LC Band-Pass Filter Calculator

Step 1

Specification

Set the −3 dB passband and the attenuation you need below it, above it, or both. Each filter type shows the lowest order that meets the spec; pick one to design it.

Requirements

Passband

Centre frequency f₀ (geometric)

−3 dB bandwidth

Lower stopband frequency

Attenuation there

Upper stopband frequency

Attenuation there

Frequency unit

Source and load impedance

Leave a stopband's two fields empty if you only need rejection on one side.

Filter type

Chebyshev passband ripple

Order (number of resonators)

Topology

Chebyshev

Attenuation all three types at the orders above · dots = your spec

Step 2

Components

The ladder from step 1: every prototype element becomes a resonator tuned to f₀. Standard values are the nearest parts in the chosen E-series; step 3 shows how much they move the response.

Schematic shows

Schematic

Component values Standard series

Part	Position	g	Ideal	Standard	Error	Detuning

Detuning: how far each resonator, built from the standard parts, resonates from f₀.

Export Values

AC analysis of the filter between its source and load resistors; S21 = 2·V(out)·√(R_S/R_L).

Show the netlist

Real/imaginary S-parameters, both ports referenced to R₀.

Show the Touchstone data

About this LC band-pass filter calculator

This free calculator from NextAntennas designs passive LC band-pass filters of the Butterworth, Chebyshev and Bessel types. You give it the −3 dB passband, as a centre frequency and bandwidth or as two band edges, the attenuation you need at a stopband frequency below and above the passband, and the system impedance; it finds the filter order, computes the inductor and capacitor of every series and parallel resonator, rounds them to standard E12, E24 or E96 parts and shows the insertion loss, return loss, group delay and envelope response of the actual ladder network. It works from audio frequencies to RF, for any source and load impedance such as 50 Ω or 75 Ω.

How to use it

Specification: enter the passband, one or two stopband frequencies with the attenuation required at each, and the impedance. The three filter types show the lowest order that meets the harder of the two stopbands; choose one, set the Chebyshev ripple, and pick shunt-resonator-first or series-resonator-first topology. You can also force the order and see how far it is from the spec.
Components: a ladder schematic with every resonator's inductor and capacitor, the prototype g-values, the nearest E12, E24 or E96 parts with their rounding error and the detuning of each rounded resonator, with a SPICE netlist and Touchstone (.s2p) S-parameters to copy into a circuit simulator.
Response: |S21| against your attenuation masks, return loss, group delay and the envelope of a switched-on carrier, each with ideal and standard values overlaid, plus the shift in the band edges and stopband attenuation caused by rounding.

Band-pass filter design equations

The band-pass filter is derived from a low-pass prototype normalized to a −3 dB cutoff of 1 rad/s. A frequency f maps to the prototype frequency Ω = (f² − f₀²)/(f·BW), where f₀ = √(f₁f₂) is the geometric centre and BW = f₂ − f₁ the −3 dB bandwidth. Each series inductor g of the prototype becomes a series resonator with L = g·R₀/(2π·BW) and C = BW/(2π·f₀²·g·R₀); each shunt capacitor g becomes a parallel resonator with C = g/(2π·BW·R₀) and L = BW·R₀/(2π·f₀²·g). The order follows from the prototype's attenuation at |Ω| for each stopband frequency.

Frequently asked questions

What order band-pass filter do I need?

The order N is the number of resonators; the ladder has 2N inductors and capacitors. Enter the −3 dB passband, a stopband frequency below and/or above it and the attenuation you need there, and the calculator maps each stopband to the equivalent low-pass frequency Ω = |f² − f₀²| / (f·BW) and finds the lowest order of each filter type that meets the more demanding side. For a 90.5–110.5 MHz passband with 40 dB at 70 MHz and 140 MHz, a Butterworth needs 4 resonators and a 0.5 dB Chebyshev needs 5.

Why is the response not symmetrical about the centre frequency?

