Filter Frequency Response — Reading the Plot

01 — What the Plot Shows

What the Plot Shows

A filter frequency response plot is almost always |S₂₁| (transmission) in dB on a vertical axis, against frequency — usually log-scaled — on the horizontal axis. Two sign conventions are both common: some plots show gain (0 dB in the passband, negative going down as loss increases); others show insertion loss / attenuation as a positive number that increases downward or upward depending on the tool. Always check the axis label before reading a number off a plot — a "20 dB" stopband spec means different things on a gain-labelled vs attenuation-labelled axis.

Reading the Y-Axis

Gain convention: IL(f) = 20·log₁₀|S₂₁(f)| — 0 dB = lossless, more negative = more attenuation
Attenuation convention: A(f) = −20·log₁₀|S₂₁(f)| — positive number, larger = more attenuation

Both describe the same physical curve, just flipped in sign. This page uses the gain convention (0 dB at the top) to match the calculators elsewhere on RFLab.

Log frequency axis: because filter behaviour spans decades of frequency, the X-axis is almost always logarithmic. Equal spacing on a log axis means equal ratios of frequency, not equal differences — this is why rolloff is quoted in dB/decade or dB/octave rather than dB/Hz.

02 — Passband

Passband — Insertion Loss & Ripple

The passband is the frequency region where the filter is meant to pass the signal with minimal loss. What "flat" means inside that band depends entirely on the approximation used to design the filter.

Insertion Loss (IL)

Two Different "Insertion Loss" Numbers

Synthesis-model IL — the normalised response of an ideal, lossless LC ladder. In the passband this is 0 dB minus any designed-in ripple.
Real-world IL — the actual measured loss, which is always worse than the model because inductors and capacitors have finite Q. From the filter theory page: IL_min ≈ (4.343/Q_L)·Σg_k dB at band centre.

A plot showing 0.05 dB "ripple" from simulation and a measured 0.6 dB passband loss on the bench are not contradicting each other — one is the ideal model, the other includes component loss.

Ripple — Flat vs Equiripple

Approximation	Passband shape	What you'll see on the plot
Butterworth	Maximally flat	Monotonic, smooth roll from 0 dB down to −3.01 dB exactly at f_c — no ripple, no bumps.
Chebyshev	Equiripple	Oscillates between 0 dB and −A_p dB a number of times equal to the filter order — count the ripples to sanity-check the order.
Bessel	Flat (looser)	Also monotonic like Butterworth, but noticeably softer near f_c — the price paid for linear phase.

Quick check: if you can count N distinct ripple peaks in the passband on a measured or simulated plot, that is strong visual confirmation the filter is order N Chebyshev.

03 — The Rolloff Region

The Rolloff Region

Between the passband and the stopband sits the transition (rolloff) region, where attenuation increases with frequency. Its steepness — set by the filter order — is the single most important number for reading how selective a filter is.

Asymptotic Rolloff Rate

All-pole responses (Butterworth, Chebyshev, Bessel) roll off at −20n dB/decade ≡ −6n dB/octave far from cutoff, where n is the filter order.

n=3 → −60 dB/decade (−18 dB/oct) · n=5 → −100 dB/decade (−30 dB/oct) · n=7 → −140 dB/decade (−42 dB/oct)

Common measurement mistake: reading the slope right next to f_c and expecting it to already equal −20n dB/decade. The response hasn't reached its asymptotic slope that close in — measure the drop over at least one full decade (or use two well-separated points well into the stopband) to get an honest slope reading.

Reading Rolloff From Two Plot Points

1

Pick two points at least a decade apart in the transition/stopband, e.g. 2f_c = −24 dB and 20f_c = −84 dB (read straight off the Y-axis).

2

Δ(dB) = −84 − (−24) = −60 dB over log₁₀(20/2) = 1 decade

3

Slope = −60 dB/decade ⇒ n = 60/20 = 3

✓ A −60 dB/decade slope confirms a 3rd-order filter, independent of whether you know the schematic.

04 — Stopband

Stopband — Attenuation & Selectivity

The stopband is where the filter is specified to reject the signal. Two numbers matter: how much attenuation is achieved (A_s), and how close to the passband that attenuation starts (the selectivity factor).

Selectivity Factor

Ω_s = f_s/f_c (LP/HP) — how many multiples of cutoff away the required stopband attenuation must be reached.
A smaller Ω_s (stopband edge close to cutoff) demands a steeper rolloff → higher order, for the same A_s.

