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The Central Object

Among Bello's four functions, the spreading function $h(\tau, \nu)$ has a privileged role. It is simultaneously (i) the most parsimonious description of the physical channel — a handful of Dirac impulses — and (ii) the object that diagonalizes the input-output map in the DD domain, as we will prove in Chapter 4. This section derives $h(\tau, \nu)$ from first principles, examines its support structure, and establishes the physical and mathematical constraints it obeys.

Definition:
Spreading Function

The delay-Doppler spreading function $h(\tau, \nu)$ represents the complex amplitude of the channel's response at delay $\tau$ and Doppler shift $\nu$ . Operationally, if a signal $x(t)$ is transmitted, the received signal is $y(t) \;=\; \iint h(\tau, \nu)\,x(t - \tau)\,e^{j 2\pi \nu t}\,d\tau\,d\nu \;+\; \mathbf{w}(t).$ The kernel $h(\tau, \nu)$ acts by delaying the signal by $\tau$ and Doppler-shifting it by $\nu$ , summed over all delay-Doppler pairs with the appropriate complex amplitude.

The exponential $e^{j 2\pi \nu t}$ is the Doppler modulation seen at the receiver. The sign convention matches Hadani et al. (2017); some earlier references use $e^{-j 2\pi \nu t}$ — check before applying formulas.

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Theorem: Discrete Form of the Spreading Function

For the physical multipath channel $h(\tau, t) = \sum_{i=1}^P h_i\,e^{j2\pi\nu_i t}\,\delta(\tau - \tau_i)$ , the spreading function is $h(\tau, \nu) \;=\; \sum_{i=1}^P h_i\,\delta(\tau - \tau_i)\,\delta(\nu - \nu_i).$ The spreading function is the sum of $P$ two-dimensional Dirac impulses, one for each resolvable physical path.

Each path has one delay and one Doppler, hence one point in the plane. With $P$ paths, the spreading function has support of cardinality $P$ .

Proof

Apply the definition

By DDelay-Doppler Spreading Function, $h(\tau, \nu) \;=\; \int h(\tau, t)\,e^{-j 2\pi \nu t}\,dt.$

Substitute the multipath model

$h(\tau, \nu) \;=\; \int \left(\sum_i h_i\,e^{j2\pi\nu_i t}\,\delta(\tau - \tau_i)\right) e^{-j 2\pi \nu t}\,dt.$ $

Pull the delay $\delta$ out of the time integral

The $\delta(\tau - \tau_i)$ factors do not depend on $t$ : $h(\tau, \nu) \;=\; \sum_i h_i\,\delta(\tau - \tau_i) \int e^{-j 2\pi (\nu - \nu_i) t}\,dt.$

Evaluate the remaining integral

The integral $\int e^{-j 2\pi (\nu - \nu_i) t}\,dt = \delta(\nu - \nu_i)$ . Substituting yields $h(\tau, \nu) \;=\; \sum_{i=1}^P h_i\,\delta(\tau - \tau_i)\,\delta(\nu - \nu_i). \;\blacksquare$

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Key Takeaway

$P$ parameters, not $MN$ . The spreading function of a physical multipath channel is described by only $3P$ real parameters — $P$ delays, $P$ Dopplers, and $P$ complex gains. For typical channels $P \leq 20$ . In contrast, the time-varying transfer function $H(f, t)$ evaluated on an $M \times N$ time-frequency grid requires $MN$ complex samples, with $MN$ often $10^4$ or more. The difference between tractable estimation and hopeless undersampling lives in that contrast.

Visualizing $h(\tau, \nu)$ as Sparse Spikes

Each path of the physical channel contributes one impulse in the delay-Doppler plane. Vary the number of paths $P$ , the maximum delay spread, and the maximum Doppler spread. Observe that the support of $h(\tau, \nu)$ is a finite set of points; the impulse magnitudes reflect the relative path powers.

Parameters

Number of paths

P

5

Max delay

\tau_{\max}

(

\mu

s)5

Max Doppler

\nu_{\max}

(Hz)1000

Realization seed42

Example: Counting Parameters in a Realistic Channel

A typical urban macro cell at 3.5 GHz has $P = 8$ significant paths. The delay spread is $\tau_{\max} = 4\,\mu\text{s}$ , the Doppler spread $f_D = 300$ Hz (vehicular mobility). The system uses an OTFS frame with $M = 512$ delay bins and $N = 128$ Doppler bins. Count the number of real parameters needed to describe the channel in (a) the DD domain, (b) the time-frequency domain, (c) the time-varying impulse response on a sampled grid.

