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i have some doubts with the implementation of the IDFT in OFDM systems. The question concerns the expression of the IDFT of the OFDM signal. During the symbols period \$T_s\$ we have the following base-band OFDM signal: $$x(t)=\sum_{m=0}^{N-1}X_m e^{j2 \pi \frac{m}{T_s} t},\ \ \ 0\leq t \leq T_s$$ If we sample the signal at time instants \$t = k\,Ts/N\$, we have: $$x_k=x \left(k \frac{T_s}{N}\right) =\sum_{m=0}^{N-1}X_m e^{j\frac{2 \pi }{N}m\,k}\ \ \ \ \text{with }k=0,1,\dotso,N-1$$ Now, except for a multiplying constant (\$1/N\$), the above formula is the equation of an N-point inverse discrete Fourier transform (IDFT).

In the OFDM Transmitter and Receiver we implement the respectively the IDFT and DFT to convert the symbol \$\{X_0,X_1,\dotso,X_{N-1}\}\$ to the time domain symbol sequence \$\{x_0,x_1,\dotso,x_{N-1}\}\$ and viceversa.

What I was wondering is if the \$1/N\$ factor that is absent in the previous expression of the IDFT that I have reported is due only to a convention of normalization of the power of the transmitted signal. Furthermore, is the absence of this factor an error?

Null
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Marco Toni
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1 Answers1

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What I was wondering is if the 1/N factor that is absent in the previous expression of the IDFT that I have reported is due only to a convention of normalization of the power of the transmitted signal. Furthermore, is the absence of this factor an error?

People can get very lazy about normalisation of FFTs and IFFTs.

It's an error in a maths exam answer. It has little effect when we are talking about a system that will have other amplitude normalisations applied to it in many other places, or when the measurements it makes are ratios, like SNR or EVM.

Personally, the first thing I do when I come to a new implementation of an FFT or IFFT is run a test tone through it and see whether the author has applied 1, 1/N, 1/sqrt(N), or some other normalisation to it. I'm often surprised.

Neil_UK
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  • So if we consider signal without normlized power and if we keep the same number of point N, the presence of the factor 1/N in the IDFT (with 1 in the DFT) or the factor 1/sqrt(N) in the IDFT and DFT is negligible, right? – Marco Toni Jun 14 '21 at 10:53
  • as Neil says, depends. If your system afterwards doesn't care about absolute amplitude, then, well, *any* posiitve scaling factor is negligible. If the thing afterwards is a measurement device for observing physical powers (and not just, say, SNRs, which are unaffected by scaling), then it matters. OFDM receivers don't measure absolute powers. They measure symbols, and maybe error vector magnitudes, typically. – Marcus Müller Jun 14 '21 at 11:27
  • Ok it's all clear... so since in this case i'm not interested about the absolute power and since i'm only explaining the idea behind the OFDM i can neglect the scaling factor – Marco Toni Jun 14 '21 at 14:44