ofdmtext.txt

Orthogonal-Frequency-Division Multiplexing (OFDM) is a method of modulating data which
is designed to exploit the fact that narrow-band signals can easily be equalized. This modulation
scheme is currently used in WiFi and LTE systems amongst others. OFDM systems utilize the
Inverse-Discrete-Fourier Trnasform (IDFT), and the Discrete-Fourier Transform (DFT).

II. OVERVIEW OF OFDM

When communications signals have a narrow bandwidth compared to the coherence bandwidth
of the channel (recall the coherence-bandwidth of a channel is loosely defined as the bandwidth
over which the frequency response of a channel can be approximated as constant), the channel
frequency response can be approximated as a single complex coefficient. In most modern systems
however, the bandwidths required to support the high data rate demands are too large for this
approximation to hold true. One possible way to get around this is to divide a wide bandwidth
into multiple subchannels, where the bandwidth of each subchannel is narrow enough for this
approximation to hold.
Recall that the transmitted signal for a pulse-amplitude modulation (PAM) system can be
written as follows


akp(t − kT) (1)
where T is the symbol period, p(t) is the pulse (e.g. a rectangular pulse of width T), and ak is
the k-th symbol.
Let us now extend this idea to a system where there are multiple PAM/QAM signals modulated

by complex exponentials of different frequencies. Note that these frequencies are called sub-
carrier frequencies. In almost all implementations, the modulation to subcarrier frequencies is

done digitally using an IDFT. The modulation to the carrier frequency is typically done in
the analog domain. For instance, in Wifi systems, the subcarriers are spaced by approximately

250kHz and occupy a range of approximately 16MHz. The carrier frequency on the other hand
is approximately 2.4GHz. In the following, we shall describe the basic idea of OFDM using
continuous-time signals in order to develop intuition. We shall follow this with a Discrete-time
development.
Suppose that you have L PAM/QAM signals, each modulated by a complex exponential, which
move the PAM signals into different frequencies. Let’s use akl to represent the PAM/QAM signal
amplitude on the l-th subcarrier on the k-th symbol. Again akl could take values of ±1 or ±1±j,
for instance. For ease of exposition, let us assume that akl can take values of ±1 here. The new
transmitted signal is then


where f is the spacing between the subcarriers.
Consider the case where the pulse p(t) is a rectangle of width T with unit height. Such a
pulse and its Fourier transform are shown in Figure 1.

0 500 1000 1500 2000 2500 3000 3500 4000 4500
0
0.2
0.4
0.6
0.8
1

0 500 1000 1500 2000 2500 3000 3500 4000 4500
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
S Fig. 1. Rectangular Pulse and its Fourier Transform  G

Suppose that f

. Then, the Fourier transform of the zeroth symbol i.e. the term inside

the first summation of (2), with k = 0 is given as follows:

The term in the second parenthesis on the RHS of the previous equation is a sequence of L
impulses separated by 1
T
. Notice from Figure 1, that the rectangular pulse transforms into a sinc

function with zero crossings every 1
T
. Thus X0(f) appears as shown in Figure 2. Note that for

clarity, we have made all the a0l

terms positive here, but they could be negative or complex.

Fig. 2. Fourier transform of a single OFDM Symbol

Observe that at the peak of each main-lobe, each sinc function aligns with the zeros of every
other sinc function – this is the ”orthogonal” part of OFDM, since each subcarrier is orthogonal
to all other subcarriers if sampled at the peak of the main-lobe. If 1
T
is small enough, then the
signal from each sub-carrier can be considered narrow-band, since the mainlobe (where most of
the energy in a sinc lies), of the sinc function associated with the signal on each subcarrier is
narrow. Additionally, observe that the summation in (3), has a form very similar to that of the
Inverse Discrete-Fourier Transform (IDFT). In real-world implementations of OFDM systems,
which are done digitally, the IDFT is used to generate time domain waveforms through the IFFT

operation, which is an efficient way to compute the IDFT. Therefore, one needs to understand
the DFT and its inverse in order to be able to understand implementations of OFDM.

III. DISCRETE FOURIER TRANSFORM

Consider a discrete-time signal x[m] which is zero when m  0 and M  L. We shall refer
to a signal such as this as a length-L signal. The L-point discrete Fourier transform (DFT) of
this signal is denoted by Xk and is defined as follows:

Xk =
X
L−1
l=0
w[l]e

L (5)

Recall that the discrete-time Fourier transform of x[m] is defined as follows


where the second equality is due to the fact that x[m] is zero for m  0 and m  L − 1. By
comparing (6) and (5), we see that the DFT can be interpreted as a sampled version of the
DTFT, where the samples are taken every 2π
L
, because Xk is equal to X

The L-point DFT of a signal x[m] is periodic with period L. In other words, if L is even, then


The inverse operation is the L-point IDFT is defined as follows,

x[m] = 1
L
X
L−1
k=0
Xke
2πj mk
L (9)

which recovers x[m] from Xk for m = 0, 1, 2,L − 1.
Just like the continuous-time and discrete-time Fourier transform have a number of useful
properties, the DFT does too. Recall that for the DTFT, multiplication in the time domain
becomes circular (or periodic) convolution in the frequency domain. Circular convolution is
equivalent to periodically replicating one of the two functions to be circularly convolved, and
then performing regular convolution with the resulting functions.

When two discrete-time signals are circularly convolved with period L, their DFTs are mul-
tiplied. This property is central to OFDM. Assume the following

• Define the L-point periodic extension of x[m] as x  Where

Equation (11) is the main idea behind OFDM. The data are encoded in the frequency domain.
e.g. Xk = ±1 to represent 1 bit per value of k, or Xk = ±1 ± j, for 2 bits per value of k.
Note that this corresponds to BPSK and QPSK/4QAM. The effect of the channel becomes a
multiplication in the frequency domain. Also note that higher order constellations are possible
as well.