0.25)^{-n},&n < 0\end{cases} $

Find the inverse Z-transform of $ X(z)=\frac{1-\frac{1}{4}z^{-1}}{1-\frac{1}{9}z^{-1}} $ ROC: $ |z|>\frac{1}{3} $

Comment on the causality of the system whose transfer function is given by $ H(z)=\frac{3-4z^{-1}}{1-3.5z^{-1}+1.5z^{-2}}, |z|>3 $

Derive the condition for BIBO stability.

Consider a continuous-time system with input $ x(t) $ and output $ y(t) $ is related by $ y(t)=x(\sin(t)) $. Is the system (i) causal and (ii) linear?

Sketch the signal $ x(t)=\delta(\cos t) $.

Find even and odd components of signal $ x(t)=e^{-2t}\cos(t) $.

Compute the Fourier transform of the rectangular pulse train signal shown in Fig. 1.

Compute the step response of the LTI system $ H(s)=\frac{6(s+1)}{s(s+3)}; Re\\{s\\}>0 $

State and prove Parseval's theorem.

A system is defined as $ y(n)=x(n^{2}) $. Check whether the system is linear or non-linear, time-varying or time-invariant, causal or non-causal and memory less or memory type.

Compute the convolution $ y[n]=x[n]*h[n] $ where $ x[n]=u(n-1) $ and $ h[n]=\alpha^{n}u(n-1) $

Compute the Nyquist sampling rate for the signal $ g(t)=10\cos(50\pi)\cos^{2}(150\pi t) $

Consider the causal difference equation $ y[n]-0.8y[n-1]=2x[n] $ where the input signal is $ x[n]=(\frac{1}{2})^{n}u(n) $ with $ y[-1]=0 $. Find the output response $ y[n] $.

Compute the inverse Z-transform of $ X(z)=\frac{z}{(1-0.5z^{-1})}; |z|<0.5 $ using the power series expansion method. Find the signal $ x[n] $.

Consider the stable LTI system defined by its transfer function $ H(z)=\frac{z^{2}+z-2}{z^{2}+z+0.5} $. Sketch the pole-zero plot for this transfer function and give its ROC. Is the system causal? Sketch the direct form realization of this system.

Let $ x[n] $ be an arbitrary function with even and odd parts as $ x_{e}[n] $, $ x_{o}[n] $, respectively. Show that $ \sum_{n=-\infty}^{\infty}x^{2}[n]=\sum_{n=-\infty}^{\infty}x_{e}^{2}[n]+\sum_{n=-\infty}^{\infty}x_{o}^{2}[n] $

Perform the convolution operation between $ x[n]=\\{0,0,0,0,(2),-3,1,0,0\\} $ and $ h[n]=\\{0,0,0,1,(2),2,0,0,0\\} $ using graphical method.

The signal $ x(t) $ is shown in Fig. 2. Sketch the signals for $ \alpha=\frac{1}{2} $ and $ \alpha=2 $.

Answer any seven of the following as directed :

Consider a causal LTI system that is represented by the difference equation $y[n] - \frac{3}{4}y[n-1] + \frac{1}{8}y[n-2] = 2x[n]$ Find the frequency response $H(e^{j\omega})$ and the impulse response $h[n]$ of the system.

Find the inverse DTFT of $X(e^{j\omega}) = \delta(\omega), -\pi < \omega \le \pi$.

Find the Fourier transform of $x(n) = a^{|n|}, |a| < 1$.

Find the Z-transform of $x(n) = \begin{cases} (0.5)^n u(n), & n > 0 \ (0.25)^{-n}, & n < 0 \end{cases}$

Find the inverse Z-transform of $X(z) = \frac{1 - \frac{1}{4}z^{-1}}{1 - \frac{1}{9}z^{-1}}$ ROC : $|z| > \frac{1}{3}$

Comment on the causality of the system whose transfer function is given by $H(z) = \frac{3 - 4z^{-1}}{1 - 3.5z^{-1} + 1.5z^{-2}}, |z| > 3$

Derive the condition for BIBO stability.

Consider a continuous-time system with input $x(t)$ and output $y(t)$ is related by $y(t) = x(\sin(t))$. Is the system (i) causal and (ii) linear?

Sketch the signal $x(t) = \delta(\cos t)$.

Find even and odd components of signal $x(t) = e^{-2t} \cos(t)$.

Compute the Fourier transform of the signal (periodic rectangular pulse train) shown in Fig. 1.

Compute the step response of the LTI system $H(s) = \frac{6(s+1)}{s(s+3)}; $\text{Re}\{s\} > 0

State and prove Parseval's theorem.

A system is defined as $y(n) = x(n^2)$. Check whether the system is linear or non-linear, time-varying or time-invariant, causal or non-causal and memory less or memory type.

Compute the convolution $y[n] = x[n] * h[n]$ where $x[n] = u(n-1)$ and $h[n] = \alpha^n u(n-1)$.

Compute the Nyquist sampling rate for the signal $g(t) = 10\cos(50\pi t)\cos^2(150\pi t)$.

Consider the causal difference equation $y[n] - 0.8y[n-1] = 2x[n]$ where the input signal is $x[n] = (\frac{1}{2})^n u(n)$ with $y[-1] = 0$. Find the output response $y[n]$.

Compute the inverse Z-transform of $X(z) = \frac{z}{(1-0.5z^{-1})}; |z| < 0.5$. Find the signal $x[n]$ using the power series expansion method.