A lumped LC band-pass filter is symmetrical on a logarithmic frequency axis, around the geometric centre f₀ = √(f₁·f₂), not the arithmetic mean. A frequency a given number of megahertz below the passband is therefore attenuated more than the same offset above it: for a 3rd-order Butterworth centred at 100 MHz with 20 MHz bandwidth, 80 MHz is 21.2 dB down but 120 MHz only 15.9 dB. Enter both stopbands and the calculator sizes the filter for whichever side is harder.

Should I enter the centre frequency or the band edges?

Either. With centre and bandwidth, the centre is taken as the geometric centre and the −3 dB edges are placed so that f₂ − f₁ equals the bandwidth and √(f₁·f₂) equals f₀. With band edges, the calculator works out the geometric centre and bandwidth for you. Switching between the two modes carries the current band across.

Butterworth, Chebyshev or Bessel: which should I use?

Butterworth has the flattest passband and is the usual default. Chebyshev accepts a small passband ripple in exchange for steeper skirts, so it meets a stopband requirement with fewer resonators; use it when rejection close to the passband matters most. Bessel keeps the group delay nearly constant across the passband, so RF bursts and modulated signals pass with little overshoot or ringing, at the cost of much gentler skirts.

What passband ripple should I choose for a Chebyshev filter?

Ripple sets both the passband flatness and the worst-case return loss: 0.01 dB ripple gives 26.4 dB return loss, 0.1 dB gives 16.4 dB, 0.5 dB gives 9.6 dB and 1 dB gives 6.9 dB. For RF filters between 50 Ω stages, 0.01–0.1 dB is common because it keeps the match good. Larger ripple buys steeper skirts and slightly fewer resonators.

Why can't an even-order Chebyshev band-pass filter be matched to 50 Ω at both ends?

An even-order Chebyshev response has a ripple trough at the centre frequency, where the band-pass ladder behaves like the low-pass prototype at DC. A lossless ladder can only produce that loss through a mismatch between source and load, so the load must differ from the source by the factor coth²(β/4): about 1.98 for 0.5 dB ripple, which means 99 Ω or 25 Ω against a 50 Ω source. Auto mode therefore uses odd orders; if you force an even order, the calculator reports the load resistance it needs.

Shunt resonator first or series resonator first: which topology is better?

Both give the same response with ideal parts. Starting with a shunt parallel-LC resonator puts a low impedance to ground at frequencies far from f₀ at the input, and for odd orders uses fewer series resonators, whose large inductors and small capacitors are usually the hardest parts to source. Starting with a series resonator presents a high impedance to the source outside the passband, which some amplifier output networks need.

How are the band-pass component values calculated?

Each element of the normalized low-pass prototype becomes a resonator tuned to f₀. A series inductor g becomes a series LC with L = g·R₀/(2π·BW) and C = BW/(2π·f₀²·g·R₀); a shunt capacitor g becomes a parallel LC with C = g/(2π·BW·R₀) and L = BW·R₀/(2π·f₀²·g). Here BW = f₂ − f₁ is the −3 dB bandwidth in hertz.

Why are my inductor values so different, and what about narrow filters?

In the series resonators the inductance scales with R₀/BW, in the shunt resonators with R₀·BW/f₀², so their ratio grows as 1/FBW², where FBW is the fractional bandwidth. At 20% bandwidth the spread is about 25:1; at 2% it is about 2500:1, with tiny capacitors and very large inductors, and the response becomes very sensitive to tolerances. Below about 5–10% fractional bandwidth a capacitively coupled resonator filter is usually easier to build, and the calculator warns you when the band is that narrow.

Can I use standard E12 or E24 component values?

Yes. Choose E12, E24 or E96 and step 2 shows how far each resonator, built from the rounded parts, is tuned away from f₀, while step 3 overlays the response with the rounded values on the ideal one and reports the shift in the band edges and stopband attenuation. Wide-band Butterworth filters tolerate rounding well; narrow and high-order Chebyshev filters usually need E96 capacitors, trimmers or tunable inductors.

