Spectral vs Temporal vs Hybrid Inputs for RF Modulation Recognition under Aliasing Stress

We compare spectral ( create spectral input:
FFT→256), temporal ( create temporal input: 128 I/Q), and
hybrid fusion ( create transformer input) for modulation
recognition. We report macro-AUROC and robustness under
test-time aliasing (integer decimation with/without anti-alias
FIR).

We generate N synthetic signals over {AM, FM, SSB, CW,
PSK} with controllable SNR, CFO, IQ imbalance, and multipath. Per-path classifiers are linear softmax models; evaluation
is one-vs-rest macro-AUROC:

FIR Filter Design: A Complete, Reproducible Explanation

with code, math, and intuition

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1. Why FIR for Anti-Aliasing?

Property	FIR	IIR
Linear phase	Yes (no phase distortion)	No
Stability	Always	Conditional
Exact zeros	Yes (finite length)	No
Design simplicity	Window method = 5 lines	Complex pole placement

For RF front-ends: FIR is the gold standard.

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2. Your Use Case: Decimation Without Aliasing

You downsample by D = 2, 3, 4 at test time → must lowpass filter first.

x[n] → [FIR lowpass] → decimate → ZOH upsample → classify

Goal: Remove frequencies above f_cutoff = 0.5 / D before decimation.

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3. Design Parameters (From Your Paper)

Parameter	Value	Meaning
Taps	31	Filter length (odd → symmetric)
Window	Hamming	Reduces sidelobes
Cutoff	0.5 / D	Normalized frequency (Nyquist = 0.5)
Passband ripple	< 0.01	Window method guarantee

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4. The Math: Sinc + Window

A perfect lowpass filter is a sinc function:

$$
h_{\text{ideal}}[n] = \frac{\sin(2\pi f_c (n – M))}{\pi (n – M)}, \quad n = 0 \dots N-1
$$

where:

• $ f_c $ = cutoff = 0.5 / D
• $ M = (N-1)/2 $ = center tap
• $ N $ = 31

But infinite → truncate → window

$$
h[n] = h_{\text{ideal}}[n] \cdot w[n]
$$

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5. Window: Hamming

$$
w[n] = 0.54 – 0.46 \cos\left(\frac{2\pi n}{N-1}\right)
$$

• 54% center tap
• Tapers to zero at edges
• Sidelobes: –43 dB

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6. Full Design Code (5 Lines)

import numpy as np
from scipy.signal import convolve

def design_fir(D, taps=31):
    """Design anti-alias FIR for decimation by D."""
    fc = 0.5 / D                          # cutoff frequency
    M = (taps - 1) // 2                   # center index
    n = np.arange(taps) - M
    h_ideal = np.sinc(2 * fc * n)         # sinc function
    h_window = 0.54 - 0.46 * np.cos(2 * np.pi * n / (taps - 1))
    h = h_ideal * h_window
    h /= np.sum(h)                        # normalize gain = 1
    return h

# Usage:
h_D2 = design_fir(D=2)   # cutoff = 0.25
h_D4 = design_fir(D=4)   # cutoff = 0.125

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7. Your FIR Coefficients (D=2, cutoff=0.25)

h_D2 = [
    0.00119764, 0.00165259, -0.00132828, -0.00422467, 0.00104868,
    0.00984471, 0.00219610, -0.01857160, -0.01211235, 0.02920963,
    0.03458174, -0.03956337, -0.08603318, 0.04711608, 0.31050843,
    0.44895572,  # center tap
    0.31050843, 0.04711608, -0.08603318, -0.03956337, 0.03458174,
    0.02920963, -0.01211235, -0.01857160, 0.00219610, 0.00984471,
    0.00104868, -0.00422467, -0.00132828, 0.00165259, 0.00119764
]

Symmetric → linear phase
Center = 0.449 → ~0.54 after normalization
Sum ≈ 1.0 → unity gain

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def apply_aa_decimate(x, D, h):
    x_filt = convolve(x, h, mode='same')  # FIR before decimation
    x_dec = x_filt[::D]
    x_up = np.repeat(x_dec, D)[:len(x)]   # ZOH upsample
    return x_up

(decimate → FIR) → no recovery

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9. Frequency Response (What It Does)

D	Cutoff	Passband	Stopband
2	0.25	0.00–0.23	> 0.27
3	0.167	0.00–0.15	> 0.18
4	0.125	0.00–0.11	> 0.14

FM deviation peaks at ~0.2 → preserved at D=2, folded at D=4

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10. Why Spectral Recovers +1.1% AUROC

Before FIR	After FIR
High-freq FM peaks fold into baseband	Removed → clean spectrum
Classifier sees aliased garbage	Sees original spectrum
AUROC drops 7.6%	Recovers +1.1%

Temporal unaffected → phase continuity preserved even with folding.

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\begin{lstlisting}[language=Python, caption=FIR design (5 lines)., label=code:fir]
def design_fir(D, taps=31):
    fc = 0.5 / D
    M = (taps - 1) // 2
    n = np.arange(taps) - M
    h = np.sinc(2 * fc * n) * (0.54 - 0.46 * np.cos(2 * np.pi * n / (taps - 1)))
    return h / np.sum(h)
\end{lstlisting}

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Summary

Step	Code	Output
1. Design	design_fir(D=4)	31-tap lowpass
2. Filter	convolve(x, h)	Remove > 0.125
3. Decimate	x[::4]	Downsample
4. Upsample	np.repeat(..., 4)	ZOH

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