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Hello I have the following question in regards to what the units should be for the y axis for a continuous time Fourier transform and that for a discrete time Fourier transform.

In my understanding if my time domain signal is a a voltage over time signal when I take the continuous Fourier transform the x axis should be Hz and the y axis should be V*s or V/Hz.

In the other scenario for the discretization of a signal if I go ahead and take the discrete time Fourier transform the y axis should be just V and the x axis should be Hz.

Could anyone let me know if these are correct assumptions?

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    $\begingroup$ To relate the two, you need to express the Continuous Fourier Transform as a Riemann sum. $\endgroup$ Mar 15 at 5:49
  • $\begingroup$ I see, are my statements incorrect though in regards to the units for the continuous and discrete fourier transforms? $\endgroup$ Mar 15 at 6:36
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    $\begingroup$ No, they're not incorrect. Well, with the DFT, there are no x-axis units, they are integer indices. But to connect the DFT to the CFT and understand how the units translate, one way to understand that is to express the CFT as a Riemann sum. The "$\mathrm{d}t$" becomes "$\Delta t$" which is $T = \frac{1}{f_\mathrm{s}}$ and $f_\mathrm{s}$ is the sample rate. $\endgroup$ Mar 15 at 16:02

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That is by and large correct. The continuous Fourier Transform adds a $\frac{1}{\text{Hz}}$ to the original units. The discrete Fourier Transform has the same units.

Some of this a matter of preference. For example the frequency axis can be displayed linear or logarithmic units (the latter one omitting DC). For the DFT you can show either actual Hz, normalize to the sampling frequency or Nyquist frequency, normalized radial frequency.

The y-axis is also frequently displayed in logarithmic units using the decibels (dB). This requires a choice of proper reference for 0 dB.

That's why proper labelling of axes is so important: There is different ways to visualize the information without good labelling it can be very misleading.

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