Orthogonality of OFDM

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OFDM, known as Orthogonal Frequency Division Multiplexing, is a digital modulation technique that divides a wideband signal into several narrowband signals. By doing so, it elongates the symbol duration of each narrowband signal compared to the original wideband signal, effectively minimizing the impact of time dispersion caused by multipath delay spread.

OFDM is categorized as a form of multicarrier modulation (MCM), where multiple user symbols are transmitted simultaneously through distinct subcarriers having overlapping frequency bands, ensuring they remain orthogonal to each other.

OFDM implements the same number of channels as the traditional Frequency Division Multiplexing (FDM). Since the channels (subcarriers) are arranged in overlapping manner, OFDM significantly reduces the bandwidth requirement.

OFDM equation

Consider an OFDM system that transmits a user symbol stream \(s_i\) (rate \(R_u\)) over a set of \(N\) subcarriers. Therefore, the symbol rate of each subcarrier is \(R_s = \frac{R_u}{N}\) and the symbol duration is \(T_s = \frac{N}{R_u}\).

The incoming symbol stream is split into \(N\) symbols streams and each of the symbol stream is multiplied by a function \(\Phi_k\) taken from a family of orthonormal functions \(\Phi_k, k \in \left\{0,1, \cdots, N-1 \right\}\)

In OFDM, these orthogonormal functions are complex exponentials

\[\Phi_k (t) = \begin{cases} e^{j 2 \pi f_k t}, & \quad for \; t \in \left[ 0, T_s\right] \\ 0, & \quad otherwise \end{cases} \quad \quad \quad (1) \]

For simplicity lets assume BPSK modulation for the user symbol \(s_i \in \left\{-1,1 \right\} \) and \(g_i\) is the individual gain of each subchannels. The OFDM symbol is formed by multiplexing the symbols on each subchannels and combining them.

\[S (t) =\frac{1}{N} \sum_{k=0}^{N-1} s_k \cdot g_k \cdot \Phi_k(t) \quad \quad \quad (2)\]

The individual subcarriers are

\[s_n (t) = s_k \cdot g_k \cdot e^{j 2 \pi f_k t} \quad \quad \quad (3)\]

For a consecutive stream of input symbols \(m = 0,1, \cdots\) the OFDM equation is given by

\[S(t) = \sum_{m = 0 }^{ \infty} \left[\frac{1}{N} \sum_{k=0}^{N-1} s_{k,m} \cdot g_{k,m} \cdot \Phi_{k}(t – m T_s) \right] \quad \quad \quad (4)\]

With \(g_{k,m} = 1\), the OFDM equation is given by

\[S(t) = \sum_{m = 0 }^{ \infty} \left[\frac{1}{N} \sum_{k=0}^{N-1} s_{k,m} \cdot e^{j 2 \pi f_k \left(t – m T_s \right)} \right] \quad \quad \quad (5)\]

Orthogonality

The functions \(\Phi\) by which the symbols on the subcarriers are multiplied are orthonormal over the symbol period \(T_s\). That is

\[ \left< \Phi_p (t), \Phi_q (t) \right> = \frac{1}{T_s} \int_{0}^{Ts} \Phi_p (t) \cdot \Phi^*_q (t) dt = \delta_{p,q} \quad \quad \quad (6) \]

where, \(\delta_{p,q}\) is the Kronecker delta given by

\[\delta_{p,q} = \begin{cases} 1, & \quad p=q \\ 0, & \quad otherwise \end{cases} \]

The right hand side of equation (5) will be equal to 0 (satisfying orthogonality) if and only if \(2 \pi \left(f_p−f_q \right)T_s=2 \pi k\) where \(k\) is a non-zero integer. This implies that the distance between the two subcarriers, for them to be orthogonal, must be

\[\Delta f = f_p – f_q = \frac{k}{T_s} \quad \quad (7)\]

Hence, the smallest distance between two subcarriers, for them to be orthogonal, must be

\[ \Delta f = \frac{1}{T_s} \quad \quad (8)\]

This implies that each subcarrier frequency experiences \(k\) additional full cycles per symbol period compared to the previous carrier. For example, in Figure 1 that plots the real and imaginary parts of three OFDM subcarriers (with \(k=1\)), each successive subcarrier contain additional full cycle per symbol period compared to the previous carrier.

orthogonal subcarriers of OFDM generated using 5G 3GPP air interface parameters
Figure 1: Three orthogonal subcarriers of OFDM

With \(N\) subcarriers, the total bandwidth occupied by one OFDM symbol will be \(B \approx N \cdot \Delta f \; (Hz) \)

OFDM spectrum illustrating 12 subcarriers
Figure 2: OFDM spectrum illustrating 12 subcarriers

Benefits of orthogonality

Orthogonality ensures that each subcarrier’s frequency is precisely spaced and aligned with the others. This property prevents interference between subcarriers, even in a multipath channel, which greatly improves the system’s robustness against fading and other channel impairments.

The orthogonality property allows subcarriers to be placed close together without causing mutual interference. As a result, OFDM can efficiently utilize the available spectrum, enabling high data rates and maximizing spectral efficiency, making it ideal for high-speed data transmission in wireless communication systems.

Reference

[1] Chakravarthy, A. S. Nunez, and J. P. Stephens, “TDCS, OFDM, and MC-CDMA: a brief tutorial,” IEEE Radio Commun., vol. 43, pp. 11-16, Sept. 2005.

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