Question

Prove that the equation x(t) = Acos(wt+Φ) corresponds to a periodic signal under what condition? a. x ( t + Φ) = x(t) b. x ( t - Φ) = x(t) c. x ( t + w) = x(t) d. x ( t - w) = x(t)

Answer 1

Main Answer:

The equation
`\(x(t) = A \cos(wt + \Phi)\)` corresponds to a periodic signal under the condition c.
`\(x(t + w) = x(t)\)`.

Therefore, the correct option is: c. x ( t + w) = x(t).

Step-by-step explanation:

The given equation represents a sinusoidal signal with amplitude (A), angular frequency (w), initial phase (Phi), and time t. For this equation to describe a periodic signal, it must repeat its values over regular intervals.

The condition c, \(x(t + w) = x(t)\), ensures periodicity by stating that the function's value at time \(t + w\) is equal to its value at time \(t\). This indicates that the signal completes one full period at each interval of \(w\), confirming its periodic nature.

Conditions a, b, and d do not guarantee periodicity. Condition a (\(x(t + \Phi) = x(t)\)) involves a phase shift but does not ensure the signal repeats after a fixed interval. Similarly, conditions b (\(x(t - \Phi) = x(t)\)) and d (\(x(t - w) = x(t)\)) involve phase-related shifts but do not guarantee the signal's periodic behavior.

The key to periodicity lies in the equality of the function's values at regularly spaced intervals, as expressed in condition c. Understanding these conditions is fundamental in analyzing and describing the periodic nature of sinusoidal signals in various fields, including physics, engineering, and signal processing.

Therefore, the correct option is: c. x ( t + w) = x(t).