Ferkans — Interactive Telecom Tutor

From Narrowband to Wideband

Everything we have done in the preceding chapters — channel hardening, favorable propagation, linear precoding, uplink detection — assumed a flat-fading channel: one $N_t \times K$ matrix $\mathbf{H}$ describing the entire band. In practice, 5G NR allocates bandwidths from 20 MHz to 400 MHz, and at sub-6 GHz the RMS delay spread $\sigma_\tau$ is typically 100–500 ns. The coherence bandwidth $B_c \approx 1/(5\sigma_\tau)$ can be as low as 400 kHz — far smaller than the system bandwidth. The channel is frequency-selective, and we need OFDM to handle it.

The golden thread of this chapter is simple: OFDM converts the wideband problem into $N$ parallel narrowband problems, each of which we already know how to solve. But this decomposition comes at a price — pilot overhead, interpolation error, and the need to manage resources across both space and frequency.

Definition:
Frequency-Selective MIMO Channel

Consider a base station with $N_t$ antennas serving $K$ single-antenna users. The wideband channel between the base station and user $k$ is modeled as a tapped delay line with $L$ resolvable taps:

$\mathbf{h}_k(t) = \sum_{\ell=0}^{L-1} \mathbf{h}_{k,\ell} \, \delta(t - \tau_\ell)$

where $\mathbf{h}_{k,\ell} \in \mathbb{C}^{N_t}$ is the $\ell$ -th tap coefficient vector and $\tau_\ell$ is the $\ell$ -th path delay. The maximum excess delay satisfies $\tau_{L-1} - \tau_0 \leq T_{\text{cp}}$ (the cyclic prefix absorbs all multipath).

The number of taps $L$ is related to the delay spread and bandwidth: $L \approx \lceil W \cdot \sigma_\tau \rceil + 1$ , where $W$ is the total signal bandwidth. For 100 MHz at sub-6 GHz with $\sigma_\tau = 300\,\text{ns}$ , we get $L \approx 31$ taps.

Delay Tap

A discrete-time representation of a multipath component in the frequency-selective channel. Each tap $\mathbf{h}_{k,\ell}$ captures the combined effect of all physical paths arriving at delay $\tau_\ell$ .

Definition:
Per-Subcarrier Channel Matrix

After OFDM demodulation with $N$ subcarriers and subcarrier spacing $\Delta f$ , the channel at subcarrier $k$ for user $j$ is the $N$ -point DFT of the delay-domain taps:

$\mathbf{h}_j[k] = \sum_{\ell=0}^{L-1} \mathbf{h}_{j,\ell} \, e^{-j 2\pi k \Delta f \tau_\ell}, \quad k = 0, 1, \ldots, N-1$

Stacking all $K$ users, the $N_t \times K$ channel matrix at subcarrier $k$ is:

$\mathbf{H}[k] = \bigl[\mathbf{h}_1[k], \, \mathbf{h}_2[k], \, \ldots, \, \mathbf{h}_{K}[k]\bigr]$

Each $\mathbf{H}[k]$ is a flat-fading channel matrix — the entire narrowband massive MIMO toolkit applies per subcarrier.

Cyclic Prefix

A copy of the last $N_{\text{cp}}$ samples of the OFDM symbol prepended to the transmitted block. It converts the linear channel convolution into a circular one, enabling the DFT-based decomposition into parallel flat-fading subchannels. The CP must satisfy $T_{\text{cp}} \geq \tau_{L-1} - \tau_0$ .

Related: Delay Tap

Theorem: OFDM Diagonalization of the Wideband Channel

Let the frequency-selective MIMO channel between the base station and user $k$ have $L$ taps $\{\mathbf{h}_{k,\ell}\}_{\ell=0}^{L-1}$ with delays $\{\tau_\ell\}$ . If the cyclic prefix duration satisfies $T_{\text{cp}} \geq \tau_{L-1}$ , then after OFDM processing (DFT at the receiver), the input-output relation at subcarrier $k$ is

$y_k[n] = \mathbf{h}_k^H[n] \, \mathbf{x}[n] + w_k[n], \quad k = 0, \ldots, N-1$

where $\mathbf{x}[n] \in \mathbb{C}^{N_t}$ is the transmitted vector at subcarrier $k$ in OFDM symbol $n$ , and $w_k[n] \sim \mathcal{CN}(0, \sigma^2)$ . The noise samples are i.i.d. across subcarriers and OFDM symbols.

The cyclic prefix makes the channel convolution circular, and the DFT diagonalizes any circulant matrix. The result is $N$ parallel single-tap channels — each identical in structure to the narrowband model we already analyzed.

Proof

Circular convolution

With the CP of length $\geq L-1$ , the linear convolution between the transmitted OFDM symbol (including CP) and the channel impulse response is equivalent to a circular convolution of the $N$ -sample data block.

DFT diagonalizes circulant matrices

A circular convolution in the time domain corresponds to element-wise multiplication in the frequency domain. Applying the $N$ -point DFT at the receiver yields

$Y_k = \underbrace{\sum_{\ell=0}^{L-1} h_{k,\ell} \, e^{-j2\pi k \ell / N}}_{H[k]} \cdot X_k + W_k$

for each subcarrier $k$ , where $H[k]$ is the channel frequency response.

Extension to MIMO

For the MIMO case with $N_t$ antennas, each antenna experiences the same diagonalization independently. The $N_t$ -dimensional received vector at subcarrier $k$ is the superposition of contributions from all transmit antennas, yielding the matrix form $\mathbf{y}[k] = \mathbf{H}^{H}[k] \, \mathbf{x}[k] + \mathbf{w}[k]$ .

