Max Math¶

Foundations For Temporal Aggregation Of Per-Subcarrier RxMER Measurements.

Overview¶

This note defines the mathematical operations used by the Multi-Capture RxMER Min/Avg/Max analysis.

The goal is to combine multiple downstream OFDM RxMER captures, taken at different times, into a small set of summary curves that describe, per subcarrier:

The minimum observed RxMER over time
The arithmetic mean (time-average) RxMER
The maximum observed RxMER over time

These statistics are computed independently for each active subcarrier and can be plotted as three curves over the same frequency or subcarrier index axis.

Notation¶

Let:

\(K \in \mathbb{N}\) be the number of RxMER captures.
\(N \in \mathbb{N}\) be the number of active subcarriers in the downstream OFDM channel.
\(k \in \{1, \dots, K\}\) index the capture (time dimension).
\(n \in \{1, \dots, N\}\) index the subcarrier.

For each capture \(k\) and subcarrier \(n\), define

\[ R_{k,n} \in \mathbb{R} \]

as the RxMER value, in dB, reported for subcarrier \(n\) in capture \(k\).

We can view the complete dataset as a real-valued matrix

\[ R \in \mathbb{R}^{K \times N}, \quad R = \begin{bmatrix} R_{1,1} & \dots & R_{1,N} \\ \vdots & \ddots & \vdots \\ R_{K,1} & \dots & R_{K,N} \end{bmatrix}. \]

Each row corresponds to a single RxMER capture, and each column corresponds to a given subcarrier.

Per-Subcarrier Min / Average / Max¶

For a fixed subcarrier \(n\), the multi-capture time series is

\[ \{ R_{1,n}, R_{2,n}, \dots, R_{K,n} \}. \]

From this 1D series in time, we define:

Minimum RxMER¶

\[ R^{\min}_n = \min_{1 \le k \le K} \, R_{k,n}. \]

This is the worst-case (lowest) RxMER observed over all captures for subcarrier \(n\).

Average (Mean) RxMER¶

\[ R^{\text{avg}}_n = \frac{1}{K} \sum_{k=1}^{K} R_{k,n}. \]

This is the arithmetic mean over captures for subcarrier \(n\), treating all captures equally.

Maximum RxMER¶

\[ R^{\max}_n = \max_{1 \le k \le K} \, R_{k,n}. \]

This is the best-case (highest) RxMER observed over all captures for subcarrier \(n\).

Optional Range (Span)¶

Sometimes it is useful to quantify the temporal variation at each subcarrier using the range:

\[ \Delta R_n = R^{\max}_n - R^{\min}_n. \]

A large \(\Delta R_n\) indicates that subcarrier \(n\) shows significant instability over time, even if the average RxMER is acceptable.

Vector Form¶

Define the three per-subcarrier vectors

\[ \mathbf{R}^{\min} = \left( R^{\min}_1, R^{\min}_2, \dots, R^{\min}_N \right), \]

\[ \mathbf{R}^{\text{avg}} = \left( R^{\text{avg}}_1, R^{\text{avg}}_2, \dots, R^{\text{avg}}_N \right), \]

\[ \mathbf{R}^{\max} = \left( R^{\max}_1, R^{\max}_2, \dots, R^{\max}_N \right). \]

In matrix notation, the per-subcarrier mean vector \(\mathbf{R}^{\text{avg}}\) can be written as

\[ \mathbf{R}^{\text{avg}} = \frac{1}{K} \, \mathbf{1}_K^\top R, \]

where \(\mathbf{1}_K \in \mathbb{R}^K\) is an all-ones column vector and the result is a \(1 \times N\) row vector of per-subcarrier averages.

The min and max vectors are obtained by column-wise reduction:

\[ R^{\min}_n = \min_{k} R_{k,n}, \quad R^{\max}_n = \max_{k} R_{k,n} \quad \text{for } n = 1, \dots, N. \]

Optional Higher-Order Statistics¶

While the core analysis focuses on Min/Avg/Max, higher-order statistics can be derived per subcarrier:

Per-Subcarrier Variance¶

\[ \sigma^2_n = \frac{1}{K} \sum_{k=1}^{K} \left( R_{k,n} - R^{\text{avg}}_n \right)^2. \]

Per-Subcarrier Standard Deviation¶

\[ \sigma_n = \sqrt{\sigma^2_n}. \]

These metrics measure temporal dispersion of RxMER at each subcarrier and can highlight carriers that are noise-sensitive or intermittently impacted by interference.

Mapping To Modulation Capacity (Optional)¶

If you define a mapping

\[ f : \mathbb{R} \to \mathbb{R}, \]

that converts RxMER (in dB) to an effective metric such as bits per symbol or Shannon capacity per subcarrier, then you can apply the same Min/Avg/Max process in that transformed space.

For example, for a per-subcarrier function \(f(R_{k,n})\), define

\[ B_{k,n} = f(R_{k,n}), \]

and compute

\[ B^{\min}_n = \min_k B_{k,n}, \quad B^{\text{avg}}_n = \frac{1}{K} \sum_{k=1}^{K} B_{k,n}, \quad B^{\max}_n = \max_k B_{k,n}. \]

This preserves the same aggregation structure but on a modulation-oriented scale instead of the raw RxMER.

Practical Interpretation¶

\(\mathbf{R}^{\min}\) shows the worst-case RxMER profile experienced across all captures (worst day).
\(\mathbf{R}^{\text{avg}\!}\) shows the typical RxMER profile (average day).
\(\mathbf{R}^{\max}\) shows the best-case profile (best day).
\(\Delta R_n\), \(\sigma_n\) identify carriers whose RxMER is unstable over time.

Plotting the three curves on the same axis (e.g., frequency or subcarrier index) provides an immediate visual indication of both spatial variation (across frequency) and temporal variation (spread between min and max) in the downstream OFDM RxMER.