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in many texts about modulators I have seen there is a component called "Serial to Parallel Converter". For instance, let's consider this scheme and let's focus on the second block (reference):

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The text about that block says:

The input serial data stream is formatted into the word size required for transmission, e.g. 2 bits/word for QPSK, and shifted into a parallel format. The data is then transmitted in parallel by assigning each data word to one carrier in the transmission.

Somewhere I have read also that this device converts a signal from serial to parallel by halving the transmission rate.

But,

  1. I do not understand what serial to parallel conversion exactly means, and if I have to see it from and electronic and circuital point of view (from a signal between a wire and GND we get two signals between two wires and GND) or from a signal theory point of view (but I do not know how to see it).
  2. Which is the math relationship of the output signals (between them and between them and the input signal)?
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So, I myself have taught based on material that uses that scheme, "P/S" and "S/P" after and before the transforms.

Personally, it's nonsense.

What the author tries to say is:

The IFFT is a mapping of sample vectors to vectors. So, you need a vector as input, not a stream of samples.

What they instead say is:

We use the terminology from very basic digital logic to deal with complex values.

That's not only a bit awkward, if you asked me, but also inaccurate:

  1. modern FFT implementations in hardware actually do take in stream data, so that operation isn't there in practical hardware implementations
  2. in software implementations, you basically never even deal with samples coming in serially – they always appear en block, in some memory location, so nothing's "serial" to even begin with
  3. S/P nor P/S even say that you should be fully accumulating one vector of $N$ samples, then move on to the next vector, and have zero overlap between these – when applied to FIFOs (which is where the term comes from), that's usually not how they work.

So, simply think of S/P as "get $N$ samples, present these $N$ to the next block as vector".

And P/S is "take this $N$ long vector, and give one sample after the other".

So, these blocks do exactly nothing to your signal - it's just a reinterpretation, if you will, between things that are logically "one after the other", and things that are logically "a vector of $N$ values".

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