Noise independence

Since the DFT is a unitary transform and the time-domain noise is i.i.d. $\mathcal{CN}(0, \sigma^2)$ , the frequency-domain noise remains i.i.d. $\mathcal{CN}(0, \sigma^2)$ across subcarriers. $\blacksquare$

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Key Takeaway

OFDM converts the wideband massive MIMO problem into $N$ independent narrowband problems. Every technique from Chapters 1–9 — channel hardening, favorable propagation, MRT, ZF, MMSE — applies per subcarrier. The new challenge is managing the $N \times K$ dimensional estimation and precoding problem efficiently.

Historical Note: OFDM Meets MIMO

1966–2018

OFDM was proposed by Robert W. Chang at Bell Labs in 1966, but the practical DFT-based implementation came from Weinstein and Ebert in 1971. The marriage of OFDM and MIMO was championed in the early 2000s by multiple groups. The IEEE 802.11n standard (2009) was the first commercial MIMO-OFDM system. 3GPP adopted OFDM for the LTE downlink in Release 8 (2008), and massive MIMO-OFDM became the backbone of 5G NR from Release 15 (2018).

Definition:
Channel Frequency Correlation

The channel frequency response $\mathbf{h}_k[n]$ at subcarrier $n$ exhibits correlation across subcarriers determined by the power delay profile. The frequency-domain correlation function for user $k$ is

$r_k[\Delta k] = \mathbb{E}\bigl[\mathbf{h}_k[n]^H \mathbf{h}_k[n + \Delta k]\bigr] = \sum_{\ell=0}^{L-1} \mathbb{E}\bigl[\|\mathbf{h}_{k,\ell}\|^2\bigr] \, e^{-j 2\pi \Delta k \Delta f \tau_\ell}$

Two subcarriers separated by $\Delta k$ are approximately uncorrelated when $\Delta k \cdot \Delta f \gg B_c$ .

This frequency correlation is what makes interpolation-based channel estimation possible: we only need to estimate the channel at a subset of subcarriers and interpolate the rest.

Example: Typical 5G NR OFDM Parameters

A 5G NR base station operates at $f_0 = 3.5\,\text{GHz}$ with $W = 100\,\text{MHz}$ , subcarrier spacing $\Delta f = 30\,\text{kHz}$ , and $N = 3276$ active subcarriers. The channel has RMS delay spread $\sigma_\tau = 300\,\text{ns}$ .

(a) How many delay taps $L$ does the channel have?

(b) What is the coherence bandwidth, and how many subcarriers fit within one coherence bandwidth?

(c) If pilot subcarriers are spaced every $D$ subcarriers, what is the maximum $D$ that avoids aliasing?

Solution

Number of delay taps

The total OFDM symbol duration (excluding CP) is $T = 1/\Delta f = 33.3\,\mu\text{s}$ . The number of resolvable taps is

$L = \lceil W \cdot \sigma_\tau \rceil + 1 = \lceil 100 \times 10^6 \times 300 \times 10^{-9} \rceil + 1 = 31$

Coherence bandwidth

$B_c \approx \frac{1}{5 \sigma_\tau} = \frac{1}{5 \times 300 \times 10^{-9}} = 667\,\text{kHz}$ . The number of subcarriers within one coherence bandwidth is $\lfloor B_c / \Delta f \rfloor = \lfloor 667/30 \rfloor = 22$ subcarriers.

Maximum pilot spacing

By the Nyquist sampling theorem in frequency, to avoid aliasing of the $L$ -tap channel we need at least $L$ pilot subcarriers uniformly spaced across the $N$ subcarriers:

$D \leq \lfloor N / L \rfloor = \lfloor 3276 / 31 \rfloor = 105$

In practice, $D$ is set well below this (e.g., $D = 4$ to $12$ ) to provide an SNR margin for estimation.

Wideband Massive MIMO Channel Response

Visualize the frequency-selective channel: magnitude of channel coefficients across subcarriers for different delay spreads and antenna counts.

Parameters

N_t

64

Number of BS antennas

N

512

Number of OFDM subcarriers

\sigma_\tau

(ns)300

RMS delay spread in nanoseconds

User index0

Which user channel to display

Common Mistake: Cyclic Prefix Overhead Is Often Forgotten

Mistake:

When computing the spectral efficiency of massive MIMO-OFDM, many analyses use the narrowband per-subcarrier rate and simply multiply by $N$ , ignoring the cyclic prefix overhead.

Correction:

The true spectral efficiency must account for the CP fraction. If $T_{\text{cp}}$ is the CP duration and $T = 1/\Delta f$ the useful symbol duration, the efficiency loss is

$\eta_{\text{CP}} = \frac{T}{T + T_{\text{cp}}}$

For 5G NR normal CP at $\Delta f = 30\,\text{kHz}$ : $T = 33.3\,\mu\text{s}$ , $T_{\text{cp}} = 2.34\,\mu\text{s}$ , giving $\eta_{\text{CP}} \approx 0.934$ — a 6.6% loss.

Coherence Bandwidth

The frequency separation $B_c$ over which the channel frequency response remains approximately constant. Inversely proportional to the RMS delay spread: $B_c \approx 1/(5\sigma_\tau)$ . Subcarriers separated by less than $B_c$ experience nearly identical fading.

Related: Delay Tap, Cyclic Prefix

Quick Check

Why does OFDM convert a frequency-selective MIMO channel into $N$ flat-fading channels?

The DFT is a unitary transform that preserves signal energy

The cyclic prefix makes the channel convolution circular, and the DFT diagonalizes circulant matrices

The channel taps are i.i.d. Gaussian, which makes the DFT output independent

The subcarrier spacing is chosen to equal the coherence bandwidth