Question

Choose the correct statement with reference to standard test signals

• A. Unit impulse signal is obtained by integrating unit step signal
• B. Unit step signal is obtained by differentiating unit impulse signal
• C. Unit ramp signal is obtained by differentiating unit step signal
• D. Unit parabolic signal is obtained by integrating unit ramp signal

Hint:

Remember the simple mnemonic order: Impulse $\rightarrow$ Step $\rightarrow$ Ramp $\rightarrow$ Parabolic. Moving from left to right requires Integration, whereas moving backwards from right to left requires Differentiation.

Answer 1

The Correct Option is D

Concept: Standard test signals in signal processing and control systems follow a strict hierarchical derivative and integral relationship chain. These signals are mathematically defined for \(t \ge 0\) as:
• Unit Impulse Signal: \(\delta(t)\)
• Unit Step Signal: \(u(t) = 1\)
• Unit Ramp Signal: \(r(t) = t\)
• Unit Parabolic Signal: \(p(t) = \frac{t^2}{2}\) The relationships via integration can be arranged sequentially as: \[ \delta(t) \xrightarrow{\int} u(t) \xrightarrow{\int} r(t) \xrightarrow{\int} p(t) \]

Step 1: Verify each mathematical transition via integration. Let's systematically perform the integration step-by-step:
• Integrating the unit step signal \(u(t)\): \[ \int u(t)\,dt = \int 1\,dt = t = r(t) \quad \text{(Unit Ramp Signal)} \] This disproves statement (A).
• Integrating the unit ramp signal \(r(t)\): \[ \int r(t)\,dt = \int t\,dt = \frac{t^2}{2} = p(t) \quad \text{(Unit Parabolic Signal)} \] This exactly matches statement (D).

Step 2: Evaluate alternative descriptions. Conversely, if we move down the chain via differentiation: \[ \frac{d}{dt}[p(t)] = r(t), \quad \frac{d}{dt}[r(t)] = u(t), \quad \frac{d}{dt}[u(t)] = \delta(t) \] Looking at Option (B), differentiating an impulse gives a doublet, not a step. Looking at Option (C), differentiating a step yields an impulse. Hence, statement (D) is uniquely correct.