71. Consider the differential equation $${{dx} \over {dt}} = 10 - 0.2x$$ with initial conduction x(0) = 1. The response x(t) for t > 0 is
72. A linear phase channel with phase delay Tp and group delay Tg must have
73. The pole-zero pattern of a certain filter is shown in figure. The filter must be of the following type
74. The ROC of z-transform of the discrete time sequence
$$x\left( n \right) = {\left( {{1 \over 3}} \right)^n}u\left( n \right) - {\left( {{1 \over 2}} \right)^n}u\left( { - n - 1} \right)$$ is
$$x\left( n \right) = {\left( {{1 \over 3}} \right)^n}u\left( n \right) - {\left( {{1 \over 2}} \right)^n}u\left( { - n - 1} \right)$$ is
75. It is desired to find three-tap causal filter which gives zero signal as an output to and input of the form
\[x\left[ n \right] = {c_1}\exp \left( { - \frac{{j\pi n}}{2}} \right) + {c_2}\exp \left( {\frac{{j\pi n}}{2}} \right),\]
Where c1 and c2 are arbitrary real numbers. The desired three-tap filter is given by
h[0] = 1, h[1] = a, h[2] = b and h[n] = 0 for n < 0 or n > 2.
What are the values of the filter taps a and b if the output is y[n] = 0 for all n, when x[n] is as given above?
\[\xrightarrow{{x\left[ n \right]}}\boxed{\begin{array}{*{20}{c}}
{n = 0} \\
\downarrow \\
{h\left[ n \right] = \left\{ {1,a,b} \right\}}
\end{array}}\xrightarrow{{y\left[ n \right] = 0}}\]
\[x\left[ n \right] = {c_1}\exp \left( { - \frac{{j\pi n}}{2}} \right) + {c_2}\exp \left( {\frac{{j\pi n}}{2}} \right),\]
Where c1 and c2 are arbitrary real numbers. The desired three-tap filter is given by
h[0] = 1, h[1] = a, h[2] = b and h[n] = 0 for n < 0 or n > 2.
What are the values of the filter taps a and b if the output is y[n] = 0 for all n, when x[n] is as given above?
\[\xrightarrow{{x\left[ n \right]}}\boxed{\begin{array}{*{20}{c}} {n = 0} \\ \downarrow \\ {h\left[ n \right] = \left\{ {1,a,b} \right\}} \end{array}}\xrightarrow{{y\left[ n \right] = 0}}\]
76. Let δ(t) denote the delta function. The value of the integral $$\int\limits_{ - \infty }^\infty {\delta \left( t \right)} \cos \left( {{{3t} \over 2}} \right)dt$$ is
77. If a signal f(t) has energy E, the energy of the signal f(2t) is equal to
78. The input-output relationship of a causal stable LTI system is given as
y[n] = αy[n - 1] + βx[n]. If the impulse response h[n] of this system satisfies the condition
$$\sum\limits_{n = 0}^\infty {h\left[ n \right] = 2,} $$ the relationship between α and β is
y[n] = αy[n - 1] + βx[n]. If the impulse response h[n] of this system satisfies the condition
$$\sum\limits_{n = 0}^\infty {h\left[ n \right] = 2,} $$ the relationship between α and β is
79. The bilateral Laplace transform of a function
$$f\left( t \right) = \left\{ {\matrix{
{1,} & {{\rm{if}}\,a \le t \le b} \cr
0 & {{\rm{otherwise}}} \cr
} } \right.$$ is
$$f\left( t \right) = \left\{ {\matrix{ {1,} & {{\rm{if}}\,a \le t \le b} \cr 0 & {{\rm{otherwise}}} \cr } } \right.$$ is
80. {a(n)} is a real-valued periodic sequence with a period N. x(n) and X(k) form N-point Discrete Fourier Transform (DFT) pairs. The DFT Y(k) of the sequence
$$y\left( n \right) = \frac{1}{N}\sum\limits_{r = 0}^{N - 1} {x\left( r \right)} x\left( {n + r} \right)$$ is
$$y\left( n \right) = \frac{1}{N}\sum\limits_{r = 0}^{N - 1} {x\left( r \right)} x\left( {n + r} \right)$$ is
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