Question 9.5: The signal z[n] = v[n] + w[n] + y[n] is the sum of three per......
The signal z[n] = v[n] + w[n] + y[n] is the sum of three periodic signals v[n], w[n] and y[n] of fundamental periods N_1 = 2, N_2 = 3\text{ and }N_3 = 4, respectively. Determine if z[n] is periodic, and if so give its fundamental period.
Let x[n] = v[n] + w[n], so that z[n] = x[n] + y[n]. The signal x[n] is periodic since N_2/N_1 = 3/2 is a rational number with non-divisible factors, and its fundamental period is N_4 = 3N_1 = 2N_2 = 6. The signal z[n] is also periodic since
{\frac{N_{4}}{N_{3}}}={\frac{6}{4}}={\frac{3}{2}}
is rational with non-divisible factors. The fundamental period of z[n] is N = 2N_4 = 3N_3 = 12. Thus z[n] is periodic of fundamental period 12, indeed
z[n + 12] = v[n + 6N_1] + w[n + 4N_2] + y[n + 3N_3] = v[n] + w[n] + y[n] = z[n].