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201
TIFR ECE 2012 | Question: 11
A Poisson random variable $X$ is given by $\operatorname{Pr}\{X=k\}=\mathrm{e}^{-\lambda} \lambda^{k} / k !, k=0,1,2, \ldots$ for $\lambda>0$. The variance of $X$ scales as $\lambda$ $\lambda^{2}$ $\lambda^{3}$ $\sqrt{\lambda}$ None of the above
A Poisson random variable $X$ is given by $\operatorname{Pr}\{X=k\}=\mathrm{e}^{-\lambda} \lambda^{k} / k !, k=0,1,2, \ldots$ for $\lambda>0$. The variance of $X$ scales ...
75
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Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
poisson-distribution
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202
TIFR ECE 2012 | Question: 12
In modeling the number of health insurance claims filed by an individual during a three year period, an analyst makes a simplifying assumption that for all non-negative integer up to $5$. \[ p_{n+1}=\frac{1}{2} p_{n} \] where $p_{n}$ denotes the probability that a ... files more than two claims in this period? $7 / 31$ $29 / 125$ $1 / 3$ $13 / 125$ None of the above
In modeling the number of health insurance claims filed by an individual during a three year period, an analyst makes a simplifying assumption that for all non-negative i...
94
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Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
conditional-probability
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203
TIFR ECE 2012 | Question: 13
Consider a single amoeba that at each time slot splits into two with probability $p$ or dies otherwise with probability $1-p$. This process is repeated independently infinitely at each time slot, i.e. if there are any amoebas left at time slot $t$, then they all split independently into ... $\min \left\{\frac{1 \pm \sqrt{1-4 p(1-p)}}{2(1-p)}\right\}$ None of the above
Consider a single amoeba that at each time slot splits into two with probability $p$ or dies otherwise with probability $1-p$. This process is repeated independently infi...
91
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Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
independent-events
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204
TIFR ECE 2012 | Question: 14
Let $X$ and $Y$ be indepedent, identically distributed standard normal random variables, i.e., the probability density function of $X$ is \[f_{X}(x)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{x^{2}}{2}\right),-\infty<x<\infty. \] The random variable $Z$ is defined ... none of the above
Let $X$ and $Y$ be indepedent, identically distributed standard normal random variables, i.e., the probability density function of $X$ is\[f_{X}(x)=\frac{1}{\sqrt{2 \pi}}...
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Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
normal-distribution
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205
TIFR ECE 2012 | Question: 15
Consider a string of length $1 \mathrm{~m}$. Two points are chosen independently and uniformly random on it thereby dividing the string into three parts. What is the probability that the three parts can form the sides of a triangle? $1 / 4$ $1 / 3$ $1 / 2$ $2 / 3$ $3 / 4$
Consider a string of length $1 \mathrm{~m}$. Two points are chosen independently and uniformly random on it thereby dividing the string into three parts. What is the prob...
163
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Dec 8, 2022
Probability and Statistics
tifr2012
probability-and-statistics
probability
uniform-distribution
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206
TIFR ECE 2012 | Question: 16
Let $P$ be a $n \times n$ matrix such that $P^{k}=\mathbf{0}$, for some $k \in \mathbb{N}$ and where $\mathbf{0}$ is an all zeros matrix. Then at least how many eigenvalues of $P$ are zero $1$ $n-1$ $n$ $0$ None of the above
Let $P$ be a $n \times n$ matrix such that $P^{k}=\mathbf{0}$, for some $k \in \mathbb{N}$ and where $\mathbf{0}$ is an all zeros matrix. Then at least how many eigenvalu...
87
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asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
eigen-values
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207
TIFR ECE 2012 | Question: 17
Let $A=U \Lambda U^{\dagger}$ be a $n \times n$ matrix, where $U U^{\dagger}=I$. Which of the following statements is TRUE. The matrix $I+A$ has non-negative eigen values The matrix $I+A$ is symmetic $\operatorname{det}(I+A)=\operatorname{det}(I+\Lambda)$ $(a)$ and $(c)$ $(b)$ and $(c)$ $(a), (b)$ and $(c)$
Let $A=U \Lambda U^{\dagger}$ be a $n \times n$ matrix, where $U U^{\dagger}=I$. Which of the following statements is TRUE.The matrix $I+A$ has non-negative eigen valuesT...
