Final answer:
The signal x(n) = 0.6^n cos(0.7π n) is not periodic because the exponential term 0.6^n is not periodic and the cosine term does not have an integer period.
Step-by-step explanation:
To determine whether the discrete signal x(n) = 0.6^n cos(0.7 π n) is periodic and to find its period, we need to analyze both parts of the function. The 0.6^n part is not periodic since it represents an exponential decay and does not repeat. The cosine function, however, is inherently periodic. The cosine function repeats every 2π radians.
The coefficient of n inside the cosine function is 0.7π. To find the period N of the cosine part of the function, we set the argument of the cosine function equal to the argument plus an integral multiple of 2π, that is, 0.7π n = 0.7π n + 2π k, where k is an integer.
Solving for N, we find that the period N of the cosine function is N = 2π / (0.7π), which simplifies to N = 2/0.7. Because N must be an integer for the signal to be periodic, and 2/0.7 is not an integer, we conclude that x(n) is not a periodic function. Therefore, x(n) = 0.6^n cos(0.7π n) is not periodic due to the combined effect of the non-periodic exponential term and the non-integer period of the cosine term.