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Digital Signal Processing Basics

A unit step (x[n] = 1 for n ≥ 0) is applied from n = 0 to the system described by the difference equation y[n] = x[n] + 0.5y[n − 1]. With y[−1] = 0, which of the following is y[2]?

  1. A1.75
  2. B1.875
  3. C2
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Answer: A. 1.75

Step by step, y[0] = 1 + 0.5 × 0 = 1, y[1] = 1 + 0.5 × 1 = 1.5 and y[2] = 1 + 0.5 × 1.5 = 1.75. Continuing gives y[n] = 2 − 0.5ⁿ, which approaches the final value 1/(1 − 0.5) = 2. Because past outputs are used (there is feedback), this is an IIR filter whose impulse response 0.5ⁿ continues forever.

  • A. Correct. The output goes 1 → 1.5 → 1.75.
  • B. This is y[3]. y[2] is two steps after y[0].
  • C. This is the final value 1/(1 − 0.5) as n → ∞, which has not been reached at y[2].
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