Failure: rejection passes, the burst fails
The 2.750 GHz marker looks excellent. Why does the receiver still lose the wanted burst?
Continue the illustrative engineering case: a 2.450 GHz condition-monitoring gateway must pass a pulse-shaped wanted signal while attenuating a nearby blocker. The routing network from 04.2 has established the ports. Now the review sees 49.411712 dB of ideal rejection and declares victory too early.
Think about itIf the blocker exceeds its rejection target by 9.411712 dB, is this filter acceptable?
No. The default third-order Chebyshev-I has a 0.500000 dB ripple and a separate Q = 200 passband debit of 1.596032 dB. Its worst passband loss is 2.096032 dB against 2.000000 dB allowed. The exact complex waveform proxy is 3.875175% against 3.0%, and passband delay ripple is 5.653218 ns against 5.0 ns.
Chebyshev-I, n = 3, Aedge = 0.5 dB, Q = 200; 2.450 GHz / 100 MHz. Every failed axis is binding; no combined score hides it.
| Axis | Result | Requirement | Margin | Decision |
|---|---|---|---|---|
| Worst passband loss | 2.096032 dB | ≤ 2.0 dB | -0.096032 dB | reject |
| Blocker attenuation | 49.411712 dB | ≥ 40.0 dB | 9.411712 dB | pass |
| Passband delay ripple | 5.653218 ns | ≤ 5.0 ns | -0.653218 ns | reject |
| Waveform distortion proxy | 3.875175 % | ≤ 3.0 % | -0.875175 % | reject |
Required before hardware selection: physical realization, power, temperature, tolerance, size, measured response, hidden modes and mismatch-shaped waveform evidence. An accepted screen leaves these unresolved.
These are three distinct failures, not three names for attenuation. Magnitude tells how much each component is scaled; relative phase tells how those components recombine. A finite-Q screening rule adds an estimate of dissipative loss without pretending to predict the complete lossy waveform.
Two responses can share magnitude while differing in phase. The output waveform follows the complex transfer function. Inspect the full passband, the pulse-shaped record and the conditions at both R1 planes.
By the end, you will write a filter requirement, distinguish the Q definitions, compare consistently normalized responses, reject the binding failure and create a traceable selection row.
Bring Path 03’s two-port waves and S-parameters and matching under real terminations. Here those foundations become filter selection conditions.
Turn spectra into passband and stopband constraints
Start with the spectrum that must survive, then the population that must not. Give each boundary a definition and a plane before choosing a response family. A single blocker is a useful check point; a real threat specification is a set of frequencies, powers, modes and uncertainty margins.
| Quantity | Required value / definition | Plane and condition |
|---|---|---|
| Passband | 2.400000…2.500000 GHz; B = 100.000 MHz; displayed fc = 2.450 GHz | R1 component input/output, real Zref = 50 Ω |
| Wanted support | 60.000 Msymbol/s QPSK, RRC α = 0.35; ideal RC null-to-null width 81.000 MHz | High-rate Path 04 variant; normalized complex envelope |
| Loss | Worst passband GT loss + local Q debit ≤ 2.000 dB | Matched teaching default; do not substitute a typical center value |
| Blocker | 2.750000 GHz; +300.000 MHz offset; Pavs = −20.000 dBm; attenuation ≥ 40.000 dB | Source-available at input R1; load-delivered output uses GT |
| Delay / waveform | Passband delay ripple ≤ 5.000 ns; periodic distortion proxy ≤ 3.0% | Intrinsic matched H; correction contract in Section 7 |
| Analytic environment | 25 °C, small signal, source/load VSWR 1.000 | Power and environment qualification still missing |
| Hardware obligations | Power, size, temperature corners, tolerance population and reference-plane evidence | Requirements to obtain from product owner; no invented compliant limits |
The ideal wanted support leaves 9.5 MHz between each ±40.5 MHz null and the ±50 MHz passband edge. Finite truncation has spectral tails. Frequency error, manufacturing spread and environmental movement consume margin; they are not automatically covered by the 9.5 MHz arithmetic.
This high-rate variant does not replace Path 02’s 10 ksymbol/s baseline. The lesson uses no named air-interface mask. Compliance and complete receiver blocker/noise allocation belong to later paths.
Think about itDoes an 81 MHz null-to-null signal justify calling a 100 MHz filter its occupied bandwidth?
No. One number describes ideal waveform support, another names this passband span. Occupied bandwidth requires an integrated power fraction, spectral population and stated integration procedure.
Mark transition bands between the guaranteed passband and guaranteed rejection regions. A marker at +300 MHz leaves every unsampled threat—including one closer to the passband—unresolved.
Resonators, stored energy, loss, and coupling
A resonator exchanges electric and magnetic energy. Loss drains that exchange; coupling allows the source to feed it and the load to receive it. Recall resonance and ring-down, then separate intrinsic loss from deliberate external loading. [1]
| Q | Energy leaves through | Decision consequence |
|---|---|---|
| Q0 · unloaded | Internal conductor, dielectric and other intrinsic losses | Sets dissipation before intended port loading |
| Qe1, Qe2 · external | Energy extraction through each named coupling port | Port coupling can broaden a response even with a high-Q resonator |
| QL · loaded | All internal and external loss channels together | Sets this mode’s decay and approximate half-power linewidth |
With Q0 = 200 and Qe1 = Qe2 = 100, 1/QL = 0.025, so QL = 40. Near 2.450 GHz the simple isolated-mode half-power bandwidth is approximately f0/QL = 61.25 MHz. The amplitude ring-down time is 2QL/ω0 ≈ 5.197 ns; stored energy decays with half that time constant.
Think about itIf intrinsic loss vanished while both coupling ports remained at Qe = 100, would loaded Q become infinite?
