Ferkans — Interactive Telecom Tutor

From Single Carrier to Multicarrier

A wideband channel exhibits frequency selectivity: different frequency components of the signal experience different gains and phase shifts. In single-carrier systems this causes intersymbol interference (ISI), which requires complex equalisation. The elegant idea behind OFDM is to divide the wideband channel into many narrow sub-channels, each of which is approximately flat fading. This converts one difficult equalisation problem into many trivially simple ones — a single complex multiplication per subcarrier.

Definition:
Multicarrier Modulation

Multicarrier modulation transmits data by modulating $N$ parallel narrowband subcarriers simultaneously. Each subcarrier carries a low-rate data stream, so the symbol duration on each subcarrier is $N$ times longer than in an equivalent single-carrier system. If the total bandwidth is $B$ , each subcarrier occupies approximately $B/N$ Hz.

Longer symbol duration relative to channel delay spread is the key advantage: it makes each sub-channel approximately flat fading.

Definition:
Orthogonal Frequency Division Multiplexing (OFDM)

OFDM is a multicarrier modulation scheme that uses $N$ orthogonal subcarriers spaced by $\Delta f = 1/T_{\text{sym}}$ Hz. The $k$ -th subcarrier has frequency $f_k = f_0 + k \Delta f$ for $k = 0, 1, \ldots, N-1$ . The transmitted baseband signal for one OFDM symbol is:

$x(t) = \sum_{k=0}^{N-1} X[k]\, e^{j2\pi k \Delta f\, t}, \qquad 0 \leq t < T_{\text{sym}}$

where $X[k]$ is the complex data symbol (e.g., QAM) on subcarrier $k$ , and $T_{\text{sym}} = 1/\Delta f$ is the symbol duration.

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Definition:
Subcarrier Orthogonality

Two subcarriers $e^{j2\pi k \Delta f\, t}$ and $e^{j2\pi m \Delta f\, t}$ are orthogonal over the interval $[0, T_{\text{sym}})$ when $\Delta f = 1/T_{\text{sym}}$ :

$\frac{1}{T_{\text{sym}}} \int_0^{T_{\text{sym}}} e^{j2\pi k \Delta f\, t}\, e^{-j2\pi m \Delta f\, t}\, dt = \begin{cases} 1 & k = m \\ 0 & k \neq m \end{cases}$

This orthogonality allows the subcarrier spectra to overlap in frequency while remaining separable at the receiver, achieving higher spectral efficiency than guard-band-separated FDM.

The sinc-shaped spectrum of each subcarrier has nulls at the centre frequencies of all other subcarriers. This is the origin of the name "orthogonal" FDM.

Theorem: Parallel Channel Decomposition

Consider an OFDM system with $N$ subcarriers and subcarrier spacing $\Delta f$ transmitting over a frequency-selective channel with frequency response $H(f)$ . If the channel is approximately constant over each subcarrier bandwidth $\Delta f$ , i.e.,

$H(f) \approx H(f_k) \triangleq H[k], \qquad f \in [f_k - \Delta f/2,\; f_k + \Delta f/2]$

then the received symbol on each subcarrier satisfies:

$Y[k] = H[k]\, X[k] + W[k], \qquad k = 0, 1, \ldots, N-1$

where $W[k] \sim \mathcal{CN}(0, \sigma^2)$ is additive noise. The frequency-selective channel is thus decomposed into $N$ independent parallel flat-fading sub-channels, each requiring only a single-tap equaliser $\hat{X}[k] = Y[k] / H[k]$ .

By making each subcarrier narrow enough ( $\Delta f \ll B_c$ , the coherence bandwidth), the channel appears flat within each sub-band. The IDFT/DFT pair at TX/RX automatically performs the multiplexing/demultiplexing, and the cyclic prefix (Section 14.2) ensures exact diagonalisation of the channel matrix.

Proof

Channel model in frequency domain

The frequency-selective channel has impulse response $h(\tau)$ with maximum delay spread $\tau_{\max}$ . In the frequency domain, $H(f) = \int h(\tau) e^{-j2\pi f \tau}\, d\tau$ . If $\Delta f \ll B_c \approx 1/\tau_{\max}$ , then $H(f)$ is nearly constant over each subcarrier band.

Per-subcarrier input-output relation

The receiver computes the DFT of the received samples. Due to the circular convolution property (enabled by the cyclic prefix), the DFT output on subcarrier $k$ is exactly $Y[k] = H[k] X[k] + W[k]$ , where $H[k]$ is the DFT of the channel impulse response at index $k$ , and $W[k]$ is the DFT of the noise — still i.i.d. Gaussian since the DFT is unitary.

