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TIFR ECE 2023 | Question: 3
Let \[ \mathcal{P}=\left\{(x, y): x+y \geq 1,2 x+y \geq 2, x+2 y \geq 2,(x-1)^{2}+(y-1)^{2} \leq 1\right\} . \] Compute \[ \min _{(x, y) \in \mathcal{P}} 2 x+3 y \] $2$ $3$ $4$ $6$ None of the above
Let\[\mathcal{P}=\left\{(x, y): x+y \geq 1,2 x+y \geq 2, x+2 y \geq 2,(x-1)^{2}+(y-1)^{2} \leq 1\right\} .\]Compute\[\min _{(x, y) \in \mathcal{P}} 2 x+3 y\]$2$$3$$4$$6$N...
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TIFR ECE 2023 | Question: 4
Recall that the entropy (in bits) of a random variable $\mathrm{X}$ which takes values in $\mathbb{N}$, the set of natural numbers, is defined as $H(X)=\sum_{n=1}^{\infty} p_{n} \log _{2} \frac{1}{p_{n}},$ ... variable which denotes the number of tosses made. What is the entropy of $\mathrm{X}$ in bits? $1$ $2$ $4$ Infinity None of the above
Recall that the entropy (in bits) of a random variable $\mathrm{X}$ which takes values in $\mathbb{N}$, the set of natural numbers, is defined as$$H(X)=\sum_{n=1}^{\infty...
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TIFR ECE 2023 | Question: 5
Let $\mathrm{B}$ denote the unit ball in $\mathbb{R}^{2}$, and $\mathrm{Q}$ a square of side length $2$. Let $\mathrm{K}$ be the set of all vectors $z$ such that for some $x \in \mathrm{B}$ and some $y \in \mathrm{Q}, z=x+y$. The area of $\mathrm{K}$ is $4+\pi$ $6+\pi$ $8+\pi$ $10+\pi$ $12+\pi$
Let $\mathrm{B}$ denote the unit ball in $\mathbb{R}^{2}$, and $\mathrm{Q}$ a square of side length $2$. Let $\mathrm{K}$ be the set of all vectors $z$ such that for some...
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TIFR ECE 2023 | Question: 6
An ant in the plane travels in a spiral such that its position $(x(t), y(t))$ at time $t \geq 0$ is $\left(e^{t} \cos t, e^{t} \sin t\right)$. At time $t=1$, find the real part of $\ln (x(t)+i y(t))$. $-2$ $1$ $0$ $-1$ $2$
An ant in the plane travels in a spiral such that its position $(x(t), y(t))$ at time $t \geq 0$ is $\left(e^{t} \cos t, e^{t} \sin t\right)$. At time $t=1$, find the rea...
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TIFR ECE 2023 | Question: 12
Consider a disk $D$ of radius $1$ centered at the origin. Let $X$ be a point uniformly distributed on $D$ and let the distance of $X$ from the origin be $R$. Let $A$ be the (random) area of the disk with radius $R$ centered at the origin. Then $\mathbb{E}[A]$ is $\frac{\pi}{3}$ $\frac{\pi}{6}$ $\frac{\pi}{4}$ $\frac{\pi}{2}$ None of the above
Consider a disk $D$ of radius $1$ centered at the origin. Let $X$ be a point uniformly distributed on $D$ and let the distance of $X$ from the origin be $R$. Let $A$ be t...
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TIFR ECE 2023 | Question: 15
Let $\left\{x_{n}\right\}_{n \geq 0}$ be a sequence of real numbers which satisfy $x_{n+1}\left(1+x_{n+1}\right) \leq x_{n}\left(1+x_{n}\right), \quad n \geq 0 .$ Choose the correct option from the following. ... always bounded but does not necessarily converge. The sequence always converges to a non-zero limit. The sequence always converges to zero. None of the above.
Let $\left\{x_{n}\right\}_{n \geq 0}$ be a sequence of real numbers which satisfy$$x_{n+1}\left(1+x_{n+1}\right) \leq x_{n}\left(1+x_{n}\right), \quad n \geq 0 .$$Choose ...
