# Step-by-Step Solution: Periodicity of \( u(n) \) ## 1. Understanding the Signal The signal \( u(n) \) is the **unit step sequence** defined as: \[ u(n) = \begin{cases} 1 & n \geq 0 \\ 0 & n < 0 \end{cases} \] For the given range \( 0 \leq n \leq \infty \), \( u(n) = 1 \) for all \( n \). --- ## 2. Definition of Periodicity A discrete-time signal \( x(n) \) is **periodic** if there exists a positive integer \( N \) such that: \[ x(n + N) = x(n) \quad \text{for all } n \] The smallest such \( N \) is called the **fundamental period**. --- ## 3. Applying the Definition to \( u(n) \) For \( n \geq 0 \): \[ u(n) = 1 \] So, \[ u(n+N) = 1 \] for any \( N \geq 1 \). Therefore, \[ u(n+N) = u(n) = 1 \quad \text{for all } n \geq 0 \text{ and any } N \geq 1 \] --- ## 4. Conclusion - **Is \( u(n) \) periodic?** Yes, **for \( n \geq 0 \), \( u(n) \) is a constant sequence (all ones)**, which is periodic for any period \( N \). - **Fundamental period:** Since the signal is constant, **any integer \( N \geq 1 \) can be considered a period**. The smallest is \( N = 1 \). --- ## 5. Final Answer > **For \( 0 \leq n \leq \infty \), the signal \( u(n) \) is periodic with fundamental period \( N = 1 \).** --- ### Illustration **Unit step sequence for \( n \geq 0 \):**  *Alt text: Stem plot of u(n) showing value 1 for all n >= 0*

# Step-by-Step Solution: Periodicity of \( u(n) \) ## 1. Understanding the Signal The signal \( u(n) \) is the **unit step sequence** defined as: \[ u(n) = \begin{cases} 1 & n \geq 0 \\ 0 & n < 0 \end{cases} \] In the specified range \( 0 \leq n \leq \infty \), \( u(n) \) is equal to 1 for all \( n \). --- ## 2. Definition of Periodicity A discrete-time signal \( x(n) \) is **periodic** if there exists a positive integer \( N \) such that: \[ x(n + N) = x(n) \quad \text{for all } n \] The smallest such \( N \) is called the **fundamental period**. --- ## 3. Applying the Definition to \( u(n) \) For \( n \geq 0 \): \[ u(n) = 1 \] Thus, for any positive integer \( N \): \[ u(n + N) = 1 \] This implies: \[ u(n + N) = u(n) = 1 \quad \text{for all } n \geq 0 \text{ and any } N \geq 1 \] --- ## 4. Conclusion - **Is \( u(n) \) periodic?** Yes, \( u(n) \) is a constant sequence (all ones) for \( n \geq 0 \), which makes it periodic for any period \( N \). - **Fundamental period:** Since the signal is constant, **any integer \( N \geq 1 \) can be considered a period**. The smallest period is \( N = 1 \). --- ## 5. Final Answer > **For \( 0 \leq n \leq \infty \), the signal \( u(n) \) is periodic with a fundamental period \( N = 1 \).** --- ### Illustration **Unit step sequence for \( n \geq 0 \):**  *Alt text: Stem plot of \( u(n) \) showing value 1 for all \( n \geq 0*