Ferkans — Interactive Telecom Tutor

The Per-Subcarrier Paradigm

With the OFDM diagonalization in hand, the wideband massive MIMO problem reduces to applying narrowband processing at each of the $N$ subcarriers independently. This is conceptually clean — but the computational and estimation costs scale with $N$ , which can be several thousand. Understanding how channel hardening and favorable propagation behave across the frequency dimension is essential for practical system design.

Definition:
Per-Subcarrier Linear Precoding

At OFDM subcarrier $k$ , the base station transmits

$\mathbf{x}[k] = \mathbf{W}[k] \, \mathbf{s}[k]$

where $\mathbf{s}[k] = [s_1[k], \ldots, s_{K}[k]]^\mathsf{T}$ contains the data symbols for the $K$ users and $\mathbf{W}[k] \in \mathbb{C}^{N_t \times K}$ is the precoding matrix. The three standard choices are:

MRT: $\mathbf{W}_{\text{MRT}}[k] = \mathbf{H}[k] / \sqrt{K}$
ZF: $\mathbf{W}_{\text{ZF}}[k] = \mathbf{H}[k](\mathbf{H}^{H}[k]\mathbf{H}[k])^{-1}$ , normalized per column
MMSE: $\mathbf{W}_{\text{MMSE}}[k] = \mathbf{H}[k](\mathbf{H}^{H}[k]\mathbf{H}[k] + \alpha_k \mathbf{I})^{-1}$

where $\alpha_k = K\sigma^2/P_t$ is the regularization parameter at subcarrier $k$ .

,

Theorem: Per-Subcarrier SINR with Linear Precoding

Consider downlink transmission at subcarrier $k$ with perfect CSI. User $j$ receives

$y_j[k] = \mathbf{h}_j^H[k] \, \mathbf{W}[k] \, \mathbf{s}[k] + w_j[k]$

The SINR of user $j$ at subcarrier $k$ under linear precoding $\mathbf{W}[k]$ is

$\text{SINR}_j[k] = \frac{p_j[k] \, |\mathbf{h}_j^H[k] \, \mathbf{v}_{j}[k]|^2} {\sum_{i \neq j} p_i[k] \, |\mathbf{h}_j^H[k] \, \mathbf{v}_{i}[k]|^2 + \sigma^2}$

where $\mathbf{v}_{j}[k]$ is the $j$ -th column of $\mathbf{W}[k]$ (normalized to unit norm) and $p_j[k]$ is the power allocated to user $j$ at subcarrier $k$ .

This is exactly the narrowband SINR expression from Chapter 6, replicated at each subcarrier. The key insight is that the channel vectors $\mathbf{h}_j[k]$ vary across $k$ — so the interference pattern changes from subcarrier to subcarrier.

Proof

Signal and interference decomposition

User $j$ observes $y_j[k] = \mathbf{h}_j^H[k] \mathbf{v}_{j}[k] \sqrt{p_j[k]} s_j[k] + \sum_{i \neq j} \mathbf{h}_j^H[k] \mathbf{v}_{i}[k] \sqrt{p_i[k]} s_i[k] + w_j[k]$ . The first term is the desired signal; the sum is multi-user interference.

SINR computation

Taking the ratio of desired signal power to interference-plus-noise power:

$\text{SINR}_j[k] = \frac{p_j[k] |\mathbf{h}_j^H[k] \mathbf{v}_{j}[k]|^2} {\sum_{i \neq j} p_i[k] |\mathbf{h}_j^H[k] \mathbf{v}_{i}[k]|^2 + \sigma^2}$

For ZF precoding, $\mathbf{h}_j^H[k] \mathbf{v}_{i}[k] = 0$ for $i \neq j$ , and the SINR simplifies to $\text{SINR}_j^{\text{ZF}}[k] = p_j[k] |\mathbf{h}_j^H[k] \mathbf{v}_{j}[k]|^2 / \sigma^2$ .

Achievable rate

The per-subcarrier rate for user $j$ is $R_j[k] = \log_2(1 + \text{SINR}_j[k])$ bits/s/Hz. The total rate for user $j$ across all $N$ subcarriers, accounting for CP overhead, is

$R_j = \frac{T}{T + T_{\text{cp}}} \sum_{k=0}^{N-1} \log_2(1 + \text{SINR}_j[k]) \cdot \Delta f$

in bits/s. $\blacksquare$

Per-Subcarrier SINR: MRT vs. ZF vs. MMSE

Compare the SINR across subcarriers for MRT, ZF, and MMSE precoding. Observe how ZF eliminates inter-user interference but suffers at subcarriers where $\mathbf{H}[k]$ is ill-conditioned.