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TIFR ECE 2015 | Question: 1
For a time-invariant system, the impulse response completely describes the system if the system is causal and non-linear non-causal and non-linear causal and linear All of the above None of the above
For a time-invariant system, the impulse response completely describes the system if the system iscausal and non-linearnon-causal and non-linearcausal and linearAll of th...
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TIFR ECE 2015 | Question: 3
Let $h(t)$ be the impulse response of an ideal low-pass filter with cut-off frequency $5 \mathrm{kHz} .\; \mathrm{Let}\; g[n]= h(n T)$, for integer $n$, be a sampled version of $h(t)$ ... -time filter with $g[n]$ as its unit impulse response is a low-pass filter high-pass filter band-pass filter band-stop filter all-pass filter
Let $h(t)$ be the impulse response of an ideal low-pass filter with cut-off frequency $5 \mathrm{kHz} .\; \mathrm{Let}\; g[n]= h(n T)$, for integer $n$, be a sampled vers...
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TIFR ECE 2015 | Question: 4
The capacity of a certain additive white Gaussian noise channel of bandwidth $1 \mathrm{~MHz}$ is $\mathrm{known}$ to be $8 \text{ Mbps}$ when the average transmit power constraint is $50 \mathrm{~mW}$. Which of the following statements can we make about the capacity $C$ ... $C=8$ $8 < C < 16$ $C=16$ $C>16$ There is not enough information to determine $C$
The capacity of a certain additive white Gaussian noise channel of bandwidth $1 \mathrm{~MHz}$ is $\mathrm{known}$ to be $8 \text{ Mbps}$ when the average transmit power ...
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TIFR ECE 2015 | Question: 5
What is the following passive circuit? Low-pass filter High-pass filter Band-pass filter Band-stop filter All-pass filter
What is the following passive circuit?Low-pass filterHigh-pass filterBand-pass filterBand-stop filterAll-pass filter
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TIFR ECE 2015 | Question: 12
Consider the following optimization problem \[ \max (2 x+3 y) \] subject to the following three constraints \[ \begin{aligned} x+y & \leq 5, \\ x+2 y & \leq 10, \text { and } \\ x & <3 . \end{aligned} \] Let $z^{*}$ be the ... $(x, y)$ that satisfy the above three constraints such that $2 x+3 y$ equals $z^{*}$.
Consider the following optimization problem\[\max (2 x+3 y)\]subject to the following three constraints\[\begin{aligned}x+y & \leq 5, \\x+2 y & \leq 10, \text { and } \\x...
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TIFR ECE 2014 | Question: 4
A system accepts a sequence of real numbers $x[n]$ as input and outputs \[ y[n]=\left\{\begin{array}{ll} 0.5 x[n]-0.25 x[n-1], & n \text { even } \\ 0.75 x[n], & n \text { odd } \end{array}\right. \] The system is non-linear. non-causal. time-invariant. All of the above. None of the above.
A system accepts a sequence of real numbers $x[n]$ as input and outputs\[y[n]=\left\{\begin{array}{ll}0.5 x[n]-0.25 x[n-1], & n \text { even } \\0.75 x[n], & n \text { od...
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TIFR ECE 2014 | Question: 9
Consider the following input $x(t)$ and output $y(t)$ pairs for two different systems. $x(t)=\sin (t), y(t)=\cos (t),$ $x(t)=t+\sin (t), y(t)=2 t+\sin (t-1).$ Which of these systems could possibly be linear and time invariant? Choose the most appropriate answer ... i) nor (ii). neither, but a system with $x(t)=\sin (2 t), y(t)=\sin (t) \cos (t) \operatorname{could~be.~}$
Consider the following input $x(t)$ and output $y(t)$ pairs for two different systems.$x(t)=\sin (t), y(t)=\cos (t),$$x(t)=t+\sin (t), y(t)=2 t+\sin (t-1).$Which of these...
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TIFR ECE 2014 | Question: 10
Consider the two quadrature amplitude modulation $\text{(QAM)}$ constellations below. Suppose that the channel has additive white Gaussian noise channel and no intersymbol interference. The constellation points are picked equally likely. Let $P\text{(QAM)}$ denote the ... .
Consider the two quadrature amplitude modulation $\text{(QAM)}$ constellations below. Suppose that the channel has additive white Gaussian noise channel and no intersymbo...
