Ferkans — Interactive Telecom Tutor

Starting From PEP

Diversity analysis in wireless begins with the pairwise error probability (PEP): the probability that the detector prefers a wrong codeword $\hat{\mathbf{x}}$ over the transmitted $\mathbf{x}$ at a given SNR. The union bound converts PEP into a BER upper bound, and the asymptotic SNR scaling of PEP reveals the diversity order. For OTFS, this line of reasoning culminates in the full-DD-diversity result of Section 2.

The point is that PEP for OTFS depends on the sum $\|\mathbf{H}_{DD}\,\Delta\mathbf{x}\|^2$ over all DD cells — a quadratic form in the path gains $h_i$ . The distribution of this quadratic form determines the diversity order. For independent Rayleigh paths, the diversity is the number of resolvable paths.

Definition:
Pairwise Error Probability

For an ML detector with transmit $\mathbf{x}$ and candidate $\hat{\mathbf{x}}$ , the pairwise error probability is $\mathrm{PEP}(\mathbf{x} \to \hat{\mathbf{x}}) \;=\; \Pr\!\left(\|\mathbf{y} - \mathbf{H}_{DD}\hat{\mathbf{x}}\|^2 \leq \|\mathbf{y} - \mathbf{H}_{DD}\mathbf{x}\|^2\right).$ Conditioned on the channel realization, the PEP is the Gaussian Q-function: $\mathrm{PEP}(\mathbf{x} \to \hat{\mathbf{x}} | \mathbf{H}_{DD}) \;=\; Q\!\left(\sqrt{\frac{\|\mathbf{H}_{DD}\,\Delta\mathbf{x}\|^2}{2\,\sigma^2}}\right),$ where $\Delta\mathbf{x} = \mathbf{x} - \hat{\mathbf{x}}$ .

Theorem: Diversity Order from PEP

The diversity order of a detector is the high-SNR scaling exponent of the BER: $d \;=\; -\lim_{\mathrm{SNR} \to \infty} \frac{\log\,\mathrm{BER}(\mathrm{SNR})}{\log\,\mathrm{SNR}}.$ Equivalently, $\mathrm{BER} \sim c \cdot \mathrm{SNR}^{-d}$ for some constant $c$ . A detector with diversity $d$ doubles the SNR gain per $3d$ -dB SNR increase.

Higher $d$ means the BER drops faster as SNR grows. Single-path Rayleigh: $d = 1$ . $P$ -path with diversity-reaching detector: $d = P$ . The detector's ability to exploit all paths is what translates physical diversity into BER gain.

Proof

PEP-to-BER bound

Union bound: $\mathrm{BER} \leq \sum_{\hat{\mathbf{x}} \neq \mathbf{x}} \mathrm{PEP}(\mathbf{x} \to \hat{\mathbf{x}})$ .

Dominant pair

At high SNR, one pair $(\mathbf{x}, \hat{\mathbf{x}})$ dominates — the one with smallest $\|\mathbf{H}\,\Delta\mathbf{x}\|^2$ . This smallest quadratic form sets the diversity.

Asymptotic SNR scaling

$\mathrm{PEP} = \mathbb{E}[Q(\sqrt{\|\mathbf{H}\Delta\mathbf{x}\|^2/(2\sigma^2)})]$ . For $\|\mathbf{H}\Delta\mathbf{x}\|^2$ distributed as chi-squared with $2d$ degrees of freedom, $\mathrm{PEP} \sim 1/\mathrm{SNR}^d$ . The diversity order $d$ equals the number of independent fading summands in the quadratic form. $\blacksquare$

Key Takeaway

Diversity order = rank of the channel action on the error vector. The quadratic form $\|\mathbf{H}_{DD}\,\Delta\mathbf{x}\|^2$ is a weighted sum of path gains $|h_i|^2$ . If the weights are non-degenerate (no linear dependence), the diversity is $P$ . For OTFS, this is the generic case — the DD channel structure naturally gives non-degenerate weights for any non-zero error vector.

Definition:
Independent Rayleigh Path Gains

We assume the path gains $\{h_i\}_{i=1}^P$ are independent and complex Gaussian: $h_i \sim \mathcal{CN}(0, |h_i|^2_{\text{avg}}), \qquad \sum_i |h_i|^2_{\text{avg}} = 1.$ Path delays $\tau_i$ and Dopplers $\nu_i$ are deterministic (geometry of the physical scene). The randomness is entirely in the complex amplitudes, which model the path's random phase and multipath scattering.

Theorem: PEP Scaling Under Rayleigh Path Gains

Under the independent Rayleigh path gain assumption, the PEP averaged over channel realizations scales as $\mathbb{E}_{\mathbf{h}}[\mathrm{PEP}(\mathbf{x} \to \hat{\mathbf{x}})] \;\sim\; \frac{C(\mathbf{x}, \hat{\mathbf{x}})}{\mathrm{SNR}^{d(\mathbf{x}, \hat{\mathbf{x}})}},$ at high SNR, where $d(\mathbf{x}, \hat{\mathbf{x}}) =$ rank of the matrix $\mathbf{E}(\mathbf{x}, \hat{\mathbf{x}}) = \mathbf{D}^{1/2}\,\mathbf{A}(\Delta\mathbf{x})\,\mathbf{D}^{1/2}$ with $\mathbf{D} = \text{diag}(|h_1|^2_{\text{avg}}, \ldots, |h_P|^2_{\text{avg}})$ and $\mathbf{A}(\Delta\mathbf{x})$ the "error covariance matrix" built from the error vector $\Delta\mathbf{x}$ .

