Question

Consider the two systems

(a) For M=2 and L=3, and any arbitrary x[n], will y_A[n]=y_B[n]? If your answer is yes, justify your answer. If your answer is no, clearly explain or give a counterexample


x[n] ʷA[ⁿ] ʸ A[ⁿ]
System A : >---------------↓M----------------L↑----------->"

← x[n] ʷB[ⁿ] ʸ A[ⁿ]
System B : >---------------L↑-----------------↓M------------>"

Answer 1

For the given systems, y_A[n] and y_B[n] will not be equal. In system A, the input signal x[n] is downscaled by a factor of 2 and then upscaled by a factor of 3. In system B, the input signal x[n] is upscaled by a factor of 3 and then downscaled by a factor of 2.

Step-by-step explanation:

For the given systems, y_A[n] and y_B[n] will not be equal.

In system A, the input signal x[n] is downscaled by a factor of M=2 and then upscaled by a factor of L=3. In system B, the input signal x[n] is upscaled by a factor of L=3 and then downscaled by a factor of M=2.

Let's consider an arbitrary input signal x[n] = [1, 2, 3].

In system A, y_A[n] will be obtained as follows:

1. Downscale x[n] by M=2: [1, 2, 3] → [1, 3]
2. Upscale the downscaled signal by L=3: [1, 3] → [1, 1, 1, 3, 3, 3]

In system B, y_B[n] will be obtained as follows:

1. Upscale x[n] by L=3: [1, 2, 3] → [1, 1, 1, 2, 2, 2, 3, 3, 3]
2. Downscale the upscaled signal by M=2: [1, 1, 1, 2, 2, 2, 3, 3, 3] → [1, 2, 3]

Therefore, y_A[n] = [1, 1, 1, 3, 3, 3] and y_B[n] = [1, 2, 3]. Since they are not equal, the answer is no, y_A[n] and y_B[n] will not be equal for any arbitrary input signal x[n].