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TIFR ECE 2014 | Question: 11
It is known that the signal $x(t)$, where $t$ denotes time, belongs to the following class: \[ \left\{A \sin \left(2 \pi f_{0} t+\theta\right): f_{0}=1 \mathrm{~Hz}, 0 \leq A \leq 1,0<\theta \leq \pi\right\} \] If you ... how many samples are required to determine the signal? $1$ sample. $2$ samples. $1$ sample per second. $2$ samples per second. None of the above.
It is known that the signal $x(t)$, where $t$ denotes time, belongs to the following class:\[\left\{A \sin \left(2 \pi f_{0} t+\theta\right): f_{0}=1 \mathrm{~Hz}, 0 \leq...
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TIFR ECE 2014 | Question: 15
You are allotted a rectangular room of a fixed height. You have decided to paint the three walls and put wallpaper on the fourth one. Walls can be painted at a cost of Rs. $10$ per meter and the wall paper can be put at the rate of Rs $20$ per meter for that ... $200$ square meter room? $400 \times \sqrt{3} $ $400$ $400 \times \sqrt{2}$ $200 \times \sqrt{3}$ $500$
You are allotted a rectangular room of a fixed height. You have decided to paint the three walls and put wallpaper on the fourth one. Walls can be painted at a cost of Rs...
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TIFR ECE 2014 | Question: 19
Consider a $2^{k} \times N$ binary matrix $A=\left\{a_{\ell, k}\right\}, a_{\ell, k} \in\{0,1\}$. For rows $i$ and $j$, let the Hamming distance be $d_{i, j}=\sum_{\ell=1}^{N}\left|a_{i, \ell}-a_{j, \ell}\right|$. Let $D_{\min }=\min _{i, j} d_{i, j}$. ... $D_{\min } \leq N-k+1$. $D_{\min } \leq N-k$. $D_{\min } \leq N-k-1$. $D_{\min } \leq N-k-2$. None of the above.
Consider a $2^{k} \times N$ binary matrix $A=\left\{a_{\ell, k}\right\}, a_{\ell, k} \in\{0,1\}$. For rows $i$ and $j$, let the Hamming distance be $d_{i, j}=\sum_{\ell=1...
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TIFR ECE 2013 | Question: 1
The unit step response of a discrete-time, linear, time-invariant system is \[ y[n]=\left\{\begin{array}{rl} 0, & n<0 \\ 1, & n \geq 0 \text { and } n \text { even } \\ -1, & n \geq 0 \text { and } ... the system is bounded-input, bounded-output $\text{(BIBO)}$ stable there is not enough information to determine $\text{(BIBO)}$ stability none of the above
The unit step response of a discrete-time, linear, time-invariant system is\[y[n]=\left\{\begin{array}{rl}0, & n<0 \\1, & n \geq 0 \text { and } n \text { even } \\-1, & ...
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TIFR ECE 2013 | Question: 2
The output $\{y(n)\}$ of a discrete time system with input $\{x(n)\}$ is given by \[ y(n)=\sum_{k=0}^{N-1} a^{k} x(n-k) . \] The difference equation for the inverse system is given by $y(n)=x(n)-a x(n-1)$ ... $(a)$ above, otherwise the inverse does not exist If $|a|<1$, then the answer is $(b)$ above, otherwise the inverse does not exist None of the above
The output $\{y(n)\}$ of a discrete time system with input $\{x(n)\}$ is given by\[y(n)=\sum_{k=0}^{N-1} a^{k} x(n-k) .\]The difference equation for the inverse system is...
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TIFR ECE 2013 | Question: 3
$X$ and $Y$ are jointly Gaussian random variables with zero mean. A constant-pdf contour is where the joint density function takes on the same value. If the constant-pdf contours of $X, Y$ are as shown above, which of the following could their covariance matrix $\mathbf{K}$ ... $\mathbf{K}=\left[\begin{array}{cc}1 & -0.5 \\ -0.5 & 2\end{array}\right]$
$X$ and $Y$ are jointly Gaussian random variables with zero mean.A constant-pdf contour is where the joint density function takes on the same value. If the constant-pdf c...