The diversity order of the detector is $d = \min_{\mathbf{x} \neq \hat{\mathbf{x}}} d(\mathbf{x}, \hat{\mathbf{x}}).$

For a generic error vector $\Delta\mathbf{x}$ (no special structure that would collapse rank), $\mathbf{A}(\Delta\mathbf{x})$ has full rank $P$ , giving diversity $P$ . The minimum over all error pairs gives the worst-case, which is also typically $P$ except in pathological cases (like $\Delta\mathbf{x}$ being supported on only one DD cell; see exercises).

Proof

Quadratic form analysis

$\|\mathbf{H}_{DD}\Delta\mathbf{x}\|^2 = \sum_i |h_i|^2\,\phi_i(\Delta\mathbf{x}) + \text{cross terms}$ , where $\phi_i$ depends on the error vector. Writing as a matrix form: $\Delta\mathbf{x}^H \mathbf{H}_{DD}^{H} \mathbf{H}_{DD} \Delta\mathbf{x}$ .

Rank of the error matrix

Define $\mathbf{A}(\Delta\mathbf{x})$ such that the quadratic form is $\mathbf{h}^H \mathbf{A} \mathbf{h}$ with $\mathbf{h} = (h_1, \ldots, h_P)^T$ . The eigenvalues of $\mathbf{A}$ determine the scaling.

Gaussian quadratic form

For $\mathbf{h} \sim \mathcal{CN}(0, \mathbf{D})$ , the PEP $\mathbb{E}[Q(\sqrt{\mathbf{h}^H\mathbf{A}\mathbf{h}/(2\sigma^2)})]$ scales as $1/\text{det}(\mathbf{I} + \mathrm{SNR}\mathbf{D}^{1/2}\mathbf{A}\mathbf{D}^{1/2})$ at high SNR. Since $\text{det}(\mathbf{I} + \rho \mathbf{B}) \sim \rho^d$ for $\mathbf{B}$ of rank $d$ , we get PEP $\sim 1/\mathrm{SNR}^d$ .

Full diversity at generic error

$\mathbf{A}$ has full rank $P$ whenever the shift indices $(\ell_i, k_i)$ are distinct and the error vector has support spanning their translations. This is the generic case. $\blacksquare$

Example: Error Vector Supported on One DD Cell

An error vector has a single non-zero: $\Delta x_{DD}[\ell_0, k_0] = 1$ , all other cells zero. The channel has $P = 2$ paths at $(\ell_1, k_1)$ and $(\ell_2, k_2)$ . What is the diversity $d(\mathbf{x}, \hat{\mathbf{x}})$ ?

Solution

Error spread by channel

$\mathbf{H}_{DD}\Delta\mathbf{x}$ has non-zeros at $(\ell_0 + \ell_1, k_0 + k_1)$ and $(\ell_0 + \ell_2, k_0 + k_2)$ with amplitudes $h_1$ and $h_2$ . $\|\mathbf{H}_{DD}\Delta\mathbf{x}\|^2 = |h_1|^2 + |h_2|^2$ .

Matrix $\mathbf{A}$

$\mathbf{A}(\Delta\mathbf{x})$ has quadratic form $|h_1|^2 + |h_2|^2$ , equivalent to $\mathbf{A} = \text{diag}(1, 1)$ . Rank: 2.

Diversity

$d(\mathbf{x}, \hat{\mathbf{x}}) = 2 = P$ . Full diversity for this error pair.

Generic case

For any single-cell error vector, the diversity is exactly $P$ . The only error vectors that lose diversity are those with special cancellation structure — extremely rare.

PEP vs SNR for Different Error Vectors

Plot PEP as a function of SNR for (a) a single-cell error, (b) a two-cell error, and (c) a many-cell error. Observe that all three have slope $-P$ at high SNR — diversity is error-vector- independent for generic errors.

Parameters

P

4

Error vector type

M

16

N

16

Different Detectors, Same Diversity?

At first glance it seems puzzling that MMSE has diversity 1 while ML has diversity $P$ . Both act on the same channel and see the same quadratic form. The resolution: MMSE linearizes and discards off-diagonal terms that encode the path diversity; ML retains the full joint structure.

Concretely, the MMSE per-cell output is $\lambda_{n, m}^* y / |\lambda_{n, m}|^2$ — a single-path Rayleigh-like coefficient. ML minimizes over the joint constellation, exploiting the coupling across cells that the channel creates. This joint treatment is what extracts the diversity.

Message passing (Ch. 8) preserves the joint treatment via iterative marginalization — and hence achieves the full $P$ .

Pairwise Error Probability and Diversity