1. The auto-correlation function of an energy signal has
2. Let x(t) be a periodic function with period T = 10. The Fourier series coefficients for this series are denoted by $${a_k}$$ , that is
$$x\left( t \right) = \sum\limits_{k = - \infty }^\infty {{a_k}{e^{jk{{2\pi } \over T}t}}} .$$
The same function x(t) can also be considered as a periodic function with period T' = 40. Let bk be the Fourier series coefficients when period is taken as T'. If $$\sum\limits_{k = - \infty }^\infty {\left| {{a_k}} \right|} = 16,$$ then $$\sum\limits_{k = - \infty }^\infty {\left| {{b_k}} \right|} $$ is equal to
$$x\left( t \right) = \sum\limits_{k = - \infty }^\infty {{a_k}{e^{jk{{2\pi } \over T}t}}} .$$
The same function x(t) can also be considered as a periodic function with period T' = 40. Let bk be the Fourier series coefficients when period is taken as T'. If $$\sum\limits_{k = - \infty }^\infty {\left| {{a_k}} \right|} = 16,$$ then $$\sum\limits_{k = - \infty }^\infty {\left| {{b_k}} \right|} $$ is equal to
3. A periodic signal x(t) of period T0 is given by
$$x\left( t \right) = \left\{ {\matrix{
{1,} & {\left| t \right| < {T_1}} \cr
{0,} & {{T_1} < \left| t \right| < {{{T_0}} \over 2}} \cr
} } \right.$$
the dc component of x(t) is
$$x\left( t \right) = \left\{ {\matrix{ {1,} & {\left| t \right| < {T_1}} \cr {0,} & {{T_1} < \left| t \right| < {{{T_0}} \over 2}} \cr } } \right.$$
the dc component of x(t) is
4. The Laplace transform of a unit ramp function starting at t = a, is
5. Let P be linearity, Q be time-invariance, R be causality and S be stability. A discrete-time system has the input-output relationship,
$$y\left( n \right) = \left\{ \matrix{
\matrix{
{x\left( n \right),} & {n \ge 1} \cr
} \hfill \cr
\matrix{
{0,} & {n = 0} \cr
} \hfill \cr
\matrix{
{x\left( {n + 1} \right),} & {n \le - 1} \cr
} \hfill \cr} \right.$$
where x(n) is the input and y(n) is the output.
The above system has the properties
$$y\left( n \right) = \left\{ \matrix{ \matrix{ {x\left( n \right),} & {n \ge 1} \cr } \hfill \cr \matrix{ {0,} & {n = 0} \cr } \hfill \cr \matrix{ {x\left( {n + 1} \right),} & {n \le - 1} \cr } \hfill \cr} \right.$$
where x(n) is the input and y(n) is the output.
The above system has the properties
6. Which of the following cannot be the Fourier series expansion of a periodic signal?
7. Let x(t) be a wide sense stationary (WSS) random with power spectral density Sx(f). If Y(t) is the process defined as y(t) = x(2t - 1), the power spectral density SY(f) is
8. The Fourier series expansion of a real periodic signal with fundamental frequency f0 is given by
$${g_p}\left( t \right) = \sum\limits_{n = - \infty }^\infty {{c_n}{e^{j2\pi {f_0}t}}} ;$$
it is given that c3 = 3 + j5. Then c3 is
$${g_p}\left( t \right) = \sum\limits_{n = - \infty }^\infty {{c_n}{e^{j2\pi {f_0}t}}} ;$$
it is given that c3 = 3 + j5. Then c3 is
9. The input x(t) and output y(t) of a system are related as $$y\left( t \right) = \int\limits_{ - \infty }^t {x\left( \tau \right)} \cos \left( {3\tau } \right)d\tau .$$ The system is
10. Consider the sequence x[n] = anu[n] + bnu[n], where u[n] denotes the unit-step sequence and 0 < |a| < |b| < 1. The region of convergence (ROC) of the z-transform of x[n] is
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