Domain-Specific Plot Types

Specialized Plots for Communications and RF

Every engineering domain has its own plot types. Wireless engineers use Smith charts for impedance matching, polar plots for antenna patterns, eye diagrams for ISI analysis, and waterfall plots for spectrum monitoring. These are all built on top of Matplotlib's primitives.

Definition:
Smith Chart

The Smith chart maps impedance $Z_L$ to reflection coefficient $\Gamma$ via:

$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$

where $Z_0$ is the characteristic impedance (typically 50 $\Omega$ ). The chart is a unit disk in the $\Gamma$ -plane with circles of constant resistance and reactance.

# Plot a point at Z_L = 25 + j50 ohm
Z0 = 50
Z_L = 25 + 50j
Gamma = (Z_L - Z0) / (Z_L + Z0)
ax.plot(Gamma.real, Gamma.imag, 'ro', ms=8)

The scikit-rf package provides production-quality Smith charts: import skrf; skrf.plotting.smith().

Definition:
Eye Diagram

An eye diagram overlays many symbol periods of a received signal to reveal inter-symbol interference (ISI), timing jitter, and noise:

# Overlay consecutive symbol periods
symbols_per_trace = 2
n_samples_per_symbol = 16
segment_len = symbols_per_trace * n_samples_per_symbol

for i in range(0, len(signal) - segment_len, n_samples_per_symbol):
    segment = signal[i:i + segment_len]
    ax.plot(t_segment, segment, color='blue', alpha=0.02, lw=0.5)

A "wide open eye" indicates low ISI; a "closed eye" indicates severe distortion.

Definition:
Polar Plots for Antenna Patterns

ax = fig.add_subplot(111, projection='polar') creates a polar axes for antenna radiation patterns:

theta = np.linspace(0, 2*np.pi, 361)
pattern_db = 20 * np.log10(np.abs(np.sinc(2*np.sin(theta))) + 1e-10)

fig, ax = plt.subplots(subplot_kw={'projection': 'polar'})
ax.plot(theta, np.clip(pattern_db, -30, 0))
ax.set_rmax(0)
ax.set_rmin(-30)
ax.set_title('Antenna Pattern (dBi)')

Theorem: Eye Opening and BER

The vertical eye opening $d_{\text{eye}}$ at the optimal sampling instant is related to the noise margin. For binary signaling with noise variance $\sigma^2$ :

$P_b \approx Q\!\left(\frac{d_{\text{eye}}}{2\sigma}\right)$

A wider eye opening means lower BER. The horizontal eye opening indicates timing margin for the clock recovery circuit.

The eye diagram is a visual summary of the channel's impact on the signal. If you can "see through" the eye, the receiver can reliably decide the transmitted bit.

Example: Eye Diagram with Raised Cosine Pulse Shaping

Generate an eye diagram for BPSK with raised cosine pulse shaping at different rolloff factors.

Solution

Implementation

from scipy.signal import firwin
import numpy as np
import matplotlib.pyplot as plt

sps = 16      # samples per symbol
n_sym = 200
rng = np.random.default_rng(42)
bits = 2 * rng.integers(0, 2, n_sym) - 1

# Upsample + pulse shape
upsampled = np.zeros(n_sym * sps)
upsampled[::sps] = bits
h = firwin(6*sps + 1, 1/sps)
signal = np.convolve(upsampled, h, mode='same')

# Add noise
snr_db = 20
noise = rng.standard_normal(len(signal)) * 10**(-snr_db/20)
rx = signal + noise

# Plot eye diagram
fig, ax = plt.subplots(figsize=(6, 4))
seg_len = 2 * sps
t = np.linspace(0, 2, seg_len)
for i in range(0, len(rx) - seg_len, sps):
    ax.plot(t, rx[i:i+seg_len], color='#2563EB',
            alpha=0.03, lw=0.5)
ax.set(xlabel='Time (symbol periods)', ylabel='Amplitude',
       title='Eye Diagram (BPSK, raised cosine)')
ax.grid(True, alpha=0.3)

Example: Waterfall Spectrum Plot

Create a waterfall (spectrogram-style) plot showing spectral occupation over time for a frequency-hopping signal.