No. With Q0 → ∞, QL approaches 50 because energy still leaves through the ports. External loading is not the same as heating inside the resonator.
Order counts poles in a defined prototype. Realization also needs adequate resonator Q, coupling range, parasitic control, operating power, size and stable geometry. The same order may be easy at one bandwidth and infeasible at another.
Go deeperWhy a narrow bandwidth spends more of a finite-Q budget
The same fractional dissipation matters more when a mode must store energy for many cycles before delivering it. Our later debit scales as n/(Q·FBW), capturing that screening trend. It has no coupling coefficients and therefore cannot replace a physical loss calculation.
Response transformations and filter-order intuition
A normalized low-pass prototype removes units so we can examine response shape once. A frequency transformation then asks where that shape should act. Keep normalized s separate from physical s in rad/s. [2] [3]
| Desired response | Substitution into normalized HLP | Meaning |
|---|---|---|
| Low-pass | sLP = s/ωc | Keep DC; scale the specified edge |
| High-pass | sLP = ωc/s | Exchange low- and high-frequency roles |
| Band-pass | sLP = (s² + ω0²)/(Bω s) | Bω=ω2−ω1; ω0=√(ω1ω2); each prototype pole produces two physical poles |
| Band-stop | sLP = Bω s/(s² + ω0²) | Map prototype rejection toward the center notch |
The allocator uses a deliberately simpler local mapping. Near resonance, (f²−f0²)/(Bf) ≈ 2(f−f0)/B. We use the displayed arithmetic center fc and impose B/fc ≤ 0.20. Even inside this guard, it remains a teaching approximation, not an error guarantee for a physical band-pass design.
A proper physical band-pass transform of 2.400 and 2.500 GHz would use geometric center √(2.4×2.5) ≈ 2.449490 GHz. The difference is small here but real. The interaction does not silently switch between those conventions.
More order can buy rejection while spending delay flatness, loss, tolerance and size. A steeper curve that fails the wanted signal is not an improvement in this decision.
Butterworth, Chebyshev, elliptic, and Bessel tradeoffs
Native prototypes normalize different properties. Before comparing them, state what “one unit of bandwidth” means. The static cases below all use n = 3, Q = 200, B = 100 MHz and an equal-spec outer-edge attenuation of 0.5 dB with a unity passband peak.
| Family | Native |Ω|=1 attenuation, n=3 | Equal-spec rule | What is optimized |
|---|---|---|---|
| Butterworth | 3.010300 dB | Hcompare(jΩ)=Hnative(jαΩ); α=(10^(Aedge/10)−1)^(1/(2n)) | Maximally flat magnitude near DC |
| Chebyshev-I | 0.500000 dB for Rp=0.5 | Rp=Aedge; α=1; unity ripple peak; even n DC remains −Rp | Equiripple magnitude trades phase/transient behavior for transition selectivity |
| Delay-normalized Bessel | 0.902973 dB (approximately) | Positive α found by bracketed bisection so the outer edge is Aedge | Maximally flat DC delay; gradual transition under the same edge spec |
The complex response is evaluated from stable left-half-plane poles, not reconstructed from this power equation. For third order, the poles are −0.313228243170137 ± j1.021927491047360 and −0.626456486340275, with numerator K = 0.715693790310798.
Go deeperInspect every pole and the reverse Bessel polynomial
Butterworth uses pk = −sin(θk) + jcos(θk) and K = ∏(−pk). All real parts are negative. For Bessel, evaluate the polynomial directly:
For n = 3: θ3(s) = 15 + 15s + 6s² + s³. The equal-spec scale is α = 0.750824662261… for Aedge = 0.5 dB. Bisection starts at [0,1], doubles the upper bound up to 2²⁰, then resolves α to max(10⁻¹²,10⁻¹²α). Failure to bracket is invalid, not a guessed curve.
| Family / n / Q | Aedge / αfamily | Native edge (dB) | Pass loss (dB) | Blocker (dB) | Delay ripple (ns) | Waveform (%) | Screen |
|---|---|---|---|---|---|---|---|
| Butterworth / 3 / 200 | 0.5 dB / 0.704267401345 | 3.010300 | 2.096032 | 37.554093 | 1.485420 | 4.965292 | Reject: worst passband loss and blocker attenuation and waveform distortion proxy |
| Chebyshev-I / 3 / 200 | 0.5 dB / 1.000000000000 | 0.500000 | 2.096032 | 49.411712 | 5.653218 | 3.875175 | Reject: worst passband loss and passband delay ripple and waveform distortion proxy |
| Bessel / 3 / 200 | 0.5 dB / 0.750824662261 | 0.902973 | 2.096032 | 17.258452 | 0.001691 | 3.560812 | Reject: worst passband loss and blocker attenuation and waveform distortion proxy |
The sample-domain proxy is not a universal family ranking. Integer-only timing correction leaves a residual fractional-sample delay, and the actual waveform spectrum weights the result. Bessel’s flatter delay alone does not promise the smallest proxy for every order and normalization.
Finite-frequency transmission zeros can shorten a transition at the cost of passband/stopband ripple and demanding phase, tolerance and implementation behavior. A deep zero is followed by a response rise; it does not establish uniformly infinite rejection. No elliptic curve or solver is included. [2]
Think about itAt the same 0.5 dB outer edge and third order, what does Bessel give up to improve delay flatness?
Here it gives up blocker attenuation: 17.258452 dB versus Chebyshev’s 49.411712 dB at Ω = 6. Both retain the same declared outer-edge attenuation, so the comparison has a common requirement.
