Let x[n] be a discrete-time signal, and let yi[n] = x[2n] and y2[n] = { x[n/2], 0, n even n odd The signals y, [n] and y₂[n] respectively represent in some sense the speeded up and slowed down versions of x[n]. However, it should be noted that the discrete-time notions of speeded up and slowed down have subtle differences with respect to their continuous-time counterparts. Consider the following statements: (1) If x[n] is periodic, then y₁[n] is periodic. (2) If y₁[n] is periodic, then x[n] is periodic. (3) If x[n] is periodic, then y₂[n] is periodic. (4) If y₂[n] is periodic, then x[n] is periodic. For each of these statements, determine whether it is true, and if so, determine the relationship between the fundamental periods of the two signals considered in the statement. If the statement is not true, produce a counterexample to it.

To determine the truth of each statement and the relationship between the fundamental periods of the signals, let's analyze them one by one:

(1) If x[n] is periodic, then y₁[n] is periodic. To determine the truth of this statement, we need to consider the relationship between the fundamental periods of x[n] and y₁[n]. If x[n] is periodic with period N, then y₁[n] can be obtained by taking every other sample of x[n]. In this case, the fundamental period of y₁[n] will be N/2. Therefore, the statement is true, and the fundamental period of y₁[n] is half the fundamental period of x[n].

(2) If y₁[n] is periodic, then x[n] is periodic. To determine the truth of this statement, we need to consider the relationship between the fundamental periods of y₁[n] and x[n]. If y₁[n] is periodic with period M, then x[n] can be obtained by repeating the samples of y₁[n]. In this case, the fundamental period of x[n] will be 2M. Therefore, the statement is true, and the fundamental period of x[n] is twice the fundamental period of y₁[n].

(3) If x[n] is periodic, then y₂[n] is periodic. To determine the truth of this statement, we need to consider the relationship between the fundamental periods of x[n] and y₂[n]. If x[n] is periodic with period N, then y₂[n] can be obtained by taking every other sample of x[n/2]. In this case, the fundamental period of y₂[n] will be 2N. Therefore, the statement is true, and the fundamental period of y₂[n] is twice the fundamental period of x[n].

(4) If y₂[n] is periodic, then x[n] is periodic. To determine the truth of this statement, we need to consider the relationship between the fundamental periods of y₂[n] and x[n]. If y₂[n] is periodic with period M, then x[n] can be obtained by repeating the samples of y₂[n] at every even index and inserting zeros at every odd index. In this case, the fundamental period of x[n] will be 2M. Therefore, the statement is true, and the fundamental period of x[n] is twice the fundamental period of y₂[n].

In summary:

• Statement (1) is true, and the fundamental period of y₁[n] is half the fundamental period of x[n].
• Statement (2) is true, and the fundamental period of x[n] is twice the fundamental period of y₁[n].
• Statement (3) is true, and the fundamental period of y₂[n] is twice the fundamental period of x[n].
• Statement (4) is true, and the fundamental period of x[n] is twice the fundamental period of y₂[n].