Which of the following is the fundamental period (the smallest positive integer N such that x[n + N] = x[n] for every integer n) of the discrete-time signal x[n] = cos(0.3πn)?

Answer: B. 20 samples

A discrete-time sinusoid cos(ω₀n) is periodic only when ω₀/2π is a rational number k/N (in lowest terms), and its fundamental period is then N. Here ω₀/2π = 0.3π/2π = 0.15 = 3/20, so N = 20 samples (exactly 3 cycles fit in them). Thinking in continuous time, 2π/ω₀ ≈ 6.67 is not an integer, so the sample positions shift from one cycle to the next. When ω₀/2π is irrational, as for cos(n) with ω₀/2π = 1/(2π), the same sequence of values never returns, so the signal is not periodic.

• A. 2π/ω₀ = 2/0.3 ≈ 6.67 is the continuous-time period. Sample indices are integers, so the values do not repeat every 6.67 samples.
• B. Correct. 0.3π/2π = 3/20, so after 20 samples (3 cycles) the sequence returns to its start.
• C. ω₀/2π = 0.15 is rational, so the signal is periodic. It is not periodic only when ω₀/2π is irrational.