What is the envelope response?

It is the amplitude of the output when a carrier at the centre frequency is switched on at t = 0, normalized to its final value. It shows how fast the filter passes the leading edge of an RF burst and how much it overshoots or rings, the band-pass counterpart of the step response of a low-pass filter. The calculator computes it from the actual ladder, so it includes the effect of rounding to standard values.

How accurate is this filter calculator?

For ideal components the results are exact. The low-pass prototype values come from the closed-form Butterworth and Chebyshev equations and an 80-digit Darlington synthesis for Bessel, and agree with the tables of Matthaei and Zverev to every printed digit. Every plotted response is computed from the actual band-pass ladder, and the validation table on this page shows that it matches the transformed prototype response to better than 10⁻⁹ dB at fractional bandwidths from 1% to 100%.

Is the calculator free?

Yes. It is free, needs no sign-up and runs entirely in your browser; nothing you enter is uploaded.

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Engineering Theory

Theory Behind the Equations

How the calculator turns a passband and stopband requirements into resonator values: the normalized low-pass prototype, the Butterworth, Chebyshev and Bessel–Thomson approximations, the low-pass to band-pass frequency transformation, impedance and frequency scaling, and the analysis of the finished network.

1. The Doubly Terminated LC Band-Pass Ladder

A lumped band-pass filter for RF and signal work is usually an LC ladder of resonators: series LC branches in the signal path alternating with parallel LC branches to ground, driven from a source resistance R_S and loaded by R_L. Every resonator is tuned to the centre frequency f₀. There, the series branches are short circuits and the parallel branches are open circuits, so the ladder vanishes and the source sees its load directly. Away from f₀ the series branches block and the parallel branches short to ground. With N resonators (2N components) the response falls by 20·N dB per decade far below and far above the passband, so N is the order of the filter.

The ladder is lossless, so every bit of power that does not reach the load is reflected back to the source. Its performance is described by the transducer gain |S21|², the fraction of the available source power delivered to the load, and the reflection |S11|², linked by the power balance of a lossless two-port. Doubly terminated ladders are also insensitive to component tolerances in the passband (Orchard's argument): at frequencies of full power transfer the loss is at a minimum, so its first-order sensitivity to any element is zero.

Lossless two-port

{| S_{21} (j ω) |}^{2} + {| S_{11} (j ω) |}^{2} = 1

A_{dB} = - 10 \log_{10} {| S_{21} |}^{2}

A_dB = insertion loss (attenuation) · |S21|² = transducer power gain

2. The Normalized Low-Pass Prototype and g-Values

Every band-pass design starts from a low-pass prototype normalized to a 1 Ω source and a cutoff of 1 rad/s. Its element values g₁ … g_N are listed from the source end; g_k is an inductance in henries for a series element and a capacitance in farads for a shunt element. Following Matthaei, Young and Jones, g₀ is the source resistance and g_N+1 the load: a resistance if g_N is a shunt capacitor, a conductance if g_N is a series inductor.

The two prototype topologies are duals with the same g-values and exactly the same response. In the band-pass filter they become shunt-resonator-first and series-resonator-first ladders. All the values in this calculator are normalized so that the prototype's −3 dB point is at exactly 1 rad/s, whatever the filter type, which puts the band-pass −3 dB points exactly on f₁ and f₂.

