The question consists of two statements, one labeled as ‘Statement (I)’ and the other labeled as ‘Statement (II)’. You are to examine these two statements carefully and select the answers to these items using the codes given below:

**Statement (I):** Discrete-time signal is derived from the continuous-time signal by the sampling process.

**Statement (II):** Energy signal has zero average power and power signal has zero energy

Option 3 : Statement (I) is true, but Statement (II) is false

Sampling is a process where a continuous-time signal is converted to its equivalent discrete-time signal. It is given by

**t = N × t.**

**Where N is no of samples**

**t is time**

**Power and Energy Signals:**

**Energy Signal:**

A signal is said to be a power/energy signal if the total energy/power transmitted is finite.

i.e. **0 < E < ∞ and 0 < P < ∞**

The energy of a signal for a continuous-time signal is defined as:

\(E = \mathop \smallint \limits_{ - T}^T {x^2}\left( t \right)dt\) (finite duration)

\(E = \mathop {\lim }\limits_{T \to \infty } \mathop \smallint \limits_{ - T}^T {x^2}\left( t \right)dt\) (infinite duration)

For a discrete-time signal, this is defined as:

\({E_{x\left[ n \right]}} = \mathop {\lim }\limits_{N \to \infty } \mathop \sum \limits_{n = - N}^N {\left| {x\left[ n \right]} \right|^2}\)

Note:

- Power of energy signal = 0
- The energy of power signal = ∞

**Average power for a periodic signal:**

,\(p = \frac{1}{T}\mathop \smallint \limits_{T/2}^{T/2} {x^2}\left( t \right)dt\)

**For discrete signal, it is defined as:**

\({P_{avg\;x\left[ n \right]}} = \mathop {\lim }\limits_{N \to \infty } \dfrac{1}{{2N + 1}}\mathop \sum \limits_{n = - N}^N {\left| {x\left[ n \right]} \right|^2}\)

**A signal x(t) is said to be an energy signal if and only if the energy of that signal is finite, and so power is equal to zero.****A signal x(t) is said to be a power signal if and only if its average power is finite, thus implying that energy is infinite.****Signals that satisfy neither property are referred to as neither energy signals nor power signals.****Energy and power signals are mutually exclusive. Energy signal has zero average power and power signal has infinite energy.**