To find the fundamental period of the signal

$x(t) = \sin(2\pi f_0 t) + \sin\left(2\pi \frac{5}{4} f_0 t\right) + \sin\left(2\pi \frac{3}{2} f_0 t\right),$

we need to find the fundamental period of each individual sinusoid and then determine the overall period.

Step-by-Step Process:

1. Each term in the expression is a sinusoid of the form $\sin(2 \pi f t)$, where $f$ is the frequency.

   • For $\sin(2 \pi f_0 t)$, the frequency is $f_0$, so the period is: $T_1 = \frac{1}{f_0}$

   • For $\sin\left(2\pi \frac{5}{4} f_0 t\right)$, the frequency is $\frac{5}{4} f_0$, so the period is: $T_2 = \frac{1}{\frac{5}{4} f_0} = \frac{4}{5 f_0}$

   • For $\sin\left(2\pi \frac{3}{2} f_0 t\right)$, the frequency is $\frac{3}{2} f_0$, so the period is: $T_3 = \frac{1}{\frac{3}{2} f_0} = \frac{2}{3 f_0}$

2. The fundamental period of the entire signal $x(t)$ will be the least common multiple (LCM) of the individual periods $T_1$, $T_2$, and $T_3$.

   The individual periods are: $T_1 = \frac{1}{f_0}, \quad T_2 = \frac{4}{5 f_0}, \quad T_3 = \frac{2}{3 f_0}$

   To find the fundamental period, we need to compute the LCM of the numerators $1$, $4$, and $2$ and the greatest common divisor (GCD) of the denominators $f_0$, $5 f_0$, and $3 f_0$.

   The LCM of the numerators $1, 4, 2$ is $4$, and the GCD of the denominators $1, 5, 3$ is $1$. Therefore, the fundamental period is: $T = \frac{4}{f_0}$

Conclusion:

The fundamental period of the signal is $T = \frac{4}{f_0}$.

Do you want further details or explanations?

Here are 5 related questions to deepen your understanding:

1. How is the fundamental period of a signal defined in general?
2. What happens to the fundamental period if the frequencies are irrational multiples of each other?
3. How do you find the least common multiple (LCM) of fractions?
4. What role does the GCD play when determining the fundamental period?
5. How would the period change if one of the frequencies was non-harmonic?

Tip: When signals involve harmonically related frequencies, finding the LCM of their periods is the key to determining the overall fundamental period.