Digital Signal Processing Basics
Which of the following gives the gain magnitudes at DC and at the Nyquist frequency of the filter y[n] = x[n] − x[n − 1] (the difference from the previous input)?
- A0 at DC and 2 at the Nyquist frequency
- B2 at DC and 0 at the Nyquist frequency
- C1 at DC and 1 at the Nyquist frequency
Show answer and explanation
Answer: A. 0 at DC and 2 at the Nyquist frequency
The transfer function is H(z) = 1 − z⁻¹; DC corresponds to z = 1 and the Nyquist frequency to z = −1. At DC, 1 − 1 = 0, so a constant value is completely removed, and at the Nyquist frequency the gain is 1 − (−1) = 2. This is a high-pass characteristic that blocks low frequencies and passes high ones, with a zero at z = 1. Changing it to y[n] = x[n] + x[n − 1] gives the opposite, a low-pass characteristic with a gain of 2 at DC and 0 at the Nyquist frequency.
- A. Correct. At z = 1, 1 − 1 = 0, and at z = −1, 1 + 1 = 2.
- B. This is the characteristic of y[n] = x[n] + x[n − 1] (the sum of neighbors).
- C. Taking the difference cancels DC to 0, and the gain at the Nyquist frequency is 2.