92
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asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
eigen-values
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208
TIFR ECE 2012 | Question: 18
Under a certain coordinate transformation from $(x, y)$ to $(u, v)$ the circle $x^{2}+y^{2}=1$ shown below on the left side was transformed into the ellipse shown on the right side. If the transformation is of the form \[ \left[\begin{array}{l} u \\ v \end{array}\right]=\mathbf{A}\ ... \] $A_{1}$ only $A_{2}$ only $A_{1}$ or $A_{2}$ $A_{1}$ or $A_{3}$ $A_{2}$ or $A_{3}$
Under a certain coordinate transformation from $(x, y)$ to $(u, v)$ the circle $x^{2}+y^{2}=1$ shown below on the left side was transformed into the ellipse shown on the ...
80
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Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
matrices
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209
TIFR ECE 2012 | Question: 19
$X$ and $Y$ are two $3$ by $3$ matrices. If \[ X Y=\left(\begin{array}{rrr} 1 & 3 & -2 \\ -4 & 2 & 5 \\ 2 & -8 & -1 \end{array}\right) \] then $X$ has rank $2$ at least one of $X, Y$ is not invertible $X$ can't be an invertible matrix $X$ and $Y$ could both be invertible. None of the above
$X$ and $Y$ are two $3$ by $3$ matrices. If\[X Y=\left(\begin{array}{rrr}1 & 3 & -2 \\-4 & 2 & 5 \\2 & -8 & -1\end{array}\right)\]then$X$ has rank $2$at least one of $X, ...
114
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asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
determinant
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210
TIFR ECE 2012 | Question: 20
Let $A$ be a $2 \times 2$ matrix with all entries equal to $1.$ Define $B=\sum_{n=0}^{\infty} A^{n} / n !$. Then $B=e^{2} A / 2$ $B=\left(\begin{array}{cc}1+e & e \\e & 1+e\end{array}\right)$ ... $B=\left(\begin{array}{cc}1+e^{2} & e^{2} \\e^{2} & 1+e^{2}\end{array}\right)$ None of the above
Let $A$ be a $2 \times 2$ matrix with all entries equal to $1.$ Define $B=\sum_{n=0}^{\infty} A^{n} / n !$. Then$B=e^{2} A / 2$$B=\left(\begin{array}{cc}1+e & e \\e & 1+e...
88
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asked
Dec 8, 2022
Linear Algebra
tifr2012
linear-algebra
matrices
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211
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)$$h(t, \tau)=h(\t...
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Others
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212
TIFR ECE 2011 | Question: 2
The minimum number of unit delay elements required for realizing an infinite impulse response $\text{(IIR)}$ filter is/are $0$ $1$ $\infty$. $>1$. None of the above.
The minimum number of unit delay elements required for realizing an infinite impulse response $\text{(IIR)}$ filter is/are$0$$1$$\infty$.$>1$.None of the above.
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Others
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213
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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Others
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214
TIFR ECE 2011 | Question: 4
Let $\lim _{n \rightarrow \infty} x_{n}=x$. Then which of the following is $\text{TRUE.}$ There exists an $n_{0}$, such that for all $n>n_{0},\left|x_{n}-x\right|=0$. There exists an $n_{0}$ ... $n>n_{0},\left|\frac{x_{n}}{x}\right| \leq \epsilon$ for any $\epsilon>0$. None of the above.
Let $\lim _{n \rightarrow \infty} x_{n}=x$. Then which of the following is $\text{TRUE.}$There exists an $n_{0}$, such that for all $n>n_{0},\left|x_{n}-x\right|=0$.There...
72
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Dec 5, 2022
Calculus
tifr2011
calculus
limits
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215
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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Others
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216
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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217
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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Others
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218
TIFR ECE 2011 | Question: 8
Let $f(x, y)$ be a function in two variables $x, y$. Then which of the following is true $\max _{x} \min _{y} f(x, y) \leq \min _{y} \max _{x} f(x, y)$. $\max _{x} \min _{y} f(x, y) \geq \min _{y} \max _{x} f(x, y)$ ... $\max _{x} \min _{y} f(x, y)=\min _{y} \max _{x} f(x, y)+\min _{y} \min _{x} f(x, y)$. None of the above.
Let $f(x, y)$ be a function in two variables $x, y$. Then which of the following is true$\max _{x} \min _{y} f(x, y) \leq \min _{y} \max _{x} f(x, y)$.$\max _{x} \min _{y...