Solution

DD representation

$P = 8$ paths, each with a complex gain (2 real numbers), delay (1), and Doppler (1). Total: $8 \times 4 = 32$ real numbers.

TF representation

The channel on the TF grid is $H[\ell, k]$ for $\ell = 0, \ldots, M-1$ and $k = 0, \ldots, N-1$ , each a complex scalar. Total: $M N \times 2 = 2 \cdot 512 \cdot 128 = 131{,}072$ real numbers.

Time-varying impulse response

Sampling $h(\tau, t)$ at delay resolution $\Delta\tau = 1/W \approx 10$ ns and time resolution $\Delta t = 10\,\mu\text{s}$ (to resolve Doppler up to $1/(2\Delta t) = 50$ kHz, well beyond the actual $f_D$ ) over one frame gives roughly $L_\tau = 400$ delay taps and $L_t = 500$ time samples per tap. With $L_\tau L_t = 2 \times 10^5$ complex entries, this is $\sim 4 \times 10^5$ real numbers.

The ratio

The DD description is four orders of magnitude more compact than the TF description and five orders of magnitude more compact than the time-varying impulse response on a uniform grid. This is the estimation lever OTFS pulls: we fit $8$ complex gains plus $16$ real support parameters, not $10^5$ samples. Intuitively, what happens is this — the channel only has a small number of physical degrees of freedom, and the DD domain exposes them directly.

🚨Critical Engineering Note

Sparsity Drives Pilot Overhead

The $P$ -parameter structure of $h(\tau, \nu)$ has direct consequences for pilot-aided channel estimation. In the TF domain, the number of independent pilots required to estimate $H(f, t)$ on an $M \times N$ grid at the Nyquist rate scales as $MN / (\tau_{\max} f_D \, T W)$ ; for typical system parameters this is $5$ – $10\%$ of all resources. In the DD domain, by contrast, a single pilot impulse suffices to identify all $P$ paths provided guard intervals accommodate $\tau_{\max}$ and $f_D$ . The resulting net pilot overhead can drop below $1\%$ — a concrete and measurable gain that motivates much of the cell-free OTFS literature (Mohammadi et al. 2023).

Practical Constraints

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Embedded pilot requires guard region of size $2\,l_{\max} + 1$ in delay and $2\,k_{\max} + 1$ in Doppler
•
Guard size is fixed by physical channel spread, not by grid size
•
Pilot power must clear threshold for reliable path detection (typically 20–30 dB)

Causality and the Delay Axis

The physical channel is causal: paths with $\tau_i < 0$ are unphysical. This means $h(\tau, \nu) = 0$ for $\tau < 0$ . On the Doppler axis there is no causality constraint; $\nu_i$ can be positive (approaching reflector) or negative (receding). The support of $h(\tau, \nu)$ therefore lives in the half-plane $\{\tau \geq 0\} \times \{\nu \in \mathbb{R}\}$ , and in practice is bounded: $\tau \in [0, \tau_{\max}]$ and $\nu \in [-f_D, +f_D]$ .

Three Paths Becoming Three DD Spikes

A three-path channel starts as three overlapping time-varying echoes (left). We Fourier-transform in time to obtain the spreading function (right): three isolated impulses at

(\tau_i, \nu_i)

. The dense temporal behavior and the sparse DD representation describe the same channel.

Common Mistake: Units in the DD Plane

Mistake:

Treating $(\tau, \nu)$ as if they were two samples of the same variable and plotting them without attention to scale. Delays are in microseconds; Doppler shifts are in hertz. A naive scatter plot with $\tau$ and $\nu$ on the same axis is meaningless.

Correction:

Always annotate the DD plot with physical units: $\tau$ in $\mu$ s (or ns for wideband), $\nu$ in Hz (or kHz at mmWave). In the discrete DD grid, the axes become indices $\ell \in \{0, \ldots, M-1\}$ and $k \in \{0, \ldots, N-1\}$ , with physical spacings $\Delta\tau = 1/W$ and $\Delta\nu = 1/T$ . We will formalize the discrete grid in Chapter 3.

Spreading function

The delay-Doppler representation of a linear time-varying channel, defined as $h(\tau, \nu) = \int h(\tau, t)\,e^{-j 2\pi \nu t}\,dt$ . For a physical multipath channel with $P$ paths, it is a sum of $P$ 2D Dirac impulses. This object is the central channel representation in OTFS.

Bello functions

The four equivalent descriptions of a linear time-varying channel — $h(\tau, t)$ , $H(f, t)$ , $h(\tau, \nu)$ , $H(f, \nu)$ — introduced by P. A. Bello in 1963 and related pairwise by 1D Fourier transforms.

The Spreading Function