Reading	What it tells you
Attenuation floor doesn't keep dropping past −60 to −80 dB	You're seeing the simulator's/measurement's noise floor or component parasitics (leakage across the board), not the ideal response — real filters rarely exceed 60–100 dB of measurable rejection.
A narrow spike back up in an otherwise deep stopband	A transmission zero has been placed there deliberately (elliptic-type designs) — or it's a parasitic resonance in a physical layout that wasn't intended.
Stopband attenuation ripples instead of monotonically increasing	Normal for Chebyshev/Elliptic-derived stopbands with finite transmission zeros; unexpected for a plain Butterworth/Bessel design — check the approximation type if you see this.

05 — Group Delay & Phase

Group Delay & Phase

The magnitude plot only tells half the story. A filter also imposes a frequency-dependent time delay, which distorts wideband signals even when the magnitude response looks perfectly flat.

Group Delay

τ(ω) = −dφ/dω — the delay experienced by a narrowband envelope centred at ω
Constant τ across the passband ⇒ no phase distortion of a modulated signal. Varying τ ⇒ different frequency components of the signal arrive at different times — waveform/eye distortion.

Approximation	Group delay shape	Why
Bessel	Nearly flat across the passband	Designed explicitly for maximally-flat group delay — linear phase by construction.
Butterworth	Gentle peak near f_c	Moderate phase curvature near cutoff from the maximally-flat magnitude constraint.
Chebyshev	Sharp peak near f_c, worst of the three	The steeper the magnitude rolloff, the more phase has to curve near cutoff — a consequence of the Bode gain–phase relationship.

Why it matters: a wideband QAM or OFDM signal passed through a Chebyshev filter with a sharp band-edge group-delay peak will show constellation smearing and EVM degradation even if the magnitude response perfectly meets its IL/ripple/attenuation spec. Check GD flatness explicitly for anything beyond narrowband modulation.

Group Delay Analyser — measure GD from a real .s2p file →

06 — LP vs HP vs BP vs BS

Reading Each Filter Shape

The four basic filter types are all built from the same lowpass prototype, but they look very different on a response plot. Recognising the shape instantly tells you the filter's job.

Lowpass (LP)

Passes DC→fc, attenuates above

Highpass (HP)

Attenuates below fc, passes above

Bandpass (BP)

Passes a band around f₀, rejects outside

Bandstop (BS)

Notches out a band around f₀, passes elsewhere

All four are derived from the same lowpass prototype g-values by frequency transformation — see RF Filter Theory for the full derivation of each transform.

07 — Interactive Explorer

Interactive Filter Response Explorer

Switch filter type and approximation below and watch the annotated magnitude and group-delay plots update — the −3 dB marker, cutoff/centre-frequency line, and rolloff readout all recompute live. Uses the same transfer functions as the LC Filter Design Calculator.

Filter Response Explorer

Type

Approximation

Order n

Ripple A_p

dB

Cutoff f_c

MHz

Magnitude Response — Annotated

Dashed amber = −3 dB reference · dashed blue = cutoff/centre frequency · shaded region = passband. The rolloff readout below is measured between two points a decade apart, well into the transition region — not read off points right next to f_c.

Group Delay

Group delay computed by numerical differentiation of the modelled phase response (−dφ/dω).

08 — Comparing Approximations

Comparing Approximation Types

Switch the Explorer above between Butterworth, Chebyshev and Bessel at the same order and cutoff to see these differences directly, rather than from a table.

Parameter	Butterworth	Chebyshev	Bessel
Passband	Flat, monotonic	Equiripple	Flat, softer than Butterworth
−3 dB point	Always exactly at f_c	At or beyond f_c, depends on ripple	Beyond f_c, depends on order
Rolloff steepness (same n)	Baseline	Steepest	Gentlest
Group delay flatness	Moderate	Worst near f_c	Best (design goal)
Best used for	General purpose	Sharp selectivity, narrowband	Pulse/wideband, linear phase

Common misconception: assuming the −3 dB point always sits at the stated cutoff frequency. That's only guaranteed for Butterworth. For Chebyshev, f_c is defined as the ripple band edge (where the response last touches −A_p dB) — the actual −3 dB point is beyond it if A_p < 3 dB.

09 — Common Reading Mistakes

Common Reading Mistakes

Mistake	Why it's wrong
Reading rolloff slope from two points near f_c	The response hasn't reached its asymptotic −20n dB/decade slope that close in — always measure over at least a decade in the transition/stopband.
Assuming −3 dB = f_c for every approximation	Only true for Butterworth by definition. Chebyshev and Bessel cutoffs are defined differently.
Ignoring group delay because the magnitude plot "looks fine"	A perfectly flat magnitude response can still carry a sharply peaked group delay near the band edge, corrupting wideband modulated signals.
Treating simulator IL (0.0X dB) as the real-world number	The synthesis model is lossless by definition. Real components add IL proportional to 1/Q_L — always budget real component loss separately.
Confusing gain-convention and attenuation-convention Y-axes	A "40 dB down" spec means the same magnitude either way, but the sign and direction on the plot differ — always check the axis label first.