Solution

Implementation

fig, ax = plt.subplots(figsize=(8, 5))

n_hops = 20
n_freq = 256
rng = np.random.default_rng(42)
freqs = np.linspace(-500, 500, n_freq)

waterfall = np.zeros((n_hops, n_freq))
for i in range(n_hops):
    center = rng.choice([-300, -100, 100, 300])
    bw = 50
    mask = np.abs(freqs - center) < bw
    waterfall[i, mask] = 1.0 + 0.2 * rng.standard_normal(mask.sum())
    waterfall[i] += 0.05 * rng.standard_normal(n_freq)

ax.imshow(waterfall, aspect='auto', origin='lower',
          extent=[freqs[0], freqs[-1], 0, n_hops],
          cmap='inferno')
ax.set(xlabel='Frequency (kHz)', ylabel='Hop Index',
       title='Frequency Hopping Waterfall')

Eye Diagram Explorer

Adjust SNR and rolloff factor to see how they affect the eye opening.

Parameters

Antenna Pattern Viewer

Explore how array size and element spacing affect the antenna radiation pattern.

Parameters

Domain-Specific Plot Gallery — Four domain-specific plot types: Smith chart, eye diagram, polar radiation pattern, and waterfall spectrum.

Domain-Specific Plot Types

Plot Type	Shows	Axes	Library
Smith Chart	Impedance/reflection coefficient	Complex plane (unit disk)	scikit-rf
Eye Diagram	ISI, timing jitter, noise margin	Time vs amplitude (overlaid)	Matplotlib
Polar Pattern	Antenna radiation pattern	Angle vs gain (dB)	Matplotlib polar
Waterfall	Spectrum over time	Frequency vs time	Matplotlib imshow
Constellation	Modulation symbol positions	I vs Q plane	Matplotlib scatter

Why This Matters: Eye Diagrams in 5G NR

In 5G NR, eye diagrams are critical for verifying transmitter signal quality at the antenna port. The eye opening must exceed thresholds specified in 3GPP TS 38.101. Excessive ISI from digital-to-analog converter (DAC) impairments or PA nonlinearity closes the eye, increasing EVM beyond the allowed limit.

Common Mistake: Polar Plot Angle Units

Mistake:

Passing angles in degrees to ax.plot(theta, r) on a polar axes. Matplotlib polar axes expect radians.

Correction:

Convert degrees to radians: ax.plot(np.radians(theta_deg), r).

Quick Check

What does a 'closed eye' in an eye diagram indicate?

Low noise

Perfect channel equalization

Severe inter-symbol interference (ISI)

High data rate

Correction:

Severe inter-symbol interference (ISI)

When the eye closes, traces from adjacent symbols overlap, making reliable bit decisions impossible.

Historical Note: The Smith Chart (1939)

1939

Phillip Hagar Smith invented the Smith chart in 1939 while working at Bell Labs. Before computers, RF engineers used the Smith chart as a graphical calculator for impedance matching and transmission line analysis. Despite modern simulation tools, the Smith chart remains the standard visualization for S-parameters and impedance data in RF engineering.

Historical Note: Eye Diagram Origins

1960s

Eye diagrams emerged from oscilloscope measurements in the 1960s at Bell Labs. Engineers triggered the oscilloscope at the symbol rate and overlaid thousands of traces. The resulting pattern resembled a human eye, giving the technique its name. Today, digital oscilloscopes and simulation software generate eye diagrams automatically.

Smith Chart

A graphical representation of the complex reflection coefficient plane, used for impedance matching and S-parameter visualization in RF engineering.

Eye Diagram

A visualization created by overlaying many consecutive symbol periods of a signal, revealing ISI, noise margin, and timing jitter.

Inter-Symbol Interference (ISI)

Distortion where energy from one symbol period leaks into adjacent symbol periods, caused by multipath or band-limited channels.

Animations Chapter Summary

Domain-Specific Plot Types

Specialized Plots for Communications and RF

Definition: Smith Chart

Definition: Eye Diagram

Definition: Polar Plots for Antenna Patterns

Theorem: Eye Opening and BER

Example: Eye Diagram with Raised Cosine Pulse Shaping

Implementation

Example: Waterfall Spectrum Plot

Implementation

Eye Diagram Explorer

Parameters

Antenna Pattern Viewer

Parameters

Domain-Specific Plot Gallery

Domain-Specific Plot Types

Why This Matters: Eye Diagrams in 5G NR

Common Mistake: Polar Plot Angle Units

Quick Check

Historical Note: The Smith Chart (1939)

Historical Note: Eye Diagram Origins

Smith Chart

Eye Diagram

Inter-Symbol Interference (ISI)

Definition:
Smith Chart

Definition:
Eye Diagram

Definition:
Polar Plots for Antenna Patterns