Insertion loss, return loss, rejection, and finite-Q floor
Separate what is reflected from what is dissipated. A lossless mismatched network can reject a signal while creating no internal heat. “Insertion loss” also needs a before/after reference; it is not automatically equal to every negative S21 number.
| Term | Definition and condition |
|---|---|
| Transducer gain GT / loss | Pload/Pavs; loss = −10log10(GT). Source-available input and delivered load power, same declared conditions. |
| Insertion loss | 10log10(Pload,before/Pload,after), with source, load and planes held fixed. Equals −20log10|S21| for a matched, unity-through reference. |
| Return loss RL | −20log10|Γ| at a named plane. S11-based RL assumes port 2 matched. Γ=0 is a perfect match, not an infinite output power. |
| Rejection | Attenuation at a specified frequency/region and reference. Here absolute −10log10(GTideal), not attenuation relative to the ripple edge. |
| Shape factor | Bandwidth at one attenuation divided by bandwidth at another, e.g. B60dB/B3dB. Both reference levels and crossings must be explicit. |
| 3 dB bandwidth | Width between specified crossings 3 dB below a declared passband reference; not automatically this lesson’s Aedge span. |
| Occupied / channel / measurement bandwidth | An integrated-power fraction / assigned channel interval / instrument resolution or integration setting. Three distinct definitions. |
| Noise-equivalent bandwidth | For a two-sided envelope, ∫|H(ν)|² dν / |H|max², over stated bounds. Include or exclude discrete spurs explicitly; no ENBW value is inferred from one edge. |
Return loss describes a reflected wave relative to the incident wave at a plane. Insertion loss describes changed delivered power after inserting a network. Dissipation, reflection and coupling to other ports must be separated to explain why delivery changed.
With −20.000 dBm source-available blocker power and matched ports, its ideal residual is −69.411712 dBm. Do not add passband loss to claim a still better stopband residual.
For loss alone at the fixed default response and matched terminations, Q must exceed 212.804296…; integer Q = 213 clears that screen. It leaves the delay and waveform failures intact. The “finite-Q floor” here means a passband loss burden. It does not define a universal out-of-band rejection floor.
At other physical temperatures the matched passive attenuator relation is F = 1 + (L−1)Tp/T0, where L is linear dissipative loss. In the ideal two-port below, ripple is reflected, so it is not added as thermal noise. This is not a complete filter noise model; Path 01 Noise owns the noise definitions.
Parasitic input-to-output coupling, finite loss and higher modes create paths absent from the prototype. A calculated −100 dB tail is not evidence that a package, PCB or measurement fixture achieves it.
Phase and group delay can damage the waveform
A constant delay translates a waveform. Delay that changes across occupied frequencies rearranges the relative timing of its components. Recovering a burst therefore needs more than removing one common phase or cable delay. [4]
Unwrap from the lowest to highest frequency, repeatedly adding/subtracting 2π so each increment lies in (−π,+π]; an exact −π tie becomes +π. The passband grid is Ωi = −1 + i/1000 for i = 0…2000. Here Δf = 50 kHz and Δω = 2π×50 kHz.
Phase/delay is suppressed where |H| < 10⁻¹². Polynomial conditioning κeval = Σ|ak||s|ᵏ / |Σak sᵏ| is inspected above 10¹⁰ and invalid above 10¹⁴. A plotted stopband floor is not evidence of physical rejection. The canonical defined delay range is 6.128549…11.781767 ns, giving 5.653218 ns ripple.
A constant component can be aligned away; frequency-dependent delay remains. Magnitude variation and residual timing error also contribute to waveform distortion, so delay ripple alone cannot predict the exact proxy.
A fixed record, a disclosed correction
Use 256 Gray QPSK symbols from PRBS-9 x⁹+x⁵+1, restarted at 0x1FF. Emit bit 0; feed bit 0 XOR bit 4 into bit 8 after a right shift. Consume b0 as the first/MSB bit of each pair: 00, 01, 11, 10 map to (1+j), (−1+j), (−1−j), (1−j), divided by √2.
Place symbol m at index 8m of 2048 samples. Linearly convolve with a unit-energy 65-tap symmetric RRC, α=0.35 and span 8 symbols. Of the 2112 convolution samples retain indices 32…2079 inclusive, then normalize mean |x|² to one. This valid record is treated as one periodic steady-state record, not a simulated burst-start transient.
Go deeperRRC taps, signed DFT bins and alignment algorithm
Use h(0)=1+α(4/π−1). At t=±1/(4α), use (α/√2)[(1+2/π)sin(π/(4α))+(1−2/π)cos(π/(4α))]. The analytic limits prevent a numerical 0/0.
Test every circular integer lag ℓ=−1024…1023, using yℓ[n]=y[(n+ℓ) mod 2048]. Maximize |Σyℓ conj(x)|; ties take the smallest |ℓ|, then negative lag. Use all 2048 pairs with no additional transient exclusion, fractional delay, timing recovery, adaptive equalizer or DPD.
| Quantity | Default result |
|---|---|
| Best circular lag | +3 samples = +6.250000 ns |
| Fitted gain c | 0.968095425927111 − j0.001498446673922 |
| Distortion proxy | 3.875175% > 3.0% local criterion |
| Waveform frequency domain | −240…+240 MHz, upper endpoint excluded; every fc+ν must be positive |
| Separate blocker | +300 MHz is outside this waveform Nyquist interval; analytic check only |
Reference normalization, symbol selection, timing correction and equalization determine EVM interpretation. This metric uses waveform samples and one fixed correction. No named standard or compliance limit is implemented. [5]
The allocator now combines these definitions. Its entire default result is also available without JavaScript and in print.
Filter Threat Allocator
Choose a response that meets the wanted-signal and blocker requirements together. You have now seen the loss and complex-response definitions; use them to explain each decision.
- Predict whether third-order Chebyshev passes 40 dB at +300 MHz; compare the result with its Q = 200 loss.
