Final answer:
The period of the signal sin(0.5t + 0.1) + cos(0.25t) is determined by examining the periodic nature of the sine and cosine functions individually and then finding their combined period, which in this case is 8π radians.
Step-by-step explanation:
To determine the period of the input signal xx(t) = sin(0.5t + 0.1) + cos(0.25t), we need to examine the properties of the sine and cosine functions separately. Both functions are periodic, with the sine function oscillating between +1 and -1 every 2π radians, as does the cosine function.
The period of a sine or cosine function is given by 2π divided by the coefficient of t (the angular frequency ω). So for sin(0.5t + 0.1), the period is 2π / 0.5 which equals 4π radians. For cos(0.25t), the period is 2π / 0.25 which equals 8π radians.
Since the periods of these two functions are not the same, they will not complete their cycles simultaneously. However, if we want to find a common period where both functions repeat their values, we must find the least common multiple (LCM) of 4π and 8π, which is 8π radians. Thus, the combined period of the signal xx(t) is 8π radians.