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TIFR ECE 2013 | Question: 5
Let $x(n)=\sin (2 \pi k n / N), n=0,1, \ldots, N-1$, where $2 k \neq N$ and $0<k \leq N-1$. Then the circular convolution of $\{x(n)\}$ with itself is $N \cos (4 \pi k n / N)$ $N \sin (4 \pi k n / N)$ $-N \cos (2 \pi k n / N) / 2$ $-N \sin (2 \pi k n / N) / 2$ None of the above
Let $x(n)=\sin (2 \pi k n / N), n=0,1, \ldots, N-1$, where $2 k \neq N$ and $0<k \leq N-1$. Then the circular convolution of $\{x(n)\}$ with itself is$N \cos (4 \pi k n /...
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TIFR ECE 2013 | Question: 6
The two-dimensional Fourier transform of a function $f(t, s)$ is given by \[ F(\omega, \theta)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(t, s) \exp (-j \omega t) \exp (-j \theta s) d t d s . \] Let $\delta(t)$ be the delta function and let $u(t)=0$ ... $\exp (-(t+s)) u(t+s)$ $\exp (-t) u(t) \delta(s)$ $\exp (-t) \delta(t+s)$ None of the above
The two-dimensional Fourier transform of a function $f(t, s)$ is given by\[F(\omega, \theta)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(t, s) \exp (-j \omega t) \e...
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TIFR ECE 2013 | Question: 7
The $Z$-transform of $\{x(n)\}$ is defined as $X(z)=\sum_{n} x(n) z^{-n}$ (for those $z$ for which the series converges). Let $u(n)=1$ for $n \geq 0$ and $u(n)=0$ for $n<0$. The inverse $Z$-transform of $X(z)=$ ... is (a), otherwise the inverse is not well-defined If $|a|<1$, then the answer is (b), otherwise the inverse is not well-defined None of the above
The $Z$-transform of $\{x(n)\}$ is defined as $X(z)=\sum_{n} x(n) z^{-n}$ (for those $z$ for which the series converges). Let $u(n)=1$ for $n \geq 0$ and $u(n)=0$ for $n<...
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TIFR ECE 2013 | Question: 8
The following circuit with an ideal operational amplifier is A low pass filter A high pass filter A bandpass filter A bandstop filter An all pass amplifier
The following circuit with an ideal operational amplifier isA low pass filterA high pass filterA bandpass filterA bandstop filterAn all pass amplifier
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TIFR ECE 2013 | Question: 15
Consider a sequence of non-negative numbers $\left\{x_{n}: n=1,2, \ldots\right\}$. Which of the following statements cannot be true? $\sum_{n=1}^{\infty} x_{n}=\infty$ but $x_{n}$ decreases to zero as $n$ increases. $\sum_{n=1}^{\infty} x_{n}<\infty$ ... and each $x_{n} \leq 1 / n^{2}$. $\sum_{n=1}^{\infty} x_{n}<\infty$ and each $x_{n}>x_{n+1}$.
Consider a sequence of non-negative numbers $\left\{x_{n}: n=1,2, \ldots\right\}$. Which of the following statements cannot be true?$\sum_{n=1}^{\infty} x_{n}=\infty$ but...
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TIFR ECE 2012 | Question: 3
A sequence of numbers $\left(x_{n}: n=1,2,3, \ldots\right)$ is said to have a limit $x$, if given any number $\epsilon>0$, there exists an integer $n_{\epsilon}$ ... $6$ and has a limit that equals $6$ . None of the above statements are true.
A sequence of numbers $\left(x_{n}: n=1,2,3, \ldots\right)$ is said to have a limit $x$, if given any number $\epsilon>0$, there exists an integer $n_{\epsilon}$ such tha...
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TIFR ECE 2012 | Question: 4
The signal $x_{n}=0$ for $n<0$ and $x_{n}=a^{n} / n$ ! for $n \geq 0$. Its $z$-transform $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$ is $1 /\left(z^{-1}-a\right)$, region of convergence $\text{(ROC)}$: $|z| \leq 1 / a$ ... $|z|>a$ Item $(a)$ if $a>1$, Item $(b)$ if $a<1$ $\exp \left(a z^{-1}\right)$, $\text{ROC}$: entire complex plane.
The signal $x_{n}=0$ for $n<0$ and $x_{n}=a^{n} / n$ ! for $n \geq 0$. Its $z$-transform $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$ is$1 /\left(z^{-1}-a\right)$, regio...
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TIFR ECE 2012 | Question: 5
Consider a periodic square wave $f(t)$ with a period of $1$ second such that $f(t)=1$ for $t \in[0,1 / 2)$ and $f(t)=-1$ for $t \in[1 / 2,1)$. It is passed through an ideal low-pass filter with cutoff at $2 \mathrm{~Hz}$. Then the output is $\sin (2 \pi t)$ ... $\sin (2 \pi t)-\cos (2 \pi t)$ None of the above
Consider a periodic square wave $f(t)$ with a period of $1$ second such that $f(t)=1$ for $t \in[0,1 / 2)$ and $f(t)=-1$ for $t \in[1 / 2,1)$. It is passed through an ide...
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TIFR ECE 2012 | Question: 6
Let $u(t)$ be the unit step function that takes value $1$ for $t \geq 0$ and is zero otherwise. Let $f(t)=e^{-t} u(t)$ and $g(t)=u(t) u(1-t)$. Then the convolution of $f(t)$ and $g(t)$ is $(e-1) e^{-t} u(t)$ $1-e^{-t}$ for $0 \leq t \leq 1,(e-1) e^{-t}$ for $t \geq 1$ and zero otherwise $t e^{-t} u(t)$ The convolution integral is not well defined None of the above
Let $u(t)$ be the unit step function that takes value $1$ for $t \geq 0$ and is zero otherwise. Let $f(t)=e^{-t} u(t)$ and $g(t)=u(t) u(1-t)$. Then the convolution of $f(...
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TIFR ECE 2012 | Question: 7
A linear time-invariant system has a transfer function $H(s)=1 /(1+s)$. If the input to the system is $\cos (t)$, the output is $\left(e^{j t}+e^{-j t}\right) / 2$ where $j=\sqrt{-1}$ $\cos (t) / 2$ $(\cos (t)+\sin (t)) / 2 \sqrt{ }$ $\sin (t) / 2$. The system is unstable and the output is not well-defined.
A linear time-invariant system has a transfer function $H(s)=1 /(1+s)$. If the input to the system is $\cos (t)$, the output is$\left(e^{j t}+e^{-j t}\right) / 2$ where $...
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TIFR ECE 2012 | Question: 8
The input to a series $\text{RLC}$ circuit is a sinusoidal voltage source and the output is the current in the circuit. Which of the following is true about the magnitude frequency response of this system? Dependending on the values of $\text{R, L}$ ... $1 /(2 \pi \sqrt{\text{LC}})$.
The input to a series $\text{RLC}$ circuit is a sinusoidal voltage source and the output is the current in the circuit. Which of the following is true about the magnitude...
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TIFR ECE 2012 | Question: 9
$x(t)$ is a signal of bandwidth $4 \mathrm{~kHz}$. It was sampled at a rate of $16 \mathrm{~kHz}$. \[ x_{n}=x(n T), \quad n \text { integer, } \quad T=\frac{1}{16} \mathrm{~ms} . \] Due to a data handling error alternate samples were erased ... $y(t)$ over a low pass filter of bandwidth $4\text{ KHz}$ any of the above none of the above
$x(t)$ is a signal of bandwidth $4 \mathrm{~kHz}$. It was sampled at a rate of $16 \mathrm{~kHz}$.\[x_{n}=x(n T), \quad n \text { integer, } \quad T=\frac{1}{16} \mathrm{...
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TIFR ECE 2011 | Question: 1
Output of a linear system with input $x(t)$ is given by \[y(t)=\int_{-\infty}^{\infty} h(t, \tau) x(\tau)+1.\] The system is linear if $h(t, \tau)=h(t-\tau)$ $h(t, \tau)=h(\tau)$ $h(t, \tau)=h(t)$ $h(t, \tau)=$ constant None of the above.
Output of a linear system with input $x(t)$ is given by\[y(t)=\int_{-\infty}^{\infty} h(t, \tau) x(\tau)+1.\]The system is linear if$h(t, \tau)=h(t-\tau)$$h(t, \tau)=h(\t...
