Let's analyze the question: **Given:** \( x(t) = e^{j 2\pi C t} \) We are to choose the correct completion: - **(A)** for any real value of \( C \) including 0. - **(B)** for any real value of \( C \ne 0 \). - **(C)** for any value of \( C \), either real or complex. - **(D)** Never. Let's determine **when** \( x(t) \) is periodic. --- ## Step 1: Definition of Periodicity A signal \( x(t) \) is **periodic** with period \( T \) if \[ x(t+T) = x(t) \quad \forall t \] For our signal: \[ x(t+T) = e^{j 2\pi C (t+T)} = e^{j 2\pi C t} \cdot e^{j 2\pi C T} = x(t) \cdot e^{j 2\pi C T} \] For equality, the extra factor must be 1: \[ e^{j 2\pi C T} = 1 \] \[ \implies 2\pi C T = 2\pi k, \quad k \in \mathbb{Z} \] \[ \implies C T = k \] \[ \implies T = \frac{k}{C} \] where \( k \) is any integer. --- ## Step 2: Analyze the Value of \( C \) Let's consider the possible values of \( C \): ### **Case 1: \( C = 0 \)** \[ T = \frac{k}{0} \] which is undefined for \( k \ne 0 \). But the signal becomes: \[ x(t) = e^{j 2\pi \cdot 0 \cdot t} = e^0 = 1 \] This is a **constant**, and a constant is considered periodic with **any period** (since \( 1 = 1 \)). ### **Case 2: \( C \) is a nonzero real number** There exists a period \( T = \frac{1}{C} \) (taking \( k = 1 \)), so the signal is **periodic**. ### **Case 3: \( C \) is a complex number (not real)** Let \( C = a + jb \), with \( b \ne 0 \), then \[ x(t) = e^{j 2\pi (a + jb) t} = e^{j 2\pi a t} \cdot e^{-2\pi b t} \] The second term, \( e^{-2\pi b t} \), is **not periodic** (it's an exponential decay/growth unless \( b = 0 \)). So, for complex \( C \), \( x(t) \) is **not periodic** unless \( b = 0 \), i.e., \( C \) is real. --- ## Step 3: Check the Options - **(A) For any real value of C including 0.** This is correct. For \( C = 0 \), \( x(t) = 1 \) (constant, periodic). For \( C \ne 0 \), \( T = 1/|C| \) is a period. - **(B) For any real value of \( C \ne 0 \).** This is not correct because it excludes \( C = 0 \), which, as shown, is periodic (constant). - **(C) For any value of \( C \), either real or complex.** Not correct. If \( C \) is not real (\( b \ne 0 \)), the signal is not periodic. - **(D) Never.** Not correct, as shown above. --- ## **Final Answer** \[ \boxed{\text{The signal } x(t) = e^{j 2\pi C t} \text{ is periodic for any real value of } C \text{ including } 0.} \] ### **Explanation of why the others are wrong:** - (B) is wrong because it excludes \( C = 0 \), but the constant signal (when \( C=0 \)) is periodic. - (C) is wrong because for complex \( C \), the signal is not periodic due to the exponential growth/decay factor. - (D) is wrong because for real \( C \), the signal is periodic (including \( C=0 \)).