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For OFDM, will the following relation always hold:

subcarrier_spacing * sampling_time = 1/N

N: number of subcarriers which is the size of IFFT/FFT

This will imply that sampling frequency is exactly equal to the bandwidth of the OFDM.

Thanks.

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2 Answers 2

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subcarrier_spacing * sampling_time = 1/N

doing minimal rearranging:

subcarrier_spacing · sampling_time · N = 1
subcarrier_spacing · N = 1/sampling_time
subcarrier_spacing · N = f_sample

Yes, that is the very definition of "subcarrier spacing". Remember that the DFT is a square transform, so the vectors going in have to be exactly the same size as those coming out.

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If the OP is referring to the sampling frequency of the time domain waveform in the OFDM transmitter (after the IFFT), then these relationships will not always hold. There are many practical reasons in an OFDM implementation to have a higher sampling frequency than the OFDM bandwidth. This higher sampling frequency could be implemented in the FFT mapping by using more bins that are filled with zeros outside of the designated bandwidth and sub-carrier assignments, or interpolated after computing an inverse FFT to create the time domain waveform to be transmitted.

The reasons for this are:

  • We must have transition bandwidth prior to the Digital to Analog converter to reject the images in the analog reconstruction of the waveform.

  • Significant excess bandwidth in the digital domain is needed for many approaches to power amplifier linearization (predistortion).

  • Excess bandwidth can also be used for crest factor reduction techniques.

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  • $\begingroup$ Thanks. This is really helpful. Could you please provide pointers to how excess bandwidth helps in the three points you have mentioned? $\endgroup$ Commented Jun 14, 2023 at 11:26
  • $\begingroup$ Regarding the first point, see this post and learn further about images from D/A conversion and the job of the filter after the D/A, and how excess bandwidth simplifies that filter: dsp.stackexchange.com/questions/70212/d-a-converter-with/… $\endgroup$ Commented Jun 14, 2023 at 11:33
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    $\begingroup$ A simplified way to understand the second point, power amplifier (and other non-linearities) produce harmonics. Harmonics appear at higher frequencies (dominantly 3rd order and next after that 5th order which means 3 to 5x the bandwidth!). To properly reproduce a waveform that would be opposite that distortion (pre-distortion) we need to have the bandwidth to create those opposite harmonics that would cancel. $\endgroup$ Commented Jun 14, 2023 at 11:34
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    $\begingroup$ and the third point "crest factor reduction" are techniques to reduce the peak to average ratio in OFDM and other waveforms. This helps significantly to increase efficiency in power amplifiers and thus reduce DC power consumption. We can use excess bandwidth in the digital domain to move energy that would otherwise be at peaks in the waveform to out of band, when we have room there to put energy compared to out-of-band emission limits. These are all motivating reasons to increase the sampling rate. Another not mentioned is combining multiple carriers digitally. $\endgroup$ Commented Jun 14, 2023 at 11:39

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