- Change only order, then only Q. Name the constraints that moved.
- Compare Bessel and Chebyshev at equal order and Aedge. Check delay and the periodic burst.
- Introduce independent source/load phase. Choose the next technology and evidence request.
Chebyshev-I, n = 3, Aedge = 0.5 dB, Q = 200; 2.450 GHz / 100 MHz. Every failed axis is binding; no combined score hides it.
| Axis | Result | Requirement | Margin | Decision |
|---|---|---|---|---|
| Worst passband loss | 2.096032 dB | ≤ 2.0 dB | -0.096032 dB | reject |
| Blocker attenuation | 49.411712 dB | ≥ 40.0 dB | 9.411712 dB | pass |
| Passband delay ripple | 5.653218 ns | ≤ 5.0 ns | -0.653218 ns | reject |
| Waveform distortion proxy | 3.875175 % | ≤ 3.0 % | -0.875175 % | reject |
Required before hardware selection: physical realization, power, temperature, tolerance, size, measured response, hidden modes and mismatch-shaped waveform evidence. An accepted screen leaves these unresolved.
| Exact row | RF (GHz) / Ω | H real / imag | |H| (dB) | Phase (°) | Delay (ns) | GT loss (dB) | Pass loss + Q (dB) | Algebraic RL (dB) |
|---|---|---|---|---|---|---|---|---|
| Lower edge | 2.400000 / -1.000 | -0.668997142 / 0.666103417 | -0.500000 | 135.124184 unwrapped | 11.781767 · forward endpoint | 0.500000 | 2.096032 | 9.635745 |
| Center | 2.450000 / 0.000 | 1.000000000 / 0.000000000 | 0.000000 | 0.000000 unwrapped | 6.826553 · central | 0.000000 | 1.596032 | perfect algebraic match |
| Upper edge | 2.500000 / 1.000 | -0.668997142 / -0.666103417 | -0.500000 | -135.124184 unwrapped | 11.781767 · backward endpoint | 0.500000 | 2.096032 | 9.635745 |
| Actual signed blocker | 2.750000 / 6.000 | -0.000710197 / 0.003308509 | -49.411712 | 102.115126 wrapped | outside passband derivative grid | 49.411712 | not defined here | 0.000050 |
| Worst passband loss | 2.475000 / 0.500 | 0.501188918 / -0.800037878 | -0.500000 | -57.934647 unwrapped | 6.207742 · central | 0.500000 | 2.096032 | 9.635745 |
| Quantity | Value | Condition |
|---|---|---|
| Ideal matched blocker | 49.411712 dB | Intrinsic prototype |H|²; no Q debit |
| Current-termination blocker | 49.411712 dB | GTideal of the algebraic two-port at current ΓS/ΓL |
| Load-delivered blocker | -69.411712 dBm | -20.0 dBm source-available at R1; no Q debit |
| Finite-Q passband debit | 1.596032 dB | Local screening rule; does not define a complex lossy filter |
| Dissipative NF | 1.596032 dB | Separate matched attenuator-equivalent debit, Tp=T0=290 K |
| Linear F / Te | 1.4441198 / 128.7947 K | log10F=0.159603; ideal ripple reflects, not heat |
| Minimum Q for loss alone | 212.804; choose integer ≥ 213 | Fixed family/order/edge and terminations; other requirements still apply |
What the burst sees
3.875175% sample-domain distortion proxy · limit 3.0%. Intrinsic matched H only; no Q debit and no mismatch shaping.
| Sample / ns | Input I / Q | Aligned output I / Q | Corrected output I / Q |
|---|---|---|---|
| 0 / 0.000000 | -0.736386 / -0.739278 | -0.423050 / -0.380180 | -0.436384 / -0.393384 |
| 1 / 2.083333 | -0.772998 / -0.785499 | -0.605165 / -0.621916 | -0.624113 / -0.643378 |
| 2 / 4.166667 | -0.788233 / -0.807086 | -0.762037 / -0.786601 | -0.785892 / -0.813740 |
| 3 / 6.250000 | -0.787204 / -0.806759 | -0.851284 / -0.855638 | -0.877969 / -0.885195 |
Explore LC or LTCC only with measured use-frequency Q and loss. Compare ceramic/acoustic and cavity options against the same waveform.
Q is a requested screening parameter, not a measured technology capability. Increasing Q changes this loss debit and its noise consequence; it does not repair intrinsic delay or waveform error.
Inspect model, grids and waveform contract
κeval > 10¹⁰: inspect the affected required metric; > 10¹⁴ or denominator ≤ 10⁻¹²: invalid. A plot-only deep-null floor does not reject a candidate. Required metric unavailability is explicit. Valid endpoint derivatives do not trigger inspect.
Filter selection / evidence row
Copy the applied case with its result, model, planes and unresolved verification points. Print preserves the decision and readable tables.