Parameters

N_t

64

Number of BS antennas

K

8

Number of users

N

256

Number of OFDM subcarriers

\text{SNR}

(dB)10

Transmit SNR in dB

\sigma_\tau

(ns)300

RMS delay spread in nanoseconds

Precoder

Theorem: Channel Hardening Across Subcarriers

As $N_t \to \infty$ with i.i.d. Rayleigh fading taps, channel hardening holds at every subcarrier simultaneously:

$\frac{1}{N_t} \|\mathbf{h}_k[n]\|^2 \xrightarrow{\text{a.s.}} \sum_{\ell=0}^{L-1} \sigma_\ell^2 \triangleq \beta_{k}, \quad \forall k$

where $\sigma_\ell^2 = \mathbb{E}[\|\mathbf{h}_{k,\ell}\|^2/N_t]$ is the average tap power. Moreover, the effective channel gain $\frac{1}{N_t}\|\mathbf{h}_k[n]\|^2$ becomes deterministic and identical across all subcarriers — the frequency selectivity is "averaged out" by the large array.

Each antenna sees an independent realization of the frequency-selective channel. Averaging over $N_t$ antennas, the law of large numbers kicks in at every frequency point simultaneously — the effective channel gain converges to the total average path gain, which is independent of frequency.

Proof

Expand the channel norm

$\frac{1}{N_t}\|\mathbf{h}_k[n]\|^2 = \frac{1}{N_t} \sum_{m=1}^{N_t} |h_k^{(m)}[n]|^2$ where $h_k^{(m)}[n] = \sum_{\ell} h_{k,\ell}^{(m)} e^{-j2\pi k\Delta f \tau_\ell}$ is the channel from antenna $m$ at subcarrier $k$ .

Apply the SLLN

The random variables $\{|h_k^{(m)}[n]|^2\}_{m=1}^{N_t}$ are i.i.d. with mean $\sum_\ell \sigma_\ell^2$ (by Parseval's theorem applied to the DFT of the tap coefficients). By the strong law of large numbers, $\frac{1}{N_t} \sum_m |h_k^{(m)}[n]|^2 \to \sum_\ell \sigma_\ell^2$ a.s.

Frequency independence

The limit $\sum_\ell \sigma_\ell^2$ does not depend on $k$ . Thus the hardened channel gain is identical at all subcarriers, and the per-subcarrier SNR becomes deterministic. $\blacksquare$

Key Takeaway

In massive MIMO-OFDM, channel hardening acts in both space and frequency: the effective channel gain becomes deterministic and frequency-flat as $N_t$ grows. This means that resource allocation across subcarriers becomes nearly trivial in the massive regime — equal power allocation across frequency is near-optimal.

Example: Sum Rate of Wideband Massive MIMO

A massive MIMO-OFDM system has $N_t = 128$ antennas, $K = 16$ users, $N = 1024$ subcarriers with $\Delta f = 30\,\text{kHz}$ , and operates at $\text{SNR} = 10\,\text{dB}$ . Under ZF precoding with perfect CSI and i.i.d. Rayleigh fading taps ( $L = 16$ , uniform power delay profile), estimate the per-user rate and the system sum spectral efficiency.

Solution

Per-subcarrier ZF SINR

With i.i.d. Rayleigh fading and ZF precoding, the per-subcarrier SINR for user $j$ at subcarrier $k$ follows a Gamma distribution. In the massive MIMO regime ( $N_t \gg K$ ), the ZF SINR concentrates around

$\text{SINR}_j^{\text{ZF}}[k] \approx (N_t - K) \cdot \frac{\text{SNR}}{K} = (128 - 16) \cdot \frac{10}{16} = 70$

which is $10\log_{10}(70) = 18.5\,\text{dB}$ .

Per-user rate

Each user achieves approximately

$R_j \approx N \cdot \Delta f \cdot \log_2(1 + 70) \approx 1024 \times 30\text{k} \times 6.15 = 189\,\text{Mbit/s}$

After CP overhead ( $\eta_{\text{CP}} \approx 0.93$ ): $R_j \approx 176\,\text{Mbit/s}$ .

Sum spectral efficiency

The sum spectral efficiency (bits/s/Hz per cell) is

$\eta_{\text{sum}} = K \cdot \log_2(1 + 70) \approx 16 \times 6.15 = 98.4\,\text{bits/s/Hz}$

This is the compelling promise of massive MIMO-OFDM: nearly 100 bits/s/Hz from a single cell with 128 antennas and 16 users.

Common Mistake: ZF Matrix Inversion at Every Subcarrier

Mistake:

Naively applying ZF precoding requires computing $(\mathbf{H}^{H}[k]\mathbf{H}[k])^{-1}$ at each of the $N$ subcarriers independently. With $N = 3276$ and $K = 16$ , this means 3276 matrix inversions of size $16 \times 16$ per OFDM symbol — and the channel changes every coherence time.