92
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asked
Dec 5, 2022
Calculus
tifr2011
calculus
maxima-minima
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219
TIFR ECE 2011 | Question: 9
Consider two independent random variables $X$ and $Y$ having probability density functions uniform in the interval $[-1,1]$. The probability that $X^{2}+Y^{2}>1$ is $\pi / 4$ $1-\pi / 4$ $\pi / 2-1$ Probability that $X^{2}+Y^{2}<0.5$ None of the above
Consider two independent random variables $X$ and $Y$ having probability density functions uniform in the interval $[-1,1]$. The probability that $X^{2}+Y^{2}>1$ is$\pi /...
93
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asked
Dec 5, 2022
Probability and Statistics
tifr2011
probability-and-statistics
probability
probability-density-function
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0
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220
TIFR ECE 2011 | Question: 10
Let $f(x)=|x|$, for $x \in(-\infty, \infty)$. Then $f(x)$ is not continuous but differentiable. $f(x)$ is continuous and differentiable. $f(x)$ is continuous but not differentiable. $f(x)$ is neither continuous nor differentiable. None of the above.
Let $f(x)=|x|$, for $x \in(-\infty, \infty)$. Then$f(x)$ is not continuous but differentiable.$f(x)$ is continuous and differentiable.$f(x)$ is continuous but not differe...
101
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asked
Dec 5, 2022
Calculus
tifr2011
calculus
continuity-and-differentiability
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0
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221
TIFR ECE 2011 | Question: 11
What is the value of $\lambda$ such that $\operatorname{Prob}\{X>\operatorname{mean}\{X\}\}=1 / e$, where $\text{PDF}$ of $X$ is $p_{X}(x)=\lambda e^{-\lambda x}, x \geq 0, \lambda>0?$ $1$ $1 / e$ $1 / \sqrt{e}$ $1 / e^{2}$ All of the above
What is the value of $\lambda$ such that $\operatorname{Prob}\{X>\operatorname{mean}\{X\}\}=1 / e$, where $\text{PDF}$ of $X$ is $p_{X}(x)=\lambda e^{-\lambda x}, x \geq ...
90
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asked
Dec 5, 2022
Probability and Statistics
tifr2011
probability-and-statistics
probability
poisson-distribution
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0
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222
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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Others
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223
TIFR ECE 2011 | Question: 13
If $a_k$ is an increasing function of $k$, i.e. $a_1<a_2<\ldots<a_k \ldots$. Then which of the following is $\text{TRUE.}$ $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{a_{k}}=\infty$ ... . Either $(a)$ or $(b)$. $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{a_{k}}=0$. None of the above.
If $a_k$ is an increasing function of $k$, i.e. $a_1<a_2<\ldots<a_k \ldots$. Then which of the following is $\text{TRUE.}$$\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \fr...
107
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asked
Dec 5, 2022
Calculus
tifr2011
calculus
limits
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224
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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Others
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225
TIFR ECE 2011 | Question: 15
Consider a channel where $x_{n} \in\{0,1\}$ is the input and $y_{n}=x_{n} * z_{n}$ is the output, where $*$ is $\text{EX-OR}$ operation, and $P\left(z_{n}=x_{n-1}\right)=P\left(z_{n}=y_{n-1}\right)=\frac{1}{2}$ ... $\frac{1}{2}.$ $1$. $<1.$ $\geq 0.$ Both $(c)$ and $(d)$.
Consider a channel where $x_{n} \in\{0,1\}$ is the input and $y_{n}=x_{n} * z_{n}$ is the output, where $*$ is $\text{EX-OR}$ operation, and $P\left(z_{n}=x_{n-1}\right)=...
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226
TIFR ECE 2011 | Question: 16
Consider a triangular shaped pulse $x$ of base $2 T$ and unit height centered at $0$ , i.e. $x(t)=0$ for $|t|>T, x(t)=1-|t|$ for $t \in[-T, T]$. Then if $x$ is convolved with itself, the output is Square shape. Triangular shape. Bell shape. Inverted $\text{U}$ shape. None of the above.
Consider a triangular shaped pulse $x$ of base $2 T$ and unit height centered at $0$ , i.e. $x(t)=0$ for $|t|>T, x(t)=1-|t|$ for $t \in[-T, T]$. Then if $x$ is convolved ...
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Others
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227
TIFR ECE 2011 | Question: 17
Let $x[n]$ and $y[n]$ be the input and output of a linear time invariant $\text{(LTI)}$ system. Then which of following system is $\text{LTI}$. $z[n]=y[n]+c$ for a constant $c$. $z[n]=x[n] y[n]$. $z[n]=y[n]+x[n]+c$ for a constant $c$. $z[n]=y[n]+x[n]$. None of the above.
