Elliptic Lowpass 2nd Order: Maximum Stopband Attenuation at Minimum Filter Order
Elliptic (Cauer) filters achieve the steepest possible transition from passband to stopband for a given filter order. They pay for this with equiripple in both passband and stopband — but when embedded in a control loop or DSP pipeline, that trade-off is often a clear win. Two biquad multiplies, and you can place a transmission zero exactly where your interference lives.
Transfer Function — H(z)
Design parameters: fs = 44 100 Hz, fc = 1 000 Hz, rp = 0.5 dB, rs = 40 dB.
Applying scipy.signal.ellip(2, 0.5, 40, fc, fs=fs) with bilinear transform:
Unnormalized coefficients (scipy output, a₀ = 1.0):
| Coefficient | Unnormalized | Normalized (÷ a₀) |
|---|---|---|
| b₀ | 0.0156130224 | 0.0156130224 |
| b₁ | −0.0048446366 | −0.0048446366 |
| b₂ | 0.0156130224 | 0.0156130224 |
| a₀ | 1.0000000000 | — |
| a₁ | −1.7895617657 | −1.7895617657 |
| a₂ | 0.8175063706 | 0.8175063706 |
Since a₀ = 1.0 exactly (scipy.signal.ellip normalizes automatically), the final biquad form is:
$$H(z) = \frac{0.01561 - 0.00484,z^{-1} + 0.01561,z^{-2}}{1 - 1.78956,z^{-1} + 0.81751,z^{-2}}$$
Difference equation:
y[n] = 0.01561·x[n] − 0.00484·x[n−1] + 0.01561·x[n−2]
+ 1.78956·y[n−1] − 0.81751·y[n−2]
The symmetric b coefficients (b₀ = b₂) confirm the transmission zeros sit on the unit circle — a z-domain property unique to elliptic (and Type I FIR) designs. Those zeros fall at ±9 932 Hz, producing >116 dB of attenuation at that frequency.
Frequency Response
Bode plot: magnitude (top) and phase (bottom). fc = 1 kHz marker shown in red.
Quantitative observations:
Passband ripple ≤ 0.5 dB up to 1 000 Hz (by design, rp constraint).
−3 dB crossover at ≈ 1 381 Hz — above fc because fc defines the rp-dB edge, not the −3 dB point.
At 2 kHz: −8.7 dB | at 3 kHz: −16.4 dB | at 5 kHz: −27.2 dB.
Transmission zero at 9 932 Hz: >116 dB attenuation — effectively infinite for 16-bit and 24-bit systems.
Phase rolls off steeply around the zero; group delay is non-monotonic. Caution for timing-sensitive applications.
Python Implementation
from scipy import signal
B0, B1, B2 = 0.0156130224, -0.0048446366, 0.0156130224
A1, A2 = -1.7895617657, 0.8175063706
def filter_signal(x, fs=44100.0):
"""2nd-order elliptic lowpass, fc=1 kHz, rp=0.5 dB, rs=40 dB."""
return signal.lfilter([B0, B1, B2], [1.0, A1, A2], x)
To retarget fc, redesign with signal.ellip(2, 0.5, 40, fc, fs=fs) — do not attempt to frequency-scale these coefficients manually.
C Implementation — Direct Form II Transposed
#define B0 ( 0.0156130224f)
#define B1 (-0.0048446366f)
#define B2 ( 0.0156130224f)
#define A1 (-1.7895617657f)
#define A2 ( 0.8175063706f)
typedef struct { float w1, w2; } FilterState;
float filter_process(FilterState *s, float x) {
float y = B0*x + s->w1;
s->w1 = B1*x - A1*y + s->w2;
s->w2 = B2*x - A2*y;
return y;
}
Direct Form II Transposed keeps internal state bounded and minimises round-off noise propagation — the right choice for the aggressive coefficient ratios elliptic filters produce. Fixed-point note: a₁ = −1.789 has |a₁| > 1; use Q14 (not Q15) for a-coefficients to avoid overflow.
MATLAB Implementation
fs = 44100; fc = 1000;
[b, a] = ellip(2, 0.5, 40, fc/(fs/2));
figure; freqz(b, a, 2048, fs); % magnitude + phase
figure; zplane(b, a); % zeros ON unit circle confirmed
zplane is the fastest sanity check: zeros exactly on the unit circle confirm transmission nulls. Both poles inside the unit circle confirms stability.
Design Trade-offs
| Property | Elliptic 2nd Order | Butterworth 2nd Order |
|---|---|---|
| Passband flatness | Equiripple (rp = 0.5 dB) | Maximally flat |
| Rolloff at 2×fc | −8.7 dB | −6.0 dB |
| Rolloff at 3×fc | −16.4 dB | −12.0 dB |
| Transmission zeros | Yes (9 932 Hz, >116 dB null) | None |
| Group delay | Non-monotonic | Monotonic |
| Coefficient sensitivity | Moderate | Low |
The elliptic filter wins when you need maximum stopband rejection from minimum multiply budget — shutting down a specific harmonic, switching noise spike, or line frequency artifact with just one biquad. It loses when predictable group delay matters: sensor fusion, synchronization loops, or waveform-preserving applications should reach for Bessel or Butterworth instead.
Key Takeaways
Steepest rolloff for any filter order: 2nd-order elliptic often beats 4th-order Butterworth in the transition band.
Transmission zeros (b₀ = b₂, unit-circle zeros) create theoretically infinite attenuation at specific frequencies.
9 932 Hz null exceeds 116 dB — aim this at a known interference tone (e.g., switching harmonics at ~10 kHz).
Use Direct Form II Transposed in C/embedded: minimises state overflow and noise for aggressive a-coefficient values.
Fixed-point caution: |a₁| > 1 requires Q14 scaling; validate headroom before deploying on integer DSPs.
Related Posts
- Butterworth Highpass Filter 2nd Order: Maximally Flat, But At What Phase Cost? — contrasts maximally-flat magnitude with the equiripple trade-off explored here.
Which interference frequency in your system would benefit from an elliptic transmission zero? Drop the Hz value in the comments.