Final answer:
Signal (a) x(k) = cos(.02πk) is periodic with a period M of 100. Signal (b) x(k) = sin(.1k) cos(.2k) is classified as aperiodic unless further simplification reveals a common period.
Step-by-step explanation:
To classify each of the given signals as periodic or aperiodic and to find the period for the periodic signals, we'll analyze each signal separately.
(a) x(k) = cos(.02πk)
This is a periodic signal. A cosine wave is inherently periodic, and the period, M, can be found as the inverse of the frequency multiplied by 2π. The frequency in this case is 0.01 cycles per unit (k), derived by dividing the coefficient of k inside the cosine (0.02π) by 2π. Therefore, the period M is 1/0.01 = 100. So, x(k) = cos(.02πk) has a period of M = 100.
(b) x(k) = sin(.1k) cos(.2k)
Since the signal is a product of two cosine functions with different frequencies, it cannot be straightforwardly classified as periodic without further manipulation. Typically, this would be considered aperiodic unless it can be simplified to a form that reveals a common period for both components. Without such simplification, we will regard x(k) as aperiodic.