Let $x[n]$ and $y[n]$ be the input and output of a linear time invariant $\text{(LTI)}$ system. Then which of following system is $\text{LTI}$.$z[n]=y[n]+c$ for a constan...
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Others
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228
TIFR ECE 2011 | Question: 18
Which of the following statements is TRUE. The cascade of a non-causal linear time invariant $\text{(LTI)}$ system with a causal $\text{LTI}$ system can be causal. If $h[n] \leq 2$ for all $n$, then the $\text{LTI}$ system with $h[n]$ as its impulse response is stable and ... $u[n]=1, n \geq 0, u[n]=0, n<0$, the $\text{LTI}$ system is stable. Both $(c)$ and $(d)$.
Which of the following statements is TRUE.The cascade of a non-causal linear time invariant $\text{(LTI)}$ system with a causal $\text{LTI}$ system can be causal.If $h[n]...
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Others
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229
TIFR ECE 2011 | Question: 19
Let $R_{X}(\tau)$ be the autocorrelation function of a zero mean stationary random process $X(t)$. Which of following statements is FALSE. If $R_{X}(\tau)=0, \forall \tau, X(n)$ and $X(m), n \neq m$ are independent. $R_{X}(\tau)=R_{X}(-\tau)$. $R_{X}(0)=E\left[X^{2}\right]$, where $E$ denotes the expectation. $R_{X}(0) \geq R_{X}(\tau), \forall \tau.$ None of the above.
Let $R_{X}(\tau)$ be the autocorrelation function of a zero mean stationary random process $X(t)$. Which of following statements is FALSE.If $R_{X}(\tau)=0, \forall \tau,...
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TIFR ECE 2011 | Question: 20
Let $x(t)$ be a signal whose Fourier transform $X(f)$ is zero for $|f|>W$. Using a sampler with sampling frequency $4 W$, which of the following filters can be used to exactly reconstruct $x(t)$ ... $\text{[-3W} \;5 \mathrm{W}]$. All the above. None of the above.
Let $x(t)$ be a signal whose Fourier transform $X(f)$ is zero for $|f|>W$. Using a sampler with sampling frequency $4 W$, which of the following filters can be used to ex...
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231
TIFR ECE 2010 | Question: 1
A linear system could be a composition of Two non-linear systems a non-causal non-linear system and a linear system a time varying non-linear system and a time varying linear system All of the above None of the above
A linear system could be a composition ofTwo non-linear systemsa non-causal non-linear system and a linear systema time varying non-linear system and a time varying linea...
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Nov 30, 2022
Others
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TIFR ECE 2010 | Question: 2
For $x \in[0, \pi / 2], \alpha$ for which $\sin (x) \geq x-\alpha x^{3}$ is $\alpha>1 /(2 \pi)$ $\alpha \geq 1 / 6$ $\alpha \leq 1 /(2 \pi)$ $\alpha=1 / 4$ None of the above
For $x \in[0, \pi / 2], \alpha$ for which $\sin (x) \geq x-\alpha x^{3}$ is$\alpha>1 /(2 \pi)$$\alpha \geq 1 / 6$$\alpha \leq 1 /(2 \pi)$$\alpha=1 / 4$None of the above
96
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Nov 30, 2022
Calculus
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calculus
maxima-minima
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233
TIFR ECE 2010 | Question: 3
Consider two independent random variables $\text{X}$ and $\text{Y}$ having probability density functions uniform in the interval $[0,1]$. When $\alpha \geq 1$, the probability that $\max (\text{X, Y})>\alpha \min (\text{X, Y})$ is $1 /(2 \alpha)$ $\exp (1-\alpha)$ $1 / \alpha$ $1 / \alpha^{2}$ $1 / \alpha^{3}$
Consider two independent random variables $\text{X}$ and $\text{Y}$ having probability density functions uniform in the interval $[0,1]$. When $\alpha \geq 1$, the probab...
92
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Nov 30, 2022
Probability and Statistics
tifr2010
probability-and-statistics
probability
probability-density-function
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234
TIFR ECE 2010 | Question: 4
Let $Y_{n}=s_{n}+W_{n}$ where $\left\{s_{n}\right\}$ is the desired signal bandlimited to $[-W, W]$ and $\left\{W_{n}\right\}$ is a noise component, which is sparse (that is, only few samples are non-zero), bursty (that is, runs of non-zero samples are ... of $\left\{Y_{n+k}\right\}_{k=-K}^{K}$ for suitably chosen $K$ Both $a)$ and $b)$ are better than the other options
Let $Y_{n}=s_{n}+W_{n}$ where $\left\{s_{n}\right\}$ is the desired signal bandlimited to $[-W, W]$ and $\left\{W_{n}\right\}$ is a noise component, which is sparse (that...
