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\begin{document}
\title{Generating Real-Valued OFDM Signals\\with the Discrete Fourier Transform}
\author{Steve C. Thompson\footnote{With the Center for Wireless Communications,
  Department of Electrical and Computer Engineering, University of California at
San Diego, La Jolla, CA.  Feedback is welcome, contact Steve at sct@ucsd.edu.
If you find this useful please reference as: S. C. Thompson, ``Generating
Real-Valued OFDM Signals with the Discrete Fourier Transform.''
[Online]. Available:
http://zeidler.ucsd.edu/$\sim$sct/pubs/t5.pdf.}}
\date{\today\footnote{Originally written on July 15, 2005; last
      updated \today.}}
\maketitle

\begin{abstract}
For some applications, a real-valued OFDM signal is required.  This can be done
by taking a DFT of a conjugate symmetric vector.  The spectral efficiency of the
real-valued OFDM signal is the same as the spectral efficiency of the
complex-valued OFDM signal.
\end{abstract}

\section{Signal Description}
The baseband OFDM signal is typically written as
\begin{equation}
\label{eqn:xt_complex}
x(t)=\sum_{k=0}^{N-1}X_ke^{j2\pi kt/T}, \quad 0 \le t < T,
\end{equation}
where $N$ is the number of subcarriers, $\{X_k\}_{k=1}^{N-1}$ are the data
symbols and $T$ is the block period.
Sampling $x(t)$ at $N$ equally spaced intervals over $0\le t < T$ yields the
sequence,
\begin{equation}
x[n]=x(t)|_{t=nT/N}=\sum_{k=0}^{N-1}X_ke^{j2\pi kn/N}, \quad n=0, 1, \ldots,
  N-1,
\end{equation}
which is the inverse discrete Fourier transform (IDFT) of the vector
$\mathbf{X}=[X_0, X_1,\ldots,X_{N-1}]$.  The sequence is complex-valued in
general.  However it can be made real-valued by making $\mathbf{X}$ conjugate
symmetric:
\begin{equation}
X_{N/2+k} = X_{N/2-k}^\ast, \quad k=1,2,\ldots,N/2-1,
\end{equation}
and
\begin{equation}
X_0=X_{N/2}=0.
\end{equation}
The IDFT is then
\begin{equation}
\label{eqn:xn2}
\begin{split}
x[n]&=\sum_{k=1}^{N-1} X_k e^{j2\pi kn/N}\\
&=\sum_{k=1}^{N/2-1} X_{N/2-k} e^{j2\pi (N/2-k)n/N} + X_{N/2+k} e^{j2\pi
  (N/2+k)n/N}\\
&=\sum_{k=1}^{N/2-1} X_{N/2-k} e^{j2\pi (N/2-k)n/N} + X_{N/2-k}^\ast e^{j2\pi
  (N/2+k)n/N},
\end{split}
\end{equation}
$n=0,1,\ldots,N-1$. But since
\begin{equation}
\begin{split}
e^{j2\pi (N/2+k)n/N} &= e^{j2\pi (N/2+k)n/N}e^{-j2\pi Nn/N}\\
&= e^{j2\pi (-N/2+k)n/N}\\
&= e^{-j2\pi (N/2-k)n/N},
\end{split}
\end{equation}
\eqref{eqn:xn2} can be written as
\begin{equation}
x[n] =\sum_{k=1}^{N/2-1} X_{N/2-k} e^{j2\pi (N/2-k)n/N} + X_{N/2-k}^\ast e^{-j2\pi
  (N/2-k)n/N},
\end{equation}
$n=0,1,\ldots,N-1$. Using the identity $A+A^\ast=2\Re\{A\}$,
\begin{equation}
\begin{split}
x[n]&=2\Re\left\{\sum_{k=1}^{N/2-1} X_{N/2-k} e^{j2\pi (N/2-k)n/N}\right\}\\
&=2\Re\left\{\sum_{k=1}^{N/2-1} X_k e^{j2\pi kn/N}\right\}, \quad
n=0,1,\ldots,N-1.
\end{split}
\end{equation}
And since $\Re\{AB\}=\Re\{A\}\Re\{B\}-\Im\{A\}\Im\{B\}$,
\begin{equation}
x[n]=2\sum_{k=1}^{N/2-1} \Re\{X_k\} \cos(2\pi kn/N) - \Im\{X_k\} \sin(2\pi kn/N),
\end{equation}
$n=0,1,\ldots,N-1$.  Thus, $x[n]$ is real.  Passing the sequence through a D/A
converter yields the continuous-time real-valued OFDM signal:
\begin{equation}
\label{eqn:xreal}
x(t)=2\sum_{k=1}^{N/2-1} \Re\{X_k\} \cos(2\pi kt/T)
    - \Im\{X_k\} \sin(2\pi kt/T).
\end{equation}

