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Seeing the Channel

Visualizing channel responses—impulse response, power delay profile, Doppler spectrum, and scattering function—provides intuition that numbers alone cannot convey. This section builds publication-quality channel visualizations.

Definition:
Power Delay Profile

The power delay profile (PDP) gives the average power at each excess delay $\tau$ :

$P(\tau) = E[|h(\tau)|^2]$

Key statistics derived from the PDP:

Mean excess delay: $\bar{\tau} = \frac{\sum_k P_k \tau_k}{\sum_k P_k}$
RMS delay spread: $\tau_{\text{rms}} = \sqrt{\overline{\tau^2} - \bar{\tau}^2}$

def pdp_stats(delays, powers):
    total = np.sum(powers)
    mean_delay = np.sum(delays * powers) / total
    rms_delay = np.sqrt(np.sum(delays**2 * powers)/total - mean_delay**2)
    return mean_delay, rms_delay

Definition:
Scattering Function

The scattering function $S(\tau, \nu)$ describes the channel's power distribution in both delay and Doppler dimensions:

$S(\tau, \nu) = |H(\tau, \nu)|^2$

where $H(\tau, \nu)$ is the delay-Doppler spreading function. The PDP is the Doppler-marginal, and the Doppler spectrum is the delay-marginal of $S(\tau, \nu)$ .

Example: Generating a 3GPP TDL Channel

Implement the 3GPP TDL-A channel model and visualize its impulse response and power delay profile.

Solution

TDL-A tap definition

# 3GPP TDL-A (TR 38.901, Table 7.7.2-2)
tdl_a = {
    'delays_ns': [0, 30, 70, 90, 110, 190, 410],
    'powers_dB': [0, -1, -2, -3, -8, -17.2, -20.8],
}
delays = np.array(tdl_a['delays_ns']) * 1e-9
powers = 10**(np.array(tdl_a['powers_dB'])/10)
powers /= np.sum(powers)  # normalize

Generate channel realization

rng = np.random.default_rng(42)
n_taps = len(delays)
h_taps = np.sqrt(powers) * (rng.standard_normal(n_taps) + 1j*rng.standard_normal(n_taps)) / np.sqrt(2)

Channel Impulse Response Viewer

Visualize channel impulse responses, power delay profiles, and frequency responses for different channel models.

Parameters

Wireless Channel Modeling Overview — Overview of wireless channel effects: path loss, shadow fading, and multipath fading at different spatial scales.

Quick Check

If the RMS delay spread is 100 ns, what is the approximate coherence bandwidth?

200 kHz

2 MHz

20 MHz

100 MHz

Correction:

2 MHz

Correct: $B_c \approx 1/(5\tau_{\text{rms}}) = 1/(5 \times 100\text{ns}) = 2$ MHz.

Key Takeaway

The power delay profile and Doppler spectrum are the two fundamental channel characterizations. They determine the coherence bandwidth ( $B_c \approx 1/5\tau_{\text{rms}}$ ) and coherence time ( $T_c \approx 0.423/f_d$ ), which dictate OFDM system design parameters.

Power Delay Profile

The average power as a function of excess delay, showing the multipath structure of the channel.

RMS Delay Spread

The second central moment of the power delay profile; characterizes the temporal dispersion of the channel.

Channel Visualization