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Fundamentals of Digital Signal Processing

IFFT and FFT

Explore how IFFT converts data symbols into time-domain signals for OFDM transmission and how FFT reverses the process to recover data on reception. Understand the role of modulation types like BPSK and QPSK in shaping signal properties, and see the mathematical operations that enable effective signal processing in digital communication systems.

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In the OFDM block diagram at the transmitter side, the next operation is an IDFT.

IFFT

As we have learned earlier, an IDFT (or DFT) is implemented in an efficient manner through an IFFT (or FFT) algorithm. Therefore, a block of NN data symbols is an input in an IFFT block that generates the corresponding time-domain signal.

The following code helps us understand this concept:

Python 3.10.4
import numpy as np
import matplotlib.pyplot as pl
figWidth = 20
figHeight = 10
# Generating the signal
fs = 1000 # sample rate
Ts = 1/fs # sample time
N = 64 # Block length and FFT size
# Binary PSK or ASK
constellation = np.array([-1, 1])
bits = np.random.randint(2, size=(N))
symbols = constellation[bits]
# ignore the pilots and null subcarriers
modulated_signal = np.sqrt(N)*np.fft.ifft(symbols, N)
# Plotting the signal
fig, axs = pl.subplots(2, figsize=(figWidth, figHeight))
axs[0].plot(range(N), np.real(modulated_signal), color='b', linewidth=2)
axs[1].plot(range(N), np.imag(modulated_signal), color='b', linewidth=2)
for ax in axs:
  ax.axhline(y=0, color='k', linewidth=1)
  ax.axvline(x=0, color='k', linewidth=1)
  ax.set_xlim(0,N)
  ax.set_xticks(np.linspace(0,N,9))
  ax.set_ylim(-2, 2)
  ax.set_yticks(np.arange(-2, 2+1, 1))
  ax.tick_params(labelsize=18)
  ax.set_xlabel('Time', fontsize=18)
  ax.grid()
axs[0].set_ylabel("Re $x(t)$", fontsize=18)
axs[1].set_ylabel("Im $x(t)$", fontsize=18)
pl.savefig('output/modulated-signal.png', bbox_inches='tight')

Remember that in the lesson on symmetry, we learned that the conjugate symmetric output signal (around sample 3232 ...