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TIFR ECE 2011 | Question: 2
The minimum number of unit delay elements required for realizing an infinite impulse response $\text{(IIR)}$ filter is/are $0$ $1$ $\infty$. $>1$. None of the above.
The minimum number of unit delay elements required for realizing an infinite impulse response $\text{(IIR)}$ filter is/are$0$$1$$\infty$.$>1$.None of the above.
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TIFR ECE 2011 | Question: 3
The Fourier transform of \[x(t)=\frac{t^{n-1}}{(n-1) !} \mathrm{e}^{-a t} u(t), \quad a>0\] $(\jmath=\sqrt{-1}, u(t)=1$ for $t \geq 0, u(t)=0, t<0)$ is $(a+\jmath \omega)^{n}$ $\sum_{k=1}^{n} \frac{(a+\jmath \omega)^{k}}{k !}$ $na\jmath \omega$ $\frac{1}{(a+\jmath \omega)^{n}}$ None of the above.
The Fourier transform of\[x(t)=\frac{t^{n-1}}{(n-1) !} \mathrm{e}^{-a t} u(t), \quad a>0\]$(\jmath=\sqrt{-1}, u(t)=1$ for $t \geq 0, u(t)=0, t<0)$ is$(a+\jmath \omega)^{n...
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TIFR ECE 2011 | Question: 5
Consider a system with input $x(t)$ and the output $y(t)$ is given by \[y(t)=x(t)-0.5 x(t-1)-0.5 x(t-2)+1 .\] The system is Linear Non-causal Time varying All of the above None of the above
Consider a system with input $x(t)$ and the output $y(t)$ is given by\[y(t)=x(t)-0.5 x(t-1)-0.5 x(t-2)+1 .\]The system isLinearNon-causalTime varyingAll of the aboveNone ...
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TIFR ECE 2011 | Question: 6
Let $\mathrm{H}(\mathrm{z})$ be the $z$-transform of the transfer function corresponding to an input output relation $y(n)-\frac{1}{2} y(n-1)=x(n)+\frac{1}{3} x(n-1)$. Then which of the following is TRUE The $\operatorname{ROC}$ ... $|z|<\frac{1}{2}$. $\operatorname{Both}$ (a) and (b). System is necessarily causal. None of the above.
Let $\mathrm{H}(\mathrm{z})$ be the $z$-transform of the transfer function corresponding to an input output relation $y(n)-\frac{1}{2} y(n-1)=x(n)+\frac{1}{3} x(n-1)$. Th...
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TIFR ECE 2011 | Question: 7
Assume you are using a binary code error correcting code $C$. If the minimum Hamming distance between any two codewords of $C$ is $3$. Then We can correct and detect $2$ bit errors. We can correct $1$ bit errors and detect $2$ bit errors. We can correct $2$ bit errors and detect $1$ bit errors. We can correct $1$ bit errors and detect $1$ bit errors. None of the above.
Assume you are using a binary code error correcting code $C$. If the minimum Hamming distance between any two codewords of $C$ is $3$. ThenWe can correct and detect $2$ b...
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TIFR ECE 2011 | Question: 12
Consider two communication systems $C_{1}$ and $C_{2}$ that use pulse amplitude modulation $\text{(PAM)}$, $P A M_{1}$ and $P A M_{2}$. Let the distance between any two points of $P A M_{1}$ be $d$, and $P A M_{2}$ be $2 d$, respectively. Assume that $C_{1}$ ... $P_{1}=P_{2}$. $P_{1} < P_{2}$ $P_{1}>P_{2}$. $P_{1}=P_{2}+\frac{1}{2}$ None of the above.
Consider two communication systems $C_{1}$ and $C_{2}$ that use pulse amplitude modulation $\text{(PAM)}$, $P A M_{1}$ and $P A M_{2}$. Let the distance between any two p...
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TIFR ECE 2011 | Question: 14
In household electrical wiring which configuration is used to connect different electrical equipments. Series. Parallel Combination of series and parallel. Any of the above. None of the above.
In household electrical wiring which configuration is used to connect different electrical equipments.Series.ParallelCombination of series and parallel.Any of the above.N...
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