FIR filter design¶

Motivation¶

In the chaotic communication scheme, a finite impulse response (FIR) filter is inserted inside the Hénon feedback loop. This turns the standard 2‑D map into a higher‑dimensional system whose dynamics are shaped by the filter coefficients.

The motivation is two‑fold:

Band‑limiting. Chaotic signals produced by the unfiltered Hénon map have a broad, flat spectrum. Real communication channels have limited bandwidth ; the FIR filter constrains the chaotic signal to occupy only the desired frequency band.
Controllable chaos. By varying the filter order and cutoff, we can study which filter parameters preserve chaos and which suppress it : the core empirical question of the PFC.

`firwin` vs. MATLAB `fir1`¶

The project uses scipy.signal.firwin() rather than implementing its own FIR design. There is a documented off‑by‑one difference relative to MATLAB’s fir1:

MATLAB: fir1(N, wc) produces N+1 coefficients.
SciPy: firwin(N, wc) produces N coefficients.

To obtain the same filter as MATLAB’s fir1(N, wc), call firwin(N+1, wc). The chaotic-pfc codebase consistently uses firwin(Nss, wc) where Nss is the number of filter taps (coefficients), which matches the convention used in the academic article.

The fs=2 default in firwin means wc is in units of \(\pi\) rad/sample — consistent with the normalisation \(\omega_c / \pi\) used throughout the sweep literature.

Filter types¶

Four pass‑zero configurations are supported:

Type	Pass‑band	Stop‑band
`lowpass`	\([0, \omega_c]\)	\((\omega_c, \pi]\)
`highpass`	\((\omega_c, \pi]\)	\([0, \omega_c]\)
`bandpass`	\([\omega_c - \Delta/2,\; \omega_c + \Delta/2]\)	remainder
`bandstop`	remainder	\([\omega_c - \Delta/2,\; \omega_c + \Delta/2]\)

For bandpass and bandstop, the bandwidth \(\Delta\) defaults to \(0.0375\) — a narrow band typical of the project’s experimental setup.

Lowpass is the default because it is the most common configuration in communication engineering and serves as the baseline for all comparisons.

Windows¶

The FIR design uses one of seven window functions to shape the filter’s frequency response:

Window	Key	Sidelobe attenuation	Ripple
Hamming	`hamming`	~53 dB	Moderate
Hann	`hann`	~44 dB	Moderate
Blackman	`blackman`	~74 dB	Low
Blackman-Harris	`blackmanharris`	~92 dB	Very low
Rectangular	`boxcar`	~13 dB	High
Bartlett	`bartlett`	~25 dB	Moderate
Kaiser	`kaiser`	Adjustable via \(\beta\)	Adjustable

Kaiser window deserves special attention. The shape parameter \(\beta\) controls the trade‑off between main‑lobe width and sidelobe level. The project includes a dedicated \(\beta\)-sweep (chaotic-pfc run sweep beta-sweep) that evaluates Lyapunov exponents across \(\beta \in [2, 14]\) in steps of \(0.5\) for each filter type.

Stability and the fixed point¶

When an FIR filter is inserted into the Hénon feedback loop, the Jacobian matrix grows from \(2 \times 2\) to \((2 + N_c) \times (2 + N_c)\), where \(N_c\) is the number of filter coefficients that feed back into the dynamics.

The fixed point of the combined map‑plus‑filter system is computed by solving a linear system involving the filter coefficients. For the system to have a bounded attractor, the fixed point must be finite and the filter must be stable (all poles inside the unit circle).

The function lyapunov_max() handles the 4‑D pole‑filtered variant, which introduces an additional stability parameter (pole radius \(r\)) that must satisfy \(0 \le r < 1\).

Known limitations¶

Cutoff extremes. \(\omega_c \to 0\) produces a filter that passes almost nothing (all‑stop); \(\omega_c \to 1\) passes almost everything (all‑pass). Both extremes degenerate the filter and produce behaviour indistinguishable from the unfiltered map — the sweep grid excludes \(\omega_c = 0\) and \(\omega_c = 1\).
Order limits. For \(N_z \ge 40\), the filter design becomes numerically ill‑conditioned (the window function has extreme values at the filter edges). The sweep currently tops at \(N_z = 41\).
DC gain. FIR filters designed with firwin have a DC gain of approximately 1.0 for lowpass , but not exactly, due to window tapering. The chaotic map is sensitive to the exact coefficient magnitudes, which is why the FIR bank is precomputed once rather than reconstructed per grid point.