Final answer:
To determine the periods of x1 and x2, one must calculate the period of each individual sinusoidal component by leveraging the relationship between frequency and angular frequency. The period T is found using T = 1/f, where f is the frequency, and for a sinusoidal function represented as A sin(2πft + φ), the frequency is derived from the coefficient of t. The individual periods of the components are calculated and may not present a clear overall repeating period.
Step-by-step explanation:
To find the period of the given signals, we need to analyze the angular frequencies inherent in their sinusoidal formulas. The general form of a sinusoidal function is A sin(2πft + φ), where A is the amplitude, f is the frequency, and φ is the phase shift. The period T of the function is given by T = 1/f.
For the first signal, x1, we have three components:
- 5sin(π/416t) with a period of T1 = 2 * 416/π
- sin(π/1144t) with a period of T2 = 2 * 1144/π
- 10sin(π/1560t) with a period of T3 = 2 * 1560/π
- For the second signal, x2, we have three components:
- 20sin(π/784t) with a period of T4 = 2 * 784/π
- 5sin(π/1568t) with a period of T5 = 2 * 1568/π
- 30sin(π/2156t) with a period of T6 = 2 * 2156/π
The overall period of a signal composed of multiple sinusoidal components is the least common multiple of the individual periods when they are rational numbers. However, if the periods do not share a common multiple or it is not a rational number, the signal may not have a repeating period. The individual periods here are based on the relationship of the frequency to 2π.