Correction:

In practice, the channel varies smoothly across adjacent subcarriers (the frequency-domain correlation). One can: (a) compute ZF only at pilot subcarriers and interpolate the precoder, (b) use recursive matrix update formulas that exploit the smooth variation, or (c) use MMSE precoding with a fixed regularization parameter across nearby subcarriers. Modern 5G NR baseband processors use specialized ASIC pipelines for this.

Wideband Precoding Strategies

Strategy	Computation per OFDM symbol	CSI requirement	Performance
Per-subcarrier ZF	$O(N K^{2} N_t)$	Full $\mathbf{H}[k]$ at every $k$	Optimal interference cancellation
Per-subcarrier MRT	$O(N K N_t)$	Full $\mathbf{H}[k]$ at every $k$	No matrix inversion, suboptimal at low $N_t/K$
Interpolated ZF	$O(N_p \K^{2} \N_t + N \K \N_t)$	CSI at pilot subcarriers only	Near-optimal if $N_p \geq L$
Block-diagonalization	$O(N_b \K^{2} \N_t)$	CSI at block centers	Groups subcarriers into blocks

Per-Subcarrier Processing

The approach of applying independent spatial processing (precoding or combining) at each OFDM subcarrier. This is optimal when the channel is estimated perfectly at every subcarrier, but incurs high computational cost proportional to the number of subcarriers $N$ .

🚨Critical Engineering Note

Baseband Processing Complexity in Massive MIMO-OFDM

The computational burden of per-subcarrier processing is the dominant cost in massive MIMO base stations. For $N_t = 64$ , $K = 16$ , and $N = 3276$ subcarriers at $\Delta f = 30\,\text{kHz}$ , ZF precoding requires roughly $3276 \times 16^2 \times 64 \approx 5.4 \times 10^7$ complex multiply-accumulate operations per OFDM symbol (every $35.7\,\mu\text{s}$ ). This translates to $\sim 1.5\,\text{TMAC/s}$ — well within the capability of modern ASIC designs (e.g., Xilinx RFSoC or custom baseband chips) but far beyond what a general-purpose CPU can handle in real time.

Practical Constraints

•
Latency budget: entire downlink processing must complete within one OFDM symbol duration
•
Power consumption scales linearly with $N_t \times N$
•
3GPP TS 38.211 specifies the OFDM numerology and resource grid structure

📋 Ref: 3GPP TS 38.211, Section 5.3

Quick Check

In massive MIMO-OFDM with i.i.d. Rayleigh fading taps, what happens to the per-subcarrier channel gain $\frac{1}{N_t}\|\mathbf{h}_k[n]\|^2$ as $N_t \to \infty$ ?

It converges to a different constant at each subcarrier

It converges to the same constant at all subcarriers

It diverges to infinity at all subcarriers

It converges to zero because the power is spread across more antennas

Correction:

It converges to the same constant at all subcarriers

By Parseval's theorem, the mean channel power at every subcarrier equals the total tap power, independent of $k$ . Channel hardening removes both spatial and frequency randomness.

Why Water-Filling Is Unnecessary in Massive MIMO-OFDM

In classical OFDM (single-antenna or small MIMO), the channel gain varies significantly across subcarriers, and water-filling power allocation provides substantial gains. In massive MIMO-OFDM, channel hardening makes the effective channel gain nearly constant across all subcarriers. Equal power allocation across frequency is therefore near-optimal — the water-filling gains vanish as $N_t / K$ grows. This is a major simplification: the scheduler only needs to allocate resources in the spatial domain, not jointly in space and frequency.

Per-Subcarrier Processing

The Per-Subcarrier Paradigm

Definition: Per-Subcarrier Linear Precoding

Theorem: Per-Subcarrier SINR with Linear Precoding

Signal and interference decomposition

SINR computation

Achievable rate

Per-Subcarrier SINR: MRT vs. ZF vs. MMSE

Parameters

Theorem: Channel Hardening Across Subcarriers

Expand the channel norm

Apply the SLLN

Frequency independence

Key Takeaway

Example: Sum Rate of Wideband Massive MIMO

Per-subcarrier ZF SINR

Per-user rate

Sum spectral efficiency

Common Mistake: ZF Matrix Inversion at Every Subcarrier

Wideband Precoding Strategies

Per-Subcarrier Processing

Baseband Processing Complexity in Massive MIMO-OFDM

Quick Check

Why Water-Filling Is Unnecessary in Massive MIMO-OFDM

Definition:
Per-Subcarrier Linear Precoding