Solution: Unit Impulse Signal Statements Analysis

This solution evaluates the given statements about the unit impulse signal in both discrete-time and continuous-time domains.

Statement 1: Discrete-Time Unit Impulse and Step Difference

The discrete-time unit impulse signal, denoted as $\delta[n]$, is defined as:

• $\delta[n] = 1$ for $n = 0$
• $\delta[n] = 0$ for $n \neq 0$

The discrete-time unit step signal, denoted as $u[n]$, is defined as:

• $u[n] = 1$ for $n \geq 0$
• $u[n] = 0$ for $n < 0$

The first difference of the unit step signal is $u[n] - u[n-1]$. Let's evaluate this difference:

• For $n=0$: $u[0] - u[-1] = 1 - 0 = 1$, which matches $\delta[0]$.
• For $n>0$: $u[n] - u[n-1] = 1 - 1 = 0$, which matches $\delta[n]$.
• For $n<0$: $u[n] - u[n-1] = 0 - 0 = 0$, which matches $\delta[n]$.

Therefore, $\delta[n] = u[n] - u[n-1]$. Statement 1 is correct.

Statement 2: Amplitude of Unit Impulse at n=1

As defined above, the discrete-time unit impulse signal $\delta[n]$ has an amplitude of 1 only at $n=0$. At $n=1$, the value is $\delta[1] = 0$. Thus, Statement 2 is incorrect.

Statement 3: Continuous-Time Unit Impulse and Step Derivative

The continuous-time unit impulse signal is the Dirac delta function, $\delta(t)$. The continuous-time unit step function is the Heaviside function, $u(t)$.

The Heaviside function $u(t)$ is discontinuous at $t=0$. In the context of distributions, the derivative of the Heaviside step function $u(t)$ is defined as the Dirac delta function $\delta(t)$. Mathematically, this is represented as:

$ \frac{du(t)}{dt} = \delta(t) $

Statement 3 is correct.

Conclusion

Based on the analysis, statements 1 and 3 are correct, while statement 2 is incorrect.

The correct option is the one that includes statements 1 and 3 only.