Read or manually copy the complete record
{
"axes": [
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{
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{
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{
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"name": "Waveform distortion proxy",
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"unit": "%",
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"decision": "Reject: worst passband loss and passband delay ripple and waveform distortion proxy",
"delayRangeNs": [
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"11.781767"
],
"errors": {},
"fixture": {
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"convolutionLength": "2112.000000",
"correction": "c=Σy_lag conj(x)/Σ|x|²; all 2048 pairs; no fractional delay, receiver matched filter, timing recovery, equalizer or DPD",
"denominatorRelativeFloor": "0.000000",
"grid": "q=k if k<1024, else k-2048; ν=q*480e6/2048; [-240,+240) MHz",
"id": "p04-qpsk-rrc-periodic-v1",
"mapping": "00,01,11,10 → (1+j),(-1+j),(-1-j),(1-j), divided by sqrt(2)",
"metric": "100 sqrt(Σ|y_lag-cx|²/Σ|cx|²); sample-domain distortion proxy, no standard limit",
"normalization": "RRC tap energy 1; valid record mean |x|²=1",
"nullToNullHz": "81000000.000000",
"prbs": "x^9+x^5+1; emit bit0; feedback bit0 XOR bit4; shift right into bit8; b0 is MSB",
"record": "periodic steady state; no additional transient exclusion",
"retainedIndices": "32…2079 inclusive",
"rollOff": "0.350000",
"sampleRateHz": "480000000.000000",
"samples": "2048.000000",
"seed": "0x1FF",
"spanSymbols": "8.000000",
"sps": "8.000000",
"symbolRateHz": "60000000.000000",
"symbols": "256.000000",
"taps": "65.000000",
"transform": "X=DFT(x), Y=Hcompare(j2ν/B)X, y=IDFT(Y); circular, no padding"
},
"ilQDb": "1.596032",
"inputs": {
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"centerGHz": "2.450000",
"delayNs": "5.000000",
"distortionPercent": "3.000000",
"edgeDb": "0.500000",
"family": "Chebyshev-I",
"loadPhase": "0.000000",
"loadVswr": "1.000000",
"lossDb": "2.000000",
"offsetGHz": "0.300000",
"order": "3.000000",
"q": "200.000000",
"rejectionDb": "40.000000",
"sourcePhase": "0.000000",
"sourceVswr": "1.000000",
"spanMHz": "100.000000",
"waveform": "canonical"
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"keyRows": [
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"floor": false,
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{
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{
"name": "Upper edge",
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"delayNs": "11.781767",
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"floor": false,
"gt": "0.891251",
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"phaseDeg": "-135.124184",
"returnLossDb": "9.635745",
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{
"name": "Actual signed blocker",
"row": {
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"condition": "1.281458",
"delayNs": null,
"derivative": "outside passband derivative grid",
"floor": false,
"gt": "0.000011",
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"hz": "2750000000.000000",
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"omega": "6.000000",
"phaseDeg": "102.115126",
"returnLossDb": "0.000050",
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{
"name": "Worst passband loss",
"row": {
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"condition": "2.534450",
"delayNs": "6.207742",
"derivative": "central",
"floor": false,
"gt": "0.891251",
"h": {
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"hz": "2475000000.000000",
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"omega": "0.500000",
"phaseDeg": "-57.934647",
"returnLossDb": "9.635745",
"sigma": "1.000000",
"totalPassLossDb": "2.096032"
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}
],
"match": "Algebraic two-port only; intrinsic matched waveform",
"minimumQForLoss": "212.804296",
"model": {
"bessel": "θn(s)=Σk (2n-k)!/[2^(n-k)k!(n-k)!] s^k; α bracket [0,1], double to 2^20, bisect abs/rel 1e-12",
"conditions": "25 °C, small signal; separate dissipative attenuator-equivalent NF at Tp=T0=290 K",
"differentiation": "unwrap low→high increments (-π,+π], -π tie→+π; central interior, one-sided first-order endpoints; Δω=2πB/2000; no interpolation/smoothing",
"evidence": "Illustrative engineering case; Derived analytic prototype; Simulated periodic waveform",
"id": "p04-m03-filter-threat-v1",
"loss": "ILQ=(10/ln10)n/(Q*FBW), scalar passband screening debit only; Lpass=-10log10(GTideal)+ILQ",
"mismatch": "lossless reciprocal algebraic S11=a, S21=S12=H, S22=-a H/conj(H); a=sqrt(1-|H|²). No hardware S11 prediction",
"native": "Butterworth -3.010300 dB edge; Chebyshev-I -Rp edge; Bessel θn(0)/θn(s), DC delay=1",
"noise": "NFdB=ILQ, log10F=NFdB/10; F/Te tagged beyond numeric range when log10F>300",
"normalization": "Equal-spec unity passband peak; outer |Ω|=1 attenuation=Aedge. Chebyshev even n retains -Rp DC",
"numerics": {
"denominatorFloor": "0.000000",
"floor": "0.000000",
"inspectCondition": "10000000000.000000",
"invalidCondition": "100000000000000.000000",
"passivityTolerance": "0.000000"
},
"passbandGrid": "Ωi=-1+i/1000, i=0…2000; fi=fc+BΩi/2; exact center and edges",
"phasor": "e^(+jωt), H(+jΩ); positive-real DC gain; positive envelope offset raises RF frequency",
"planes": "R1 input/output; real Zref=50 Ω; no fixture, board, or access lines",
"plotGrid": "union passband + 1025 log magnitudes 1…max(8,2|offset|/B), signed mirrors, exact blocker; deduplicate; omit nonpositive physical plot-only rows",
"prototype": "p04-m03-filter-prototype-v1",
"rounding": "p04-m03-decimal-v1: fixed 6 metric decimals, 12 complex/α decimals; normalize displayed -0; JSON LF, stable field order",
"transform": "Ω=2(f-fc)/B; local narrowband mapping, B/fc<=0.20; not physical bandpass synthesis",