3. The Low-Pass to Band-Pass Transformation

Replacing the prototype frequency variable by a reactance function turns the low-pass response into a band-pass one. The substitution maps the prototype's DC to the centre frequency ω₀ and its two cutoff points ±1 rad/s to the band edges ω₁ and ω₂:

Frequency transformation

Ω = \frac{1}{FBW} (\frac{f}{f_{0}} - \frac{f_{0}}{f}) = \frac{f^{2} - f_{0}^{2}}{f \cdot BW}

f_{0} = \sqrt{f_{1} f_{2}}, BW = f_{2} - f_{1}, FBW = \frac{BW}{f_{0}}

The band-pass attenuation at f equals the prototype attenuation at |Ω|; f₁ ↦ Ω = −1, f₀ ↦ 0, f₂ ↦ +1

Under this substitution a series inductor g becomes an impedance s·g/FBW + 1/(s·FBW/g), a series LC resonator, and a shunt capacitor becomes a parallel LC resonator. Because the transformation is exact, every property of the prototype carries over: ripple, return loss, equal or unequal terminations, and the attenuation at any frequency. The order needed for a stopband at f_s is the low-pass order needed at Ω_s = |Ω(f_s)|; with stopbands on both sides, the calculator takes the larger of the two orders.

4. Impedance Scaling and the Resonator Values

Multiplying all impedances by R₀ moves the terminations to R₀, and the transformation above sets the frequencies. Each prototype element then becomes one resonator tuned to f₀:

Series resonator (from a series inductor g)

L_{s} = \frac{g R_{0}}{2 π BW}

C_{s} = \frac{BW}{2 π f_{0}^{2} g R_{0}}

Parallel resonator (from a shunt capacitor g)

C_{p} = \frac{g}{2 π BW R_{0}}

L_{p} = \frac{BW R_{0}}{2 π f_{0}^{2} g}

R_{L} = g_{N + 1} R_{0} or \frac{R_{0}}{g_{N + 1}}

Example: f₀ = 100 MHz, BW = 20 MHz, 50 Ω, g = 1 → series 397.9 nH with 6.366 pF; parallel 159.2 pF with 15.92 nH. Every pair satisfies L·C = 1/(2πf₀)²

The series inductors scale as 1/BW and the shunt inductors as BW, so the ratio between them is about 1/FBW². That spread, rather than the theory, is what limits lumped ladder band-pass filters in practice; section 13 returns to it.

5. Geometric Symmetry and the Band Edges

Since Ω(f) depends on f/f₀ − f₀/f, two frequencies with the same ratio to f₀, such as f₀/k and k·f₀, have the same attenuation. The response is symmetrical on a logarithmic frequency axis around the geometric centre √(f₁f₂), which always lies below the arithmetic mean of the band edges. On a linear axis the lower skirt is steeper than the upper one: for a 3rd-order Butterworth filter centred at 100 MHz with 20 MHz bandwidth, 80 MHz is 21.2 dB down but 120 MHz only 15.9 dB.

When you enter a centre frequency and bandwidth, the calculator treats the centre as geometric and places the edges at f₁ = (√(BW² + 4f₀²) − BW)/2 and f₂ = f₁ + BW. When you enter the edges, it computes f₀ and BW from them. Either way the −3 dB points of the designed filter fall exactly on f₁ and f₂.

6. Butterworth: the Maximally Flat Response

The Butterworth approximation makes the first 2N−1 derivatives of the prototype's |S21|² zero at DC, which in the band-pass filter means a passband as flat as possible around f₀ with no ripple. The response is monotonic on each side, and its element values have a closed form:

Butterworth response and elements

{| S_{21} |}^{2} = \frac{1}{1 + Ω^{2 N}}

g_{k} = 2 \sin \frac{(2 k - 1) π}{2 N}

Ω from the transformation in section 3 · k = 1 … N · g_N+1 = 1 (equal terminations for every order)

Setting the attenuation at Ω_s equal to the requirement A_s and solving for N gives the order directly; the calculator rounds up to the next integer and repeats this for each stopband side:

