Never be afraid to get dirty, but be sufficiently sure-footed to avoid the abyss of contamination.

The Question

The Company asked the Question. I stumbled but eventually answered the Question. Or so I thought. While at dinner with a friend tonight, he asked me about the Question. When I told it to him, I initially tried guiding him to the Answer, but as he moved forward, I realized something. I had forgotten to take into account an equation in my solution, which meant my solution was wrong. I worked it out and figured out the Answer a few minutes later.

Here’s the Question:

Please design a discrete time LTI filter with the following three properties:

  1. For any input that is periodic with period 3, the output should be constant.
  2. If the input x[n] = n^2 (i.e. x[n] is a parabola), then the output y[n] = n^2.
  3. The impulse response h[n] of the filter should be even symmetric.
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