How to visualize Cyclic Prefix in OFDM system

[engineer_eda]

Hi,

Let’s suppose that after IFFT we have N-subcarriers which are generating 1 OFDM symbol.

X0, X1, X2,...,X{N-1}

And after IFFT we want to add Cyclic Prefix at the beginning of the OFDM symbol by copying it from the end of the OFDM symbol.

Does it mean we actually adding the Cyclic Prefix to all of the subcarriers in the OFDM symbol (N sinusoids after IFFT)?

Thanks in advance!

 

[engineer_eda]

I would appreciate if anyone help me on this query.

 

[rcdaniels]

I'm somewhat confused by your question.

For clarity, I'm going to go over some basics which you probably already know in an attempt to find your gap in understanding.

You must first define the OFDM symbol in the frequency domain on N tones or 'subcarriers'. After the N-point IDFT, in general, each of the N time-domain samples is mapped from all subcarriers. This is easy to see by looking at the (I)DFT transformation math.

After the transformation, we can add the cyclic prefix as you suggested.

The cyclic prefix, just like the original OFDM symbol (sans prefix), contains data for all subcarriers...but this data is incomplete. For example (and you can test this on your own), if you take the N-point FFT of the cyclic prefix alone (zero padded to N-length), you may see a distorted version of the original OFDM subcarrier profile (although this will depend on how large the cyclic prefix is relative to the OFDM symbol).

Intuitively, this is because the zero padding can be thought of as a time domain window, which is equivalent to convolving with a sinc function in the frequency domain.

So in short, yes, the cyclic prefix is a representation of all subcarriers. However, it is not ADDING to subcarriers in the OFDM symbol like you say above. A subcarrier is defined only over a single N-length block of time-domain samples for which the DFT transformation is taken. If the cyclic prefix is not within the block for which the DFT is taken (which is the nominal case), it does not contribute to the subcarriers (although the multipath excess delay taps will add part of the cyclic prefix to the OFDM symbol).

 

[permute]

The fast explanation is that the FFT is a cyclic convolution. For example, people sometimes try to filter data by taking an N-point FFT adjusting coefficients, and then doing the inverse FFT. This would be nice if it always worked. But even a simple problem, like modeling a sample delay, is impossible. The FFT takes in N samples, and outputs N samples, while true convolution takes in N, and outputs N+k. (k being the FIR filter length).

The cyclic convolution of a unit delay would have placed the last sample at the front of the output (eg, FFT, apply "delay" coefficients, and then do inverse FFT). This means the FFT + EQ + iFFT will let you model a filter/channel only when the last samples in the time domain smear across the first ones.

So basically, the CP attempts this. You can output N+k samples, pass it through the channel. There is some convolution, and thus energy in N+k+k samples*. samples k to N+k are ideal though. the first k samples have smeared into this window, and the last original samples (N-k to N) smear into samples N to N+k (which are the CP and copies of the first k samples). This looks exactly like the cyclic convolution that the FFT is so good with. (the first k samples have whatever the last symbol was smeared into them, and samples N+k to N+2k smear into the next symbol)

* assuming N + cyclic prefix + channel

 

[switychandra@gmail.com]

i am unable to understand your question