"waveformPath": "intrinsic matched equal-spec H; no finite-Q debit or mismatch-shaped modulation"
},
"noise": {
"factor": "1.444120",
"linearStatus": "defined",
"log10F": "0.159603",
"nfDb": "1.596032",
"teK": "128.794742"
},
"power": "not qualified",
"prototype": {
"alpha": "1.000000000000",
"coefficients": [
"0.715694",
"1.534895",
"1.252913",
"1.000000"
],
"edgeDb": "0.500000",
"family": "Chebyshev-I",
"normalization": "Equal-spec: unity passband peak; Aedge=0.5 dB at |Ω|=1; α=1; Chebyshev even-order DC remains -Rp",
"numerator": "0.715694",
"order": "3.000000",
"poles": [
{
"im": "1.021927491047",
"re": "-0.313228243170"
},
{
"im": "0.000000000000",
"re": "-0.626456486340"
},
{
"im": "-1.021927491047",
"re": "-0.313228243170"
}
]
},
"size": "allocation required",
"status": "reject",
"technologyRegion": "Explore LC or LTCC only with measured use-frequency Q and loss. Compare ceramic/acoustic and cavity options against the same waveform.",
"temperature": "25 °C analytic; corners unverified",
"verificationPoints": [
"Physical resonator/coupling realization and use-frequency Q/loss model",
"Condition-matched measured complex response, S11/S22 and mismatch-shaped waveform",
"Passband/stopband populations, tolerance and temperature corners",
"Average/peak/burst power, acoustic or contact nonlinearity, heating and lifetime",
"Package/board/fixture planes, size allocation, parasitic feedthrough and hidden modes"
],
"warnings": [],
"waveform": {
"c": {
"im": "-0.001498446674",
"re": "0.968095425927"
},
"lag": "3.000000",
"percent": "3.875175",
"reason": "Simulated intrinsic matched H; periodic sample-domain proxy"
}
}
Choose an RF filter technology
Choose a feasible technology region before requesting a product. Frequency alone is insufficient: bandwidth, realizable Q, loss, tuning, power and physical volume must fit the same operating conditions. Analog Devices’ block overview provides context for where filters sit in an RF chain. [6]
| Technology | Useful region / trade | Evidence needed before selection |
|---|---|---|
| Lumped LC | Compact and tunable at electrically small dimensions; component Q, SRF, pads and tolerance constrain narrow GHz responses | Use-frequency component models, assembly parasitics, voltage/current peaks, tuning range and temperature coefficients |
| Distributed / cavity | Geometry stores EM energy; attractive Q and power regions can require more space; re-entrant/higher modes matter | Connector/board planes, material loss, coupling, dimensions, tuning and wide frequency scans at power/temperature corners |
| Ceramic / LTCC | Ceramic resonators or integrated multilayer LC structures; compact integration does not imply the same Q or mode behavior | Identify actual topology; package land pattern, dielectric temperature behavior, fixture treatment and resonator versus LTCC mechanism |
| SAW / BAW | Acoustic resonances enable compact selective responses; material/cut/mode affect Q, thermal drift and nonlinear/power behavior | Exact acoustic technology, matching network, temperature, average/peak waveform, harmonics and application-board evidence |
| Active analog | Useful at suitable baseband/IF with gain and tunability; amplifier bandwidth, noise, headroom and stability limit performance | Bias, source/load, dynamic range, noise, distortion, supply and amplifier excess phase |
| Digital | Precisely adjustable response after conversion; computation and latency buy flexibility | Sample rate, alias protection, converter overload and arithmetic limits. It cannot undo clipping before the ADC |
The following exact examples teach how to preserve conditions. They are not purchasing recommendations, and their generic attributes are not used as coefficients in the allocator.
| Example / technology | Inspected definition | Why it is not an automatic candidate |
|---|---|---|
| Mini-Circuits SLP-1000+ · lumped LC low-pass [7] | DC–900 MHz, loss <1 dB; nominal 990 MHz 3 dB cutoff; separate >20/>40 dB regions; 50 Ω SMA interfaces | Different frequency and response type. The 0 dBm characterization does not establish behavior at every power. |
| Mini-Circuits BFCN-2450+ · ceramic LTCC band-pass [8] | 2400–2550 MHz passband; separate lower/upper stop regions; 50 Ω package, board conditions matter | A ceramic package is not a generic unloaded-Q promise; do not transfer typical group delay into a full-temperature guarantee. |
| Mini-Circuits ZVBP-2450-S+ · cavity band-pass [9] | 2400–2500 MHz; max passband loss 1.3 dB; minimum 40 dB at 2635–2780 MHz, at 25 °C | Conditions cover our passband and 2.75 GHz marker; modulated waveform, power corners and physical allocation still need verification. |
| Qorvo 885136 · BAW band-pass [10] | Rev. F, May 2017; channel-specific 2.4 GHz passbands and adjacent rejection bands; actual matching/board conditions apply | Its selected-channel spans do not equal our 2400–2500 MHz requirement. Stopband attenuation within that interval can conflict with the wanted band. |
Its limits belong to specified conditions. Moving the plane or changing the matching network may alter loss and phase; larger drive can create heating or nonlinear response. Preserve the document’s test conditions, not just its product title.
A cavity example may provide a useful evidence starting point for the gateway; an acoustic or LTCC option may better fit a small node. Neither decision follows from the prototype family name alone.
Mismatch, tolerance, temperature, power, and hidden modes
A scalar H cannot determine an arbitrary filter’s return loss. To expose termination dependence without inventing measured S11, this lesson uses a disclosed lossless reciprocal algebraic two-port. It is a teaching embedding, not a physical synthesis.
Ideal ripple/rejection returns energy as reflection. The separate Q debit is kept outside this matrix. Each evaluated row checks σmax ≤ 1+10⁻¹². A roundoff-only negative 1−|t|² sets a to zero without renormalizing H.