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Others
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TIFR ECE 2010 | Question: 5
Let $Y(t)=\sum_{n=-\infty}^{\infty} x_{n} h(t-n T)$. We sample $Y(t)$ at time instants $n T / 2$ and let $Y_{n}=Y(n T / 2)$. Which of the following is true? $\left\{Y_{n}\right\}$ can be interpreted as the output of a discrete time, ... of a discrete time, linear, time-invariant system with input $\left\{X_{n}\right\}$. Both $a)$ and $b)$ above Both $b)$ and $c)$ above
Let $Y(t)=\sum_{n=-\infty}^{\infty} x_{n} h(t-n T)$. We sample $Y(t)$ at time instants $n T / 2$ and let $Y_{n}=Y(n T / 2)$. Which of the following is true?$\left\{Y_{n}\...
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TIFR ECE 2010 | Question: 6
If we convolve $\sin (t) / t$ with itself, then we get $C \sin (t) / t$ for some constant $C$ $C \cos (t) / t$ for some constant $C$ $C \cos (t) / t^{2}$ for some constant $C$ $C_{1} \sin (t) / t^{2}+C_{2} \cos (t) / t^{2}$ for some constants $C_{1}, C_{2}$ None of the above
If we convolve $\sin (t) / t$ with itself, then we get$C \sin (t) / t$ for some constant $C$$C \cos (t) / t$ for some constant $C$$C \cos (t) / t^{2}$ for some constant $...
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237
TIFR ECE 2010 | Question: 7
A voltage source with internal resistance $\text{R}$ is connected to an inductor $\text{L}$ and a capacitor $\text{C}$ connected in parallel. The output is the common voltage across the inductor and the capacitor. What is the nature of the transfer ... depending upon the values of $\text{L}$ and $\text{C}$. The circuit is not stable and no transfer function exists.
A voltage source with internal resistance $\text{R}$ is connected to an inductor $\text{L}$ and a capacitor $\text{C}$ connected in parallel. The output is the common vol...
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Nov 30, 2022
Others
tifr2010
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1
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238
TIFR ECE 2010 | Question: 8
Consider a discrete time channel with binary inputs and binary outputs. Let $x_{n}$ denote the input bit at time $n$ and $y_{k}$ denote the output bit at time $\text{k}$. The channel operation is such that to produce the output $y_{n}$ it drops one ... we do not make any error If $R<1 / 2$, then there exists a scheme with zero error All of the above None of the above
Consider a discrete time channel with binary inputs and binary outputs. Let $x_{n}$ denote the input bit at time $n$ and $y_{k}$ denote the output bit at time $\text{k}$....
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Others
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1
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239
TIFR ECE 2010 | Question: 9
The $z$-transform of a sequence $\left\{x_{n}\right\}_{n=-\infty}^{\infty}$ is defined to be $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$. The $z$-transform of the sequence $y_{n}=x_{2 n+1}$ is $Y(z)=z(X(z)-X(-z)) / 2$ ... $Y(z)=z(X(\sqrt{z})-X(-\sqrt{z})) / 2$ $Y(z)=(X(\sqrt{z})-X(-\sqrt{z})) / 2$
The $z$-transform of a sequence $\left\{x_{n}\right\}_{n=-\infty}^{\infty}$ is defined to be $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$. The $z$-transform of the seque...
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Others
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240
TIFR ECE 2010 | Question: 10
$\text{H}$ is a circulant matrix (row $n$ is obtained by circularly shifting row $1$ to the right by $n$ positions) and $\text{F}$ is the $\text{DFT}$ matrix. Which of the following is true? $F H F^{H}$ is circulant, where $F^{H}$ is the inverse $\text{DFT}$ matrix. $F H F^{H}$ is tridiagonal $F H F^{H}$ is diagonal $F H F^{H}$ has real entries None of the above
$\text{H}$ is a circulant matrix (row $n$ is obtained by circularly shifting row $1$ to the right by $n$ positions) and $\text{F}$ is the $\text{DFT}$ matrix. Which of th...
78
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Nov 30, 2022
Linear Algebra
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matrices
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