# Problem

Given $$X[k] = \sum_{n=0}^{N-1} x[n]e^{-j2\pi kn/N}, k = 0, ..., N-1$$

What are the units on the x-axis and y-axis? Note that for the x-axis there are two answers.

# Attempted solution

My first thought is frequency is on the x-axis and amplitude is on the y-axis. However, it confuses me that there are two answers for the x-axis. What could be the other one?

Actually there are more than two answers for the DFT’s x-axis. I’ve seen spectral plots where an N-point DFT’s positive-frequency axis range is labeled:

• Zero -to- N/2 (Frequency axis value is measured in DFT bins)

• Zero -to- Fs/2 (Frequency axis value is measured in Hz where Fs is the data sample rate measured in Hz)

• Zero -to- 1/2 (Frequency axis value is multiplied by Fs)

• Zero -to- 1 (Frequency axis value is normalized to Fs/2 Hz. MATLAB’s ‘freqz()’ command uses this convention.)

• Zero -to- pi (Frequency axis value is measured in radians/sample)

As for the y-axis, be careful about your terminology! An “amplitude” value can be a positive or negative number. A “magnitude” value can only be a positive number. It takes two curves to plot a DFT result’s amplitude versus frequency. One curve for the DFT’s real part and another curve for the DFT’s imaginary part. Much more common is to plot a DFT’s result’s magnitude versus frequency which only requires one curve. It’s fairly common for people to use the phrase “DFT amplitude” when they really mean “DFT magnitude.”

• And as Richard well knows when normalized to Fs (spanning zero to 1/2) it is the frequency axis value measured in cycles/sample. (For comparison to the case above "Frequency axis value is measured in radians/sample") Commented Jun 15, 2022 at 1:14