Assume the number of points N is large enough. In a DFT with a rectangular window, a sine wave lies exactly halfway between two adjacent bins. Which of the following is closest to the drop in the amplitude read at the nearest bin, compared with a sine wave exactly at a bin center?

Answer: B. About 3.9 dB

With a rectangular window, a sine wave offset by δ bins from a bin center is read with sin(πδ)/(πδ) times its amplitude, and for δ = 0.5 this is sin(π/2)/(π/2) = 2/π ≈ 0.637. 20 log₁₀(0.637) ≈ −3.9 dB, so the amplitude can be read up to about 36% too low. This drop is called scalloping loss, and with a Hann window it is smaller, about 1.4 dB. For measurements that need accurate amplitudes, flat-top windows, whose main lobe has a flat top, are also used.

• A. This is the drop with a Hann window (about 1.42 dB). With a rectangular window the drop is larger.
• B. Correct. 2/π ≈ 0.637, about −3.9 dB.
• C. This is the value when the amplitude halves (about −6.0 dB). Halfway between bins the amplitude is 2/π ≈ 0.64 times.