Min-Avg-Max Complex Statistics Guide¶

This document describes a purely mathematical view of the Min-Avg-Max statistics computed from per-subcarrier complex channel-estimation coefficients across multiple snapshots.

We assume a downstream OFDM channel with:

\(K\) subcarriers (frequency bins)
\(M\) snapshots (independent captures of the same channel)

All statistics are computed per subcarrier by aggregating across snapshots.

1. Complex Channel Matrix¶

Let the complex channel estimate for snapshot \(m\) and subcarrier \(k\) be

\[ H^{(m)}[k] \in \mathbb{C}, \quad m = 0,\dots,M-1, \quad k = 0,\dots,K-1. \]

We can write each coefficient in terms of its real and imaginary parts:

\[ H^{(m)}[k] = \Re\{H^{(m)}[k]\} + j\,\Im\{H^{(m)}[k]\}, \]

and define the real and imaginary matrices

\[ R_{m,k} = \Re\{H^{(m)}[k]\}, \qquad I_{m,k} = \Im\{H^{(m)}[k]\}, \]

which form two real-valued arrays of size \(M \times K\).

2. Per-Subcarrier Real-Part Statistics¶

For each subcarrier \(k\), the minimum, average, and maximum of the real part across all snapshots are defined as:

\[ \begin{aligned} r_{\min}[k] &= \min_{0 \le m < M} R_{m,k}, \\ r_{\text{avg}}[k] &= \frac{1}{M} \sum_{m=0}^{M-1} R_{m,k}, \\ r_{\max}[k] &= \max_{0 \le m < M} R_{m,k}. \end{aligned} \]

The sequences

\[ \{r_{\min}[k]\}_{k=0}^{K-1},\quad \{r_{\text{avg}}[k]\}_{k=0}^{K-1},\quad \{r_{\max}[k]\}_{k=0}^{K-1} \]

form the Min-Avg-Max real-part profiles across the OFDM band.

3. Per-Subcarrier Imaginary-Part Statistics¶

Similarly, for the imaginary component we define

\[ \begin{aligned} i_{\min}[k] &= \min_{0 \le m < M} I_{m,k}, \\[4pt] i_{\text{avg}}[k] &= \frac{1}{M} \sum_{m=0}^{M-1} I_{m,k}, \\[4pt] i_{\max}[k] &= \max_{0 \le m < M} I_{m,k}. \end{aligned} \]

The sequences

\[ \{i_{\min}[k]\}_{k=0}^{K-1},\quad \{i_{\text{avg}}[k]\}_{k=0}^{K-1},\quad \{i_{\max}[k]\}_{k=0}^{K-1} \]

are the Min-Avg-Max imaginary-part profiles.

These statistics characterize how the in-phase and quadrature components of the channel vary across snapshots for each subcarrier.

4. Magnitude Statistics¶

4.1 Instantaneous Magnitude¶

For each snapshot and subcarrier, the instantaneous magnitude of the complex channel is

\[ A^{(m)}[k] = \left|H^{(m)}[k]\right| = \sqrt{\bigl(\Re\{H^{(m)}[k]\}\bigr)^2 + \bigl(\Im\{H^{(m)}[k]\}\bigr)^2}. \]

This yields a real-valued magnitude matrix \(\{A^{(m)}[k]\}\) of size \(M \times K\).

4.2 Min and Max Magnitude Across Snapshots¶

For each subcarrier, the minimum and maximum magnitude observed across all snapshots are

\[ \begin{aligned} a_{\min}[k] &= \min_{0 \le m < M} A^{(m)}[k], \\[4pt] a_{\max}[k] &= \max_{0 \le m < M} A^{(m)}[k]. \end{aligned} \]

These describe the envelope of how strong or weak the channel can become at each subcarrier.

4.3 Coherent Average Magnitude¶

Instead of averaging magnitudes directly, a coherent average is formed by first averaging the complex channel across snapshots, then taking the magnitude:

Complex average per subcarrier:

\[ \overline{H}[k] = \frac{1}{M} \sum_{m=0}^{M-1} H^{(m)}[k]. \]

Magnitude of the averaged complex channel:

\[ a_{\text{avg}}[k] = \left|\overline{H}[k]\right| = \left|\frac{1}{M} \sum_{m=0}^{M-1} H^{(m)}[k]\right|. \]

This is not, in general, equal to the average of magnitudes:

\[ a_{\text{avg}}[k] \neq \frac{1}{M} \sum_{m=0}^{M-1} \left|H^{(m)}[k]\right| \]

unless all \(H^{(m)}[k]\) share the same phase. The coherent average

\[ a_{\text{avg}}[k] = \left|\overline{H}[k]\right| \]

naturally accounts for phase alignment and destructive/constructive interference, making it consistent with other phase-sensitive analyses (for example, group delay or echo detection).

5. Summary of Per-Subcarrier Outputs¶

For each subcarrier \(k\), the Min-Avg-Max complex statistics provide:

Real part: \(r_{\min}[k],\ r_{\text{avg}}[k],\ r_{\max}[k]\).
Imaginary part: \(i_{\min}[k],\ i_{\text{avg}}[k],\ i_{\max}[k]\).
Magnitude: \(a_{\min}[k],\ a_{\text{avg}}[k],\ a_{\max}[k]\).

Plotted versus subcarrier frequency \(f_k\), these profiles show:

The range of channel variation across snapshots (min vs max),
The typical coherent channel strength per subcarrier (average magnitude),
And how the real and imaginary components evolve across the OFDM band.

This provides a compact but information-rich view of channel stability and variability suitable for diagnostics, visualization, and downstream PNM analyses.