Independence of sub-channels

Since $\mathbf{W} = \mathbf{F} \mathbf{w}$ and $\mathbf{F}$ is unitary, $\mathbf{W}$ has the same covariance as $\mathbf{w}$ : $\sigma^2 \mathbf{I}$ . Therefore the noise terms $W[k]$ are independent across subcarriers, making the $N$ sub-channels statistically independent. $\blacksquare$

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OFDM Subcarrier Spectra

Visualise the overlapping sinc spectra of OFDM subcarriers. Each subcarrier has a sinc-shaped spectrum centred at $k \Delta f$ . Observe how the nulls of each subcarrier align with the peaks of its neighbours, ensuring orthogonality despite spectral overlap.

Parameters

Number of subcarriers

N

8

Highlighted subcarrier index3

Show composite spectrum

Example: OFDM System Parameters for LTE

An LTE system uses $N = 2048$ subcarriers with subcarrier spacing $\Delta f = 15$ kHz. The cyclic prefix is $N_{\text{cp}} = 144$ samples (normal CP).

(a) Find the useful symbol duration $T_{\text{sym}}$ .

(b) Find the total OFDM symbol duration $T_{\text{total}}$ .

(c) What is the system bandwidth?

(d) Calculate the CP overhead as a fraction of the total symbol duration.

Solution

Useful symbol duration

$T_{\text{sym}} = \frac{1}{\Delta f} = \frac{1}{15 \times 10^3} \approx 66.7\;\mu\text{s}$ $

Total symbol duration

The CP duration is $T_{\text{cp}} = N_{\text{cp}} \cdot T_s$ where $T_s = 1/(N \Delta f) = 1/(2048 \times 15000) \approx 32.6$ ns.

$T_{\text{cp}} = 144 \times 32.6\;\text{ns} \approx 4.69\;\mu\text{s}$

$T_{\text{total}} = T_{\text{sym}} + T_{\text{cp}} \approx 66.7 + 4.69 = 71.4\;\mu\text{s}$

System bandwidth

$B = N \cdot \Delta f = 2048 \times 15\;\text{kHz} = 30.72\;\text{MHz}$ $

Note: only 1200 subcarriers (20 MHz) are used for data in LTE; the rest are guard bands.

CP overhead

$\text{CP overhead} = \frac{T_{\text{cp}}}{T_{\text{total}}} = \frac{4.69}{71.4} \approx 6.6\%$ $This is the price paid for ISI-free transmission.$ \blacksquare$

Historical Note: Origins of OFDM

1966-2009

The concept of multicarrier modulation was first proposed by Robert W. Chang of Bell Labs in 1966, who showed that data could be transmitted simultaneously on overlapping orthogonal subcarriers without mutual interference. The practical breakthrough came in 1971 when Weinstein and Ebert demonstrated that OFDM modulation and demodulation could be efficiently implemented using the DFT, eliminating the need for banks of oscillators. The addition of the cyclic prefix by Peled and Ruiz in 1980 completed the modern OFDM architecture. OFDM was adopted for DAB (1995), DVB-T (1997), IEEE 802.11a/g Wi-Fi (1999), and 4G LTE (2009), making it arguably the most successful modulation technology of the 21st century.

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Quick Check

Why must the subcarrier spacing in OFDM satisfy $\Delta f = 1/T_{\text{sym}}$ ?

To maximise the total bandwidth of the system

To ensure orthogonality between subcarriers over the symbol interval $[0, T_{\text{sym}})$

To minimise the peak-to-average power ratio

To make the cyclic prefix as short as possible

Correction:

To ensure orthogonality between subcarriers over the symbol interval

[0, T_{\text{sym}})

With $\Delta f = 1/T_{\text{sym}}$ , each subcarrier completes an integer number of cycles more than its neighbours during $T_{\text{sym}}$ , making $\int_0^{T_{\text{sym}}} e^{j2\pi(k-m)\Delta f\, t}\,dt = 0$ for $k \neq m$ .

Why This Matters: OFDM as the Foundation for MIMO-OFDM

OFDM's parallel flat-fading decomposition is the key enabler for MIMO-OFDM: by converting the frequency-selective MIMO channel into $N$ independent narrowband MIMO channels, OFDM allows standard MIMO techniques (spatial multiplexing, beamforming, space-time coding) to be applied per subcarrier without wideband matrix equalisers. This is why every modern MIMO system — from 802.11n Wi-Fi to 5G NR massive MIMO — is built on OFDM. Chapters 15--18 develop the MIMO theory that operates on top of the per-subcarrier flat-fading model established here.

OFDM

Orthogonal Frequency Division Multiplexing — a multicarrier modulation scheme that divides a wideband channel into $N$ narrowband orthogonal subcarriers, each experiencing flat fading.

Subcarrier

One of the $N$ narrowband orthogonal frequency channels in an OFDM system, spaced by $\Delta f = 1/T_{\text{sym}}$ Hz.

Subcarrier Spacing

The frequency separation $\Delta f$ between adjacent OFDM subcarriers. Must satisfy $\Delta f = 1/T_{\text{sym}}$ for orthogonality. In LTE, $\Delta f = 15$ kHz; in 5G NR, $\Delta f \in \{15, 30, 60, 120, 240\}$ kHz.

Principle of OFDM