Butterworth order

N \geq \frac{\log_{10} (10^{A_{s} / 10} - 1)}{2 \log_{10} Ω_{s}}

Passband 90.5–110.5 MHz, 40 dB at 140 MHz → Ω_s = 3.43 → N ≥ 3.73 → N = 4

7. Chebyshev: the Equiripple Response

A Chebyshev (type I) filter lets the passband loss ripple between 0 and A_r dB in exchange for the steepest skirts any all-pole filter of the same order can achieve. The response uses the Chebyshev polynomial T_N, which oscillates between ±1 inside |x| ≤ 1 and grows as cosh(N·arccosh x) outside it:

Chebyshev response

{| S_{21} |}^{2} = \frac{1}{1 + ε^{2} T_{N}^{2} (Ω / Ω_{p})}

ε = \sqrt{10^{A_{r} / 10} - 1}

A_r = passband ripple in dB · Ω_p = ripple band edge of the prototype

The element values follow from the classic recursion of Matthaei, Young and Jones, written here for a ripple band edge of 1 rad/s:

Chebyshev g-values

β = \ln (\coth \frac{A_{r}}{17.37})

γ = \sinh \frac{β}{2 N}

a_{k} = \sin \frac{(2 k - 1) π}{2 N}

b_{k} = γ^{2} + \sin^{2} \frac{k π}{N}

g_{1} = \frac{2 a_{1}}{γ}

g_{k} = \frac{4 a_{k - 1} a_{k}}{b_{k - 1} g_{k - 1}}

17.37 = 40/ln 10 (the calculator uses the exact constant) · k = 2 … N

8. Ripple Bandwidth Versus the −3 dB Bandwidth

Published Chebyshev tables are normalized to the ripple band edge, the last frequency where the loss equals A_r, not to the −3 dB point. Used directly with a −3 dB bandwidth, they make the real passband too wide; for a 0.1 dB, 3rd-order filter the −3 dB bandwidth comes out 39% too wide. The −3 dB point of the prototype lies where ε²T_N² = 1, which gives the ratio between the two bandwidths:

Ripple bandwidth for a given −3 dB bandwidth

\frac{{BW}_{ripple}}{{BW}_{- 3 dB}} = \frac{1}{\cosh (\frac{1}{N} arccosh \frac{1}{ε})}

Ripple-normalized g-values are divided by this ratio before the transformation · both bands share the geometric centre f₀

The calculator applies this correction for every order and ripple, so the −3 dB edges of the designed filter land exactly on f₁ and f₂, and step 1 reports the ripple band separately. Because the ratio changes with N, the Chebyshev order cannot be solved in closed form against a −3 dB spec; the calculator evaluates the exact attenuation of each order in turn.

9. Even-Order Chebyshev Filters and Unequal Terminations

At Ω = 0, the centre frequency of the band-pass filter, T_N(0) is 0 for odd N but ±1 for even N. An even-order Chebyshev response therefore has A_r dB of loss at f₀. At f₀ the resonator ladder is transparent, a plain wire from source to load, so this loss can only come from a deliberate mismatch between the source and load resistances:

Load for even N

g_{N + 1} = \coth^{2} \frac{β}{4}

\frac{4 R_{S} R_{L}}{{(R_{S} + R_{L})}^{2}} = \frac{1}{1 + ε^{2}}

0.5 dB ripple → g_N+1 = 1.9841: a 50 Ω source needs a 99.2 Ω or 25.2 Ω load

In a 50 Ω system this is usually unacceptable, so in Auto mode the calculator only uses odd Chebyshev orders. If you force an even order, it designs the filter for the required load, shows that load in the schematic and the netlist, and warns you. Exporting the S-parameters with both ports at R₀ then shows the mismatch you would see with an equal load.

