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if any discrete time signal is multipled by unit step then is it periodic

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if any discrete time signal is multipled by unit step then is it periodic

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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.

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