51. Match List-I with List-II and select the correct answer using the options given below:
List-I (x[n])
List-II (X(z))
a. $${a^n}u\left[ n \right]$$
1. $$\frac{{az}}{{{{\left( {z - a} \right)}^2}}}$$
b. $${a^{n - 2}}u\left[ {n - 2} \right]$$
2. $$\frac{{z{e^{ - j}}}}{{z{e^{ - j}} - a}}$$
c. $${e^{jn}}{a^n}$$
3. $$\frac{z}{{z - a}}$$
d. $$n.{a^n}u\left[ n \right]$$
4. $$\frac{{{z^{ - 1}}}}{{z - a}}$$
| List-I (x[n]) | List-II (X(z)) |
| a. $${a^n}u\left[ n \right]$$ | 1. $$\frac{{az}}{{{{\left( {z - a} \right)}^2}}}$$ |
| b. $${a^{n - 2}}u\left[ {n - 2} \right]$$ | 2. $$\frac{{z{e^{ - j}}}}{{z{e^{ - j}} - a}}$$ |
| c. $${e^{jn}}{a^n}$$ | 3. $$\frac{z}{{z - a}}$$ |
| d. $$n.{a^n}u\left[ n \right]$$ | 4. $$\frac{{{z^{ - 1}}}}{{z - a}}$$ |
52. The minimum number of delay elements required in realizing a digital filter with the transfer function $$H\left( z \right) = \frac{{1 + a{z^{ - 1}} + b{z^{ - 2}}}}{{1 + c{z^{ - 1}} + d{z^{ - 2}} + e{z^{ - 3}}}}$$ is
53. An input x[n] with length 3 is applied to a linear time invariant system having an impulse response h[n] of length 5 and Y(ω) is the DTFT of the output y[n] of the system. If |h[n]| ≤ L and |x[n]| ≤ L for all n, the maximum value of Y(0) can be:
54. The complex exponential power form of Fourier series of x(t) is: $$x\left( t \right) = \sum\nolimits_{k = - \infty }^\infty {{a_k}.{e^{j\frac{{2\pi }}{{{T_0}}}.kt}}} $$
If $$x\left( t \right) = \sum\nolimits_{b = - \infty }^\infty \delta \left( {t - b} \right),$$ then the value of ak is:
If $$x\left( t \right) = \sum\nolimits_{b = - \infty }^\infty \delta \left( {t - b} \right),$$ then the value of ak is:
55. If x(t) = 10 rect $$\left( {\frac{{\text{t}}}{2}} \right),$$ the zero-frequency value of its spectrum is given by
56. FIR filters can be designed using
57. The auto correlation function Rx(τ) satisfies which one of the following properties?
58. If a random process X(t) is ergodic then, statistical averages
59. Which one of the following systems described by the following input-output relation is non-linear?
60. Let h(t) denote the impulse response of a causal system with transfer function $$\frac{1}{{{\text{s}} + 1}}.$$ Consider the following three statements:
S1: The system is stable.
S2: $$\frac{{{\text{h}}\left( {{\text{t}} + 1} \right)}}{{{\text{h}}\left( {\text{t}} \right)}}$$ is independent of t for t > 0.
S3: A non-causal system with the same transfer function is stable.
For the above system,
S1: The system is stable.
S2: $$\frac{{{\text{h}}\left( {{\text{t}} + 1} \right)}}{{{\text{h}}\left( {\text{t}} \right)}}$$ is independent of t for t > 0.
S3: A non-causal system with the same transfer function is stable.
For the above system,
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