For an ideal through H=1 with ΓS=ΓL=+0.5, GT=1. Rotate only ΓL to −0.5 and GT=0.36, or 4.436975 dB loss. Both terminations still have VSWR 3. Phase changes delivered power even when the scalar VSWR labels do not move. The allocator keeps its waveform metric intrinsic and matched; it does not claim to simulate the mismatch-shaped burst.
| Mechanism | Observable consequence | Discriminating evidence |
|---|---|---|
| Tolerance / correlation | Resonances and couplings move together or apart; ripple, match and skirts change | Bounded corners or declared population/distribution/correlation; actual lot and assembly data |
| Temperature | Resonance, loss and dimensions drift; margins migrate across the band | Hot/cold complex response at stable DUT temperature, power and mounting state |
| Power / heating | Internal field/current peaks exceed small-signal conditions; compression or mixing may appear | Average, peak, duty, pulse length, mismatch and thermal conditions; acoustic/contact nonlinear characterization |
| Parasitic coupling | A path around resonators limits observed rejection | Board/package/fixture separation and response at multiple planes |
| Higher / hidden modes | Out-of-band transmission reappears beyond the first rejection region | Wideband response through relevant blocker/harmonic populations; probe mode and environment sensitivity |
Go deeperA first-order frequency sensitivity check
For an isolated LC mode, f0=1/(2π√LC) gives Δf0/f0≈−½(ΔL/L+ΔC/C). If both change +1%, the first-order shift is −1%, or −24.5 MHz at 2.450 GHz. The exact common-scale result is f0/1.01, a −24.257426 MHz shift. Couplings and parasitics make a real multi-resonator response more complicated.
A sweep explores a specified set of values. Yield needs a defensible population, distributions, correlation, process and measurement uncertainty. No synthetic tolerance population or production-yield metric is claimed here.
A receiver can fail from other frequencies, intermodulation, noise, compression, mixing, or input clipping. This marker answers one linear transfer question. System immunity is a later allocation and verification task.
Select the node coexistence filter
Select a bounded numerical candidate, then specify what would falsify it. Do not soften the requirement until a preferred response appears to pass. The default Chebyshev and third-order Bessel are both rejected under the canonical requirements.
| Family / n / Q | Aedge / αfamily | Native edge (dB) | Pass loss (dB) | Blocker (dB) | Delay ripple (ns) | Waveform (%) | Screen |
|---|---|---|---|---|---|---|---|
| Chebyshev-I / 3 / 200 | 0.5 dB / 1.000000000000 | 0.500000 | 2.096032 | 49.411712 | 5.653218 | 3.875175 | Reject: worst passband loss and passband delay ripple and waveform distortion proxy |
| Bessel / 3 / 200 | 0.5 dB / 0.750824662261 | 0.902973 | 2.096032 | 17.258452 | 0.001691 | 3.560812 | Reject: worst passband loss and blocker attenuation and waveform distortion proxy |
| Butterworth / 4 / 500 | 0.4 dB / 0.746540970272 | 3.010300 | 1.251217 | 52.096419 | 2.467315 | 1.866213 | Accept for this bounded prototype screen only |
The fourth-order Butterworth candidate changes two response choices explicitly: n=4 and Aedge=0.4 dB, with Q=500. It retains the same 100 MHz passband, 2 dB loss, 40 dB rejection, 5 ns delay-ripple and 3% waveform thresholds. It passes this screen with 1.251217 dB passband loss, 52.096419 dB blocker attenuation, 2.467315 ns delay ripple and 1.866213% proxy.
Think about itDoes this passing Butterworth row qualify a 2.45 GHz hardware filter?
No. It justifies requesting a physical realization and discriminating data. The algebraic return loss, constant Q debit and periodic proxy do not qualify power, environment, tolerance, size, hidden modes or a particular manufacturer part.
| Field | Proposed entry | Verification / rejection evidence |
|---|---|---|
| Selected screen | Butterworth n=4; Aedge=0.4 dB; Q proxy=500; local Ω mapping | Prototype p04-m03-filter-prototype-v1; interaction p04-m03-filter-threat-v1 |
| Passband / loss | 2400…2500 MHz; ≤2 dB total screening loss | Measured full-band loss/match at required source/load and temperature corners |
| Stopband / threat | At least 40 dB at 2750 MHz; −20 dBm Pavs at R1 | Expand to actual blocker/harmonic population; no single-marker immunity claim |
| Delay / waveform | ≤5 ns delay ripple; ≤3% local periodic proxy | Complex response and modulated test; preserve declared correction/normalization |
| Match / planes | R1 input/output, real 50 Ω; default matched source/load | Obtain physical S11/S22 and fixture/de-embedding record; test actual Γ region |
| Power / size / temperature | Small-signal analytic screen at 25 °C; hardware allocations unresolved | Specify average/peak/duty, permitted dimensions, operating temperatures and thermal conditions before hardware selection |
| Rejected alternative A | Chebyshev-I n=3 / Aedge=0.5 / Q=200 | Rejection passes; pass loss, delay ripple and waveform fail |
| Rejected alternative B | Bessel n=3 / Aedge=0.5 / Q=200 | Delay flatness improves; blocker rejection, pass loss and waveform still fail |
| Next evidence | LC/LTCC, ceramic/acoustic or cavity physical realization study | Finite-Q physical model; corner data; power/temperature qualification; board/fixture modes |
A filter is a conditional complex mapping. Its useful response must meet wanted-signal, threat, loss, delay, power, environment and evidence requirements together. Hand the conditional filter loss and terminations to 04.4: Low-Noise Amplifiers, where active gain, noise and stability enter.
Check your understanding
Answer each question in your own words, then reveal the model answer.
01Why does the default candidate fail even with 49.411712 dB blocker attenuation?
Model answerIts 2.096032 dB worst passband loss exceeds 2 dB, its 5.653218 ns delay ripple exceeds 5 ns, and its 3.875175% periodic waveform proxy exceeds 3%. The blocker check passes but cannot erase the other failures.
02What does increasing Q fix in this lesson, and what does it leave unresolved?
Model answerIt reduces only the scalar passband loss debit and its separate matched 290 K noise consequence. Intrinsic H, phase, delay and waveform remain unchanged. It creates no physical realizability or power evidence.
