Ferkans — Interactive Telecom Tutor

From Continuous to Discrete

The continuous-time channel $h(\tau; t)$ is the physical reality, but digital communication systems work with sampled signals. This section develops the tapped-delay-line (TDL) model — the standard discrete-time representation used in simulation, receiver design, and performance analysis throughout this book.

Definition:
Tapped-Delay-Line Channel Model

Sampling the channel at rate $1/T_s$ (where $T_s$ is the symbol period), the discrete-time baseband channel is

$r[n] = \sum_{l=0}^{L-1} h_l[n]\, s[n - l] + w[n]$

where:

$h_l[n] = h(lT_s; nT_s)$ is the $l$ -th tap coefficient at time index $n$
$L = \lceil \tau_{\max} / T_s \rceil + 1$ is the number of taps
$w[n] \sim \mathcal{CN}(0, N_0)$ is AWGN

Each tap $h_l[n]$ is a complex Gaussian process (Rayleigh or Ricean fading) with:

Average power $P_l$ from the power delay profile
Doppler spectrum $S_l(\nu)$ from the scattering function
Taps at different delays are uncorrelated (US assumption)

,

Definition:
Block Fading Approximation

When $T_c \gg T_{\text{block}}$ (the channel coherence time is much longer than a block/frame duration), we model the channel as constant within each block but independently fading between blocks:

$r[n] = h\, s[n] + w[n], \quad n = 0, 1, \ldots, N-1$

where $h \sim \mathcal{CN}(0, 1)$ (Rayleigh) is constant for all $N$ symbols in the block. Each new block draws an independent $h$ .

This is the simplest fading model and is widely used for information-theoretic analysis (Chapter 9).

Jakes' Fading Simulator

Complexity:

O(MN)

per channel realization

Input:

f_D

(max Doppler shift),

T_s

(sample period),

N

(number of samples),

M

(number of oscillators)

Output:

h[0], h[1], \ldots, h[N-1]

(Rayleigh fading samples)

1. Set

N_0 = (M - 1)/2

(number of oscillator pairs)

2. For

n = 0

to

N - 1

:

a.

h_I[n] = \sqrt{2}\cos(2\pi f_D t_n) + 2\sum_{k=1}^{N_0} \cos(\beta_k)\cos(2\pi f_k t_n)

b.

h_Q[n] = \sqrt{2}\sin(2\pi f_D t_n) + 2\sum_{k=1}^{N_0} \sin(\beta_k)\cos(2\pi f_k t_n)

c.

h[n] = (h_I[n] + j\,h_Q[n]) / \sqrt{2(N_0 + 1)}

where

t_n = nT_s

,

f_k = f_D\cos(2\pi k/M)

,

\beta_k = \pi k / (N_0 + 1)

.

Modern simulators use the sum-of-sinusoids method or IFFT-based filtering of white noise through the Doppler spectrum. The IFFT method has complexity $O(N\log N)$ and produces exactly the correct autocorrelation.

Tapped-Delay-Line Channel Model

Step-by-step animation of the TDL model: the input signal passes through delay elements, each tap is multiplied by a fading coefficient

h_l[n]

, and the weighted copies are summed with additive noise to produce the received signal.

The TDL model:

r[n] = \sum_{l=0}^{L-1} h_l[n]\,s[n-l] + w[n]

. Each tap fades independently according to the power delay profile.

Fading Channel Realisation

Watch a Rayleigh fading channel evolve in time. Adjust the Doppler shift to see how faster motion causes more rapid fluctuations. The dashed line shows the mean power.

Parameters

Max Doppler shift

f_D

(Hz)50

K-factor (dB, -100 = Rayleigh)-100

Duration (ms)100

Channel Model Classification

	Slow fading ( $T_s \ll T_c$ )	Fast fading ( $T_s \gg T_c$ )
Flat fading ( $W \ll B_c$ )	Single tap, constant per symbol	Single tap, varies within symbol
Frequency-selective ( $W \gg B_c$ )	Multiple taps, constant per symbol (most common)	Multiple taps, varies within symbol (rare)

Quick Check

A channel has maximum excess delay $\tau_{\max} = 5\,\mu$ s and the system uses symbol period $T_s = 1\,\mu$ s. How many taps does the TDL model need?

5

6

10

4

Correction:

6

Correct. $L = \lceil 5/1 \rceil + 1 = 6$ taps (indices 0 to 5).

🔧Engineering Note

Fading Channel Simulation Accuracy

When simulating fading channels for system performance evaluation, several practical issues affect accuracy:

Sample rate: The fading process must be sampled at $f_s \geq 2f_D$ (Nyquist on the Doppler bandwidth). In practice, use $f_s \geq 10 f_D$ for accurate envelope statistics. Under-sampling aliases the Doppler spectrum and produces incorrect temporal correlation.
Jakes vs IFFT method: The classical Jakes simulator with $M$ oscillators has complexity $O(MN)$ but produces only approximately correct statistics (the envelope is not perfectly Rayleigh for finite $M$ , and different realisations are correlated). The IFFT-based method (filter white noise through the Doppler spectrum) gives exact statistics with $O(N\log N)$ complexity and is preferred for standards-compliant simulations.
Tap correlation: In the TDL model, taps at different delays are assumed uncorrelated (US property). This is valid when the tap spacing $T_s$ exceeds the inverse of the maximum bandwidth. For very wideband systems (e.g., $W > 400$ MHz at mmWave), sub-path resolution may violate US, requiring the full GSCM instead of simplified TDL.

Practical Constraints

•
Fading sample rate must satisfy $f_s \geq 10 f_D$
•
IFFT-based simulation preferred over Jakes for accuracy

Tapped-Delay-Line Model

Discrete-time channel model: $r[n] = \sum_l h_l[n]\,s[n-l] + w[n]$ . Each tap $h_l$ fades independently according to the PDP.

Block Fading

A channel model where $h$ is constant within a block of symbols but changes independently between blocks. Used for information-theoretic analysis.

Discrete-Time Baseband Channel Model