How to convert discrete-time QAM pulses to a continuous-time signal using the IFFT (for OFDM)?

Question

So I've run into some trouble in trying to plot the PSD of an OFDM signal. Specifically, I cant see how we go from the discrete domain to the continuous in time domain.

I have a time vector, which is sample at a rate of 10Khz, over a duration of 100s

t = 0:1/1e4:100

I then generate a signal by low-pass filtering some noise at a maximum frequency

f_max = 50;                                              % maximum freq of message signal
x     = randn(size(t));                                  % Generate some noise
x     = filter2(fir1(10*ceil(f_s/f_max),2*f_max/f_s),x); % Low pass filter the noise
x     = x-mean(x);                                       % Remove any DC offset
x     = x/sqrt(mean(x.^2));                              % Normalise the signal

This is my 'continuous in time' signal which is then sampled at a rate of 8Hz to give by discrete in time signal

f_sample          = 8;
number_of_symbols = f_sample*t(end);
sample_indices    = round((0:number_of_symbols -1)*(length(x)-1)/(number_of_symbols -1))+1;
xSampled          = x(sample_indices);

This is then quantised and QAM modulated in the discrete domain

xQuant = quantiz(xSampled,partition,codebook); % quantisation
xMod   = qammod(xQuant,M);                     % QAM modulation

This is where my problem comes. The next step for OFDM would then be to convert this frequency domain representation to a continuous in time representation using an IFFT. But lets say I have FFT_Length= 64, I'd split my QAM modulated signal into 64 vectors of length number_of_symbols/64, and this would give me 64 vectors out, each of length number_of_symbols/64, one for each subcarrier - we have remained in the discrete domain?

If this was a single carrier scheme this is when id apply pulse shaping to the QAM symbols and upsample the signal frequency to the original 10khz, however pulse shaping is not used in OFDM. I was thinking we could either upsample the QAM symbols prior to the OFDM modulation by padding with 0's for the remainder of the symbol period, or do the same but pad with the QAM value, however in both cases the resulting FFT is like a square wave - stuck at a one value for a symbol period, then switches immediately to another.

So my question:

how do I convert a series of QAM pulses to a continuous time signal using the IFFT (in OFDM)

Many thanks :-)

Answer 1

You simply take the IFFT of the same length vector to get the time domain signal. Note that relative to each sub-carrier, you have multiple samples per QAM "symbol". If you wish to interpolate to higher samples, the FFT should be zero padded in the center with the value in the shared bin at index N/2 split in half. For a simple 6 bin example note that bin 0 is the DC bin, bin 1 and 2 are the positive frequency bins, bin 3 is the Nyquist bin and bins 4 and 5 are the negative frequency bins as described below. This approach will provide "perfect" interpolation in that it will maintain all subcarrier orthogonality. Traditional interpolation with zero insert and filtering of the time domain signal would likely degrade EVM due to the inter-carrier interference the filter would introduce (but that would not be of any concern for purposes of creating a power spectral density).

Bin Index          0     1     2    3    4    5
As pos/neg         0     1     2    N   -2   -1
Samples            x[0] x[1] x[2] x[3] x[4] x[5] 

Here interpolation to a higher sampling rate would properly done with zero inserts and splitting the Nyquist bin (if non-zero already) as follows (I show a 3 zero insert but could just as well be 6 zeros to exactly interpolate by 2 or more zeros for higher interpolation factors:

Bin Index          0     1     2    3    4    5    6      7     8    9
As pos/neg         0     1     2    3    4    N   -4     -3    -2   -1
Samples            x[0] x[1] x[2] x[3]/2 0    0    0    x[3]/2 x[4] x[5] 

If there were an odd number of samples, then the Nyquist bin would not be included and in this case there would simply be the DC bin 0 followed by $(N-1)/2$ "positive frequency axis" samples, followed by the zero inserts, followed by the final $(N-1)/2 "negative frequency axis" samples (no splitting of a Nyquist bin).

The IFFT output is the OFDM signal as transmitted in the time domain. To observe the power spectral density of the actual signal as transmitted, an FFT should be done over a duration of multiple OFDM symbols, each with the cyclic prefix added, which will then provide an averaged power spectral density. The other option is to use the Welch method or other similar periodograms which serve to reduce the noise in the spectrum at the expense of resolution bandwidth (the spectrum is created multiple times with less samples in each and averaged). Below is a Python function I wrote that can be easily ported to MATLAB for simple spectrum plots which properly windows the waveform and compensates for resolution bandwidth in the FFT for reporting the levels of sinusoidal tones (so over-reports by a couple dB any spread signals that occupy multiple tones given the resolution bandwidth occupies more than one bit):

import numpy as np
import scipy.fftpack as fft
import scipy.signal as sig
import matplotlib.pyplot as plt

def winfft(vec, fs=1., full_scale=1., beta=12., n_samp=None):
    '''
    vec: input signal
    fs: sampling rate (default 1)
    full_scale: normalization (default 1)
    beta: Kaiser window beta (default 12)
    n_samp: number of samples (default vec length)
    '''

    input_len = np.shape(vec)[0]
    if n_samp == None:
        n_samp = input_len
    elif n_samp < input_len:
        vec = vec[:n_samp]
    
    # only window data, not zero padded signal:
    win_len = input_len if n_samp > input_len else n_samp 

    win = sig.kaiser(win_len, beta)
    winscale = np.sum(win)
    win_data = win * vec 

    # scaled fft
    fout = 1./(winscale*full_scale) * fft.fft(win_data, n_samp)

    fout = fft.fftshift(fout)

    # create frequency axis
    faxis = np.array(range(n_samp)) * fs / n_samp - fs / 2

    return faxis, fout

The following is an example OFDM spectrum with 256 QAM subcarriers plotted with this function after doing an interpolate by 2 using the zero insert method above: