Ferkans — Interactive Telecom Tutor

The Channel Moves — And So Must Our Model

The LTI framework of Section 4.1 assumes the system does not change over time. But a wireless channel between a moving user and a base station changes continuously: scatterers move, the propagation path lengths vary, and the received signal fades. To model this, we need linear time-variant (LTV) systems — a generalisation where the impulse response depends on both the observation time $t$ and the delay $\tau$ .

Definition:
Linear Time-Variant (LTV) System

An LTV system maps input $x(t)$ to output $y(t)$ via the time-variant impulse response $h(t, \tau)$ :

$y(t) = \int_{-\infty}^{\infty} h(t, \tau)\,x(t - \tau)\,d\tau.$

Here $h(t, \tau)$ is the response at time $t$ to an impulse applied $\tau$ seconds earlier.

$t$ : observation time
$\tau$ : delay variable

Special case: If $h(t, \tau) = h(\tau)$ (independent of $t$ ), the system is LTI and the expression reduces to standard convolution.

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Definition:
Time-Variant Transfer Function

The time-variant transfer function is the Fourier transform of $h(t, \tau)$ with respect to $\tau$ :

$H(t, f) = \int_{-\infty}^{\infty} h(t, \tau)\,e^{-j2\pi f\tau}\,d\tau.$

At each time $t$ , $H(t, f)$ describes the channel's frequency response. The input-output relation becomes

$Y(t, f) = H(t, f)\,X(f)$

only instantaneously — unlike LTI systems, different times see different frequency responses.

Definition:
Doppler Spread Function

Taking the Fourier transform of $h(t, \tau)$ with respect to $t$ yields the spreading function:

$S(\nu, \tau) = \int_{-\infty}^{\infty} h(t, \tau)\,e^{-j2\pi \nu t}\,dt$

where $\nu$ is the Doppler frequency (Hz).

The spreading function describes how the channel spreads an input impulse simultaneously in delay ( $\tau$ ) and Doppler ( $\nu$ ). For the wireless channel:

$\tau$ spread $\to$ delay spread (multipath)
$\nu$ spread $\to$ Doppler spread (mobility)

Theorem: Bello's System Functions

An LTV system is completely described by four equivalent system functions, related by Fourier transforms:

Function	Variables	FT pair
$h(t, \tau)$	time, delay	$\xleftrightarrow{\mathcal{F}_t}$
$S(\nu, \tau)$	Doppler, delay	$\xleftrightarrow{\mathcal{F}_\tau}$
$H(t, f)$	time, frequency	$\xleftrightarrow{\mathcal{F}_t}$
$B(\nu, f)$	Doppler, frequency	—

Any one determines all others. This framework, due to Bello (1963), is the standard for wireless channel characterisation.

Think of it as a 2D Fourier transform grid. The delay–time function $h(t,\tau)$ sits in one corner; successive Fourier transforms in $t$ or $\tau$ move you to the other three corners.

Proof

Relationships

$S(\nu, \tau) = \mathcal{F}_t\{h(t, \tau)\}$
$H(t, f) = \mathcal{F}_\tau\{h(t, \tau)\}$
$B(\nu, f) = \mathcal{F}_t\{H(t, f)\} = \mathcal{F}_\tau\{S(\nu, \tau)\}$

Each transform pair follows from the standard Fourier transform definitions. $\blacksquare$

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Historical Note: Philip Bello and the System Functions Framework

In 1963, Philip Bello published his landmark paper "Characterization of Randomly Time-Variant Linear Channels," establishing the four system functions framework that remains the standard model for wireless channels to this day. Bello's insight was to apply the theory of LTV systems specifically to radio channels, introducing the key assumptions (WSSUS) that make statistical characterisation tractable.

Definition:
Coherence Time and Coherence Bandwidth

The coherence time $T_c$ is the time duration over which the channel impulse response remains approximately constant:

$T_c \approx \frac{1}{B_D}$

where $B_D$ is the Doppler spread (maximum Doppler frequency).

The coherence bandwidth $B_c$ is the frequency range over which the channel frequency response is approximately flat:

$B_c \approx \frac{1}{\sigma_\tau}$

where $\sigma_\tau$ is the RMS delay spread.

These are approximate inverse relationships:

Large Doppler spread $\Rightarrow$ small coherence time (fast fading)
Large delay spread $\Rightarrow$ small coherence bandwidth (frequency-selective fading)

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Definition:
Quasi-Static (Block Fading) Approximation

If the symbol duration $T_{\text{sym}} \ll T_c$ , the channel changes negligibly over one symbol and can be treated as LTI within each symbol. This is the quasi-static or block fading model:

$h(t, \tau) \approx h(\tau) \quad \text{for } t \in [0, T_{\text{sym}}].$

The channel is modelled as a sequence of LTI systems, each with a different (random) impulse response. This simplification is valid for most current cellular systems (LTE, 5G NR) at pedestrian speeds and is widely used in information-theoretic analyses.