Now, suppose the data symbols are derived from a $M$-QAM (quadrature-amplitude
modulation) constellation; that is,
\begin{equation}
X_k = \Re\{X_k\} + j\Im\{X_k\},
\end{equation}
where
\begin{equation}
\Re\{X_k\},\Im\{X_k\} \in \{\pm1, \pm3,\ldots, \pm (\sqrt{M}-1)\},  \quad
\text{for all $k$}.
\end{equation}
In other words, the real and imaginary components are derived from
$\sqrt{M}$-PAM (pulse-amplitude modulation) constellations.  Therefore,
processing $M$-QAM data with the IDFT, \eqref{eqn:xreal} is a real-valued
$\sqrt{M}$-PAM OFDM signal.

\section{Spectral Efficiency}
Complex-valued baseband signals are transmitted as bandpass signals, centered at
a carrier frequency $f_\text{c}$ Hz.  This is the case for the complex-valued
signal in \eqref{eqn:xt_complex}.  The transmitted signal is represented as
\begin{equation}
\label{eqn:s1}
s_1(t) = \Re\left\{x(t)e^{j2\pi f_\text{c}t}\right\}. 
\end{equation}
In the frequency domain, $x(t)$ is shifted to the right by $f_\text{c}$ Hz, and
the subcarriers are centered at $f_\text{c}, f_\text{c}+1/T,
f_\text{c} + 2/T, \ldots, f_\text{c}+(N-1)/T$ Hz.  The effective bandwidth of
the signal is therefore $N/T$ Hz.  Each data symbol represents $\log_2M$ bits
(i.e., they are assumed to be selected from a $M$-ary constellation), therefore
the spectral efficiency is
\begin{equation}
\mathcal{S}_1 = \frac{\text{Bits per second (b/s)}}{\text{Bandwidth (Hz)}} =
  \frac{N\log_2M/T}{N/T} = \log_2M \: \text{ b/s/Hz}.
\end{equation}

The real-valued OFDM signal in \eqref{eqn:xreal} has the same spectral
efficiency as the complex-valued signal, so long as it is transmitted at
baseband.  Transmitting the signal as-is, $\Re\{X_k\}$, $k=1,2,\ldots,(N/2)-1$,
modulate cosine subcarriers centered at $1/T, 2/T, \ldots, (N/2-1)/T$ Hz; and
likewise, $\Im\{X_k\}$, $k=1,2,\ldots,N/2-1$, modulate sine subcarriers at
the same frequencies.  The effective bandwidth of the
signal is $(N/2-1)/T$ Hz\footnote{Only the positive frequencies, $f\ge 0$,
count.}, and since the real and imaginary parts of $M$-QAM $X_k$
represent $0.5\log_2M$ bits,  the spectral efficiency of the real-valued OFDM
signal is
\begin{equation}
\mathcal{S}_2 = \frac{\text{Bits per second (b/s)}}{\text{Bandwidth (Hz)}} =
  \frac{2\times (N/2-1)0.5\log_2M/T}{(N/2-1)/T} = \log_2M \: \text{ b/s/Hz}.
\end{equation}
Therefore the spectral efficiency is the same as for the complex case.

However, the spectral efficiency of the real-valued signal is 1/2 that of the
complex-valued signal if the real-valued signal is translated up to a carrier
frequency.  This is due to the fact that the cosine and sine subcarriers in
\eqref{eqn:xreal} have a double sideband spectrum: i.e., $\cos(2\pi kt/T)$ [or
$\sin(2\pi kt/T)$] has a spectral components at $\pm k/T$ Hz.  [This isn't the
case for the complex-valued signal, which has complex sinusoids:
$\exp(j2\pi kt/T)$ has a spectral component only at $k/T$ Hz and is thus
considered single sideband.]  The carrier frequency is typically much larger
than the signal bandwidth, so the frequency translation brings all the negative
frequencies to the positive side: $-(N/2-1)/T+f_\text{c}\gg0$.
Consequently, the passband transmission of \eqref{eqn:xreal} results in a signal
with double the bandwidth and 1/2 the spectral efficiency.
\end{document}