10. Bessel–Thomson: Maximally Flat Group Delay

The Bessel (Thomson) prototype optimizes the time domain instead of the magnitude: its group delay is as flat as possible at DC, so all frequency components of a pulse arrive together. The denominator is the reverse Bessel polynomial θ_N(s), whose integer coefficients have a closed form:

Bessel–Thomson prototype

H (s) = \frac{θ_{N} (0)}{θ_{N} (s)}

θ_{N} (s) = \sum_{k = 0}^{N} \frac{(2 N - k)!}{2^{N - k} k! (N - k)!} s^{k}

θ₃(s) = s³ + 6s² + 15s + 15 · rescaled so that the −3 dB point is at 1 rad/s

There is no closed form for the Bessel ladder elements, so they are found by Darlington synthesis: the reflection polynomial is obtained by spectral factorization of θ_N(s)θ_N(−s) − θ_N(0)², and the input impedance is expanded as a continued fraction that peels off one element at a time. That expansion is numerically ill-conditioned in double precision, so the element values were synthesized at 80 significant digits for every order from 1 to 12 and stored; they agree with Zverev's tables to every printed digit.

After the band-pass transformation the delay is no longer exactly flat. The transformation stretches the frequency axis non-linearly, so the band-pass delay is flat in terms of Ω but tilts across the passband in terms of f. For a fractional bandwidth of a few percent the tilt is negligible and the burst response is essentially as clean as the prototype's step response; for octave-wide filters the delay is visibly higher at the lower band edge. The group delay plot in step 3 shows the actual delay of your ladder.

11. Choosing the Filter Type and Order

The three types trade selectivity against time-domain behaviour. At a stopband frequency where Ω_s = 2, that is where the band between the two attenuation points is twice the −3 dB bandwidth, they give these attenuations:

Order N	Butterworth	Chebyshev 0.1 dB	Chebyshev 0.5 dB	Bessel
3	18.1 dB	21.5 dB	23.7 dB	12.0 dB
5	30.1 dB	41.1 dB	44.9 dB	14.1 dB
7	42.1 dB	62.3 dB	67.0 dB	14.0 dB
9	54.2 dB	84.2 dB	89.4 dB	13.4 dB

Chebyshev gains roughly 10 dB per added resonator close to the passband, Butterworth about 6 dB, and the Bessel attenuation near the passband barely changes with order. Far from the passband all three types roll off at the same 6N dB per octave on both sides; they differ only in how quickly they reach that asymptote.

Butterworth for a flat passband and moderate skirts, the default choice for preselectors and band filters.
Chebyshev when components are expensive or an interferer sits close to the passband, such as receiver front ends and transmitter band filters.
Bessel for pulsed and modulated signals where overshoot and ringing matter more than rejection.

12. Group Delay and Envelope Response

The group delay is the derivative of the transmission phase, computed by the calculator from the actual ladder (with ideal or standard values) at every frequency. It peaks near both band edges, more sharply the more selective the filter, and near f₀ it equals the prototype's delay at DC multiplied by 2/(2π·BW): halving the bandwidth doubles the delay.

Group delay and envelope

τ_{g} (ω) = - \frac{d \arg S_{21}}{d ω}

e (t) = \frac{1}{| S_{21} (ω_{0}) |} | \int_{0}^{t} h_{bb} (τ) d τ |

h_bb = complex-baseband impulse response, the inverse Fourier transform of S21(ω₀ + Δω), computed by FFT from the ladder

The envelope response is the band-pass counterpart of a low-pass step response: the amplitude of the output when a carrier at f₀ is switched on. For a narrow filter it is, to a very good approximation, the step response of the prototype with time stretched by 2/(2π·BW), so a 5th-order Butterworth band-pass filter overshoots by about 12.8% and a 0.5 dB Chebyshev by about 13%, while a Bessel filter overshoots by under 1%. Because it is computed from the ladder itself, it also shows the effect of resonators detuned by standard-value rounding.

13. Narrow Bandwidths, Standard Values and Real Components

Ideal values rarely match catalogue parts. The calculator rounds each element to the nearest E12, E24 or E96 value on a logarithmic scale (E12 steps are about 20%, E24 about 10%, E96 about 2.4%), shows how far each rounded resonator is tuned from f₀, and re-analyses the rounded ladder so you can see the shift in the band edges, stopband attenuation, delay and overshoot. A resonator detuned by a fraction δ shifts its centre by roughly δ/FBW of the bandwidth, so narrow filters are far more sensitive than wide ones.