Example: Doppler Spread from Vehicle Speed

A mobile terminal moves at $v = 120$ km/h. The carrier frequency is $f_c = 2$ GHz. Find the maximum Doppler frequency, the coherence time, and determine whether a symbol duration of $T_{\text{sym}} = 71\;\mu$ s (one OFDM symbol in LTE) satisfies the quasi-static assumption.

Solution

Maximum Doppler frequency

$f_D = \frac{v}{c} f_c = \frac{120/3.6}{3 \times 10^8} \times 2 \times 10^9 = \frac{33.33}{3 \times 10^8} \times 2 \times 10^9 = 222.2 \text{ Hz}.$ $

Coherence time

$T_c \approx \frac{1}{f_D} = \frac{1}{222.2} \approx 4.5 \text{ ms}.$ $

Quasi-static check

$T_{\text{sym}} = 71\;\mu\text{s} \ll T_c = 4.5\;\text{ms}$ .

The ratio $T_{\text{sym}}/T_c \approx 0.016 \ll 1$ , so the quasi-static assumption is well justified. The channel is approximately constant over each OFDM symbol. $\blacksquare$

Common Mistake: Time-Variant Impulse Response Conventions

Mistake:

Confusing $h(t, \tau)$ with $h(\tau, t)$ or treating $\tau$ as absolute time rather than delay.

Correction:

In the convention used here (and in most communications texts), $h(t, \tau)$ means: the output at time $t$ due to an impulse at time $t - \tau$ . So $\tau$ is the delay (time elapsed since the impulse), and $t$ is the observation time. Some references (e.g., Oppenheim & Willsky) define $h(t, \sigma)$ where $\sigma$ is the absolute time of the impulse. The two are related by $\sigma = t - \tau$ . Always check which convention a reference uses before borrowing results.

LTI vs. LTV Systems

Property	LTI	LTV
Impulse response	$h(\tau)$	$h(t, \tau)$
Transfer function	$H(f)$	$H(t, f)$
Input–output (time)	$y = x * h$	$y(t) = \int h(t,\tau) x(t-\tau)\,d\tau$
Input–output (freq)	$Y(f) = X(f)H(f)$	No simple product
Eigenfunctions	$e^{j2\pi ft}$	None in general
When to use	Static / slowly varying channel	Mobile / fast fading channel

Quick Check

A mobile at 60 km/h uses a 900 MHz carrier. Approximately what is the coherence time?

$0.18$ ms

$20$ ms

$200$ ms

$2$ ms

Correction:

20

ms

Correct. $v = 60/3.6 = 16.67$ m/s, $f_D = vf_c/c = 50$ Hz, $T_c \approx 1/f_D = 20$ ms.

Why This Matters: From LTV Theory to Statistical Channel Models

The LTV framework introduced here provides the mathematical language for the wireless channel. Chapter 6 (Small-Scale Fading and Statistical Channel Models) applies this framework to model real wireless channels: Rayleigh and Ricean fading statistics arise from the random time-variant impulse response $h(t, \tau)$ , the WSSUS assumption makes Bello's system functions tractable, and the coherence time/bandwidth defined here directly determine whether fading is flat or frequency-selective.

Delay–Doppler Processing in Modern Waveforms

The spreading function $S(\nu, \tau)$ places the channel in the delay–Doppler domain — exactly the domain exploited by OTFS (Orthogonal Time Frequency Space) modulation. OTFS places information symbols on a delay–Doppler grid and processes the channel in a domain where it is sparse and quasi-static even at high mobility. The OTFS book covers this in depth, building directly on the Bello system functions framework from this section.

Linear Time-Variant System

A linear system whose impulse response $h(t, \tau)$ depends on both observation time $t$ and delay $\tau$ . The wireless channel is the prototypical LTV system.

Coherence Time

The time duration over which the channel response is approximately constant. $T_c \approx 1/B_D$ where $B_D$ is the Doppler spread.

Doppler Spread

The range of Doppler frequencies introduced by relative motion between transmitter and receiver. $f_D = vf_c/c$ .

Coherence Bandwidth

The frequency range over which the channel has approximately constant gain. $B_c \approx 1/\sigma_\tau$ where $\sigma_\tau$ is the RMS delay spread.

Linear Time-Variant Systems