03Why are native Butterworth, Chebyshev and Bessel edges not an equal-bandwidth comparison?
Model answerAt normalized frequency one they optimize different definitions: Butterworth is -3.010300 dB, Chebyshev is -Rp and delay-normalized third-order Bessel is about -0.90297 dB. The equal-spec view rescales each to a unity passband peak and the common outer-edge attenuation. Even-order Chebyshev retains its -Rp DC droop.
04What sign and units must be used to compute group delay?
Model answerWith e^(+jωt), τg=-dφ/dω. Unwrap phase in radians and differentiate against angular frequency in rad/s. The result is seconds. Central interior and one-sided endpoint stencils use the physical grid; deep-null rows are suppressed.
05Why is 3.875175% not a standards-compliance EVM result?
Model answerIt is a sample-domain proxy over all 2048 periodic pairs after a best integer circular lag and one fitted complex gain. It has no fractional timing recovery, adaptive equalizer or standard-defined symbol population. The 3% threshold is a local illustrative decision criterion.
06What must accompany the accepted fourth-order Butterworth screen in the evidence pack?
Model answerThe exact family/order/edge/Q, passband and blocker population, delay/waveform conditions, source/load and planes, physical realization, power/size/temperature allocations, tolerance and hidden-mode evidence, and rejected alternatives. Accept here means the bounded numerical screen only.
Sources and further study
Accessed 6 September 2026. Theory, independent derivations and manufacturer examples are separated. All curves are drawn from the disclosed local models. No vendor response table supplies the allocator.
- MW-1 · D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012. Publisher edition and contents; chapters 4, 6, 8 and 10 orient networks, resonators, filters and noise. Publisher record consulted; full text was not available for fresh page-by-page verification. The shown equations are independently derived and numerically checked.
- H. Zumbahlen, Basic Linear Design, Analog Devices, 2006, chapter 8. Analog Filters, standard responses and transformations. Canonical design-handbook source consulted; native normalization is explicitly replaced by the disclosed common-edge rule for comparisons. No coefficient or physical synthesis tables are copied.
- SciPy official reference, v1.18.0 web documentation. Bessel prototype normalization and Low-pass to band-pass transformation. Used to cross-check the natural DC-delay convention and geometric band-pass mapping. SciPy is not an application dependency.
- H. Zumbahlen, Analog Devices. Phase Relations in Active Filters, Analog Dialogue 43, 2009. Informative phase/order and active implementation limits. The lesson’s exact derivative and waveform algorithm are local, versioned derivations.
- SIG-2 · J. G. Proakis and M. Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008. Publisher edition record, chapters 2, 4 and 5 for signal representation/pulse shaping; full text not newly accessible. SYS-3 · Rohde & Schwarz, Understanding EVM, 3683.8038.52, version 01.00, consulted for reference and normalization interpretation; no standard limit is imported.
- CIR-2 · Anton Patyuchenko, Analog Devices. RF Signal-Chain Discourse, Part 2: Essential Building Blocks, 2021. Informative signal-chain roles and filter technology context; not a component rating.
- Mini-Circuits SLP-1000+. Official sheet Rev. A, stamp 061024, inspected via official indexed text; current product page identifies lumped LC. 50 Ω SMA, FF99 package; DC–900 MHz <1 dB, nominal 990 MHz at 3 dB; >20 dB at 1340–1750 MHz, >40 dB at 1750–2000 MHz. Curves use 0 dBm; operating −55…+100 °C, 0.5 W maximum. Exact calibration plane and characterization temperature need confirmation. Different band: orientation only.
- Mini-Circuits BFCN-2450+. Official sheet Rev. OR, M143261, stamp 190725, official indexed text inspected. Ceramic LTCC band-pass, FV1206-4, 50 Ω, 2400–2550 MHz. Operating −40…+85 °C; passband input maximum 2 W at 25 °C, derated to 0.5 W at 85 °C. Preserve its separate stop regions and device/board port treatment; fixture removal and exact characterization drive were not established in the inspected excerpt. Do not infer resonator Q from the package.
- Mini-Circuits ZVBP-2450-S+. Official sheet Rev. A, ECO-000859, stamp 191202, official indexed text inspected. Cavity, 50 Ω SMA-female interfaces, QT2302. At 25 °C: 2400–2500 MHz loss ≤1.3 dB, VSWR ≤1.5; attenuation ≥40 dB at 2635–2780 MHz. Operating −55…+100 °C, 15 W maximum input. Typical delay data are not a corner guarantee; exact calibration plane and drive require confirmation. No power or waveform qualification is transferred to the synthetic candidate.
- Qorvo 885136. Official product page and manufacturer datasheet Rev. F, May 2017, distributor-hosted copy, pp. 2–5, 7 inspected. BAW, 1.1×0.9×0.50 mm; optimum 50 Ω single-ended ports and external ±3% matching, characterized on 885136-EVB. Electrical tables cover separately stated temperature regions; typical values use 25 °C. Loss integrates linear S-parameters over 19 MHz; attenuation over 5 MHz. These are not spot values across our whole band. Lifetime/power conditions specify waveform, temperature and input pin; a +39 dBm CW rating is limited to 20 ms at 25 °C. Matching may need adjustment for the actual PCB; the exact calibration/de-embedding plane remains to confirm. Technology illustration only.
Physical prototype tables, coupling matrices, EM geometry, tuning and tolerance distributions remain outside this lesson. LNA/PA/mixer/PLL design is in 04.4–04.7; receiver allocation in Path 05; named masks in Paths 07/09; measurement and group-delay uncertainty in Path 08; manufacturing/layout in Path 10. Model p04-m03-filter-prototype-v1 remains a Learn teaching interaction.