The ratio of the largest to the smallest inductor grows as about 1/FBW². Below roughly 5–10% fractional bandwidth, the series inductors become very large and their capacitors very small, comparable to stray capacitance. A capacitively coupled resonator filter, with identical parallel resonators joined by small coupling capacitors, is then usually the better choice.
Inductors have finite Q. The passband loss of a band-pass filter rises roughly in proportion to the loaded Q, f₀/BW, divided by the component Q, so narrow filters need high-Q inductors to keep the insertion loss low.
Each inductor resonates with its winding capacitance at its self-resonant frequency (SRF), and capacitors resonate with their series inductance; above those frequencies the upper stopband can collapse. Choose parts whose resonances lie well above the highest frequency you need to reject.
PCB pads and traces add a few tenths of a picofarad and nanohenry. At VHF and above, use small-case C0G/NP0 capacitors, short ground connections, and include pad parasitics in a circuit simulator using the exported netlist.

14. Passband Ripple and Return Loss

Because the ladder is lossless, any passband loss is reflection. A ripple of A_r dB means a worst-case return loss of:

Return loss at a ripple peak

RL = - 10 \log_{10} (1 - 10^{- A_{r} / 10})

0.01 dB → 26.4 dB · 0.1 dB → 16.4 dB · 0.5 dB → 9.6 dB · 1 dB → 6.9 dB

This is why RF filters between 50 Ω stages typically use 0.01–0.1 dB ripple: the match to neighbouring amplifiers and mixers matters as much as the extra selectivity. Butterworth and Bessel filters have a perfect match at f₀ and a return loss that degrades smoothly toward both band edges, as step 3 shows.

15. Accuracy and Validation

The prototype element values are compared with the standard published tables below. Each prototype is then transformed into band-pass ladders of 1%, 10% and 100% fractional bandwidth, which are re-analysed and compared with the transformed ideal response over −20 ≤ Ω ≤ 20. The table is computed live in your browser by the same code that designs your filter. Chebyshev values are shown normalized to the ripple band edge, as the tables print them; Bessel values are listed from the source end.

Type	N	Computed g₁ … g_N	Max \|Δg\| vs table	Max band-pass response error	Reference
Computing…

With ideal components the results are exact. In practice the accuracy of a built filter is set by component tolerances, finite Q and parasitics; verify critical designs in a circuit simulator with vendor models, using the SPICE netlist or Touchstone file from step 2, and by measurement on a network analyser.

References

Matthaei, G.L., Young, L. & Jones, E.M.T., Microwave Filters, Impedance-Matching Networks, and Coupling Structures, McGraw-Hill, 1964; Artech House reprint, 1980, ch. 4 and 8.
Zverev, A.I., Handbook of Filter Synthesis, Wiley, 1967.
Williams, A.B. & Taylor, F.J., Electronic Filter Design Handbook, 4th ed., McGraw-Hill, 2006.
Pozar, D.M., Microwave Engineering, 4th ed., Wiley, 2011, ch. 8.
Hong, J.-S., Microstrip Filters for RF/Microwave Applications, 2nd ed., Wiley, 2011, ch. 3.
Hunter, I.C., Theory and Design of Microwave Filters, IET, 2001.
Darlington, S., “Synthesis of Reactance 4-Poles Which Produce Prescribed Insertion Loss Characteristics,” Journal of Mathematics and Physics, vol. 18, 1939.
Thomson, W.E., “Delay Networks Having Maximally Flat Frequency Characteristics,” Proc. IEE, vol. 96, pt. III, 1949.
Orchard, H.J., “Inductorless Filters,” Electronics Letters, vol. 2, no. 6, 1966.
Bowick, C., RF Circuit Design, 2nd ed., Newnes, 2008, ch. 3.