Consider the stable LTI system defined by its transfer function $H(z) = \frac{z^2 + z - 2}{z^2 + z + 0.5}$. Sketch the pole-zero plot for this transfer function and give its ROC. Is the system causal? Sketch the direct form realization of this system.

Let $x[n]$ be an arbitrary function with even and odd parts as $x_e[n], x_o[n]$, respectively. Show that $\sum_{n=-\infty}^\infty x^2[n] = \sum_{n=-\infty}^\infty x_e^2[n] + \sum_{n=-\infty}^\infty x_o^2[n]$

Perform the convolution operation between $x[n] = \{0, 0, 0, 0, \underset{\uparrow}{2}, -3, 1, 0, 0\}$ and $h[n] = \{0, 0, 0, \underset{\uparrow}{1}, 2, 2, 0, 0, 0\}$ using graphical method.

The signal $x(t)$ is shown in Fig. 2. Sketch the signals for $\alpha = 1/2$ and $\alpha = 2$ (assuming the task is to sketch $x(\alpha t)$).

Write short notes on any four of the following :

If the Z-transform of $ x(n) $ is $ X(z) $ then show that $ Z[x_{1}(n)*x_{2}(n)]=X_{1}(z)X_{2}(z) $

If the impulse response for a system is given by $ h(n)=a^{n}u(n) $, then what is the condition for the system to be BIBO stable?

A voltage having the laplace transform $ \frac{4s^{2}+3s+2}{7s^{2}+6s+5} $ is applied across a 2H inductor. What is the current in inductor at $ t \to \infty $ assuming zero initial condition?

Differentiate between Kronecker delta function and Direc delta function.

The step response of an LTI system when the impulse response $ h(n) $ is unit step $ u(n) $ is _________.

Find the Laplace transform $ f(t)=e^{3t}\cos(2t)u(t) $ where symbols have their usual meanings.

An LTI system is described as $ 0.5\frac{d^{2}y(t)}{dt^{2}}+2.5\frac{dy(t)}{dt}+2y(t)=\delta(t) $. Find the final value of the output response where $ y(t) $ is output and $ x(t) $ is input.

The period of a sequence $ x(n)=\cos(\frac{2\pi n}{3}) $ is _________.

The final value of step response of a causal LTI system with $ H(s)=\frac{s+1}{s+4} $ is

Consider two functions $ f(t)=h(t)h(3-t) $ and $ g(t)=h(t)-h(t-3) $. Are these two functions identical? Show that $ L[f(t)]=L[g(t)] $ where L is the Laplace operator.

Let a system is described by the differential equation as $ \ddot{y}+3\dot{y}+2y=e^{-t} $; with initial condition $ y(0)=\dot{y}(0)=0 $. Compute the solution of the equation.

Let $ f(t) $ is a periodic function with periodicity T for $ t\ge0 $, then show that $ L[f(t)]=\frac{L[f_{T}(t)]}{1-e^{-sT}} $, $ s>0 $

Find the Laplace transform of Fig. 1.

State why ROC does not include any pole. Find the Z-transform of $ x(n)=\begin{cases}(0.5)^{n}u(n),&n>0\\ (0.25)^{-n},&n < 0\end{cases} $

Find the inverse Z-transform of $ X(z)=\frac{1-\frac{1}{4}z^{-1}}{1-\frac{1}{9}z^{-1}} $ where ROC: $ |z|>\frac{1}{3} $

Show that $ Z[nx(n)]=-z\frac{dX(z)}{dz} $ where $ X(z)\leftrightarrow x[n] $.

Briefly explain the causality of a system.

Find whether the signal $ x[n]=\sin(\frac{3\pi}{4}n)+\sin(\frac{\pi}{3}n) $ is periodic or aperiodic. If periodic, then what is the periodicity of x[n]?

Write down the Dirichlet condition.

Find the Fourier transform of $ x(t)=e^{-|t|}u(t) $, and hence draw the magnitude and phase spectrums.

Compute the Fourier transform of signal shown in Fig. 2: $ R_{x}(\tau)=\begin{cases}\frac{N}{2},&-B\le\tau\le B\\ 0,&elsewhere\end{cases} $

Find the convolution of the following discrete sequences: $ x(n)=\frac{1}{3}u(n) $ and $ h(n)=\frac{1}{5}u(n) $

State why the realization of an ideal low-pass filter is not possible, with proper justification.

A system is defined as $ y(n)=x(n^{2}) $. Check whether the system is linear or non-linear, time-varying or time-invariant, causal or non-causal, and memoryless or memory type.

State Parseval's theorem.

Sketch the signal $ x(t)=-2u(t-1) $.

Compute the Nyquist sampling rate for the signal $ g(t)=10\cos(50\pi)\cos^{2}(150\pi t) $

Show that $ u(n)=\sum_{k=-\infty}^{n}\delta(n) $ where symbols have their usual meanings.

What is unit doublet? Prove that $ \int_{-\infty}^{\infty}x(t)\delta^{k}(t)dt=(-1)^{k}\frac{d^{k}x(t)}{dt^{k}}|_{t=0} $ where $ x^{k} $ is k-th derivative of function $ x(t) $ and $ \delta(t) $ is Dirac delta function.

A system is described by its input-output relationship as $ y[n]=\sum_{k=-\infty}^{n}x[n-k] $. Is the system memoryless, stable, causal, time-invariant and linear?

Find the fundamental period of signal $ x[n]=e^{j7.351\pi n} $